Mass Term Effect on Fractional Quantum Hall States of Dirac Particles (Springer Theses) 9811691657, 9789811691652

Table of contents :
Supervisor’s Foreword
Acknowledgements
Contents
1 Introduction
1.1 Quantum Hall Effect and Its History
1.2 Graphene
1.3 Quantum Hall Effect in Graphene
1.3.1 Integer Quantum Hall Effect
1.3.2 Fractional Quantum Hall Effect
1.4 Recent Experiments on Quantum Hall Effect in Graphene
1.4.1 Mass Effects in Graphene
1.4.2 Fractional Quantum Hall Effects in Electron and Hole Doped Regime
1.4.3 Increasing of Excitation Gap
1.5 Purpose and Organization of This Study
References
2 Fundamental Theory of Quantum Hall Effect
2.1 Overview of Graphene
2.2 Integer Quantum Hall Effect
2.3 Fractional Quantum Hall Effect
2.3.1 Laughlin Wave Function
2.3.2 Composite Fermion Theory
2.3.3 Pfaffian State
References
3 Model and Method
3.1 Haldane Sphere
3.1.1 Landau Quantization in 2DEG System
3.1.2 Landau Quantization in Dirac Particle System
3.2 Lanczos Method
References
4 Mass Term Effects in Spinless Dirac Particle System
4.1 Model
4.2 Results and Discussion
4.2.1 Laughlin State at νn=pm1=1/3
4.2.2 Pfaffian State and Fermi Liquid at νn=+1=1/2
4.3 Summary
References
5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States
5.1 Model
5.2 Results and Discussion
5.2.1 ΔZ lle2/εlB < Δv
5.2.2 Δv lle2/εlB < ΔZ
5.2.3 Comparison with Experimental Results
5.3 Summary
References
6 Mass and Valley Effects on Excitations in Quantum Hall States
6.1 Model
6.2 Results and Discussion
6.2.1 Mass Dependence on Excited State at νnuparrow=+1=1/3
6.2.2 Mass Dependence on Excited State at νnuparrow=+1=1
6.2.3 Comparison with Previous Experiments
6.3 Summary
References
7 Conclusion

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Springer Theses Recognizing Outstanding Ph.D. Research

Kouki Yonaga

Mass Term Effect on Fractional Quantum Hall States of Dirac Particles

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Kouki Yonaga

Mass Term Effect on Fractional Quantum Hall States of Dirac Particles Doctoral Thesis accepted by Tohoku University, Sendai, Japan

Author Kouki Yonaga Department of Physics Tohoku University Sendai, Japan

Supervisor Prof. Naokazu Shibata Department of Physics Tohoku University Sendai, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-9165-2 ISBN 978-981-16-9166-9 (eBook) https://doi.org/10.1007/978-981-16-9166-9 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

Graphene, a monatomic layer material, is the closest material to a pure twodimensional system. The existence of its two triangular sublattices leads to the appearance of energy dispersion of relativistic Dirac particles, whose research has been extensively conducted from both experimental and theoretical perspectives. The characteristic of the graphene as a two-dimensional electron system is prominent as the appearance of the quantum Hall effect in which the Hall resistance is quantized only by the physical constants of Planck’s constant and elementary charge, and the property as Dirac particles appears as chiral symmetry in which positive and negative energy levels appear symmetrically. However, in recent experiments, graphene made on a substrate is often used to improve the sample quality of monatomic layer, whose chiral symmetry is broken by the substrate. The effect of this chiral symmetry breaking on the quantum Hall effect is still unknown, and clarifying the effect of this chiral symmetry breaking has essential significance in verifying the consistency of the results obtained by experiments and theories. This thesis investigates the effect of the presence of a substrate that causes chiral symmetry breaking on the fractional quantum Hall effect in graphene. It is shown that this symmetry breaking modifies the electron-electron interaction and stabilizes the valley polarized state through the many-body effect. This stabilization of the valley polarized state increases the energy of the excited state of the valley unpolarized state. As a result, the author draws the conclusion that the Laughlin state, which is a valley polarized fractional quantum Hall state, is stabilized. This result is consistent with the fact that the fractional quantum Hall effect is clearly observed experimentally even in the presence of disorder potential. These studies bring new insights into electric states in layer materials on a substrate and clarified the importance of chiral symmetry breaking on the fractional quantum Hall effect. Sendai, Japan December 2021

Naokazu Shibata

v

Parts of this thesis have been published in the following journal articles 1.

2. 3.

K. Yonaga and N. Shibata, “Ground State Phase Diagram of Twisted Three-Leg Spin Tube in Magnetic Field,” Journal of the Physical Society of Japan, vol. 84, p. 094706, 2015. K. Yonaga, K. Hasebe, and N. Shibata, “Formulation of the relativistic quantum hall effect and parity anomaly,” Phys. Rev. B, vol. 93, p. 235122, Jun 2016. K. Yonaga and N. Shibata, “Fractional quantum hall effects in graphene on a h-BN substrate,” Journal of the Physical Society of Japan, vol. 87, p. 034708, 2018.

vii

Acknowledgements

I would like to express my sincere thanks to my supervisor Associate Professor Naokazu Shibata for the continuous support of my research. In addition, I would like to thank my thesis committee: Associate Professor Kentaro Nomura and Profs. Sumio Ishihara, Riichiro Saito, and Yoshiro Hirayama, for their insightful comments. I am grateful to Dr. Koji Muraki and Dr. Kazuki Hasebe for the collaboration of this research. I would like to thank my labmates in Theoretical Condensed Matter and Statistical Physics Group at Tohoku University, for their helpful discussions. Finally, I would like to thank my family for their support and encouragement.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Quantum Hall Effect and Its History . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum Hall Effect in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Integer Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Recent Experiments on Quantum Hall Effect in Graphene . . . . . . . . . 1.4.1 Mass Effects in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Fractional Quantum Hall Effects in Electron and Hole Doped Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Increasing of Excitation Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Purpose and Organization of This Study . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 5 10 12 13 14 16 18 19

2 Fundamental Theory of Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 2.1 Overview of Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Integer Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Laughlin Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Composite Fermion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Pfaffian State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 27 28 28 30 33 34

3 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Haldane Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Landau Quantization in 2DEG System . . . . . . . . . . . . . . . . . . . 3.1.2 Landau Quantization in Dirac Particle System . . . . . . . . . . . . . 3.2 Lanczos Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 37 40 43 45

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4 Mass Term Effects in Spinless Dirac Particle System . . . . . . . . . . . . . . . . 4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Laughlin State at νn=±1 = 1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Pfaffian State and Fermi Liquid at νn=+1 = 1/2 . . . . . . . . . . . 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 51 52 57 60 60

5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States . . . 5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Z  e2 /l B < v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 v  e2 /l B < Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Comparison with Experimental Results . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 65 65 70 73 76 76

6 Mass and Valley Effects on Excitations in Quantum Hall States . . . . . . 6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Mass Dependence on Excited State at νn ↑ =+1 = 1/3 . . . . . . . 6.2.2 Mass Dependence on Excited State at νn ↑ =+1 = 1 . . . . . . . . . . 6.2.3 Comparison with Previous Experiments . . . . . . . . . . . . . . . . . . 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 80 83 85 87 88

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 1

Introduction

1.1 Quantum Hall Effect and Its History The two-dimensional electron gas (2DEG) system is realized in the inversion layer of metal-oxide-semiconductor field-effect transistors (MOSFET) or the heterojunction interface of the two different semiconductors. When we apply a magnetic field into the 2DEG system, the kinetic energy of each electron is quantized and the discrete energy levels occur. This phenomenon is called the Landau quantization, which is the basic characteristic of the 2DEG system. The discrete energy level is referred to the Landau level that has macroscopic degeneracy as shown in Fig. 1.1a. In this situation, the Hall conductivity is given as σx y = e2 ν/ h where ν is the Landau level filling factor, e is the elementary charge, and h is the Planck constant. When we keep the electron density constant, the Landau level filling factor is proportional to the magnetic field. In the classical Hall effect, the Hall conductivity grows linearly as the magnetic field increases. However, von Klitzing discovered the integer quantum Hall effect (IQHE) in 1980. He found that the Hall conductivity in MOSETs is quantified 2 as σx y = eh N , where N is a positive integer. Moreover, he also showed that the longitudinal resistivities disappear around ν = N . The previous study reported that the IQHE is caused by the Landau quantization and disorder effects [1]. Thus, in the integer quantum Hall state (IQHS), the electron localization by the disorders plays an essential role. After the discovery of the IQHE, the higher quality 2DEG systems have been realized with developing processing technologies of semiconductors. In 1982, Tsui et al. found the new plateaus in the Hall conductivity around the fractional filling factor as ν = p/q, where p and q are both prime integers. In their experiment, the Hall con2 ductivity was quantized with the fractional value as σx y = eh qp . This phenomenon is named the fractional quantum Hall effect (FQHE) [3–5]. More interestingly, the excitations in the fractional quantum Hall state (FQHS) are characterised by quasiparticles with the fractional charge [3]. The fractionally-charged quasiparticles are called anyons because they obey the fractional statistics, not boson and fermion ones [4]. The existence of fractionally-charged quasiparticles was demonstrated by the © Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_1

1

2

1 Introduction

Fig. 1.1 a Density of states in the 2DEG system without/with a magnetic field. b Hall and longitudinal resistivities in an experimental system. c QHE in the heterojunction of GaAs/GaAlAs. Reprinted figure with permission from [2]. Copyright 1987 by the American Physical Society

shot noise experiment [5]. Figure 1.1c shows the Hall conductivity and longitudinal resistivity in the heterojunction of GaAs/GaAlAs. Generally, the FQHE can be observed in clean samples. This fact means that, in the FQHE, the Coulomb interaction is essential rather than the disorder effects. In the following, we refer to the IQHE and FQHE together as the quantum Hall effect (QHE).

1.2 Graphene

3

1.2 Graphene Graphene is a single layer of carbon atoms composed of sp 2 hybridization. As shown in Fig. 1.2a, the carbon atoms of graphene are arranged in a two-dimensional honeycomb lattice. Figure 1.2a also shows the A (B) sublattice sites with the white (black) circles. The honeycomb lattice structure creates novel physical properties such as strong mechanical strength, high thermal conductivity, and good electrical conductivity. To understand these interesting properties, it is important to discuss the low-energy model in graphene. Let us consider the tight-binding model of graphene. Since graphene has the two sublattices, A and B, the tight-binding Hamiltonian is given by the 2 × 2 matrix as

Fig. 1.2 a Lattice structure of graphene. White (black) circles indicate the A (B) sites. b First Brillouin zone of graphene. Here, K + and K − correspond to K and K  in the main text, respectively. Reprinted figures a and b with permission from [6]. Copyright 2013 by IOP Publishing. c Energy dispersion and Dirac cone in graphene. Reprinted figure with permission from [7]. Copyright 2009 by the American Physical Society

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1 Introduction

Htb =



 [c†A (k), c†B (k)]

k

0 D(k) D ∗ (k) 0



 c A (k) . c B (k)

(1.1)

Figure 1.2b displays the Brillouin zone of graphene. We label the vertices of the hexagonal Brillouin zone as K and K  points that cannot be connected by a reciprocal lattice vector. Figure 1.2c shows the energy dispersion obtained by the tight-binding model. As can be seen from Fig. 1.2c, the energy gap is zero around the K and K  points. The low-energy Hamiltonian around the K and K  points in graphene are given as HD = vF ( px σx + p y σ y )  = vF

 0 px − i p y , px + i p y 0

(1.2)

where px and p y are the momenta, and σx and σ y represent the Pauli matrices. Equation (1.2) corresponds to the Dirac Hamiltonian that describes relativistic particles [8]. The energy spectrum of HD has the linear dispersion as E( p) = ±vF | p|. Here, vF represents the Fermi velocity, which is estimated to be 105 ∼ 106 [m/s] in graphene [7]. Equation (1.2) is called the massless Dirac Hamiltonian since the effective mass is zero. In addition, because the Dirac Hamiltonians appear at both of the K and K  points, the low-energy state of graphene has the valley degree of freedom. In graphene, the relativistic particles with the valley degeneracy have a great impact on its physical properties. Here, we explain the chiral symmetry that is the fundamental characteristic of the massless Dirac particles [9, 10]. When a Hamiltonian H has the chiral symmetry, the chiral operator γ satisfies the following anticommutation relation as {H, γ} = 0.

(1.3)

When we obtain an eigenstate | with an eigenvalue E, the chiral paired state |γ  is given as (1.4) |γ  = γ|. From Eq. (1.3), |γ  satisfies the following relation as H |γ  = H γ| = −γ H | = −E|γ .

(1.5)

Equation (1.5) means that, when | is the eigenstate with the eigenenergy E, |γ  with the eigenenergy −E is always the eigenstate. The tight-binding model of graphene is given as the 2 × 2 matrix with the zero diagonal elements. Therefore, the tight-binding Hamiltonian has the chiral symmetry as

1.2 Graphene

5

{Htb , σz } = 0.

(1.6)

With the similar reason, the low-energy model of graphene, HD , also satisfies Eq. (1.3) as (1.7) {HD , σz } = 0.

1.3 Quantum Hall Effect in Graphene 1.3.1 Integer Quantum Hall Effect In the 2000s, the QHE was observed in graphene and this discovery attracted a great deal of attention. As we have discussed, the low-energy state of graphene is given by the Dirac Hamiltonian and has the valley degrees of freedom. The QHE in graphene shows unique characteristics that can not be observed in the conventional 2DEG system. In the following, we focus on the chiral symmetry and valley degree of freedom and explain the previous studies on the QHE of graphene. Figure 1.3a shows the energy levels of graphene in the magnetic field, where n and ν represent the index of the Landau level and the filling factor, respectively.

Fig. 1.3 a Energy levels in graphene. Here, n and ν represent the index of the Landau level and the filling factor, respectively. b IQHE in single layer graphene. The red (green) line corresponds to the Hall conductivity (longitudinal resistivity). The inset in b shows the IQHE in bilayer graphene. Reprinted by permission from [11], Springer Nature (2005)

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1 Introduction

Fig. 1.4 Phase diagram of the IQHS in graphene. The left vertical axis represents the concentration of impurities, n s . The right vertical axis corresponds to the electron mobility μ. Reprinted figure with permission from [12]. Copyright 2006 by the American Physical Society

Each level has the four-fold degeneracy due to the spin and valley degeneracy when we neglect the Zeeman effect. The energy levels are symmetric with respect to the zero-energy point, since ideal graphene has the chiral symmetry. Thus, the Landau level filling factor, νG , in graphene is defined as νG = 4(n + 1/2) (n = 0, ±1, ±2, . . .),

(1.8)

where the factor 4 corresponds to the four-fold degeneracy. By using this definition, the Landau level filling factor is given as νG = −2 when the Fermi level lies between n = 0 and n = −1. When the Fermi level lies between n = 0 and n = +1, the filling factor is given as νG = +2. As shown in Fig. 1.3a, νG is symmetric with respect to the zero-energy point. The IQHE in graphene is also unique as well as the Landau level filling factor. In the 2DEG system, the Hall conductivity is given as σx y = e2 ν/ h, and the IQHE is observed around ν = N , where N is an integer. Unlike the 2DEG system, the Hall conductivity in graphene is quantized with a half-odd number as σx y ∝ (n + 1/2). This is because the Landau level filling factor has the 1/2-factor. Figure 1.3b shows the IQHE in graphene on the SiO2 substrate. The left and right vertical axis are the longitudinal resistivity and Hall conductivity normalized with 4e2 / h, respectively. The horizontal axis represents the carrier density where the positive (negative) n corresponds to electrons (holes). As can be seen in Fig. 1.3b, the Hall conductivity is quantized with n + 1/2 (n = 0, ±1, ±2, . . .). In addition, the longitudinal resistivity ρx x is symmetric with respect to n = 0. Because of the chiral symmetry, we can see the IQHE in both of the electron and hole regime. The Landau quantization and disorder effects play an important role in the IQHE. Moreover, the Coulomb interaction affects the spin and valley physics in the IQHS. When the QHS has the spin degree of freedom and we neglect the Zeeman effect, the

1.3 Quantum Hall Effect in Graphene

7

Hamiltonian has SU(2) symmetry. In this situation, the ferromagnetic state, called the quantum Hall ferromagnet, is realized by the exchange interaction of the Coulomb repulsion [13]. In graphene, the low-energy Hamiltonian has SU(4) symmetry due to the spin and valley degeneracy. Nomura et al. studied the quantum Hall ferromagnet with SU(4) symmetry [11]. They investigated the stability of the quantum Hall ferromagnet with using the Hartree–Fock and self-consistent Born approximation. Figure 1.4 shows the phase diagram of the IQHS in graphene. The left (right) vertical axis corresponds to the impurity density (carrier mobility). The horizontal axis is the Landau level filling factor. They confirmed the quantum Hall ferromagnet can be stably realized by the Coulomb interaction. While Nomura et al. assumed the SU(4) Hamiltonian, the Zeeman effect cannot be ignored in actual graphene. The Zeeman effect causes additional plateaus in the Hall conductivity [14]. Figure 1.5a shows the IQHE obtained by Zhang et al. We can see that the Hall plateaus appear not only around ν = 4(n + 1/2), but also ν = ±1, ±4. Here, we focus our attention on the IQHE at ν = −4. Figure 1.5b shows the longitudinal resistivity when the sample is tilted by the angle θ with fixing the perpendicular magnetic field B⊥ . As θ increases, the parallel magnetic field B becomes greater. Therefore, Fig. 1.5b represents the B dependence on the longitudinal resistivity. The total magnetic field Btot is given as the summation of B⊥ and B . The energy scale of the Landau level and the Coulomb interaction are determined by only B⊥ , whereas the Zeeman effect depends on Btot . Thus, only the Zeeman energy increases when B grows with fixing B⊥ . As shown in Fig. 1.5, Rx x at ν = 2 does not change even with increasing of B ; however, at ν = 4, the longitudinal resistivity approaches zero as B⊥ increases. These results mean that the IQHS at ν = 4 is the spin polarized state whose stability depends on the Zeeman effect. Here, we note that the Hamiltonian still has SU(2) symmetry because the valley degeneracy remains even in the magnetic field. Therefore, the valley-polarized state, where all the electrons are at either K or K  point, can be realized [15]. When the ground state is the ferromagnetic IQHS, its excited state is characterized by the skyrmion [17]. To understand the skyrmion excitation intuitively, we represent electron spins with classical vectors in Fig. 1.6. The left side in Fig. 1.6 shows the skyrmion configuration on a plane. In the skyrmion on the plane, the spins around the center point downward, and their directions gradually change to upward toward the edge. Here, let us map the spins onto a sphere as shown in the right side in Fig. 1.6. We can distinguish the skyrmion excitation by the number of times the spins cover the spherical surface. The skyrmion can be regarded as one of the toplogical excitations, which is characterised by the winding number. The skyrmion excitation can occur when a Hamiltonian has SU(N) symmetry. Off course, we can see the “valley” skyrmion in the QHE of graphene. Shibata et al. demonstrated the existence of the valley skyrmion with the density matrix renormalization group (DMRG) [18]. They assumed that the spin degeneracy was completely lifted by the Zeeman effect and the low-energy Hamiltonian has only the valley degeneracy. Figure 1.7a shows the computation results obtained by Shibata et al. The vertical axis s means the excitation gap with the valley skyrmion. The

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1 Introduction

Fig. 1.5 a IQHE in graphene when B⊥ = 45 [T]. The horizontal axis represents the gate voltage. The vertical axes represent the Hall resistivity (left) and the longitudinal resistivity (right). b Perpendicular magnetic field dependence on the longitudinal resistivity R x x at ν = −4. The left panel of b shows Rx x in B = 0 [T] (black line) and B = 15 [T] (green line) with B⊥ = 30 [T]. The right panel of b corresponds to Rx x at θ = 0◦ , 18.1◦ , 18.8◦ , 35.5◦ , 49.0◦ , and 58.4◦ in Btot = 45 [T]. Reprinted figures with permission from [14]. Copyright 2006 by the American Physical Society

1.3 Quantum Hall Effect in Graphene

9

Fig. 1.6 Skyrmion configuration on the plane and sphere. Reprinted figure with permission from [16]. Copyright 2013 by the American Physical Society

Fig. 1.7 a Number of particles dependence of the skyrmion gap (s ). Reprinted figure with permission from [18]. Copyright 2008 by the American Physical Society. b Table of the quasiparticle excitation  Q P and the skyrmion gap sk [19]. Here,  D and  P represent the excitation gap in the Dirac particle system and 2DEG system, respectively. Reprinted figure with permission from [19]. Copyright 2006 by the American Physical Society

horizontal axis is the inverse of the number of particles. Here, νn corresponds to the filling factor in nth Landau level, which is given as  1 . νn = ν − 4‘ n + 2 

(1.9)

Shibata et al. estimated s in the thermodynamic limit (1/Ne → 0). They found that s estimated by DMRG was in good agreement with the results by the Hartree–Fock approximation [19]. Thus, the previous studies concluded that the excited state of the IQHS in graphene is characterised by the valley skyrmion [17, 19].

10

1 Introduction

1.3.2 Fractional Quantum Hall Effect The FQHS is one of the quantum liquids created by the quantum many-body effects. When the strong repulsion prohibits particles from approaching each other, the FQHS can be stablely realized. Therefore, the short-range component of the Coulomb interaction is essential for the stability of the FQHS [21]. Apalkov et al. compared the FQHSs in the 2DEG system and massless Dirac particle system (grpahene) [20]. Figure 1.8a and b show the effective Coulomb potentials Vm called the pseudpotentials in n = 0, 1, and 2. The horizontal axis represents the relative angular momentum m between the two particles. In addition, ‘nonrelativistic’ and ‘relativistic’ in Fig. 1.8 correspond to the pseudopotentials in the 2DEG system and graphene, respectively. They focused on the FQHSs in n ≥ 0 since the ground states in n = −1, −2, . . . are equivalent to the ones in n = +1, +2, . . . due to the chiral symmetry. In n = 0, there is no difference between the pseudopotentials in the nonrelativistic and relativistic system. Therefore, the relativistic FQHSs are identical with the nonrelativistic ones in n = 0. Unlike n = 0, the relativistic pseudopotential is significantly different from the nonrelativistic one in n = 0. As shown in Fig. 1.8a, the relativistic Vm in n = 1 shows monotonic decay with increasing of m. Because this pseudopotential creates the strong short-range repulsion, FQHSs appear at various filling factors. In particular, Apalkov et al. found that the valley-polarized Laughlin state is the ground one νn=+1 = 1/3 [3, 20]. Here, the Laughlin state is one of the most famous FQHSs and we will explain the detail about it in the next chapter. They also showed the ground state at νn=+1 = 2/3 is the valley unpolarized FQHS [22]. The nonrelativistic pseudopotential in n = 1 has the local minimum at m = 1. Since this minimum

Fig. 1.8 Pseudopotential Vm for each Landau level. The horizontal axis represents the relative angular momentum. a Comparison between Vm in graphene (relativistic) and the 2DEG (nonrelativistic) system. b Vm for each Landau level in graphene. Reprinted figures with permission from [20]. Copyright 2006 by the American Physical Society

1.3 Quantum Hall Effect in Graphene

11

Fig. 1.9 Table of the ground states and the lowest excited states at each νn . c represents the quasiparticle excitation, which means the excited state with valley polarization. s corresponds to the excitation gap for the valley unpolarized state. Copyright 2009 The Physical Society of Japan [24]

suppresses the short-range repulsion, the ground state is not the FQHS. Thus, we cannot see the Laughlin state at νn=+1 = 1/3 in the nonrelativistic system [23]. Apalkov et al. and Töke et al. concluded that the ground state at νn=1 = 2/3 is the valley unpolarized FQHS. Unfortunately, the energy difference between the valley-polarized and unpolarized state is very small. To clarify the ground state at νn=1 = 2/3, the numerical study on large-size Dirac particle systems is required. Shibata and Nomura investigated the FQHS in the massless Dirac particle system with DMRG [24]. Figure 1.9 shows the ground state and the lowest excited state at νn obtained by DMRG. They revealed that the ground state at νn=+1 = 2/3 is the valleypolarized FQHS, not the valley-unpolarized one. Moreover, Balram et al. studied the FQHS in the large-size systems with the composite fermion diagonalization. They also reported the valley-polarized FQHS is the ground state at νn=+1 = 2/3 [25]. The excited state in the FQHS is given by the quasiparticles and quasiholes with fractional charge. Shibata et al. studied the excited states of the FQHS in graphene with DMRG. Figure 1.9 displays the excitation gap s (c ) with (without) the valley skyrmion. Here, the excitation gap corresponds to the energy required to create the quasiparticle and quasihole. They revealed that the lowest excited states in several νn s are characterized by the skyrmion excitations. Figure 1.10a shows the pair correlation functions with the quasihole excitation at νn=1 = 1/3. The value of g K K (r ) means the correlation between the particles at the K point. In addition, g K K  (r ) corresponds to the correlation between the particles at the K and K  points. While g K K (r ) shows the large value near r = 0 and decreases as r increases, g K K  (r ) becomes larger

12

1 Introduction

Fig. 1.10 a Pair correlation function g(r ) with the quasihole excitation at νn=1 = 1/3. Reprinted figures with permission from [18]. Copyright 2008 by the American Physical Society. b g(r ) with the quasihole excitation at νn=1 = 2/3 (red lines) and νn=1 = 2/5 (blue lines). The solid and doted lines correspond to g K K (r ) and g K K  (r ), respectively. Copyright 2009 The Physical Society of Japan [24]

as r increases. These results are consistent with the characteristics of the skyrmion excitation explained in Sect. 1.3.1. The lowest excited state at νn=1 = 1/3 is given by the valley skyrmion with s . Figure 1.10b shows the pair correlation functions at νn=1 = 2/3. Because both g K K (r ) and g K K  (r ) shows the large value near r = 0, the excitation at νn=1 = 2/3 is not the valley skyrmion. At νn=1 = 1/2, the unique FQHS called the Pfaffian state can be observed in the 2DEG system [26, 27]. In the Pfaffian state, all the particles are in pairs and the quasiparticles obey the noncommutative statistics. From these properties, the Pfaffian state can be applied into the topological quantum computer [28]. Unfortunately, the Pfaffian state can not be observed at νn=1 = 1/2 in graphene. Wojs et al. reported that the ground state at νn=1 = 1/2 is the Fermi liquid, not Pfaffian one [29]. This is because the strong short-range repulsion in n = 1 of graphene destroys the particle pairs.

1.4 Recent Experiments on Quantum Hall Effect in Graphene To observe the FQHE experimentally, the high mobility graphene is required. We can obtain such a high quality sample with a suspended graphene sheet as shown in Fig. 1.11a [32]. Unfortunately, we cannot apply a large gate voltage into suspended graphene to avoid electrostatic breakdown [33]. To overcome this problem, graphene on a substrate is widely used instead of the suspended sheet. In particular, graphene on the hexagonal Boron Nitride (h-BN) substrate has high mobility, and the QHE has been observed at various Landau level filling factors. h-BN is a layered material consisting of Boron and Nitrogen with the honeycomb lattice. Owing to the strong

1.4 Recent Experiments on Quantum Hall Effect in Graphene

13

Fig. 1.11 a Sketch of suspended graphene. Reprinted by permission from [30], Springer Nature 2011. b Sketch of graphene on a h-BN substrate. Reprinted by permission from [31], Springer Nature 2011

in-plane ionic bonding, h-BN has the smooth surface without dangling bonds and charge traps. As a result, graphene on h-BN shows the ultra-high carrier mobility [34]. In recent years, not only h-BN but also various other materials have been studied as the substrate for graphene [35]. Since the high quality samples have been fabricated on h-BN, it is important to discuss the substrate effects for grpahene. Unlike graphene, the unit cell of h-BN is composed of different atoms, B and N. Therefore, the sublattices A and B in graphene are affected by the different potentials from h-BN. The simplest low-energy model that includes the substrate effects is given as HD (M) = vF ( px σx + p y σ y ) + Mσz  = vF

 M px − i p y . px + i p y −M

(1.10)

The energy spectrum of Eq. (1.10) has the finite gap as 2M, which means that the Dirac particles obtain the finite mass. For this reason, M and HD (M) is called the mass term and massive Dirac Hamiltonian, respectively. When the mass term occurs, the chiral symmetry is broken as {HD (M), σz } = 0. Furthermore, the valley degeneracy is lifted in graphene on h-BN because the spatial inversion symmetry is also broken by the mass term. Thus, we can expect that the mass term has a significant impact on the QHE in graphene.

1.4.1 Mass Effects in Graphene Figure 1.12 shows the gate-voltage Vg dependence on the conductivity in graphene [36]. The conductivity has the minimum value at Vg = 0 and increases almost linearly

14

1 Introduction

Fig. 1.12 a Gate voltage dependence on the conductivity in graphene on the SiO2 . Reprinted figures with permission from [36]. Copyright 2008 by the American Physical Society

as the gate voltage grows. This is because the density of states in the massless Dirac particle system is zero at the Dirac point. Hunt et al. measured the gate-voltage dependence of the conductivity in graphene on h-BN [37]. They reported that the conductivity increases linearly as the carrier density becomes larger. In addition, they also found that there is the special region where the conductivity is zero around Vg = 0. Their experiment demonstrates that the mass term creates the finite gap in the energy spectrum of graphene on h-BN. When we consider the mass term and Zeeman effect, the spin and valley degeneracy is lifted. Figure 1.13a shows the splitting of the Landau levels caused by the mass term and the Zeeman effect. Here, we assume that the energy scale of the mass term is greater than that of the Zeeman effect. Since both of the spin and valley degeneracy are lifted, the new Hall plateaus appear at ν = 0 and ±1. Figure 1.13b displays the Hall conductivity obtained by Dean et al. [34]. They confirmed the existence of the IQHE at ν = 0, ±1 in graphene on h-BN [34]. The energy gap,  = 2M, of graphene on h-BN is estimated to be about 100−400 [K] [34, 37]. This value is comparable to the energy scale of the Coulomb interaction in the magnetic field. Therefore, the mass term affects not only the IQHE but also the FQHE which is caused by the Coulomb repulsion.

1.4.2 Fractional Quantum Hall Effects in Electron and Hole Doped Regime Hunt et al. measured the gate-voltage dependence on the capacitance in graphene on h-BN under the magnetic field [37]. They observed the dip structures where the capacitance is zero or significantly reduced. These dips correspond that the system becomes insulating, and that the QHS is realized. They confirmed the existence of

1.4 Recent Experiments on Quantum Hall Effect in Graphene

15

Fig. 1.13 a Landau level splitting. b QHE in graphene on a h-BN substrate. Reprinted by permission from [34], Springer Nature 2010

the FQHE in |ν| ≤ 2 which corresponds to the Landau level filling factor in n = 0. Moreover, the dip structures can be seen in 2 ≤ |ν| ≤ 6 corresponding to the filling factor in n = ±1. Their results indicate that the FQHE can be observed in both the electron and hole doped regime although the chiral symmetry is broken by the h-BN substrate. Dean et al. studied the QHE in graphene on h-BN with measuring the electric resistivity [31]. Figure 1.14a shows the gate-voltage dependence of the Hall and longitudinal resistivity obtained by Dean et al. As shown in Fig. 1.14a, the FQHE is clearly observed in |ν| ≤ 2. This result is consistent with Hunt et al.’s study. However, in 2 ≤ |ν| ≤ 6, the FQHE is realized in only the hole doped regime. Figure 1.14b displays the gate-voltage dependence of the longitudinal resistivity in graphene on h-BN, which is obtained by Amet et al. [38]. The blue region corresponds to the QHE, and Amet et al. reported that FQHE does not occur in ν > 2. Dean et al. discussed the disorder effects as the reason the FQHE is not observed in the electron doped regime. The mobility of the electrons is lower than that of the

16

1 Introduction

Fig. 1.14 a Gate-voltage dependence on the Hall and longitudinal resistivity. Reprinted by permission from [31], Springer Nature 2011. b Magnetic field and gate voltage dependence of the longitudinal resistivity. The blue color corresponds to the low longitudinal resistivity. Reprinted by permission from [38], Springer Nature 2015

holes in graphene on h-BN. They mentioned that the FQHS in the electron doped regime was smeared by the disorders. Unfortunately, it is difficult to conclude that the disorder effects cause the instability of the FQHS in only the electron regime. The charge gaps at νn=0 = 2/3 and νn=1 = 2/3 are comparable as shown in Fig. 1.9. If the disorder really smears the FQHS, the FQHE at νn=0 = 2/3 should not be observed. However, in the previous experiments, the FQHE has been clearly observed at νn=0 = 2/3. Thus, it is an important challenge to clarify the mechanism that smears the FQHS in ν > 2.

1.4.3 Increasing of Excitation Gap The stability of the QHS is determined by the excitation gap in the lowest excited state. In experiments, the QHS can be observed when the excitation gap is greater than the energy scale of disorders. The energy scale of the disorders, , in graphene on h-BN is approximately give as 30 [K] [39]. Shibata et al. estimated the excitation gap of the FQHS in the massless Dirac particle system (Fig. 1.9). Table 1.1 shows the excitation gap ex of the lowest excitation at νn = 1/3. Here, the lowest excited state in n = 0 (n = 1) is characterized by the valley-polarized (unpolarized) state. The lowest excitation gap in n = 0 (n = 1) gap corresponds to ex = c (ex = s ). In the following, we assume the dielectric constant in the graphene on h-BN is given as  = 5 [31, 40]. As shown in Table 1.1, ex at νn=0 = 1/3 is larger than . This result is consistent with the fact that the FQHE is observed at −2 ≤ |ν| ≤ 2. By contrast, ex is lower than , because the lowest excited state at νn=1 = 1/3 is characterized by the valley skyrmion. According to Shibata et al. study, we cannot observe the FQHE in |ν| > 2.

1.4 Recent Experiments on Quantum Hall Effect in Graphene

17

Table 1.1 Magnetic field dependence on the lowest excitation gap ex at νn=0 = 1/3 (left) and νn=1 = 1/3 (right). Here, we assume the dielectric is given by  = 5

B [T] Δex (νn=0 = 1/3) [K] 10 41 15 50 20 58

B [T] 10 15 20

Δex (νn=1 = 1/3) [K] 21 25 29

Fig. 1.15 a Magnetic field dependence on the thermal activation energy. Reprinted by permission from [38], Springer Nature 2015. b Thermal activation energy for each ν in B = 35 [T]. Reprinted by permission from [31], Springer Nature 2011

Actually, the previous experiments show the FQHE even in n = ±1. Figure 1.15a displays the magnetic field dependence of the thermal activation energies at ν = −8/3, −11/3 and −13/3 obtained by Amet et al. In Fig. 1.15a, the negative signs are omitted. Here, the thermal activation energy ac is defined as ac = ex − .

(1.11)

We can estimate ac with using the temperature dependence of the longitudinal resistivity. The thermal activation energies at ν = −11/3 and −13/3 increase monotonically with the magnetic field up to 20 [T], and then decrease. The value of ac in n = +1 is estimated to be 2 ∼ 20 [K], which is in agreement with the results by Dean et al.. Thus, although the theoretical study expects that the excitation gap in n = −1 is smaller than the energy scale of the disorders, the FQHE is clearly observed in the previous experiments. The previous theoretical studies have assumed that the lowenergy model of graphene is given by the massless Dirac Hamiltonian. In this case, the low-energy Hamiltonian has the valley degeneracy, and the lowest excited state is the valley skyrmion. However, since we cannot neglect the mass term in graphene on

18

1 Introduction

h-BN, the low-energy Hamiltonian no longer has the valley degeneracy. Therefore, it is possible that the novel excited state, which is different from the valley skyrmion, can be realized in graphene on h-BN.

1.5 Purpose and Organization of This Study We have mentioned the current status and problems about the QHE in graphene. Since graphene is fabricated on the h-BN substrate in the recent experiments, its low-energy Hamiltonian has the mass term. The energy scale of the mass term is comparable to that of the Coulomb interaction in the magnetic field. Thus, the mass term has a great impact on the quantum meany-body physics such as the QHE. In addition, when the mass term occurs in graphene, not only the chiral symmetry but also the spatial inversion symmetry is broken. Therefore, the valley physics of graphene are greatly affected by the mass term. However, the previous theoretical studies have assumed ideal graphene without the mass, and the mass term effects on the QHE have not been clarified yet. Recently, various Dirac materials with the mass term are synthesized. In particular, transition metal dichalcogenides such as MoS2 or WSe2 have the honeycomb lattice like graphene [41, 42]. Because the unit cell of these materials is composed of the different atoms, the finite gap occurs in the energy spectrum around K point (Fig. 1.16). In other words, the Dirac particles obtain the finite mass even without the substrate. It is important to study the QHE in the massive Dirac particle system for predicting the quantum many-body phenomena in the transition metal dichalcogenides.

Fig. 1.16 a Lattice structure of a transition metal dichalcogenide. b Energy bands of MoS2 and WSe2 . The red and blue lines represent the numerical results obtained by the tight-binding model and the density functional theory, respectively. Reprinted figures with permission from [42]. Copyright 2015 by the American Physical Society

1.5 Purpose and Organization of This Study

19

In the present study, we investigate the mass term effects on the QHE in the Dirac particle system. In particular, we focus on the QHE in graphene on h-BN and set the following purpose: 1. Revealing mass term dependence on FQHS We investigate the mass term dependence on the FQHS and its stability. Focusing on the previous experiments in graphene on h-BN, we reveal the properties of the ground state in the electron and hole doped region. 2. Revealing mass effects on excitation structure in FQHS We investigate the mass term dependence on the valley skyrmion and reveal the mechanism of the excitation gap increasing in the previous experiments. This thesis is organized as follows. In Chap. 2, we explain the fundamental theory of the QHE. Chapter 3 introduces the computation method in this research. In Chap. 4, we define the pseudopotential in the massive Dirac particle system. Next, we focus on graphene and reveal the relation between the mass term and the valley degree of freedom in Chap. 5. In Chap. 6, we discuss the mechanism of the gap increasing caused by the mass term. Finally, in Chap. 7, we summarize this study.

References 1. Laughlin RB (1981) Quantized hall conductivity in two dimensions. Phys Rev B 23:5632–5633 2. Willett R, Eisenstein JP, Störmer HL, Tsui DC, Gossard AC, English JH (1987) Observation of an even-denominator quantum number in the fractional quantum hall effect. Phys Rev Lett 59:1776–1779 3. Laughlin RB (1983) Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys Rev Lett 50:1395–1398 4. Einarsson T (1990) Fractional statistics on a torus. Phys Rev Lett 64:1995–1998 5. de Picciotto R, Reznikov M, Heiblum M, Umansky V, Bunin G, Mahalu D (1997) Direct observation of a fractional charge. Nature 389:162–164 6. McCann E, Koshino M (2013) The electronic properties of bilayer graphene. Rep Prog Phys 76:056503 7. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (2009) The electronic properties of graphene. Rev Mod Phys 81:109–162 8. Wallace PR (1947) The band theory of graphite. Phys Rev 71:622–634 9. Semenoff GW (1984) Condensed-matter simulation of a three-dimensional anomaly. Phys Rev Lett 53:2449–2452 10. Haldane FDM (1988) Model for a quantum hall effect without landau levels: condensed-matter realization of the “parity anomaly”. Phys Rev Lett 61:2015–2018 11. Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV, Firsov AA (2005) Two-dimensional gas of massless dirac fermions in graphene. Nature 438:197–200 12. Nomura K, MacDonald AH (2006) Quantum hall ferromagnetism in graphene. Phys Rev Lett 96:256602 13. Ezawa ZF, Tsitsishvili G (2009) Quantum hall ferromagnets. Rep Prog Phys 72:086502 14. Zhang Y, Jiang Z, Small JP, Purewal MS, Tan Y-W, Fazlollahi M, Chudow JD, Jaszczak JA, Stormer HL, Kim P (2006) Landau-level splitting in graphene in high magnetic fields. Phys Rev Lett 96:136806

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15. Fogler MM, Shklovskii BI (1995) Collapse of spin splitting in the quantum hall effect. Phys Rev B 52:17366–17378 16. Dai YY, Wang H, Tao P, Yang T, Ren WJ, Zhang ZD (2013) Skyrmion ground state and gyration of skyrmions in magnetic nanodisks without the Dzyaloshinsky-Moriya interaction. PhD thesis 17. Sondhi SL, Karlhede A, Kivelson SA, Rezayi EH (1993) Skyrmions and the crossover from the integer to fractional quantum hall effect at small zeeman energies. Phys Rev B 47:16419–16426 18. Shibata N, Nomura K (2008) Coupled charge and valley excitations in graphene quantum hall ferromagnets. Phys Rev B 77:235426 19. Yang K, Das Sarma S, MacDonald AH (2006) Collective modes and skyrmion excitations in graphene su(4) quantum hall ferromagnets. Phys Rev B 74:075423 20. Apalkov VM, Chakraborty T (2006) Fractional quantum hall states of dirac electrons in graphene. Phys Rev Lett 97:126801 21. Haldane FDM (1983) Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states. Phys Rev Lett 51:605–608 22. Wu XG, Dev G, Jain JK (1993) Mixed-spin incompressible states in the fractional quantum hall effect. Phys Rev Lett 71:153–156 23. Haldane FDM, Rezayi EH (1985) Finite-size studies of the incompressible state of the fractionally quantized hall effect and its excitations. Phys Rev Lett 54:237–240 24. Shibata N, Nomura K (2009) Fractional quantum hall effects in graphene and its bilayer. J Phys Soc Jpn 78(10):104708 25. Balram AC, T˝oke C, Wójs A, Jain JK (Aug 2015) Fractional quantum hall effect in graphene: quantitative comparison between theory and experiment. Phys Rev B 92:075410 26. Moore G, Read N (1991) Nonabelions in the fractional quantum hall effect. Nucl Phys B 360(2):362–396 27. Greiter M, Wen X, Wilczek F (1992) Paired hall states. Nucl Phys B 374(3):567–614 28. Nayak C, Simon SH, Stern A, Freedman M, Das Sarma S (2008) Non-abelian anyons and topological quantum computation. Rev Mod Phys 80:1083–1159 29. Wojs A, Moller G, Cooper NR (2011) Composite fermion dynamics in half-filled landau levels of graphene. Acta Phys Pol, A 119:592 30. Tombros N, Veligura A, Junesch J, Guimarães MHD, Vera-Marun IJ, Jonkman HT, van Wees BJ (2011) Quantized conductance of a suspended graphene nanoconstriction. Nat Phys 7:697–700 31. Dean CR, Young AF, Cadden-Zimansky P, Wang L, Ren H, Watanabe K, Taniguchi T, Kim P, Hone J, Shepard KL (2011) Multicomponent fractional quantum hall effect in graphene. Nat Phys 7:693–696 32. Bolotin K, Sikes K, Jiang Z, Klima M, Fudenberg G, Hone J, Kim P, Stormer H (2008) Ultrahigh electron mobility in suspended graphene. Solid State Commun 146(9):351–355 33. Ghahari F, Zhao Y, Cadden-Zimansky P, Bolotin K, Kim P (2011) Measurement of the ν = 1/3 fractional quantum hall energy gap in suspended graphene. Phys Rev Lett 106:046801 34. Dean CR, Young AF, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard KL, Hone J (2010) Boron nitride substrates for high-quality graphene electronics. Nat Nanotechnol 5:722–726 35. Zhou SY, Gweon GH, Fedorov, First PN, de Heer WA, Lee DH, Guinea F, Castro Neto AH, Lanzara A (2007) Substrate-induced bandgap opening in epitaxial graphene. Nat Mat 6:770– 775 36. Morozov SV, Novoselov KS, Katsnelson MI, Schedin F, Elias DC, Jaszczak JA, Geim AK (2008) Giant intrinsic carrier mobilities in graphene and its bilayer. Phys Rev Lett 100:016602 37. Hunt B, Sanchez-Yamagishi JD, Young AF, Yankowitz M, LeRoy BJ, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, Ashoori RC (2013) Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure. Science 340(6139):1427–1430 38. Amet F, Bestwick AJ, Williams JR, Balicas L, Watanabe K, Taniguchi T, Goldhaber-Gordon D (2015) Composite fermions and broken symmetries in graphene. Nat Commun 6:5838 39. Young AF, Dean CR, Wang L, Ren H, Cadden-Zimansky P, Watanabe K, Taniguchi T, Hone J, Shepard KL, Kim P (2012) Spin and valley quantum hall ferromagnetism in graphene. Nat Phys 8:550–556

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40. Ando T (2006) Screening effect and impurity scattering in monolayer graphene. J Phys Soc Jpn 75(7):074716 41. Radisavljevic B, Radenovic A, Brivio J, Giacometti V, Kis A (2011) Single-layer mos2 transistors. Nat Nanotechnol 6:147–150 42. Fang S, Kuate Defo R, Shirodkar SN, Lieu S, Tritsaris GA, Kaxiras E (2015) Ab initio tightbinding hamiltonian for transition metal dichalcogenides. Phys Rev B 92:205108

Chapter 2

Fundamental Theory of Quantum Hall Effect

2.1 Overview of Graphene Figure 2.1a shows the lattice structure of graphene. Graphene has a honeycomb lattice, and its unit cell is given by rhombus-shaped as indicated by the dotted line. The unit cell contains two carbon atoms as shown by the black (white) circle on the A (B) site. Thus, graphene has the sublattice structure. Here, the primitive translational vectors are given as  a1 =

 √  √  a 3a 3a a , , a2 = − , , 2 2 2 2

(2.1)

where a = 0.246 nm represents the lattice constant. The reciprocal lattice vectors are written as     2π 2π 2π 2π , b2 = − , √ . (2.2) ,√ b1 = a a 3a 3a Figure 2.1b shows the first Brillouin zone of graphene. The hexagonal corners at the zone boundary are called the K and K  points, which are defined as  K=

   4π 4π √ , 0 , K − √ , 0 , 3 3a 3 3a

(2.3)

respectively. The K and K  points are cannot be mutually connected by the reciprocal lattice vector. Hamiltonian of graphene is often given by the tight-binding model as H = −t

  † (c A (R)c B (R) + c†A (R + a1 )c B (R) + c†A (R + a2 )c B (R)) + h.c. , R

(2.4) where R is the position of the unit cell, and c†A(B) (R) represents the creation operator on the A (B) site in the unit cell at R. Here, t is the transfer integral and estimated to be © Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_2

23

24

2 Fundamental Theory of Quantum Hall Effect

Fig. 2.1 a Lattice structure of graphene. The dotted line depicts the unit cell. The white and black circles represent the A and B sites, respectively. b First Brillouin zone of graphene.  is the center of the Brillouin zone. The K and K  points are the independent points at the corners

about 3 [eV]. With using the Fourier transformation, the tight-binging Hamiltonian is rewritten as



 † 0 D(k) c A (k) † [c A (k), c B (k)] , (2.5) H= c B (k) D ∗ (k) 0 k

where the matrix element is D(k) = −t (1 + ei k·a1 + ei k·a2 ). In addition, the energy dispersion is given as E(k) = ±|D(k)|. Because of D(K ) = D(K  ) = 0, the energy spectrum has the electron-hole symmetry with respect to the zero energy points. With the wave vector k = K + p, we approximate D(K ) up to the 1st order of p as √ D(K + p) ≈

3 ta( px − i p y ). 2

(2.6)

The effective Hamiltonian around the K point is given as HK = vF

0 px − i p y , px + i p y 0

where vF is the Fermi velocity estimated to be vF = tonian around K  point is similarly given as HK  = vF

√ 3 ta 2

(2.7)

∼ 106 . The effective Hamil-

0 − px − i p y . − px + i p y 0

(2.8)

2.1 Overview of Graphene

25

Fig. 2.2 a Energy dispersion around the K and K  points in a zero-magnetic field. b Landau levels in a finite magnetic field

With using the Pauli matrices σ, Eqs. (2.7) and (2.8) are rewritten as HK = vF ( px σx + p y σ y )

(2.9a)

HK  = −vF ( px σx − p y σ y ).

(2.9b)

The low-energy state in graphene are described by the Dirac particle whose energy dispersion is (k) = ±v F | p| (see Fig. 2.2a). We consider the low-energy state of graphene in a magnetic field. The kinetic momentum in the magnetic field is defined as  = p + e A. Here, A is a vector potential, where Bz = ∂x A y − ∂ y A x√and [x ,  y ] = −i2 /l 2B . The value of l B is the magnetic length defined as l B = /eB. The position of the electron, r, can be separated into the center of mass coordinate X and the relative coordinate x as  r = X+x=

X+

 l 2B l 2B  , Y −  y x . 2 2

(2.10)

Although the coordinates r x and r y are commutative, the relative coordinates x and y are noncommutative because they are proportional to  y and x , respectively. The coordinates in the center of mass, X and Y , satisfy the following relation as [X, Y ] = −[x, y] = il 2B .

(2.11)

Now, we introduce the ladder operators as

and

lB lB a = √ (x − i y ), a † = √ (x + i y ) 2 2

(2.12)

1 1 b = √ (X − iY ), b† = √ (X + iY ). 2l B 2l B

(2.13)

26

2 Fundamental Theory of Quantum Hall Effect

With substituting Eq. (2.12) into HK and HK  , we obtain the low-energy Hamiltonian in the magnetic field as √ √



2vF 0 a 2vF 0 −a †  = , H . HK = K −a 0 a† 0 lB lB

(2.14)

Here, let us consider the number operator nˆ = a † a and its eigenstate |n which satisfies the relation as n|n ˆ = n|n (n = 0, 1, 2, . . .). The eigenvalues of HK and HK  are given as √ (2.15) n = ±ωc n, √ where the cyclotron frequency in graphene is given as ωc = 2vF /l B . Thus, the Landau quantization occurs in the Dirac Hamiltonian as shown in Fig. 2.2b. We redefine the single-particle energy of the nth Landau level as  n = sgn(n)ωc |n|,

(2.16)

where n is integer values as n = 0, ±1, ±2, . . .. The eigenenergy depends on only the relative motion of the electrons since the low-energy Hamiltonians includes x and  y . As a result, the macroscopic degeneracy occurs in each Landau level. Because of the noncommutation relation [X, Y ] = −[x, y] = il 2B , the uncertainty of the center of mass coordinates is defined as X Y ∼ 2πl 2B . By using this definition, the degeneracy N in the Landau level is given as S S = N = , (2.17) X Y 2πl 2B where S is the area of a whole system. The value of N corresponds to the number of cyclotron motions that can be arranged in the system with area S (Fig. 2.3). When we assume HK and HK  have the rotational symmetry, the eigenstates are distinguished by the angular momentum. The angular momentum operator is defined as (2.18) L z = (a † a − b† b).

Fig. 2.3 a Sketch of the center of mass and the relative coordinate. b Degeneracy in the Landau quantization. When a magnetic field is strong, the cyclotron radius becomes small

2.1 Overview of Graphene

27

The eigenstates of HK and HK  in n = 0 are written as

1 |n − 1, m |n, m K = √ 2 ±|n, m |n, m

K

(2.19a)



1 |n, m =√ 2 ∓|n − 1, m

(2.19b)

and, in n = 0, given as |0, m K = |0, m K  =

0 |0, m

(2.20a)

|0, m . 0

(2.20b)

2.2 Integer Quantum Hall Effect In the previous section, we explained how macroscopic degeneracy N occurs in the Landau quantization. As shown in Eq. (2.17), the degeneracy N with area S is redefined as S BS = , (2.21) N = 2 0 2πl B where 0 = h/e is the magnetic flux quantum and and B S is the total flux through the system. The Landau level filling factor is defined as ν=

Ne , N

(2.22)

where Ne is the total number of electrons. If all the electrons are fully occupied up to the N th Landau level, the filling factor ν is N . We refer to this fully occupied state as the IQHS in the following discussion. With using the Landau level filling factor, the classical Hall conductivity is given as σx y = B/en e =

e2 ν. 

(2.23)

Note that, in an extremely clean system, we cannot observe the Hall plateau. To observe the Hall plateaus, it is necessary to consider both of the Landau quantization and disorder effects. In the 2DEG system with the disorder, the degeneracy of the

28

2 Fundamental Theory of Quantum Hall Effect

Fig. 2.4 a Landau levels with disorders. b Conceptual diagram of the electron localization

Landau levels is lifted and the Ladau levels are broadened as shown in Fig. 2.4a. Here, the energy spectrum is classified into the two regions: localized and extended region. When the Fermi level lies in the extend region, the wave function is spread throughout the whole system. By contrast, when the Fermi level lies in the localized region, the motion of electrons is restricted around the disorder. Owing to the localized state, we can observe the Hall plateaus around ν = N . Because of the chiral symmetry and valley degeneracy, the IQHE in graphne has the unique characteristics. In the 2DEG system, the IQHE is observed when the filling factor ν is around an integer. In graphene, the IQHE occurs when the filling factor is   1 , (2.24) νG = 4 n + 2 where n = 0, ±1, ±2, . . .. Here, the factor 4 in Eq. (2.24) corresponds to the spin and valley degeneracy. Moreover, the factor 1/2 is derived from the chiral symmetry of graphene; the Hall conductivities σx y in n = 0 and n = −1 are given by +2e2 / h and −2e2 / h, respectively. Thus, the IQHE can be seen in both of the electron and hole region.

2.3 Fractional Quantum Hall Effect 2.3.1 Laughlin Wave Function The IQHE can be understood as the single-particle phenomenon by the Landau quantization and disorder effects. By contrast, the FQHE is the quantum many-body phenomenon caused by the quantum fluctuation and Coulomb interaction. In par-

2.3 Fractional Quantum Hall Effect

29

ticular, the Coulomb interaction is important because the kinetic energy is constant when the Landau quantization occurs. Laughlin proposed the variational wave function which is called the Laughlin wave function or Laughlin state. The Laughlin wave function is defined as N q − (r) = i> j (z i − z j ) e

N i

|z i |2 /4

,

(2.25)

where z i is given by z i = (xi − i yi )/l B which corresponds to the complex coordinate of the ith electron. The most important part of the Laughlin wave function is the Jastrow factor (z i − z j )q . We can see that the amplitude of the wave function is zero in z i = z j , The Jastrow factor corresponds that electrons avoid each other due to the Coulomb repulsion. Here, we explain the detail about the variational parameter q. The wave function (r) should be antisymmetric as (r1 , r2 ) = −(r2 , r1 ) because an electron is a fermion. Therefore, the variational parameter q should be odd. When the system has the rotational symmetry, the variational parameter is given as q = ν − 1 due to the conservation law of the angular momentum. From these facts, the value of q is uniquely determined as q = 1/ν. The previous studies reported that the Laughlin wave function is a good candidate of the ground state at ν = 1/3 and ν = 1/5. Figure 2.5a shows the pair correlation function and the density profiles of quasiparticle and quasihole obtained by the exact diagonalization [1]. The indices ‘a’, ‘b’, and ‘c’ in the figure represent the results of the pair correlation function in the ground state, the density profile of the quasiparticle, and that of the quasihole excited state at ν = 1/3, respectively. In the ground state, the correlation around r = 0 is zero. This corresponds that the electrons around r = 0 are excluded by the Pauli exclusion and strong Coulomb repulsion. Figure 2.5b shows the table of the lowest energy at

Fig. 2.5 a Pair correlation functions in the Laughlin state, which was obtained by exact diagonalization. Reprinted figure with permission from [1]. Copyright 1985 by the American Physical Society. b Table of the lowest energy E 0 by exact diagonalization. E(1/3) is the energy obtained by the Laughlin wave function. The last column represents the overlap between the ground state and Laughlin wave function. Reprinted figure with permission from [2]. Copyright 1986 by the American Physical Society

30

2 Fundamental Theory of Quantum Hall Effect

ν = 1/3 [2]. Here, E 0 and E(1/3) represent the computation results obtained by the exact diagonalization and Laughlin wave function. In addition, the last column in Fig. 2.5b shows the overlap between the ground state and the Laughlin wave function. The overlap is more than 99%, which means the ground state at ν = 1/3 is characterized by the Laughlin wave function. Due to the uniform charge distribution and finite excitation gap, the Laughlin state is called the incompressible quantum liquid. In addition, the lowest excited state of the Laughlin state is given by the quasiparticle and quasihole with fractional charge. The origin of the fractional charge is discussed in the next subsection.

2.3.2 Composite Fermion Theory The Laughlin state reproduces the FQHE at ν = 1/q. In this subsection, we explain another aspect of the FQHS based on the composite particle theory. We introduce the flux attachment proposed by Girvin et al. [3]. Let us consider a electron and m flux quanta in a closed two-dimensional system. In this situation, even if a magnetic field is zero, the electron is affected by the vector potential. The additional phase  is given by the vector potential a as =

2π 0

a · d r.

(2.26)

C

The vector potential can be expressed as a(r) = −

m0 ∇θ(r), 2π

(2.27)

where θ(r) is the angle between the position vector r and the x-axis. Girvin et al. introduce the generator J (x) as [3]

J (x) = −m

d x  ρ(x  )θ(x − x  ),

(2.28)

where ρ(x) is the electron density in the system. With using J (x), the gauge transformation for the electron field (x) is given as (x) = e−i J (x) (x).

(2.29)

Equation (2.29) can be interpreted that the m flux quanta are attached into the one electron. The new field (x) created by the gauge transformation is called the composite particle. The gauge transformation has the singular point at x = 0 because θ(x) cannot be defined at x = 0. The commutation relation between the composite particles is given as

2.3 Fractional Quantum Hall Effect

31

Fig. 2.6 a CF with the flux attachment at ν = 1/3. b Ground state at ν = 1/3 with the CF theory

{(x), † (x  )} = δ(x − x  ) {(x), (x  )} = 0 (m = even)     (x), † (x  ) = δ(x − x  ) (x), (x  ) = 0 (m = odd).

(2.30)

As shown in Eq. (2.30), the statistics of the composite particle is determined by m. When m is even (odd), the composite particle is a fermion (boson). In the following, we explain the FQHS with the composite fermion (CF). Generally, when we attach the even number of flux quanta into electrons, the CFs are expressed as composite fermion = electron + the even number of magneticflux.

(2.31)

As an example, we consider the ground state at ν = 1/3. If both the electron density and external magnetic field are spatially uniform, we have three flux quanta per one particle. With the flux attachment transformation, we can obtain the one CF with the one flux quantum. Here, we define the Landau level filling factor of the CF as ν ∗ . The situation in Fig. 2.6a corresponds to ν ∗ = 1, and the ground state is given as the IQHS of the CF as shown in Fig. 2.6b. In the CF theory, the total magnetic field is modified as (2.32) B ∗ = B − 2ρ0 0 , where ρ0 is the electron density. Since the magnetic field changes from B into B ∗ , the cyclotron energy also does from ωc into ωc∗ . Therefore, the IQHS of the CFs has the finite energy gap ωc∗ . This feature is consistent with the Laughlin wave function. Jain applied the mean-field approximation to the CF theory and revealed that the ground state at ν = 1/3 is identical with the Laughlin wave function [4]. Thus, the CF theory can explain the characteristics of the Laughlin wave function. The CF theory can be applied to more general filling factors. Here, let us consider the FQHS at ν = 2/3. In this case, there are three flux quanta per two electrons as shown in Fig. 2.7a. With using Eq. (2.31), we obtain the another representation for the CF as

32

2 Fundamental Theory of Quantum Hall Effect

Fig. 2.7 a CFs at ν = 2/3. b Ground state at even numerator filling factors

electron = composite fermion − even number of magnetic flux.

(2.33)

Equation (2.33) means that the electron can be interpreted as the CFs and the flux quanta whose direction is opposite to the external magnetic field. Because the external magnetic field cancelled by the flux quanta, we obtain the two CFs and one flux quantum. As a result, the ground state at ν = 2/3 corresponds to the IQHS of the CFs at ν ∗ = 2. In the above discussion, we have neglected the spin degree of freedom. When ν ∗ is even, the spin has a great impact on the ground state of the CFs. We represent the excitation gap between the Landau levels and the Zeeman energy with ωc∗ and Z , respectively. In ωc∗ > Z , all the CFs occupy the up-spin level as shown in Fig. 2.7b. Therefore, the ground state is the fully spin-polarized FQHS. In ωc∗ < Z , the CFs are in both of the up and down-spin levels as shown in Fig. 2.7c. In this case, the spin-unpolarized FQHS is the ground state [5]. With the CF theory, the Landau level filling factors where we can observe the FQHE are generally given as [4] ν=

ν∗ ( p = integer). 2 pν ∗ ± 1

In fact, this results is in agreement with the many experiments.

(2.34)

2.3 Fractional Quantum Hall Effect

33

Fig. 2.8 Fractional excitation at ν = 1/3 in Ne = 3 and N = 9. a and b Show the quasihole excitation with single and three flux quanta, respectively. c IQHS of the CFs at ν = 1/3 in Ne = 4

Here, we explain the mechanism of the fractionally-charged excitation in the FQHS. For simplicity, we consider the FQHS at ν = 1/3 with Ne = 3 and N = 9. In this situation, the ground state is given by the IQHS of the CFs at ν ∗ = 1. With adding one flux quantum into the ground state, the excited state is given by four flux quanta and a quasihole as shown in Fig. 2.8a. Furthermore, three quasiholes can be created by adding three flux quanta (Fig. 2.8b). Then, we add one electron into the excited state and use the flux attachment. Finally, the FQHS at ν = 1/3 is again realized with Ne = 4 and N = 12 as shown in Fig. 2.8c. Thus, the quasihole has the fractional charge −e/3 because the three quasiholes are neutralized by the single electron. With similar way, when we remove one flux quantum, we can confirm that the quasiparticle has the fractional charge e/3.

2.3.3 Pfaffian State The Pfaffian state is the unique FQHS which is beyond the CF theory. When we use the flux attachment at ν = 1/2, we obtain the CF system without the external magnetic field. In this case, the ground state can be the Fermi liquid of the CF. However, if the system has an attractive interaction, the paired state of the CFs occurs. Such a paired state is called the Pfaffian state defined as

34

2 Fundamental Theory of Quantum Hall Effect

Fig. 2.9 a Candidate of the ground state at ν = 1/2. b Pair correlation of the Pfaffian state obtained by the DMRG. Reprinted figure with permission from [6]. Copyright 2008 by the American Physical Society

 Pf (x z , x 2 , . . . , x N ) = Pf

1 zi − z j



i 0 for simplicity in our study.

3.1.1 Landau Quantization in 2DEG System First, we discuss the Landau quantization in the conventional 2DEG system on the spherical surface [3]. The spherical coordinates are defined as x = R sinθ cosφ y = R sinθ sinφ

(3.1)

z = R cosθ, where R is the radius of the sphere. When we put the magnetic monopole at the center of the sphere, the magnetic field B in the surface is given as © Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_3

37

38

3 Model and Method

Fig. 3.1 Haldane sphere with the magnetic monopole Q. Reprinted figure with permission from [1]. Copyright 2016 by the American Physical Society

B=

Q . e R2

(3.2)

Here, Q is the strength of the magnetic monopole and satisfies the below relation: 4π R 2 B = 2Q0 ,

(3.3)

where 2Q is integer and 0 is the magnetic flux quantum. The momentum of electrons without the magnetic field is pi = −i ∂x∂ i (i = x, y, z), and the angular momentum is defined as L i = i jk x j pk . In the finite magnetic field, the electron motion is defined by the dynamical momentum as i = pi + e Ai .

(3.4)

In the spherical coordinate system, i can be written as ∂ ∂r ∂ θ = −i ∂θ ∂ + Q cosθ. φ = −i ∂φ r = −i

(3.5)

Moreover, the dynamical angular momentum is i = i jk xi i .

(3.6)

3.1 Haldane Sphere

39

Here, i satisfies the bellow relation as L =  + Qer .

(3.7)

In the following discussion, we refer to the angular momentum and the dynamical angular momentum as L Q and  Q , respectively. The Hamiltonian in the spherical coordinate system is given as H=

1 −1 ∂ 2 1 ∂ 1  2 + i = − 2 , 2m i 2m ∂r 2 m R ∂r 2m R 2 Q

(3.8)

where m is the mass of the electron. As the electron motion is restricted to the surface of the sphere, we can rewrite the Hamiltonian as H0 =

1 eB (L 2 − Q 2 ). 2Q = 2 2m R 2m Q Q

(3.9)

Since Eq. (3.9) commutes with L 2 and L z , the eigenstates |Q, l, m can be written as H0 |Q, l, m = |Q, l, m

(3.10)

L 2Q |Q, l, m L zQ |Q, l, m

= l(l + 1)|Q, l, m

(3.11)

= m|Q, l, m.

(3.12)

Moreover, the eigenenergy  is =

ωc [l(l + 1) + Q 2 ], 2Q

(3.13)

where the angular momentum is given as l = n + Q (n = 0, 1, 2, . . .) to keep the eigenenergy positive. Thus, the kinetic energy on the Haldane sphere is quantized with an integer number n as ωc n = 2Q

   1 n(n + 1) n+ + . 2 Q

(3.14)

When the radius of the sphere becomes infinite as R → ∞ (or Q → ∞), the eigenenergy is going to ωc (n + 21 ) Furthermore, each single Landau level has 2l + 1 degeneracy due to −l ≤ m ≤ l. The wave function in the nth Landau level is as follows:

40

3 Model and Method



(2l + 1)(l + m)!(l − m)! imφ e 4π(l + Q)!(l − Q)!        θ 2k+Q−m θ 2l−2k+Q+m l−Q l+Q cos × (−1)k , sin k Q−m+k 2 2 k

θ, φ|n; Q, l, m = (−1)

l+m

(3.15) which is called monopole harmonics [4, 5].

3.1.2 Landau Quantization in Dirac Particle System Next, we consider the Landau quantization of the massless Dirac particles on the Haldane sphere. As discussed in Sect. 3.1.1, we assume that the monopole is placed at the center of the sphere and satisfies Eq. (3.2). The dynamic momentum of the Dirac particles on the sphere is defined as μ = eμ { pμ + (Aμ − ωμ )},

(3.16)

where eθ and eφ are given by 1/R and 1/(Rsinθ), respectively. Here, ωμ is called the spin connection defined as ωθ = 0 and ωφ = 21 σz . The spin connection represents the correction of the differential by the sphere’s curvature [6]. The dynamical angular momentum can be written as  L Q− 21 0 JQ = . (3.17) 0 L Q+ 21 The eigenvalue j ( j + 1) of J Q is j =Q−

1 + n (n = 0, 1, 2, . . .). 2

(3.18)

The massless Dirac Hamiltonian on the Haldane sphere is HD = θ σx + φ σ y  (Q+ 1 ) 1 0 −ið− 2 = , (Q− 21 ) R −ið+ 0

(3.19)

where σ is the Pauli matrix. Also, ð±Q is ð±Q = ∂θ ∓ i Q cotθ ± i

1 ∂φ sinθ

(3.20)

3.1 Haldane Sphere

41

and satisfies the below relation: ð+Q+1 ð−Q − ð−Q ð+Q+1 = −2Q ð+Q+1 ð−Q

+

ð−Q ð+Q+1

=

−2(L 2Q

(3.21a)

− Q ). 2

(3.21b)

Because H D commutes with J 2Q and JQz , the eigenstate |n; j, m, Q satisfies the following relations: H D |n; j, m, Q = n |n; j, m, Q

(3.22)

J 2Q |n; JQz |n;

j, m, Q = j ( j + 1)|n; j, m, Q

(3.23)

j, m, Q = m|n; j, m, Q.

(3.24)

From Eqs. (3.21a) and (3.21b), the Hamiltonian can be rewritten as HD2

−1 = 2 R 1 = 2 R =

1 R2

 



(Q+ 21 ) (Q− 21 ) ð+

ð−

2

1 4

−Q

0

 J 2Q +

(Q− 21 ) (Q+ 21 ) ð− 2

ð+

0 L 2Q− 1 +

0

1 − Q2 4





0

L 2Q+ 1 + 2

1 4

− Q2 (3.25)

and its eigenvalue is given as

1 1 n = ± j ( j + 1) + − Q 2 R 4 1 =± n(2Q + n). R

(3.26)

The Landau quantization in the Dirac particle system includes the positive and the negative energy levels because of the chiral symmetry. In the following, the Landau levels are represented by n = 0, ±1, ±2 . . ., and are rewritten as n =

sgn(n)  |n|(2Q + |n|). R

(3.27)

When the radius of the√Haldane sphere is going to infinity, the Landau energy level is given by n ≈ sgn(n) 2|n|/l B , which is consistent with the eigenenergy on a plane. Owing to − j ≤ m ≤ j, each Landau level has 2 j + 1 degeneracy. The eigenstates in n = ±1, ±2, . . . are

42

3 Model and Method

  1 1 1 | ± n; j, m, Q = √ |n; j, m, Q − | ↑ ∓ i|n − 1; j, m, Q + | ↓ , 2 2 2 (3.28) where θ, φ|n; j, m, Q is the monopole harmonics defined in Eq. (3.15). Here, we define | ↑ and | ↓ as     1 0 | ↑ = , | ↓ = , (3.29) 0 1 respectively. In the Dirac particle system, | ± n; j, m, Q is given by the eigenstates of n and n − 1 Landau levels in the 2DEG system. When n = 0, the eigenstate is given as 1 (3.30) |n = 0; j, m, Q = |n = 0; j, m, Q − | ↑. 2 Thus, the eigenstate in n = 0 is identical with the one in the 2DEG system as shown in Eq. (3.15). Here, we consider the effects of the mass term in the Landau quantization. The mass term is defined as (3.31) HM = Mσz and the Hamiltonian is H = H D + HM . Since the mass term does not commute with σz , the chiral symmetry is broken in the Dirac Hamiltonian. The Landau levels are given by E n = ± 2n + M 2 (n = 1, 2, . . .) E n=0 = +M

(n = 0).

(3.32a) (3.32b)

Figure 3.2 shows the mass dependence of the Landau energy levels. When M = 0, the energy levels are symmetric about the n = 0 energy level. By contrast, in M = 0, the lowest Landau level appears in only the positive side. The eigenstate in n = 0 is given as 

E n + n En   1 M |n; j, m, Q − | ↑ × 1± E n + n 2    1 M |n − 1; j, m, Q + | ↓ . ∓i 1∓ E n + n 2

|n; j, m, Q, M±En

1 = 2

(3.33)

In n = 0, the eigenstate is 1 | j, m, Q, M E0 = | j, m, Q − | ↑. 2

(3.34)

3.2 Lanczos Method

43

Fig. 3.2 Landau levels in the Dirac particle system. The vertical axises correspond to the Landau level energy. The integer n N R means the index of the Landau level n. a and b Show the energy levels in the massless and the massive Dirac Hamiltonian, respectively. c Depicts the Landau levels at the large-mass limit. Reprinted figure with permission from [7]. Copyright 2016 by the American Physical Society

3.2 Lanczos Method To obtain the eigenstates of the many body Hamiltonian, we use the Lanczos algorithm. The Lanczos method achieve the lowest eigenvalue and its eigenstates with high accuracy. For a real symmetric matrix A, the tridiagonal matrix T is given as ⎛

⎞ a1 b1 ⎜ b1 a2 b2 ⎟ ⎜ ⎟ ⎜ ⎟ T = PT AP = ⎜ ... ... ... ⎟. ⎜ ⎟ ⎝ bk−2 ak−1 bk−1 ⎠ bk−1 ak−1

(3.35)

Here, P is the orthogonal matrix and is given by column vectors x k as follows P = (x 1 x 2 · · · x N ). When P T = A P, Eq. (3.35) can be rewritten as

(3.36)

44

3 Model and Method

⎞ a1 b1 ⎟ ⎜ b1 a2 b2 ⎟ ⎜ ⎟ ⎜ .. .. .. A(x 1 x 2 · · · x N ) = (x 1 x 2 · · · x N ) ⎜ ⎟. . . . ⎟ ⎜ ⎝ bk−2 ak−1 bk−1 ⎠ bk−1 ak−1 ⎛

(3.37)

Solving the above equation yields the following recurrence formula: Ax 1 = a1 x 1 + b1 x 2 Ax 2 = b1 x 1 + a2 x 2 + b2 x 3 .. . Ax k = bk−1 x k−1 + ak x k + bk xk + 1 .. . Ax N = b N −1 x N −1 + a N x N .

(3.38)

Taking the inner product, ak is given by ak = x kT Ax k .

(3.39)

The off-diagonal element bk and the orthogonal matrix P can be determined with the recurrence formula in Eqs. (3.38) and (3.39). For k = 0, vk is given by

If k = 0, vk is

v k = Ax 1 − a1 x 1 .

(3.40)

v k+1 = Ax k − (bk x k−1 + ak x k ),

(3.41)

where vk+1 = bk x k+1 , ||x k+1 || = 1. Therefore, bk and x k+1 are bk = ||v k+1 || x k+1 = v k+1 /bk .

(3.42) (3.43)

When we assume x 1 is normalized, the initial vector can be arbitrarily chosen. However, if the ground state does not include x 1 , we cannot achieve the lowest eigenstate using the Lanczos method. In the present work, we initialize x 1 with a random vector. Since it is not necessary to calculate up to k = N , we can terminate the calculations at bk 1. Therefore, the dimension of the tridiagonal matrix T is much lower than the one of the original Hamiltonian matrix.

References

45

References 1. Arciniaga M, Peterson MR (2016) Landau level quantization for massless dirac fermions in the spherical geometry: graphene fractional quantum hall effect on the haldane sphere. Phys Rev B 94:035105 2. Haldane FDM (1983) Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states. Phys Rev Lett 51:605–608 3. Fano G, Ortolani F, Colombo E (1986) Configuration-interaction calculations on the fractional quantum hall effect. Phys Rev B 34:2670–2680 4. Wu TT, Yang CN (1976) Dirac monopole without strings: monopole harmonics. Nucl Phys B 107(3):365–380 5. Wu TT, Yang CN (1977) Some properties of monopole harmonics. Phys Rev D 16:1018–1021 6. Nakahara M (2003) Geometry, topology and physics, 2nd edn. Institute of Physics Publishing 7. Yonaga K, Hasebe K, Shibata N (2016) Formulation of the relativistic quantum hall effect and parity anomaly. Phys Rev B 93:235122

Chapter 4

Mass Term Effects in Spinless Dirac Particle System

4.1 Model We introduce the effective Hamiltonian with the pseudopotential. The Hamiltonian of the Dirac particle system in a magnetic field is H = HD (M) + HC ,

(4.1)

where HD represents the Dirac Hamiltonian on the Haldane sphere. The Coulomb interaction HC is defined as  V (ri − r j ), (4.2) HC = i< j

where V (ri − r j ) is given by 1/|r i − r j |, and ri represents the coordinates of the ith particle. We focus on the single Landau level around the Fermi level. To find the effective Coulomb interaction, we project HC onto the nth Landau level. The projected Coulomb potential is called the pseudopotential. We define the projection operator Pi,n j (m) which projects Dirac particles onto the nth Landau level with the relative angular momentum m. Note that the relative angular momentum m is discrete because the angular momentum is quantized when the Landau quantization occurs. After we project the Coulomb interaction, the effective Hamiltonian is given as HC =



V n (m)Pi,n j (m),

(4.3)

i, j

where Vmn represents the pseudopotential at the nth Landau level. In the following, we ignore the kinetic term because HD (M) is constant in the magnetic field. For an illustrative example, we consider the special pseudopotential that creates the Laughlin state at ν = 1/3. The Laughlin state is the exact solution when the pseudopotential is given as © Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_4

47

48

4 Mass Term Effects in Spinless Dirac Particle System

V (m) = 0 (m = 1) V (m) = 0 (else). Equation (4.4) means that the relative angular momenta of all the particles is greater than three. This is because all the particles are forbidden from approaching each other by the strong Coulomb repulsion. In particular, when the short-range part, V (1) − V (3), is large enough, the ground state is given by the Laughlin wave function. Here, we formulate the specific form of the pseudopotential on the Haldane sphere. We assume that the magnetic monopole of the magnitude Q is placed at the center of the sphere. We introduce the projection operator which projects the Dirac particles onto the nth Landau level. The Dirac particles on the Haldane sphere in the magnetic field have the angular momentum jz where − j ≤ jz ≤ j. The total angular momentum of the two particles, 2 j, is given as 2 j = J + m,

(4.4)

where J and m correspond to the angular momentum of the center of mass and the relative angular momentum, respectively. With the projection operator PJn (i, j), the effective Hamiltonian is  VDn (J )PJn ( p, q). (4.5) H = HD (M) + p 0 without loss of generality. When the mass term is finite, the FQHSs in n = −1 and n = +1 are not equivalent because the chiral symmetry is broken. Here, the mass term is normalized by n which is the singleparticle energy in M = 0. In M > 0, VDn=−1 has the local minimum at m = 1 whereas VDn=−1 decays monotonically. Figure 4.5a and b show the mass dependence of the excitation gap c and the overlap 0 |L  at νn=±1 = 1/3. In n = −1, the ground state is given by the Laughlin wave function because 0 |L  is almost one for all M/n . In n = +1, 0 |L  is less than one and the excitation gap decays rapidly with increasing of the mass term (Fig. 4.5b). Thus, the ground state is no longer the

56

4 Mass Term Effects in Spinless Dirac Particle System

Fig. 4.3 Ne dependence of the excitation gap at νn=+1 = 1/3. a and b show the excitation gap scaled by l B and l B , respectively. The red (black) circles in (a) and (b) represent the numerical results from the Dirac particle (2DEG) system. Adapted figures with permission from [5]. Copyright 2016 by the American Physical Society

Laughlin state in n = +1. To understand our results more clearly, we consider the pseudopotential in M → ∞. In this limit, the single-particle state in n = 1 is 1 |n; j, m, Q, M ∼ |n; j, m, Q − | ↑. 2

(4.31)

In n = −1, the single particle state is 1 |n; j, m, Q, M ∼ |n − 1; j, m, Q + | ↓. 2

(4.32)

Here, |n; j, m, Q − 21  and |n − 1; j, m, Q + 21  correspond to the eigenstate of the 2DEG system in the n = 1 and n = 0, respectively. As the mass term increases, |n; j, m, Q, M approaches the single-particle state of the 2DEG system asymptotically. Compared to Figs. 4.1 and 4.4, the pseudopotential VDn=−1 (m) in n = −1 shows

4.2 Results and Discussion

57

Fig. 4.4 Mass dependence on the pseudopotentials in n = −1 (a) and n = +1 (b). Adapted figures with permission from [5]. Copyright 2016 by the American Physical Society

n=0 the monotonic decay, which is similar to V2DEG (m). By contrast, VDn=+1 in M > 0 has n=1 the local minimum which can be seen in V2DEG (m). As a result, the Laughlin wave function cannot be the ground state in n = +1 because the short-range repulsion is reduced by the mass term.

4.2.2 Pfaffian State and Fermi Liquid at νn=+1 = 1/2 According to the CF theory, the ground state is the Pfaffian state or Fermi liquid at νn=+1 = 1/2. In the Dirac particle system at M = 0, the ground state is the gapless Fermi liquid because the paired state of the CFs is destroyed by the strong short-

58

4 Mass Term Effects in Spinless Dirac Particle System

Fig. 4.5 a Overlap between the ground state and the Laughlin wave function. b Mass dependence on the excitation gap in the thermodynamic limit. The mass term, M is normalized with n which is the single-particle energy when M = 0. The insets in (a) and (b) represent the results for 0 ≤ M/n ≤ 3. Adapted figures with permission from [5]. Copyright 2016 by the American Physical Society

range repulsion [7]. In M = 0, the pseudopotentials are dramatically affected by the mass term. As mentioned in the previous subsection, the pseudopotential in n = +1 has the local minimum at m = 1. This local minimum enhances the pairing of CFs. Therefore, we can expect the Pfaffian state appears in the massive Dirac particle system. We investigate the ground state at νn=+1 = 1/2 in the massive Dirac particle system with the exact diagonalization. Since there are two candidates for the ground states at νn=+1 = 1/2, δ in Eq. (4.20) depends on the Fermi liquid or the Pfaffian state. The Fermi liquid state appears at N = 2Ne − 2,

(4.33)

4.2 Results and Discussion

59

Fig. 4.6 Mass dependence on the lowest energy at νn=+1 = 1/2. The black (red) circles correspond to the numerical results for the Pfaffian state (Fermi liquid)

Fig. 4.7 Mass dependence on the excitation gap in the thermodynamic limit at νn=+1 = 1/2

60

4 Mass Term Effects in Spinless Dirac Particle System

and the Pfaffian state is realized at N = 2Ne − 3.

(4.34)

Figure 4.7 shows the mass dependence of the energy per particle in the thermodynamic limit. When the mass term is small, the ground state is the gapless Fermi liquid. The phase transition occurs from the Fermi liquid into the Pfaffian state around M/n ≈ 2.7. This is because the local minimum appears at m = 1 in the pseudopotential as M/n increases. We find that the Pfaffian state is enhanced by the mass term.

4.3 Summary In this chapter, we investigated the ground and its excitation gap to reveal the mass term effect on the FQHE. With the exact diagonalization, we obtained the following results: • At νn=±1 = 1/3, the ground state in n = −1 is characterized by the Laughlin wave function. In n = +1, the excitation gap decays rapidly as the mass term increases. Thus, we find that the Laughlin state is not realized in n = +1 and M > 0. • At νn=+1 = 1/2, the ground state is the Fermi liquid when the mass term is small. As the mass term increases, the phase transition occurs and the Pfaffian state can be seen. In this chapter, we have assumed M > 0: however, the results are qualitatively the same in M < 0. For example, when M < 0, the Laughlin state appears in n = +1 and cannot be realized in n = −1.

References 1. Wu TT, Yang CN (1976) Dirac monopole without strings: monopole harmonics. Nucl Phys B 107(3):365–380 2. Wu TT, Yang CN (1977) Some properties of monopole harmonics. Phys Rev D 16:1018–1021 3. Haldane FDM (1983) Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states. Phys Rev Lett 51:605–608 4. Wen XG, Zee A (1992) Shift and spin vector: new topological quantum numbers for the hall fluids. Phys Rev Lett 69:953–956 5. Yonaga K, Hasebe K, Shibata N (2016) Formulation of the relativistic quantum hall effect and parity anomaly. Phys Rev B 93:235122 6. Morf RH, d’Ambrumenil N, Das Sarma S (2002) Excitation gaps in fractional quantum hall states: an exact diagonalization study. Phys Rev B 66:075408 7. Wojs A, Moller G, Cooper NR (2011) Composite fermion dynamics in half-filled landau levels of graphene. Acta Phys Polon A 119:592

Chapter 5

Spin, Valley, and Mass Effects on Fractional Quantum Hall States

5.1 Model The lattice constants of graphene and h-BN are almost the same, and there are three types of stacking structures. We show the inequivalent stacking of graphene on h-BN in Fig. 5.1. The previous study reported that the structure (c) is the most stable [1]. In all the structures (a), (b), and (c), the carbon atoms A and B of graphene are affected by different on-site potentials from h-BN. In this chapter, we define the simple model as (5.1) Hkin = Htb + Hsub , where Htb corresponds to the tight-binding Hamiltonian of graphene. The substraint effect, Hsub , is given as Hsub = M

  † c A (R)c A (R) − c†B (R)c B (R) .

(5.2)

R

The effective Hamiltonians around the K and K  points are written as  HK = vF

   M px − i p y M − px − i p y , HK  = vF . px + i p y −M − px + i p y −M

(5.3)

This approximation is valid when the twist angle between graphene and h-BN  is small [2]. The energy dispersions of HK and HK  are given as E( p) = ±vF p 2 + M 2 . When we apply a magnetic field into HK and HK  , the effective Hamiltonians are written as √ √     2vF M a 2vF M −a †  , H , (5.4) (M) = HK (M) = K −a −M a † −M lB lB

© Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_5

61

62

5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Fig. 5.1 Graphene on h-BN. The red, blue and black circles represent nitrogen, boron, and carbon atoms, respectively. a AA-stacking. b and c AB-stacking. Reprinted figures with permission from [1]. Copyright 2007 by the American Physical Society

where a (a † ) is the annihilation (creation) operator. Here, the Landau level is given as   E nK (K ) = sgn(n) n2 + M 2 (n = ±1, ±2, . . .), (5.5) where n is the single-particle energy at M = 0. The Landau levels at K and K  points in n = 0 are as E 0K = +M  E 0K

= −M,

(5.6a) (5.6b)

respectively. As shown in Fig. 5.2, the valley degeneracy is lifted by the spatial inversion symmetry breaking. The eigenstates for HK (M) and HK  (M) in n = 0 are

5.1 Model

63

Fig. 5.2 Energy spectrums in with/without the mass term. Reprinted figure with permission from [3]. Copyright 2018 by The Physical Society of Japan

  1 (E n ± M)|n − 1, m |n, m K = √ ±n |n, m 2E n (E n ± M)   1 (E n ± M)|n, m . |n, m K  = √ 2E n (E n ± M) ∓n |n − 1, m

(5.7a) (5.7b)

In n = 0, the eigenstates are given as 

   0 |0, m |n, m K = , |n, m K  = . |0, m 0

(5.8)

The eigenstates in n = 0 are not modified by the mass term. We introduce the projected Coulomb potential to study the FQHS. In the following, we represent the spin and valley with s =↑, ↓ and σ = K , K  , respectively. The projected Coulomb interaction in the nth Landau level is given as HC =

  i< j



Vσns,σ s  (m)Pi,σ s, j,σ s  (m).

(5.9)

m s,s  =↑,↓ σ,σ  =K ,K 

Here, Piσ s, jσ  s  (m) is the projection operator onto the nth Landau level, and Vσns,σ  s  (m) is the pseudopotential between the particles with the relative angular momentum m. The total Hamiltonian is written as H = HK + HK  + HC + z Sˆz + v Tˆz ,

(5.10)

where the first and second term represent the kinetic Hamiltonians at K and K  points, respectively. In addition, z Sˆz and v Tˆz correspond to the Zeeman effect, and the valley-Zeeman effect which are defined as

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5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Sˆz =

j/2 



† † cm,σ,↓ cm,σ,↓ − cm,σ  ,↑ cm,σ  ,↑

(5.11)

 † † cm,K ,s cm,K ,s − cm,K .  ,s  cm,K  ,s 

(5.12)



m=− j/2 σ,σ  =K ,K 

Tˆz =

j/2 

m=− j/2 s,s  =↑,↓

Generally, however, it is difficult to deal with both of the spin and valley simultaneously. For simplicity, we consider the following two cases: (a) Z e2 /l B < v H=

  i< j

n Vσ,σ (m)Pi,σ s, j,σ s  (m) + z Sˆz (σ = K or K  )

(5.13)

n ˆ Vσ,σ  (m)Pi,σ s, j,σ  s (m) + v Tz (s =↑ or ↓)

(5.14)

s,s  =↑,↓

(b) v e2 /l B < Z H=





i< j

σ,σ  =K ,K 

Here, we omit s and s  since the pseudopotentials do not depend on the spin. In addition, we ignore the effects of Landau level mixing. We use the pseudopotentials on the Haldane sphere. The pseudopotentials on the sphere are defined as VKn K (m) = f 1n V n−1,n−1 (m) + 2 f 2n V n,n−1 (m) + f 3n V n,n (m)

(5.15a)

VKn  K  (m) = f 3n V n−1,n−1 (m) + 2 f 2n V n,n−1 (m) + f 1n V n,n (m) VKn K  (m) = f 2n {V n−1,n−1 (m) + V n,n (m)} + { f 1n + f 3n }V n,n−1 (m) VKn K  (m) = VKn  K (m).

(5.15b) (5.15c) (5.15d)

The structure factors are given by f 1n =

(E n + M)2 4E n2

(5.16a)

f 2n =

n2 4E n2

(5.16b)

f 3n =

n4 2 4E n (E n +

M)2

.

(5.16c)

With the exact diagonalization, we estimate the lowest energy per particle, E 0 , and the excitation gap c in the thermodynamic limit. Our results are normalized as e2 /l B = 1. In this chapter, we consider the FQHSs in only the n = ±1 where

5.1 Model

65

interesting results were observed in the previous experiments [4, 5]. In addition, we assume M > 0 and Z , v > 0 without loss of generality.

5.2 Results and Discussion 5.2.1 Z  e2 /l B < v In Z = 0, the Landau levels have the spin degeneracy. We refer to the upper and lower branch of the Landau levels as n K and n K  , respectively (see Fig. 5.3a). We introduce the Landau level filling factor νn K =±1 and νn K  =±1 for each branch. Figure 5.4a shows the pseudopotentials V n K =+1 (m) and V n K  =+1 (m). We find that V n K =+1 (m) decays monotonically as the relative angular momentum increases

Fig. 5.3 Landau level splitting of graphene; a Z e2 /l B < v b v e2 /l B < Z

Fig. 5.4 Pseudopotentials in Z e2 /l B < v . a V n (m) in n K and n K  = +1. b V n (m) in n K and n K  = −1

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5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

whereas V n K  =+1 (m) has the local minimum at m = 1. By contrast, V n K =−1 (m) has the local minimum while V n K  =−1 (m) shows the monotonic decay as shown in Fig. 5.4b. From Eq. (5.7), the eigenstates in n = +1 are 

 |0, m |1, m K ∼ 0   |1, m  |1, m K ∼ 0

(5.17a) (5.17b)

in M → ∞. Here, |0, m and |1, m correspond to the eigenstates in the 2DEG system. The pseudopotentials in this limit are given as n=0 (m) lim V n K =+1 (m) = V2DEG

(5.18a)

n=1 lim V n K  =+1 (m) = V2DEG (m).

(5.18b)

M→∞ M→∞

In n = −1, the eigenstates are written as  |1, m K ∼  |1, m K  ∼

0 |1, m

 (5.19a) 

0 , |0, m

(5.19b)

and the pseudopotentials are n=1 (m) lim V n K =−1 (m) = V2DEG

(5.20a)

n=0 lim V n K  =−1 (m) = V2DEG (m)

(5.20b)

M→∞ M→∞

in the large mass limit. The single-particle states at K and K  points are inequivalent due to the mass term, and V n K =−1 (m) and V n K  =−1 (m) have different structures. The characteristics of the FQHS are determined by the pseudopotentials, especially its short-range part. Therefore, we can expect that the different ground states can be observed in n K = ±1 and n K  = ±1. We focus on the FQHS in the positive Landau level n = +1. Figure 5.5a shows the mass dependence on the excitation gap at νn K =+1 = 1/3 and νn K  =+1 = 1/3. We also show the overlap between the eigenstate and the Laughlin wave function in Fig. 5.5a. Here, the mass term M is normalized by the absolute of the single-particle energy |n |. In addition, we assume that the FQHSs are fully spin-polarized. As shown in Fig. 5.5a, L |0  is almost one. Therefore, the ground state is characterized by the Laughlin wave function νn K =+1 = 1/3. This is because the strong short-range repulsion creates the stable Laughlin state. In n K  = +1, V n K  =+1 (1) − V n K  =+1 (3) is reduced by the mass term. As discussed in the previous section, when the shortrange repulsion is suppressed, the Laughlin state becomes unstable. In fact, Fig. 5.5a

5.2 Results and Discussion

67

Fig. 5.5 Mass dependence on the overlap between the eigenstate and Laughlin wave function (left) and the excitation gap (right). a Results at νn K =+1 = 1/3 and νn K  =+1 = 1/3. b Results at νn K =−1 = 1/3 and νn K  =−11 = 1/3. The black (red) circles correspond to the computation results at νn K =±1 = 1/3 (νn K  =±1 = 1/3)

shows that c and L |0  at νn K  =+1 = 1/3 decay with the mass increases. In the negative Landau level, the pseudopotential in n K  = −1 shows the monotonic decay. We can see the Laughlin state with the large excitation gap at νn K  =−1 = 1/3. By contrast, at νn K =−1 = 1/3, the excitation gap decays rapidly as the mass increases. We explain the computation results at νn K =+1 = 2/3 and νn K  =+1 = 2/3. At νn = 2/3, the spin-polarized Laughlin and spin-singlet states are the candidates of the ground state. At νn = 2/3 in the spherical geometry, we can obtain the Laughlin state when the magnetic flux N and the number of particles Ne are given as N =

3 Ne . 2

(5.21)

The spin-singlet state is realized when Ne and N are N =

3 Ne − 1. 2

(5.22)

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5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Fig. 5.6 a Mass dependence on the energy of the fully spin-polarized Laughlin state and the singlet state. a Results at νn K =+1 = 2/3 (left) and νn K  =+1 = 2/3 (right). b Results at νn K =−1 = 2/3 (left) and νn K  =−1 = 2/3 (right). Solid and dotted lines represent the energies of the Laughlin and singlet states, respectively

The left (right) panel in Fig. 5.6a shows that the mass dependence on the energies of the spin-polarized Laughlin and spin-singlet states at νn K =+1 = 2/3 (νn K  =+1 = 2/3). In the massless limit, the Laughlin and singlet state are almost degenerate. When the mass term is finite, the energy of the spin-singlet state becomes lower than that of the Laughlin state. According to the previous work [6], the singlet state can be stably realized when V (m = 1) − V (m = 2) is sufficiently large, whereas the Laughlin state depends on V (m = 1) − V (m = 3). In n K = +1, even when the mass term is finite, the pseudopotential decays monotonically and V n K =+1 (m = 1) − V n K =+1 (m = 2) keeps a large value as shown in Fig. 5.4a. Thus, the singlet state is the ground state at νn K =+1 = 2/3. At νn K  =+1 = 2/3, the ground state is the fully spin-polarized state, not singlet state due to V n K  =+1 (m = 1) < V n K  =+1 (m = 2). The excitation gap of this polarized state is not large because V n K  =+1 (m) has the local minimum at m = 1. In the following, we refer to this polarized state as “unstable spin-polarized state” whose excitation gap decays with increasing of the mass. In n = −1, V n K =−1 (m) has the local minimum while V n K  =−1 (m) decays monotonically. Therefore, the ground state at νn K =−1 = 2/3 and νn K  =−1 = 2/3 are the unstable spin-polarized and spin-singlet state, respectively (see Fig. 5.6b).

5.2 Results and Discussion

69

Fig. 5.7 Phase diagrams at νn K =±1 = 1/3 and νn K =±1 = 2/3. a and b Represent the results in n = +1 and n = −1, respectively. The vertical axis represents the Zeeman term, Z . The horizontal line corresponds to the mass term, M. Z and M are normalized with e2 /l B and |n |, respectively. FP(stable) is the fully spin-polarized Laughlin state with the large excitation gap. FP(unstable) is the spin-polarized state whose excitation gap decays with increasing of the mass term

In the following, we consider the Zeeman effect in Z > 0. Figure 5.7a shows Z M phase diagram in n = +1. Here, Z and M are normalized with e2 /l B and |n |, respectively. At νn K =+1 = 1/3, the ground state is the fully spin-polarized Laughlin state labeled as FP(stable) in Z > 0. At νn K =+1 = 2/3, the ground state in the small Z region is the spin-singlet state, while the Laughlin state can be seen in the large Z region. We find that the phase transition occurs from the singlet state into the spinpolarized Laughlin state. FP(unstable) in Fig. 5.7a means the unstable spin-polarized

70

5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Fig. 5.8 Pseudopotentials at M/|n | = 3 in n ↑ = +1 (a) and n ↑ = −1 (b). Black, red, and blue lines correspond to VKn K (m), VKn  K  (m) and VKn  K (m), respectively

ground state. We also find that the unstable spin-polarized state is the ground one at νn K  =+1 = 1/3 and νn K  =+1 = 2/3. Figure 5.4b shows the phase diagram in the negative Landau level. We can see the Laughlin state, which is labeled as FP(stable), at νn K  =−1 = 1/3. Moreover, the phase transition occurs from the singlet state into the Laughlin state occurs at νn K  =−1 = 2/3. As shown in Fig. 5.7a and b, the phase diagrams in n K = +1 and n K  = −1 (n K = −1 and n K  = +1) are identical.

5.2.2 v  e2 /l B < Z In this subsection, we consider another limit as v e2 /l B < Z . Due to the Zeeman effect, the Landau levels are separated into two branches, n ↑ = ±1 and n ↓ = ±1 as shown in Fig. 5.3b. In v = 0, each branch of the Landau levels has the valley degeneracy. Figure 5.8a shows the pseudopotentials at M/|n | = 3 in n ↑ = +1. Since the pseudopotentials does not depend on the spin, we show only the results of n ↑ = ±1. When the mass term is finite, as mentioned in Eq. (5.17), the eigenstate at the K point is localized whereas the wave function at K  point is spatially n =+1 n =+1 extended. The pseudopotential VK ↑K (m) decays monotonically, and VK ↑ K  (m) has the local minimum at m = 1 in M > 0. By contrast, In n = −1, VKn=−1  K  (m) shows the monotonic decay as shown in Fig. 5.8b. Here, let us consider the ground state in v = 0. Figure 5.9a shows the ground state energy for the K - and K  -polarized Laughlin states at νn ↑ =+1 = 1/3. Because of the valley degeneracy, these two states have the same energy. In M > 0, the spatial inversion symmetry is broken, and the K -polarized Laughlin state becomes the ground one. The inset in Fig. 5.9a displays the mass dependence on the excitation gap c . The excitation gap of the K -polarized state remains large, while the one of K  -polarized state decays rapidly as M increases. Thus, the Laughlin state is

5.2 Results and Discussion

71

Fig. 5.9 Mass dependence on the energy of the valley-polarized state. a and b Show the results at νn ↑ =+1 = 1/3 and νn ↑ =−1 = 1/3, respectively. The insets in a and b show the effect of mass on the excitation gap c . The black (red) line represent the valley K -polarized (K  -polarized) state

Fig. 5.10 Mass dependence on the energy at νn ↑ =+1 = 2/3 (a) and νn ↑ =−1 = 2/3 (b). Black, red, and blue lines represent the energy for the K -polarized, K  -polarized, and the valley-singlet states, respectively

stably realized in M > 0 because VKn=+1  K  (m) has the strong short-range repulsion. In n = −1, the pseudopotential VKn=−1  K  (m) has the strong short-range repulsion as shown in Fig. 5.8b. Therefore, the K  -polarized Laughlin state is the ground one in n = −1. At νn ↑ =+1 = 2/3, the valley-polarized Laughlin state with (Tz = K or K  ) and the valley-singlet state with (Tz = 0) are the candidates of the ground state. Figure 5.10a shows the mass dependence on the energy of K -polarized, K  -polarized and the valley-singlet state at νn ↑ =+1 = 2/3. To obtain the Laughlin state and valley-singlet state, we use the Landau level filling factors defined as Eqs. (5.21) and (5.22). In M = 0, the K -polarized and K  -polarized states are degenerate because of the spatial inversion symmetry. In addition, the energies of the valley-singlet and valley-

72

5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Fig. 5.11 Ground state phase diagrams in n = +1 (a) and n = −1 (b). The horizontal and vertical axis show M and v , respectively. K-FP (stable) and K’-FP (stable) are the valley K -polarized and K  -polarized Laughlin states with large excitation gap. K-FP (unstable) and K’-FP (unstable) are the K -polarized and K  -polarized states whose excitation gaps decay with increasing of the mass

polarized states are almost the same in M = 0. When the mass term is finite, the K -polarized Laughlin state becomes the ground state. In n = −1, the valley K  polarized Laughhlin state is the ground state as shown in Fig. 5.10b. Now, we discuss the ground state phase diagram in v > 0. Figure 5.11a shows the phase diagrams in n ↑ = +1 and n ↓ = +1. Here, M and v are normalized by |n | and e2 /l B , respectively. In the small v region at νn ↑ =+1 = 1/3, the ground state is the valley K -polarized Laughlin one with the large excitation gap. We label this ground state as K-FP (stable) in Fig. 5.11. As the valley-Zeeman term v increases,

5.2 Results and Discussion

73

Fig. 5.12 v dependence on the energy at νn ↑ =+1 = 2/3 in M/|n | = 3. Black, red, and blue lines correspond to the energies of the K -polarized, K  -polarized, and valley-singlet states, respectively

the phase transition occurs from the K -polarized to K  -polarized state. Since the excitation gap of the K  -polarized state decays rapidly with increasing of the mass term, we refer to this state as K’-FP(unstable) in the phase diagram. We next focus on the ground state at νn ↑ =+1 = 2/3. Here, we display the v dependence on the energy of the valley K -polarized, K  -polarized, and the valleysinglet state at M/|n | = 3 in Fig. 5.12. We find that the phase transition occurs from the K -polarized to K  -polarized state, and that the valley-singlet state is not realized in all v region. The K  -polarized state at νn ↑ =+1 = 2/3 is identical with the one at νn ↑ =+1 = 1/3. As a result, the phase diagram at νn ↑ =+1 = 2/3 also includes two region; K-FP(stable) and K’-FP(unstable). Figure 5.11b shows the M − v ground state phase diagram in the negative Landau level n = −1. The pseudopotential VKn=−1  K  (m) has the strong short-range repulsion. In v = 0, the ground states at νn ↑,↓ = 1/3 and 2/3 are given as the K  polarized Laughlin states with the large excitation gap. In this study, since we define the valley-Zeeman effect as Eqs. (5.10) and (5.12), the ground state is given by the K  -polarized Laughlin wave function even in v > 0. Therefore, the M − v phase diagram in n = −1 shows only the K  -polarized Laughlin state which is labeled as K’-FP(stable). We conclude that the ground state phase diagrams in n = +1 and n = −1 are significantly different in the limit as v e2 /l B < Z .

5.2.3 Comparison with Experimental Results In this chapter, we have investigated the mass effect on the FQHS with the spin and valley in the two cases as Z e2 /l B < v and v e2 /l B < Z . Here, we compare our results and previous experiments. The stability of the IQHE at ν = ±4 depends on the strength of the transverse magnetic field [7]. This result indicates that the Landau levels are split by the Zeeman effect. From this experiment, v

74

5 Spin, Valley, and Mass Effects on Fractional Quantum Hall States

Fig. 5.13 a Ripples and effective magnetic field induced by lattice distortions in graphene. b Modification of the hopping integral in the tight-binding model. c Dirac cone around K and K  points with lattice distortion. Reprinted figures with permission from [8]. Copyright 2015 by the American Physical Society

e2 /l B < Z is appropriate to understand the QHE in graphene. In the following discussion, we focus on the M − v phase diagram. At v = 0, we can see the Laughlin state with c ∼ 0.1e2 /l B in both n = ±1. This result means that the FQHE can be observed in both of the electron and hole doped regime of graphene. However, Dean et al. and Amet et al. reported that the FQHE of graphene on h-BN occurs clearly only in n = −1 (hole regime) [4, 5]. To explain this previous experiment, it is important to consider the valley-Zeeman effect. The M − v phase diagram in n = +1 includes the two states: K-FP(stable) and K’FP(unstable). Since the excitation gap in K’-FP(unstable) decays with increasing of the mass, the K  -polarized state is easily smeared by the disorders in graphene. Therefore, if v is large enough, we can observe the FQHE in only the hole regime. Here, we discuss the origins of the valley-Zeeman effect and estimate its energy scale. One origin of the valley-Zeeman effect is the lattice distortion and ripples of graphene as shown in Fig. 5.13a. When we consider such lattice distortion, the tight-binding model in zero magnetic field is written as H =−

 R,i

  (t + δti,R ) (c†A (R)c B (R) + c†A (R + ai )c B (R)) + h.c. .

(5.23)

5.2 Results and Discussion

75

Fig. 5.14 a Superlattice of graphene on h-BN. Here, graphene and h-BN are aligned and the ratio of lattice constants is ahBN /a = 10/9. b Brillouin zone folding in the superlattice. c Relative positions of K and K  points. Reprinted figures with permission from [2]. Copyright 2015 by the American Physical Society

In this case, the Dirac Hamiltonians around K and K  points are given as [8]: HK HK 

  0 px − i p y + a K = vF px + i p y + a K 0   0 − px − i p y + a K  = vF . − px + i p y + a K  0

(5.24) (5.25)

Thus, the lattice distortion behaves as the vector potentials a K and a K  in the effective Hamiltonians, and the particles around the K and K  points feel the effective magnetic field. Note that the effective magnetic fields at the K and K  points are in the opposite directions because graphene has the time reversal symmetry. The Landau quantization also occurs when we apply the magnetic field into distorted graphene. Here, the valley degeneracy in the Landau levels is lifted since the net magnetic fields at K and K  points are different. This energy difference between the K and K  points can be interpreted as the valley-Zeeman effect. The lattice distortion in graphene is random, and we require approximation approaches to estimate the effective magnetic fields. Abanin et al. revealed that the effective magnetic field is given as 0.1 ∼ 1 [T] with assuming random distortion [9]. This value is consistent with the another result estimated by the transport experiments [10]. The previous experimental and theoretical studies support the existence of the valley-Zeeman effect. Another origin of the valley-Zeeman effect is the superlattice effects. Although we have assumed that the lattice constants of graphene and h-BN are the same, they slightly differ by about 1.8%. Because the graphene layer is stacked with a twist on a h-BN surface, we can see the superlattice as shown in Fig. 5.14a [2]. In the

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superlattice, the Brillouin zone is folded because the unit cell becomes effectively larger, see Fig. 5.14b and c. Moon and Koshino investigated the effective model of graphene on h-BN with the superlattice effects. They revealed that the effective model around K (K  ) is given as the Dirac Hamiltonian with the additional gauge fields a K (R) (a K  (R)) [2]. They also numerically showed that the valley degeneracy in the Landau levels is lifted when a magnetic field is applied. Here, the magnitude of the effective field in the actual experiment is very small, which is about 0.03 [T]. In addition, the effective magnetic fields induced by the superlattice depend on the spatial coordinates. Therefore, to understand the FQHE in graphene on h-BN, it is necessary to study the effect of the space-dependent magnetic field.

5.3 Summary In this chapter, we have focused on the two cases: Z e2 /l B < v and v e2 /l B < Z . We investigated the ground states and excitation gaps by numerical exact diagonalization. The results are summarized as follows: • In Z e2 /l B < v , the valley degeneracy is lifted. We obtained the new Landau levels as n K = ±1 or n K  = ±1; each branch has the spin degrees of freedom. At νn K =+1 = 1/3 and νn K  =−1 = 1/3, the spin-polarized Laughlin states are realized with the large excitation gap. At νn K =+1 = 2/3 and νn K  =−1 = 2/3, we found the spin-polarized Laughlin and the spin-singlet state. • In the case of v e2 /l B < Z , the Landau levels n ↑ = ±1 and n ↓ = ±1 emerge, which have the valley degrees of freedom. In n = +1, the Laughlin state is stably realized in the region where v is small. When v becomes large, the phase transition occurs and the valley polarized state with a small excitation gap emerges. In n = −1, the ground state is the valley-polarized Laughlin state with the large excitation gap for all v . We reveal that the phase diagram in n = +1 and n = −1 is significantly different due to the mass term and the valley Zeeman effect. Our results indicate that, we can observe the FQHE clearly only in n = −1 in actual experiments. The previous studies reported that only the FQHE in n = −1 can be clearly observed in graphene on h-BN. For such a phenomenon to occur, the situation where v Z and v > 0 is appropriate. To understand the previous studies in more detail, it is necessary to study the FQHE incorporating the effect of the lattice distortion.

References 1. Giovannetti G, Khomyakov PA, Brocks G, Kelly PJ, van den Brink J (2007) Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations.

References

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Phys Rev B 76:073103 2. Moon P, Koshino M (2014) Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys Rev B 90:155406 3. Yonaga K, Shibata N (2018) Fractional quantum hall effects in graphene on a h-bn substrate. J Phys Soc Jpn 87:034708 4. Dean CR, Young AF, Cadden-Zimansky P, Wang L, Ren H, Watanabe K, Taniguchi T, Kim P, Hone J, Shepard KL (2011) Multicomponent fractional quantum hall effect in graphene. Nat Phys 7:693–696 5. Amet F, Bestwick AJ, Williams JR, Balicas L, Watanabe K, Taniguchi T, Goldhaber-Gordon D (2015) Composite fermions and broken symmetries in graphene. Nat Commun 6:5838 6. Shibata N, Nomura K (2009) Fractional quantum hall effects in graphene and its bilayer. J Phys Soc Jpn 78(10):104708 7. Zhang Y, Jiang Z, Small JP, Purewal MS, Tan Y-W, Fazlollahi M, Chudow JD, Jaszczak JA, Stormer HL, Kim P (2006) Landau-level splitting in graphene in high magnetic fields. Phys Rev Lett 96:136806 8. Yang B (2015) Dirac cone metric and the origin of the spin connections in monolayer graphene. Phys Rev B 91:241403 9. Abanin DA, Lee PA, Levitov LS (2007) Randomness-induced x y ordering in a graphene quantum hall ferromagnet. Phys Rev Lett 98:156801 10. Morozov SV, Novoselov KS, Katsnelson MI, Schedin F, Ponomarenko LA, Jiang D, Geim AK (2006) Strong suppression of weak localization in graphene. Phys Rev Lett 97:016801

Chapter 6

Mass and Valley Effects on Excitations in Quantum Hall States

6.1 Model The Landau levels n = ±1 have four-fold degeneracy due to the spin and the valley degrees of freedoms. In this chapter, we assume that this four-fold degeneracy is partially lifted by the Zeeman effect, and each level has only the valley degeneracy, as shown in Fig. 6.1. We represent the Landau level as n ↑ = ±1 and n ↓ = ±1. The Hamiltonians in n ↑ = ±1 are written as H = HK + HK  + HC   n Vσ,σ HC =  (m)Pi,σs, j,σ  s (m).

(6.1)

i< j σ,σ  =K ,K 

Here, we ignore the valley-Zeeman effect. In the following discussion, we assume that the ground state is the valley-polarized state. Even though the ground state is the polarized state, the lowest excited state can be the valley-unpolarized state. This valley-unpolarized state corresponds to the valley-skyrmion excitation. To investigate such a unique excitation, we study the excitation energy E(Ne , N , γ) of the K (K  )polarized Laughlin state, where γ is the number of electrons at the K  (K ) point. For example, γ = 0 corresponds to the excitation of the fully valley-polarized state, and γ = Ne /2 corresponds to that of the valley-unpolarized state. Thus, the value of γ determines the polarization in the excited state. Here, we define the excitation gap as ± c (Ne , γ) = E(Ne , N ± 1, γ) − E(Ne , N ),

(6.2)

where + (−) represents the creation energy of a quasiparticle (quasihole). Because the number of electrons is conserved, the excitation gap in the thermodynamic limit N → ∞ is given as (6.3) c (γ) = c (Ne → ∞, γ),

© Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_6

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Fig. 6.1 Landau level splitting by the Zeeman energy

− where c (Ne , γ) = + c (Ne , γ) + c (Ne , γ). In this chapter, we focus on only the excited state in n = +1. Since the pseudopotential is independent on electron spin, we study the excited states only in n ↑ = +1. In the following, we ignore the effects of the Landau level mixing and normalize the energy with e2 /l B .

6.2 Results and Discussion 6.2.1 Mass Dependence on Excited State at νn↑ =+1 = 1/3 Figure 6.2 shows the mass dependence on the lowest energy at νn ↑ =+1 = 1/3. Here, M is normalized with n that is the single-particle energy at M = 0. Owing to the inversion symmetry breaking, the K -polarized Laughlin state is the ground one in M > 0. Let us consider the excitation in the K -polarized Laughlin state. Figure 6.3 shows the mass and γ dependence on the excitation gap in the thermodynamic limit. Figure 6.3a and b represent the computation results in the region of 0 ≤ M/n ≤ 4 and 0 ≤ M/n ≤ 0.1, respectively. The value of (γ = 0), which corresponds to the fully valley-polarized excitation gap, is nearly independent of the mass term as shown in Fig. 6.3a. This means that the lowest excitation is characterized by γ = 0 in M/n  1. The excitation gaps for finite γ are strongly affected by the mass term. As shown in Fig. 6.3b, the lowest excited state at M/n = 0 is given as γ = Ne /2, where Ne is the total number of electrons. In this case, since the number of the electrons at K and K  points are equal, the excitation is characterized by the valley-unpolarized state. As M/n increases, the value of c (γ = Ne /2) also grows. We find that the lowest

6.2 Results and Discussion

81

Fig. 6.2 Mass dependence on E 0 at νn ↑ =+1 = 1/3. Black (K-FP) and red (K’-FP) lines represent K polarized and K  -polarized states, respectively. Adapted figure with permission from [1]. Copyright 2018 by The Physical Society of Japan

excitation shifts as c (Ne /2) → · · · → c (γ = 1) → c (γ = 0) with increasing of the mass term. To understand the mass effect on the lowest excitation, we consider the CF theory. According to the CF theory, the Laughlin state is interpreted as the IQHS of CFs as shown in Fig. 6.4a. If we ignore the valley degrees of freedom, the excited state is obtained by adding one CF into the upper Landau level, see Fig. 6.4b. When we consider the valley, we can generate the new excited state by adding one CF in another valley as shown in Fig. 6.4c. This is reason the partially valley-polarized or unpolarized states occur in the excitation of the FQHS. Figure 6.5 shows the quasiparticle excitation in the IQHS. Here, we represent the valleys, K and K  as pseudospins, ↑ and ↓, respectively. The value of m in Fig. 6.5 corresponds to the angular momentum in the Landau level. In the ground state, the pseudospins of the electrons are aligned in the same direction as shown in Fig. 6.5a. Here, we add one electron with pseudospin ↓ to generate an excited state. Figure 6.5b1 and b2 show the excited states when one electron with pseudospin ↓ is attached to m = 0 and m = 1, respectively. The electrons with pseudospin ↓ cannot move because the total angular momentum is conserved. As a result, the electrons are localized, and the total energy rises greatly due to the Coulomb interaction. Next, we consider another excited state. We add one electron with pseudospin ↓ into the ground state and invert one ↑ to ↓. We show the three excited states as shown in Fig. 6.5c1, c2, and c3. Note that total angular momentum is conserved in (c1), (c2), and (c3). Therefore, the structures of (c1), (c2), and (c3) are mixed by the quantum fluctuations, and the electrons with ↓ become spatially itinerant. Thus, the

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6 Mass and Valley Effects on Excitations in Quantum Hall States

Fig. 6.3 Mass and γ dependence on the excitation gap c (γ). a and b Correspond to the results in 0 ≤ M/n ≤ 5 and 0 ≤ M/n ≤ 0.1, respectively. Adapted figures with permission from [1]. Copyright 2018 by The Physical Society of Japan

extra valley-inversions create low-energy excited states with itinerant electrons. This is the reason the lowest excited state at M = 0 is given by c (γ = Ne /2). The excitation at γ = Ne /2 corresponds to the valley-unpolarized state with the small energy gap [2, 3]. With increasing of the mass term, c (γ = Ne /2) grows rapidly as shown in Fig. 6.3. When the mass term is finite, K and K  points are inequivalent because of the spatial inversion symetry breaking. In particular, the n =+1 n =+1 short-range repulsion the value of VK ↑ K  (1) − VK ↑ K  (3) is reduced by the mass effect. Due to decreasing of the short-range repulsion, the electrons at K  point get close easily each other. As a result, the valley-inversion creates the energy cost by the Coulomb interaction with increasing of the mass. In addition, the lowest excitation gap shifts as c (γ = Ne /2) → · · · → c (γ = 2) → c (γ = 1) → c (γ = 0).

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Fig. 6.4 a Sketch of the K -polarized Laughlin state with the CF theory. b and c Represent the quasiparticle excitation of γ = 0 and γ = 1, respectively

6.2.2

Mass Dependence on Excited State at νn↑ =+1 = 1

In the previous subsection, we have discussed the excitation in the FQHS. With similar discussion, we can see the partially valley-polarized or valley-unpolarized excitation in the IQHS. Figure 6.6 shows the effect of M/n on the energies for the K -polarized and K  -polarized states at νn = +1 = 1. We confirm that the K -polarized state is the ground state in M > 0. Figure 6.7 shows the mass dependence on the excitation gap c (γ) at νn ↑ =+1 = 1. The excitation at γ = 1 is given by Fig. 6.5b. Since we ignore the effects of Landau level mixing, the excited state with γ = 0 does not occur. As shown in Fig. 6.7a and b, the valley-unpolarized excitation (γ = Ne /2) is the lowest excitation in the massless limit. As the mass term increases, the short-range repulsion of the pseudopotential decreases and an energy cost is caused by the Coulomb repulsion. As a result, the lowest excitation changes as follows: c (γ = Ne /2) → · · · → c (γ = 2) → c (γ = 1). Thus, the lowest excited state of the IQHS is affected by the mass term, and its polarization increases with M.

84 Fig. 6.5 Sketch of the ground state and the excited states with the valley and angular momentum m. Here ↑ and ↓ correspond to the valley K and K  , respectively. a Shows the fully valley-polarized ground state. b1 and b2 Represent the excited states with adding one electrons in m = 0 and m = 1, respectively. c1, c2 and c3 Show some of the patterns of the excitation in γ = 2

Fig. 6.6 Mass dependence on E 0 at νn ↑ =+1 = 1. Black and red lines correspond to the energy for K -polarized and K  -polarized states, respectively

6 Mass and Valley Effects on Excitations in Quantum Hall States

6.2 Results and Discussion

85

Fig. 6.7 Mass dependence of the excitation gap at νn ↑ =+1 = 1. a and b Show the results for 0 ≤ M/n ≤ 5 and 0 ≤ M/n ≤ 0.1, respectively

6.2.3 Comparison with Previous Experiments According to Shibata et al.’s study, the excited state at νn ↑ =+1 = 1/3 is given by the valley-unpolarized state with the excitation gap c (γ = Ne /2) ≈ 0.05e2 /l B . The energy scale of the disorders in graphene, which is estimated to be about 30 [K], is greater than that of the valley-unpolarized excitation. Therfore, if the excitation is really characterized by the valley-unpolarized state, the FQHE cannot be observed clearly. Here, let us consider the mass effect on the excited state at νn ↑ =+1 = 1/3. In the following, we assume the dielectric constant  = 5, single-particle energy n=±1 = 400 B[T] [K], and mass term M = 200 [K], for simplicity. In this case, the lowest excited state is given by the partially-polarized state with γ = 1, and

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6 Mass and Valley Effects on Excitations in Quantum Hall States

Table 6.1 Excitation gap at νn=+1 = 1/3 for B = 10 [T]. Here, we assume that the dielectric constant  = 5, the single-particle energy n=±1 = 400 B[T] [K], and the mass term M ∼ 200 [K], for simplicity. The same data is published in [1] γ c (γ) [K] 0 1 Ne /2

45 41 21

Fig. 6.8 Magnetic field dependence on the thermal activation energy of the FQHSs in graphene on h-BN. Reprinted by permission from [4], Springer Nature 2010

its excitation gap is c (γ = 1) ≈ 0.1 e2 /l B (see Fig. 6.3). Table 6.1 shows that c (γ = 1) is greater than  all over B. This result means that the Laughlin state can be experimentally observed even in B = 10 [T]. We conclude that the mass term enhances the excitation gap of the Laughlin state. Our result is consistent with the previous experiment reported by Amet et al. [4]. In the final part of this chapter, we discuss the superlattice effect in graphene on h-BN. Figure 6.8 shows the magnetic field dependence on the thermal activation energy [4]. We focus on the experimental results at ν = −13/3 which corresponds to the FQHS at νn ↓ =−1 = 1/3. As has been discussed above, we can see the FQHE at this filling factor because the excitation gap increases by the mass √ term. Since the excitation gap is scaled to e2 /l B , ac should be proportional to B[T]. However, ac starts to decrease when the magnetic field is greater than 15 [T] as shown in Fig. 6.8. One reason for ac decay is the superlattice effect. In a strong magnetic field, the cyclotron radius becomes small and comparable to the lattice constant. In this case, the Landau quantization does not occur, and we observe the unique energy spectrum called the Hofstadter butterfly [6]. Since the lattice constants of h-BN and graphene differ by about 1.8%, the superlattice is formed. The lattice constant of the superlattice is much longer than the one of graphene. As a result,

6.2 Results and Discussion

87

Fig. 6.9 Energy spectrum in graphene on h-BN. The horizontal line corresponds to the magnetic field (T) and the total flux (/0 ) penetrating a unit cell. Reprinted figure with permission from [5]. Copyright 2015 by the American Physical Society

the number of flux quanta penetrating the hexagonal lattice increases greatly, and Hofstadter butterfly occurs in graphene on h-BN. Figure 6.9 shows the magnetic field dependence on the energy of graphene on h-BN [5]. As the magnetic field becomes larger, the degeneracy of the Landau levels is lifted. In fact, Hunt et al. have observed the Hofstadter butterfly experimentally in graphene on h-BN under the strong magnetic field. Thus, the decay of the thermal activation energy can be explained by disappearing of the Landau levels.

6.3 Summary In this chapter, we have investigated the mass effects on the excitation gap in the FQHS and IQHS. We obtain the following results: • At ν = 1/3 in n = +1, the K -polarized Laughlin state is the ground state. In the massless limit, the extra valley-inversions create the lower-energy excitated state. When the mass term is sufficiently small, the lowest excitation is given as the valley-unpolarized state called the valley-skyrmion. As the mass term increases, the total energy grows greatly by the extra valley-inversion. As a result, the lowest excited state is given as the partially valley-polarized state, and the mass term

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6 Mass and Valley Effects on Excitations in Quantum Hall States

increases the excitation gap. This gap increasing is also seen in the IQHS at ν = 1 in n = 1. • Based on our results, we have discussed the FQHE in graphene on h-BN. By comparing the energy scales of the excitation gap and the disorder, we confirmed that the Laughlin state is experimentally realized even in 10 [T]. The previous experiments reported the decay of the thermal activation energy in high magnetic fields. We proposed one senario that the Landau levels disappear by the superlattice effect.

References 1. Yonaga K, Shibata N (2018) Fractional quantum hall effects in graphene on a h-bn substrate. J Phys Soc Jpn 87:034708 2. Ezawa ZF, Tsitsishvili G (2009) Quantum hall ferromagnets. Rep Prog Phys 72:086502 3. Shibata N, Nomura K (2008) Coupled charge and valley excitations in graphene quantum hall ferromagnets. Phys Rev B 77:235426 4. Amet F, Bestwick AJ, Williams JR, Balicas L, Watanabe K, Taniguchi T, Goldhaber-Gordon D (2015) Composite fermions and broken symmetries in graphene. Nat Commun 6:5838 5. Moon P, Koshino M (2014) Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys Rev B 90:155406 6. Hofstadter DR (1976) Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Phys Rev B 14:2239–2249

Chapter 7

Conclusion

In this study, we have investigated the mass dependence on the FQHS in the Dirac particle system. We have focused on graphene on h-BN which is a typical Dirac material. We have used the exact diagonalization method, and obtained the following results: Mass Term Effects in Spinless Dirac Particle System In Chap. 4, we formulated the exact pseudopotential for the Dirac particle system on the Haldane sphere. Owing to this pseudopotential, we can estimate the ground state energy and its excitation gap accurately in the thermodynamic limit. With this exact pseudopotential, we studied the mass effects on the Laughlin and Pfaffian states. • Ground states at νn = 1/3 in n = ±1 In n = −1, the Laughlin state is the ground state with a large excitation gap when the mass term is finite. By contrast, in n = −1, the excitation gap decays rapidly and the Laughlin state becomes unstable as the mass term increases. • Ground states at νn = 1/2 in n = ±1 When the mass term is large, the ground state is given by the Pfaffian state, while the Fermi liquid is realized in the small mass region. We found that the phase transition from the Laughlin into Pfaffian state. Spin, Valley, and Mass Effects on Fractional Quantum Hall States In graphene, each landau level has the spin and valley degeneracy. In Chap. 5, we have studied the mass effects on the FQHS with the spin and valley degrees of freedom. We have consider the two cases: Z  e2 /l B < v and v  e2 /l B < Z . • Z  e2 /l B < v In this case, each branch of the Landau levels has the spin degrees of freedom. We studied the ground state and its excitation at νn=±1 = 1/3 and νn=±1 = 2/3. At νn=±1 = 1/3, we found that the Laughlin and the unstable spin-polarized state. Here, in the unstable spin-polarized state, its excitation gap decreases as the mass term increases. At νn = 2/3, the three types of ground © Springer Nature Singapore Pte Ltd. 2022 K. Yonaga, Mass Term Effect on Fractional Quantum Hall States of Dirac Particles, Springer Theses, https://doi.org/10.1007/978-981-16-9166-9_7

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7 Conclusion

states are realized: the spin-singlet, Laughlin, and unstable spin-polarized state. We showed that the phase diagrams in n = ±1 are similar. • v  e2 /l B < Z In this case, each branch of the Landau levels has the valley degrees of freedom. At νn=±1 = 1/3 and 2/3, the ground states are given by the valley-polarized states. Here, we found the two types of the ground states: the Laughlin state and unstable valley-polarized whose excitation gap is small. We also showed the valley-singlet state is not realized. In n = −1 Landau level, the phase diagram includes only the valley-polarized Laughlin state. Thus, we revealed that the phase diagrams in n = ±1 are significantly different. Mass and Valley Effects on Excitations in Quantum Hall States We studied the mass and valley effects on the excited states in Chap. 6. At the Landau level filling factor 1/3 in n = +1, the ground state is the valley-polarized Laughlin state. In the massless limit, the lowest excitation is given by the valley-unpolarized state called the valley-skyrmion. By contrast, the partially or fully valley-polarized excitation appears when the mass term is finite. We also found that the lowest excitation gap grows by the mass term and the Laughlin state becomes stable. These results are consistent with the previous experiments. Previous studies on the FQHE in graphene assumed the massless Dirac particle system. However, graphene on a h-BN substrate is widely used for studying the FQHE in recent experiments. Thus, it is necessary to investigate the quantum many-body phenomena in the massive Dirac particle system. In this study, we focused on the graphene on h-BN sand investigated the mass dependence on the FQHS. The results obtained in this study can be applied not only to graphene but also to atomic-layer materials which have been actively synthesized in recent years.