# Emergent Fermion Dynamical Symmetry for Monolayer Graphene in a Strong
Magnetic Field

Mike Guidry(1<EMAIL_ADDRESS>Lian-Ao Wu(2<EMAIL_ADDRESS>Fletcher
Williams(1<EMAIL_ADDRESS>(1)Department of Physics
and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
(2)Department of Physics, University of the Basque Country UPV/EHU, 48080
Bilbao, IKERBASQUE Basque Foundation for Science, 48013 Bilbao, EHU Quantum
Center, University of the Basque Country UPV/EHU, Leioa, Biscay 48940, Spain

###### Abstract

We review the physics of monolayer graphene in a strong magnetic field, with
emphasis on highly collective states that emerge from the weakly interacting
system because of correlations (emergent states). After reviewing the general
properties of graphene and of electrons in a magnetic field, we give a brief
introduction to the integer quantum Hall effect (IQHE) and the fractional
quantum Hall effect (FQHE) in a 2D electron gas as foundation to show that
monolayer graphene in a magnetic field exhibits both effects, but with
properties modified by the influence of the graphene crystal. After giving an
introduction to standard methods of dealing with emergent states for this
system, we show that an SO($8$) fermion dynamical symmetry governs the
emergent degrees of freedom and that the algebraic and group properties of the
dynamical symmetry provide a new view of strongly correlated states observed
in monolayer graphene subject to a strong magnetic field.

###### pacs:

71.10.-w, 71.27.+a, 74.72.-h

###### Contents

  1. I Introduction
  2. II Some material properties of graphene
  3. III Fundamental properties of graphene
  4. IV Structure of Graphene
    1. IV.1 Electronic structure
    2. IV.2 Lattice structure
    3. IV.3 Band structure
      1. IV.3.1 Tight binding approximation
      2. IV.3.2  Dispersion in graphene
    4. IV.4  Low-energy excitations
    5. IV.5  Chirality
  5. V Classical and Quantum Hall Effects
    1. V.1 The classical Hall effect
    2. V.2 Quantum Hall effects
  6. VI  Non-Relativistic Landau Levels
    1. VI.1 Schrödinger equation and solution
    2. VI.2 Properties of Landau levels
    3. VI.3 Degeneracy and level densities
  7. VII  The Integer Quantum Hall Effect
    1. VII.1 Discovery of the integer quantum Hall effect
    2. VII.2 Understanding the integer quantum Hall effect
      1. VII.2.1 Impurity scattering and mobility gaps
      2. VII.2.2 Edge states and conduction
      3. VII.2.3  Topology of the 2D Brillouin Zone
      4. VII.2.4  Topological Protection of Hall Plateaus
      5. VII.2.5 Bulk–Boundary Correspondence
      6. VII.2.6  Incompressible States
    3. VII.3 Summary: Integer Quantum Hall Effect
  8. VIII  The Fractional Quantum Hall Effect
    1. VIII.1  Properties of the fractional quantum Hall state
    2. VIII.2  The Laughlin wavefunction
  9. IX Graphene in Strong Magnetic Fields
    1. IX.1 Quantization of Dirac Landau levels
    2. IX.2 Degeneracy of Landau levels
  10. X Quantum Hall Effects in Graphene
    1. X.1 Strength of correlations in graphene
    2. X.2 The graphene integer quantum Hall effect
    3. X.3 The effect of strong correlations in graphene
    4. X.4  The graphene fractional quantum Hall effect
  11. XI SU(4) Quantum Hall Ferromagnetism
    1. XI.1 Degeneracies and filling of Landau levels
    2. XI.2 SU(4) quantum Hall states in graphene
      1. XI.2.1 Effective Hamiltonian
      2. XI.2.2  Symmetry and explicit symmetry breaking
  12. XII  Emergent States in Magnetic Fields
    1. XII.1  The role of correlations
    2. XII.2 Symmetry-breaking interactions
  13. XIII  Low-Energy Collective Modes
    1. XIII.1  Basis states for collective modes
    2. XIII.2  Most general many-body wavefunction
    3. XIII.3  Explicit vs. spontaneous symmetry breaking
  14. XIV  Fermion Dynamical Symmetry
    1. XIV.1 The microscopic dynamical symmetry method
    2. XIV.2  Dynamical symmetry solution algorithm
    3. XIV.3 Validity of the dynamical symmetry approach
  15. XV Dynamical Symmetry in Graphene
    1. XV.1 Basis states
    2. XV.2 SO(8) generators and Lie algebra
    3. XV.3 A more physical generator basis
    4. XV.4 Coupled representations for pair operators
    5. XV.5 SO(8) pair states in graphene
    6. XV.6 The SO(8) collective subspace
    7. XV.7 Structure of SO(8) pairs
      1. XV.7.1  Pauli restrictions on collective pairs
      2. XV.7.2  Pair and product wavefunctions
    8. XV.8  Beyond quantum Hall ferromagnetism
    9. XV.9 Graphene SO(8) dynamical symmetries
    10. XV.10  Generalized coherent states for graphene
      1. XV.10.1 Another useful generator basis
      2. XV.10.2 SO(8) coherent state energy surfaces
      3. XV.10.3 Physical interpretation of energy surfaces
    11. XV.11  Quantum phase transitions
    12. XV.12  Universality of Emergent States
    13. XV.13  Critical Dynamical Symmetries
  16. XVI Summary and Conclusions
  17. A  Resistivity and conductivity tensors
  18. B  Anderson localization
  19. C The Gauss–Bonnet and Chern theorems
  20. D U(4) and SU(4) subgroups of SO(8)

##  I Introduction

Graphene is a crystalline material composed of a single one-atom thick layer
of carbon; thus it represents the ideal two-dimensional material. It was first
isolated by Geim and Novoselov in 2004, by exfoliating graphite using strips
of tape. The common pencil was invented in 1564; it functions because the
graphite in its “lead” stacks layers of graphene having strong bonding within
layers but weak van der Waals bonding between layers. In writing with a
pencil, layers of graphene are sloughed off by breaking the weak bonds between
layers and are transferred to the paper; presumably some parts of a line drawn
with a pencil consist of a single layer of graphene. Thus, it is likely that
graphene has been produced since 1564 by the ordinary practice of writing with
pencils, but until 2004 it was thought that the single layer allotrope of
carbon atoms perhaps did not exist in its free state. Hence the common view
that graphene is easy to produce but difficult to detect, which delayed its
discovery for more than 400 years after it was first unknowingly produced.

This paper provides an overview of graphene that emphasizes its role as a
laboratory for studying emergent states in pesudo-relativistic matter. There
is considerable current interest in samples having two or more layers of
graphene, but this review will confine itself to a single layer (monolayer
graphene). We will begin by giving a concise overview of graphene as a
material, its basic structure at the atomic and crystalline level, and of
conventional approaches to studying its emergent states when placed in a
strong magnetic field. Then we shall introduce an alternative view of this
many-body system that is the primary emphasis of this review: emergent states
for monolayer graphene in a strong magnetic field in terms of Lie algebras,
Lie groups, and associated fermion dynamical symmetries of the Hamiltonian.
Since the Lie algebra and Lie group methodology employed in this discussion
may be less familiar than more conventional condensed matter methods for some
readers, we will include important details and proofs of various assertions in
a supplemental web document Williams et al. .

##  II Some material properties of graphene

Graphene has many unusual material properties; let us summarize a few of the
more important ones.

  * •

It is planar, with a hexagonal unit cell of area 0.052 nm2 containing two
carbon atoms. Taking the mass of the unit cell to be twice the mass of carbon,
the mass density is $\rho=0.77\mbox{\,mg\,m}^{-2}$.

  * •

A single layer of graphene is almost, but not quite, transparent, absorbing
$\sim 2.3$% of the light passing through it at optical wavelengths. Absorption
is additive for multiple layers, so the number of layers in a sample may be
inferred by how dark it appears in normal light. This method can be used to
distinguish regions of a sample having one, two, three, …, layers of graphene
after exfoliation with tape, for example.

  * •

The breaking strength of graphene is 42 N m-1, which is $\sim$100 times larger
than that of a hypothetical film of steel having the same thickness.

  * •

The thermal conductivity of graphene is dominated by phonons and is large,
with a measured value of 5000 W/mK that is ten times higher than that of
copper.

  * •

The mobility of electrons in graphene is very high. Electric fields can be
used to tune charge carriers continuously between electrons and holes, with
carrier densities as high as $10^{13}\mbox{\,cm}^{-2}$ and carrier mobilities
that can exceed 15,000 cm2V-1s-1, even at room temperature and pressure.

As a consequence of such material properties, graphene is of large interest as
a basis for new devices and other practical applications. But because of its
ideal 2D geometry, non-trivial crystal structure, and unusual electronic
properties, it also represents a novel laboratory for fundamental physics, as
elaborated in Section III.

##  III Fundamental properties of graphene

Because graphene exhibits linear dispersion near the Fermi surface (see Fig.
4), its charge-carrying electrons behave in a manner analogous to that of
relativistic electrons described by a Dirac equation, with the fermi velocity
$v_{\mbox{\scriptsize F}}$ (maximum velocity of a fermion in a system) playing
a role analogous to that of light speed, where the actual speed of light is
$c/v_{\mbox{\scriptsize F}}\sim$ 300 times larger than the graphene fermi
velocity. Just as an actual relativistic electron moving very near the speed
of light typically has a small rest mass, an electron in graphene near the
fermi surface typically displays an effective mass near zero. Thus we may view
graphene as pseudo-relativistic matter that behaves mathematically as actual
Lorentz-invariant, relativistic matter would be expected to behave.

Strongly-correlated, non-relativistic matter has been studied extensively in
phenomena such as superconductivity. However, before the isolation of graphene
it was not easy to observe highly-correlated states for relativistic fermions
because it was difficult to find them under conditions that could produce the
strong particle–particle interactions necessary to form such states. It has
been proposed that neutrinos in the hot, dense environs of a core-collapse
supernova, or in the early Universe before about one second after the Big
Bang, could undergo sufficient interactions to produce a correlated,
relativistic, many-body state but no observations yet support this conjecture.
As we now discuss, graphene affords a benchtop laboratory for studying
strongly-correlated, pseudo-relativistic matter.

In the presence of a magnetic field applied transverse to the sample, a single
layer of graphene exhibits both an integer quantum Hall effect (IQHE) and a
fractional quantum Hall effect (FQHE). The integer quantum Hall effect can
occur for weakly interacting electrons but FQHE states can exist only because
of strong correlations, so observation of a fractional quantum Hall effect for
graphene placed in a magnetic field is evidence of strongly-correlated,
pseudo-relativistic electronic states. A description of these emergent states
will be the overall focus of this review.

##  IV Structure of Graphene

Let’s consider in more detail the electronic and crystal structure of
graphene. Since graphene is pure carbon, the obvious point of departure is the
chemical bonding tendencies of carbon. In this section we adapt extensively
from the presentation of Ref. Goerbig (2011).

###  IV.1 Electronic structure

Carbon has six electrons in a $1s^{2}2s^{2}2p^{2}$ configuration, with the
$2s$ and $2p$ orbitals playing the dominant role in chemical bonding. A
signature characteristic of carbon is its tendency to hybridize the $2s$ and
$2p$ orbitals and form covalent bonds with other atoms. A common hybridization
is $sp^{n}$, in which the $\left|2s\right\rangle$ orbital and $n$ of the
$\left|2p_{i}\right\rangle$ orbitals mix. Because graphene is a planar crystal
of carbon atoms bonded to each other, the dominant hybridization is $sp^{2}$,
in which the spherically symmetric $2s$ orbital mixes with two of the $2p$
orbitals lying in the same plane (typically chosen to be $2p_{x}$ and
$2p_{y}$). This gives the orbital geometry illustrated in Fig. 1(a),

Figure 1:  (a) Carbon $sp^{2}$ hybridization. (b) Schematic structure of the
ring compound benzene; the actual ground state is a quantum superposition of
this configuration and one with the positions of the double and single bonds
exchanged. (c) Schematic picture of graphene as an extended set of benzene
rings joined together, with each benzene C–H bond replaced by a C–C bond
between adjacent rings.

with three planar bonding orbitals separated by 120∘ angles. In Fig. 1(b) the
schematic structure of benzene is indicated, with the covalent bonding at each
vertex between a carbon and two other carbons and a hydrogen corresponding to
$sp^{2}$ hybridization, with the in-plane bonds termed $\sigma$-bonds. The
double lines indicate double bonds, with the second bond created by overlap of
the non-hybridized $2p_{z}$ orbitals on adjacent carbons directed
perpendicular to the plane of the ring ($\pi$-bonds). The $\pi$-bonding
electrons are delocalized over the ring and the benzene ground state
corresponds to a quantum superposition of configurations having the double
bonds in alternating positions around the ring.

In Fig. 1(c) we indicate that the planar structure of graphene may be viewed
as similar to fitting many benzene rings together geometrically, with the
hydrogen stripped off and the bond to hydrogen at each benzene vertex replaced
by a covalent bond to a carbon in the adjacent ring. The in-plane $\sigma$
bonds between the carbon atoms are responsible for the robustness of the
lattice structure in all allotropes of carbon; they lead to a $\sigma$-band
that is completely filled. As in benzene, the $p$ orbital that does not
participate in the $sp^{2}$ hybridization (typically chosen to be $p_{z}$) is
perpendicular to the carbon-atom plane and can bind covalently with
neighboring carbon atoms, delocalizing and leading to a $\pi$-band. In pure
graphene each $p$ orbital has one extra electron, so the $\pi$-band is half
full.

We shall refer to the resulting graphene structure as the honeycomb lattice,
for obvious geometrical reasons. It is believed that all graphitic compounds
(graphite, carbon nanotubes, graphene, and fullerenes) have as their basic
building block this honeycomb graphene lattice: graphite corresponds to stacks
of graphene sheets with strong covalent bonds within the sheets and weak van
der Waals forces between the sheets, a carbon nanotube corresponds to a
graphene sheet rolled into a tube, and a fullerene is wrapped-up graphene with
pentagons inserted to allow an average spherical shape.

###  IV.2 Lattice structure

To address the direct (real-space) and reciprocal (momentum-space) graphene
lattices it is useful to define the direct lattice vectors illustrated in Fig.
2(a).

Figure 2:  (a) The direct lattice for graphene. The graphene honeycomb lattice
is not a Brevais lattice since from the crystallographic point of view two
neighboring sites are inequivalent. It may be viewed as a triangular Brevais
lattice with a two-atom basis, A and B, or as bipartite, with interlocking
triangular lattices A and B. The vectors $\bm{a}_{1}$ and $\bm{a}_{2}$ are the
basis vectors of the triangular Brevais lattice. The vectors $\bm{\delta}_{1}$
, $\bm{\delta}_{2}$, and $\bm{\delta}_{3}$ connect nearest-neighbor (NN)
carbon atoms, with a measured separation of $a=0.142$ nm. (b) Reciprocal
lattice of triangular lattice and first Brillouin zone (shaded). The primitive
lattice vectors are $\bm{a}_{1}^{*}$ and $\bm{a}_{2}^{*}$. The center $\Gamma$
and inequivalent corners K and K′ are marked. Adapted from Ref. Goerbig
(2011).

As described in the figure caption, the graphene lattice may be viewed as
bipartite, consisting of two distinct triangular sublattices (labeled A and B
in the figure, with the A sublattice indicated by solid blue circles and the B
sublattice by open red circles). Alternatively, the lattice may be viewed as
triangular, with a two-atom basis. It is common to define a quantum number
associated with this distinction between sublattices A and B that takes two
values and thus may be described by an SU($2$) symmetry that is termed the
sublattice pseudospin. The lattice vectors are

$\bm{a}_{1}=\sqrt{3}a\bm{e}_{x}\qquad\bm{a}_{2}=\frac{\sqrt{3}a}{2}\big{(}\bm{e}_{x}+\sqrt{3}\,\bm{e}_{y}\big{)},$
(1)

where the spacing between carbon atoms is $a\simeq 0.142\mbox{\,nm}$. The
corresponding reciprocal lattice and Brillouin zone are illustrated in Fig.
2(b). The lattice vectors of the reciprocal lattice are given by

$\bm{a}_{1}^{*}=\frac{2\pi}{\sqrt{3}a}\left(\bm{e}_{x}-\frac{\bm{e}_{y}}{\sqrt{3}}\right)\qquad\bm{a}_{2}^{*}=\frac{4\pi}{3a}\bm{e}_{y},$
(2)

and are illustrated in Fig. 2(b).

###  IV.3 Band structure

The carbon–carbon bonding in graphene involves in-plane $\sigma$-bonds and
out-of-plane $\pi$-bonds. The $\sigma$ electrons typically are associated with
energy bands far from the Fermi surface while the low-energy electronic
properties are primarily associated with the $\pi$ electrons. Since our
primary interest is low-energy states, our focus will be on the $\pi$
electrons. It is conventional to address the energy bands of the $\pi$
electrons in graphene using the tight-binding approximation, as formulated
originally by Wallace in 1947 Wallace (1947) to study the structure of
graphite (then of large practical interest because of the importance of
graphite in early nuclear reactors). Our discussion will follow the
presentation by Goerbig Goerbig (2011).

####  IV.3.1 Tight binding approximation

The honeycomb lattice may be viewed as triangular with two atoms per unit
cell, suggesting a trial wavefunction Goerbig (2011)

$\psi_{\bm{k}}(\bm{r})=a_{\bm{k}}\psi_{\bm{k}}^{\rm\scriptstyle(A)}(\bm{r})+b_{\bm{k}}\psi_{\bm{k}}^{\rm\scriptstyle(B)}(\bm{r}),$
(3)

where the $\psi_{\bm{k}}^{\rm\scriptstyle(A)}$ and
$\psi_{\bm{k}}^{\rm\scriptstyle(B)}$ are Bloch functions

$\displaystyle\psi_{\bm{k}}^{\rm(A)}(\bm{r})$
$\displaystyle=\sum_{\small\bm{R}_{\ell}}\phi^{\rm(A)}(\bm{r}+\bm{\delta}_{\rm
A}-\bm{R}_{\ell})e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{R}}$}_{\ell}},$ (4a)
$\displaystyle\psi_{\bm{k}}^{\rm\scriptstyle(B)}(\bm{r})$
$\displaystyle=\sum_{\small\bm{R}_{\ell}}\phi^{\mbox{\scriptsize(B)}}(\bm{r}+\bm{\delta}_{\mbox{\scriptsize
B}}-\bm{R}_{\ell})e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{R}}$}_{\ell}},$ (4b)

for the A and B lattices, respectively, $\bm{\delta}_{\mbox{\scriptsize A}}$
and $\bm{\delta}_{\mbox{\scriptsize B}}$ are the vectors that connect the
sites of the Brevais lattice with the site of the A or B atom, respectively,
in the unit cell, and $\phi^{\rm(A)}(\bm{r}+\bm{\delta}_{\mbox{\scriptsize
A}}-\bm{R}_{\ell})$ and $\phi^{\rm(B)}(\bm{r}+\bm{\delta}_{\mbox{\scriptsize
B}}-\bm{R}_{\ell})$ are atomic orbital wavefunctions defined near the A or B
atoms, respectively, at the Brevais lattice site $\bm{R}_{\ell}$. Utilizing
Eq. (3), the Shrödinger equation
$H\psi_{\bm{k}}=\epsilon_{\bm{k}}\psi_{\bm{k}}$ may be written as

$(a^{*}_{\bm{k}},b^{*}_{\bm{k}})\,H_{\bm{k}}\begin{pmatrix}a_{\bm{k}}\\\
b_{\bm{k}}\end{pmatrix}=\epsilon_{\bm{k}}(a^{*}_{\bm{k}},b^{*}_{\bm{k}})\,S_{\bm{k}}\begin{pmatrix}a_{\bm{k}}\\\
b_{\bm{k}}\end{pmatrix},$ (5)

where the Hamiltonian matrix $H_{\bm{k}}$ is

$H_{\bm{k}}=H_{\bm{k}}^{\dagger}=\begin{pmatrix}\psi_{\bm{k}}^{\rm\scriptstyle(A)^{*}}H\psi_{\bm{k}}^{\rm\scriptstyle(A)}&\psi_{\bm{k}}^{\rm\scriptstyle(A)^{*}}H\psi_{\bm{k}}^{\rm\scriptstyle(B)}\\\\[4.0pt]
\psi_{\bm{k}}^{\rm\scriptstyle(B)^{*}}H\psi_{\bm{k}}^{\rm\scriptstyle(A)}&\psi_{\bm{k}}^{\rm\scriptstyle(B)^{*}}H\psi_{\bm{k}}^{\rm\scriptstyle(B)}\end{pmatrix}$
(6)

and the overlap matrix $S_{\bm{k}}$ reflecting non-orthogonality of the trial
wavefunctions is

$S_{\bm{k}}=S_{\bm{k}}^{\dagger}=\begin{pmatrix}\psi_{\bm{k}}^{\rm\scriptstyle(A)^{*}}\psi_{\bm{k}}^{\rm\scriptstyle(A)}&\psi_{\bm{k}}^{\rm\scriptstyle(A)^{*}}\psi_{\bm{k}}^{\rm\scriptstyle(B)}\\\\[4.0pt]
\psi_{\bm{k}}^{\rm\scriptstyle(B)^{*}}\psi_{\bm{k}}^{\rm\scriptstyle(A)}&\psi_{\bm{k}}^{\rm\scriptstyle(B)^{*}}\psi_{\bm{k}}^{\rm\scriptstyle(B)}\end{pmatrix}.$
(7)

The Hamiltonian includes an atomic orbital part $H^{a}$ satisfying

$H^{a}\phi^{(j)}(\bm{r}+\bm{\delta}_{j}-\bm{R}_{\ell})=\epsilon^{(j)}\phi^{(j)}(\bm{r}+\bm{\delta}_{j}-\bm{R}_{\ell})$
(8)

and a perturbative part $\Delta V$ accounting for all other terms not
contained in the atomic-orbital Hamiltonian. Then the matrix components of Eq.
(6) for $N$ unit cells may be written as

$H_{\bm{k}}^{ij}=N\big{(}\epsilon^{(j)}s_{\bm{k}}^{(ij)}+t_{\bm{k}}^{(ij)}\big{)},$
(9)

where $i$ and $j$ denote A or B, we have utilized Eqs. (4) and defined

$s_{\bm{k}}^{ij}\equiv\sum_{\bm{R}_{\ell}}e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{R}}$}_{\ell}}\int
d^{2}\kern-0.41702ptr\,\phi^{(i)^{*}}(\bm{r})\phi^{(j)}(\bm{r}+\bm{\delta}_{ij}-\bm{R}_{\ell}))=\frac{S^{ij}_{\bm{k}}}{N}$
(10)

with $\bm{\delta}_{ij}\equiv\bm{\delta}_{j}-\bm{\delta}_{i}$, and the hopping
matrix $t_{\bm{k}}^{ij}$ is defined by

$t_{\bm{k}}^{ij}\equiv\sum_{\bm{R}_{\ell}}e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{R}}$}_{\ell}}\int
d^{2}\kern-0.41702ptr\,\phi^{(i)^{*}}(\bm{r})\,\Delta
V\,\phi^{(j)}(\bm{r}+\bm{\delta}_{ij}-\bm{R}_{\ell}).$ (11)

The electronic bands corresponding to eigenvalues of the Schrödinger equation
follow from the secular equation

$\det\,[H_{\bm{k}}-\epsilon^{\lambda}_{\bm{k}}S_{\bm{k}}]=\det\left[t^{ij}_{\bm{k}}-(\epsilon_{\bm{k}}^{\lambda}-\epsilon^{(j)})\,s_{\bm{k}}^{ij}\right]=0,$
(12)

which has two solutions labeled by the band index $\lambda$ for the case of
two atoms per unit cell. For the special case where the atoms on the different
sublattices have the same electronic configurations, the onsite energy
$\epsilon^{(i)}$ is a physically irrelevant constant for all $i$ that may be
omitted and for graphene

$\det\left[t^{ij}_{\bm{k}}-\epsilon_{\bm{k}}^{\lambda}s_{\bm{k}}^{ij}\right]=0,$
(13)

determines the bands in the tight-binding approximation.

To solve this equation we may choose the Brevais lattice vectors to correspond
to the A sublattice, with the equivalent site on the B sublattice obtained
from $\bm{\delta}_{\mbox{\scriptsize B}}=\bm{\delta}_{\mbox{\scriptsize
AB}}=\bm{\delta}_{3}$, as illustrated in Fig. 3,

Figure 3:  Vectors for a tight-binding model with nearest-neighbor (NN) and
next-nearest-neighbor (NNN) interactions in graphene Goerbig (2011).

and include nearest-neighbor (NN) and next-nearest-neighbor (NNN) terms. From
Fig. 3, the NN hopping amplitude $t$ connects points on different sublattices
and is given by

$t\equiv\int d^{2}\kern-0.41702ptr\,\phi^{A*}(\bm{r})\,\Delta
V\,\phi^{B}(\bm{r}+\bm{\delta}_{3}),$ (14)

while the NNN hopping amplitude connects points on the same sublattice and is
given by

$t_{\mbox{\scriptsize NNN}}\equiv\int
d^{2}\kern-0.41702ptr\,\phi^{A*}(\bm{r})\,\Delta
V\,\phi^{A}(\bm{r}+\bm{a}_{1}).$ (15)

The atomic orbitals are assumed normalized

$\int d^{2}\kern-0.41702ptr\,\phi^{(i)*}(\bm{r})\phi^{(i)}(\bm{r})=1,$ (16)

the overlap correction for NN site orbitals is

$s\equiv\int
d^{2}\kern-0.41702ptr\,\phi^{A^{*}}(\bm{r})\phi^{B}(\bm{r}+\bm{\delta}_{3}),$
(17)

and the overlap corrections for sites that are not nearest-neighbor are
assumed to be small and will be neglected.

For an arbitrary site on the A sublattice the off-diagonal terms of the
hopping matrix [$i\neq j$ in Eq. (11), meaning that the hopping is between the
A and B sublattices] consist of three terms corresponding to the nearest
neighbors $B_{1}$, $B_{2}$, and $B_{3}$, as illustrated in Fig. 3 (since we
are neglecting higher than NNN couplings). All of these have the same hopping
amplitude but different phases. From Fig. 3 the site $B_{3}$ is described by
the same lattice vector as the site A (shifted by $\bm{\delta}_{3}$), so the
phase factor in the hopping matrix Eq. (11) is just 1. But the site $B_{1}$ is
shifted by the vector $\bm{a}_{2}$ relative to $B_{3}$ and so contributes a
phase factor $\exp(i\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{2})$, while the site $B_{3}$ is shifted by the vector
$\bm{a}_{3}$ relative to $B_{3}$ and so contributes a phase factor
$\exp(i\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{a}}$}_{3})$.
Defining the sum of the NN phase factors at $\bm{k}$ by

$\gamma_{\bm{k}}\equiv 1+e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{2}}+e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{3}},$ (18)

the off-diagonal elements of the hopping matrix may be written as
$t_{\bm{k}}^{AB}=(t_{\bm{k}}^{BA})^{*}=t\gamma_{\bm{k}}^{*}$ (with the
convention that $t_{\bm{k}}^{AB}$ corresponds to hopping from B to A), and the
overlap matrix as

$s^{AB}_{\bm{k}}=(s^{BA}_{\bm{k}})^{*}=s\gamma_{\bm{k}}^{*}\qquad
s_{\bm{k}}^{AA}=s_{\bm{k}}^{BB}=1,$ (19)

where the values of the diagonal terms follow from the normalization (16). The
NNN hopping amplitudes yield the diagonal elements of the hopping matrix in
our approximation, since they connect sites on the same sublattice,

$\displaystyle t_{\bm{k}}^{\rm AA}$ $\displaystyle=t_{\bm{k}}^{\rm
BB}=2\,t_{\mbox{\scriptsize NNN}}\sum_{i=1}^{3}\cos(\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{a}}$}_{i})=t_{\mbox{\scriptsize
NNN}}\,\left(|\gamma_{\bm{k}}|^{2}-3\right).$ (20)

Thus, upon inserting these results in Eq. (13) the secular equation takes the
form

$\det\left[\begin{matrix}t_{\bm{k}}^{\rm\scriptstyle
AA}-\epsilon_{\bm{k}}&(t-s\epsilon_{\bm{k}})\gamma_{\bm{k}}^{*}\\\\[2.0pt]
(t-s\epsilon_{\bm{k}})\gamma_{\bm{k}}&t_{\bm{k}}^{\rm\scriptstyle
AA}-\epsilon_{\bm{k}}\end{matrix}\right]=0,$ (21)

which has two solutions labeled by the band index $\lambda=\pm 1$,

$\epsilon_{\bm{k}}^{\lambda}=\frac{t_{\bm{k}}^{\rm\scriptstyle AA}+\lambda
t\,|\gamma_{\bm{k}}|}{1+\lambda s\,|\gamma_{\bm{k}}|}.$ (22)

As a first approximation, it may be assumed that the overlap $s$ is much less
than one and that the NN interaction dominates the NNN interaction so that
$t_{\mbox{\scriptsize NNN}}\ll t$. Expanding Eq. (22) with these assumptions
gives

$\displaystyle\epsilon_{\bm{k}}^{\lambda}$ $\displaystyle\simeq
t_{\bm{k}}^{\rm\scriptstyle AA}+\lambda
t|\gamma_{\bm{k}}|-st|\gamma_{\bm{k}}|^{2}$ $\displaystyle=\lambda
t|\gamma_{\bm{k}}|+(t_{\rm\scriptstyle NNN}-st)|\gamma_{\bm{k}}|^{2},$ (23)

where a constant $-3t_{\mbox{\scriptsize NNN}}$ has been omitted. (Thus the
effect of the overlap $s$ at the present level of approximation is to
renormalize the strength of the next-nearest-neighbor hopping.) Finally,
inserting Eq. (18) into Eq. (23) yields the dispersion relation

$\begin{gathered}\epsilon_{\bm{k}}^{\lambda}=\lambda t\sqrt{f_{\bm{k}}}\ +\
(t_{\mbox{\scriptsize NNN}}-st)f_{\bm{k}},\\\ f_{\bm{k}}\equiv
3+2\sum_{i=1}^{3}\cos\,(\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{i}).\end{gathered}$ (24)

Comparison with more sophisticated numerical calculations or spectroscopic
measurements suggests that reasonable physical choices for the parameters are

$t\sim 3\mbox{\,meV}\qquad(t_{\mbox{\scriptsize NNN}}-st)\sim 0.1t\sim
0.3\mbox{\,meV},$

which justifies the expansion used in going from Eq. (22) to Eq. (23). From
the vectors in Fig. 3, the scalar products $\bm{k}\cdot\bm{a}_{i}$ in Eq. (24)
may be evaluated to give

$\begin{gathered}\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{1}=\sqrt{3}ak_{x}\qquad\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{2}=\frac{\sqrt{3}a}{2}\left(k_{x}+\sqrt{3}k_{y}\right)\\\
\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{a}}$}_{3}=\frac{\sqrt{3}a}{2}\left(-k_{x}+\sqrt{3}k_{y}\right)\end{gathered}$
(25)

and $f_{\bm{k}}$ is given explicitly by

$\displaystyle f_{\bm{k}}$
$\displaystyle=3+2\cos\left(\sqrt{3}\,ak_{x}\right)+2\cos\left(\frac{\sqrt{3}\,a}{2}\big{(}k_{x}+\sqrt{3}\,k_{y}\big{)}\right)$
$\displaystyle\quad+2\cos\left(\frac{\sqrt{3}\,a}{2}\big{(}-k_{x}+\sqrt{3}\,k_{y}\big{)}\right)$
$\displaystyle=3+2\cos(\sqrt{3}\,ak_{x})$
$\displaystyle\quad+4\cos\left(\frac{\sqrt{3}\,a}{2}\,k_{x}\right)\cos\left(\frac{\vphantom{\sqrt{3}}3a}{2}\,k_{y}\right),$
(26)

where $2\cos\theta\cos\phi=\cos(\theta-\phi)+\cos(\theta+\phi)$ was used in
the last step. Figure 4

Figure 4:  (a) Electronic dispersion [energy versus momentum $(k_{x},k_{y})$]
of graphene calculated from Eqs. (24) and (26) in a tight-binding model with
no magnetic field; figure adapted from Ref. Guidry and Sun (2022). Two
inequivalent points in the Brillouin zone (points not connected by reciprocal
lattice vectors) are labeled $K$ and $K^{\prime}$. (b) Near these K-points the
dispersion is approximately linear, leading to six Dirac cones, three labeled
by $K$ and three labeled by $K^{\prime}$. The Fermi surface for undoped
graphene lies at zero energy in this diagram (in the plane where the cones
just touch), and the level density vanishes there for undoped graphene. This
implies that the electrons near the Fermi surface are described by a Dirac
equation. (c) The six minima in the conduction band at the Dirac cones are
called valleys, which are labeled by $K$ or $K^{\prime}$. The two possible
valley labels $K$ or $K^{\prime}$ for an electron are termed valley isospin.
(d) The cyclotron mass of electrons in graphene, adapted from Ref. Novoselov
et al. (2005). Near the Dirac cones of Fig. 4(a) the electronic number density
vanishes $(n=0)$, since the level density tends to zero; the data indicate
that the electrons become essentially massless there and hence obey a massless
Dirac equation.

illustrates the electronic dispersion for graphene calculated using Eqs. (24)
and (26). The valley isospin labels in Fig. 4(c) will be discussed further
below.

####  IV.3.2  Dispersion in graphene

In Fig. 4 the valence band $\pi$, corresponding to $\lambda=-1$, and the
conduction band $\pi^{*}$, corresponding to $\lambda=+1$, meet at six discrete
points that are termed K points. As shown in Fig. 4(b), near the K points the
dispersion is approximately linear. Since a linear dispersion is
characteristic of the Dirac equation for ultrarelativistic massless electrons,
these points are also termed the Dirac points, and the dispersion in the
vicinity of a Dirac point is called a Dirac cone. In Fig. 4(d) we show
experimental evidence that the effective mass of graphene electrons approaches
zero near the Dirac points (where the electron density vanishes). As we shall
elaborate further below, the low-energy excitations of graphene are expected
to look formally like those of relativistic, massless fermions.

Each carbon atom contributes one $\pi$ electron and in the ground state the
lower band is completely full and the upper band is completely empty. Thus,
the Fermi surface lies at the Dirac points where the $\pi$ band and $\pi^{*}$
bands just touch, corresponding to $\epsilon_{\bm{k}}^{\lambda}=0$. From Eq.
(23), this condition requires the real and imaginary parts of $\gamma_{k}$ to
vanish. From Eqs. (18) and (25) we require the simultaneous conditions

$\displaystyle{\rm Re\,\,}\gamma_{\bm{k}}$
$\displaystyle=1+\cos\left(\frac{\sqrt{3}a}{2}(k_{x}+\sqrt{3}k_{y})\right)$
$\displaystyle\quad+\cos\left(\frac{\sqrt{3}a}{2}(-k_{x}+\sqrt{3}k_{y})\right)=0,$
$\displaystyle{\rm Im\,\,}\gamma_{\bm{k}}$
$\displaystyle=\sin\left(\frac{\sqrt{3}a}{2}(k_{x}+\sqrt{3}k_{y})\right)$
$\displaystyle\quad+\sin\left(\frac{\sqrt{3}a}{2}(-k_{x}+\sqrt{3}k_{y})\right)=0,$

where $e^{ix}=\cos x+i\sin x$ has been used. Since sine is an odd function the
second of these equations is satisfied if $k_{y}=0$. Inserting that into the
first of the above equations gives $k_{x}=\pm 4\pi/(3\sqrt{3}a)$. Thus there
are two solutions, K and K′ , corresponding to

$\bm{K}=+\frac{4\pi}{3\sqrt{3}a}\,\bm{e}_{x}\qquad\bm{K}^{\prime}=-\frac{4\pi}{3\sqrt{3}a}\,\bm{e}_{x}.$
(27)

As illustrated in Fig. 5,

Figure 5:  (a) The graphene A sublattice (red triangle connecting sites
labeled by K) and B sublattice (blue triangle connecting points labeled by K′)
are related by inversion. (b) The points K and K′ form inequivalent sets of
three connected by reciprocal lattice vectors.

there are two other equivalent points that are connected to the K solution by
the reciprocal lattice vectors $\bm{a}_{1}^{*}$ and $\bm{a}_{2}^{*}$ defined
in Eq. (2) and Fig. 2. Likewise, there are two other equivalent points
connected to the K′ solution by reciprocal lattice vectors. The sets K and K′
are distinct, representing the two independent solutions of Eq. (27). The
solutions come in pairs because the band Hamiltonian (6) is time-reversal
invariant ($H_{k}=H_{-k}^{*}$). Hence if $\bm{k}$ is a solution of
$\epsilon_{\bm{k}}=0$, so is $-\bm{k}$. Because of the lattice symmetry there
is only one such independent $({\rm K},{\rm K^{\prime}})$ pair in the
Brillouin zone.

Thus, the six Dirac points can be divided into two sets of three,
corresponding to the points K and K′ in Fig. 2(b). The points K are all
equivalent because they are connected by reciprocal lattice vectors; likewise
for the points K′. The inequivalent points K and K′ constitute a 2-dimensional
degree of freedom called the valley isospin (or just isospin for brevity)
$\xi=\pm 1$, so-called because a two-component wavefunction can be treated
formally as a “spin” [fundamental representation of SU(2)], similar to the
isotopic spin formalism of nuclear physics]. One says that there is a two-fold
valley degeneracy in graphene labeled by the quantum number $\xi$. This
degeneracy is expected to survive approximately for excitations having energy
much smaller than the bandwidth $\sim|t|$, since they are restricted to the
vicinity of a given K-point.

###  IV.4  Low-energy excitations

Let us now consider the low-energy excitations expected in graphene. The
Hamiltonian and dispersion relations derived above are complicated by the non-
orthogonality of the atomic wavefunctions, which necessitates the overlap
integrals (10). As noted in conjunction with Eq. (23), the primary effect of
the overlaps is to renormalize the strengths of next-nearest-neighbor hopping.
For low-energy excitations in particular, the overlap effect should be small
and it is convenient to absorb it into NNN hopping amplitudes and define an
effective tight-binding Hamiltonian that takes the form Goerbig (2011)

$H_{\bm{k}}\simeq t\begin{pmatrix}0&\gamma_{\bm{k}}^{*}\\\\[2.0pt]
\gamma_{\bm{k}}&0\end{pmatrix}+t_{\mbox{\scriptsize
NNN}}\begin{pmatrix}|\gamma_{\bm{k}}|^{2}&0\\\\[2.0pt]
0&|\gamma_{\bm{k}}|^{2}\end{pmatrix}.$ (28)

The corresponding eigenvectors are the spinors

$\psi_{\bm{k}}^{\lambda}=\begin{pmatrix}a_{\bm{k}}^{\lambda}\\\\[2.0pt]
b_{\bm{k}}^{\lambda}\end{pmatrix},$ (29)

with components that are the amplitudes for the Bloch wavefunctions of Eqs.
(4a) and (4b) on the two different sublattices A and B.

From Fig. 4, the low-energy excitations in graphene are expected to occur in
the vicinity of the Dirac points $\pm\bm{K}$. We decompose the wavevector
$\bm{k}$ as

$\bm{k}=\pm\bm{K}+\bm{q},$ (30)

where we assume that $|\bm{q}|\ll|\bm{K}|\sim a^{-1}$ and expand the energy
dispersion around $\pm\bm{K}$. The relative phase between the two sublattice
components that we used above was an arbitrary choice. As a consequence, the
dispersion relation (23) is unaffected by a change
$\gamma_{\bm{k}}\rightarrow\gamma_{\bm{k}}\exp(ig_{\bm{k}})$ for any real,
nonsingular function $g_{\bm{k}}$ and it is convenient to use this freedom to
replace Eq. (18) with the more symmetric expression

$e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}}\gamma_{\bm{k}}=e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{1}}+e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{2}}+e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}}=\sum_{i=1}^{3}e^{\pm i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i}},$ (31)

where the relationship of the vectors $\bm{\delta}_{i}$ and $\bm{a}_{i}$ that
is illustrated in Fig. 6

Figure 6:  Relationship among the vectors $\bm{\delta}_{i}$ and $\bm{a}_{i}$.
Adapted from Ref. Goerbig (2011).

has been used to write $\bm{a}_{2}+\bm{\delta}_{3}=\bm{\delta}_{1}$ and
$\bm{a}_{3}+\bm{\delta}_{3}=\bm{\delta}_{2}$ Goerbig (2011); Bena and
Montambaux (2009). Thus

$e^{i\,\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}}\gamma_{k}|_{\bm{k}=\pm\bm{K}+\bm{q}}=\sum_{i=1}^{3}e^{i\,(\pm\bm{K}+\bm{q})\cdot\bm{\delta}_{i}}=\sum_{i=1}^{3}e^{\pm
i\,\mbox{\kern 0.0pt${\bm{K}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}}e^{i\,\mbox{\kern 0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}}.$ (32)

Assuming $\bm{q}$ to be a small displacement from $\bm{K}$ and expanding the
last exponential in the preceding expression to second order gives

$\displaystyle\gamma_{\bm{q}}^{\pm}$ $\displaystyle\equiv e^{i\,\mbox{\kern
0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}}\gamma_{\bm{k}}|_{\bm{k}=\pm\bm{K}+\bm{q}}$
$\displaystyle=\sum_{i=1}^{3}e^{\pm i\,\mbox{\kern
0.0pt${\bm{K}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}}\left(1+i\,\mbox{\kern
0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}-\frac{1}{2}(\mbox{\kern
0.0pt${\bm{q}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i})^{2}\right)$
$\displaystyle=\sum_{i=1}^{3}e^{\pm i\,\mbox{\kern
0.0pt${\bm{K}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}}+i\sum_{i=1}^{3}e^{\pm i\,\mbox{\kern
0.0pt${\bm{K}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i}}(\mbox{\kern
0.0pt${\bm{q}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i})$
$\displaystyle\quad-\frac{1}{2}\sum_{i=1}^{3}e^{\pm i\,\mbox{\kern
0.0pt${\bm{K}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i}}(\mbox{\kern
0.0pt${\bm{q}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i})^{2}$
$\displaystyle\equiv(\gamma_{\bm{q}}^{\pm})^{(0)}+(\gamma_{\bm{q}}^{\pm})^{(1)}+(\gamma_{\bm{q}}^{\pm})^{(2)}.$
(33)

From the geometry in Fig. 6, the components of the vectors $\delta_{i}$ are

$\begin{gathered}\delta_{1}^{x}=\displaystyle\frac{\sqrt{3}}{2}a\qquad\delta_{2}^{x}=-\frac{\sqrt{3}}{2}a\qquad\delta_{3}^{x}=0,\\\
\delta_{1}^{y}=\displaystyle\frac{a}{2}\qquad\delta_{2}^{y}=\frac{a}{2}\qquad\delta_{3}^{y}=-a.\end{gathered}$
(34)

The zero-order term in Eq. (33) vanishes:

$(\gamma_{\bm{q}}^{\pm})^{(0)}=\sum_{i=1}^{3}e^{\pm i\mbox{\kern
0.0pt${\bm{K}}$}\cdot\mbox{\kern 0.0pt${\bm{\delta}}$}_{i}}=e^{\pm 2\pi
i/3}+e^{\mp 2\pi i/3}+1=0,$

where Eqs. (27) and (34), and that $e^{ix}+e^{-ix}=2\cos x$, were used. Thus,
if second and higher order terms are neglected,

$e^{i\mbox{\kern 0.0pt${\bm{k}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}}\gamma_{\bm{k}}|_{\bm{k}=\pm\bm{K}+\bm{q}}=(\gamma_{\bm{q}}^{\pm})^{(1)}=i\sum_{i=1}^{3}e^{\pm
i\mbox{\kern 0.0pt${\bm{K}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}}(\mbox{\kern 0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{i}).$ (35)

From Eq. (34)

$\displaystyle\mbox{\kern 0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{1}=\frac{\sqrt{3}}{2}aq_{x}+\frac{a}{2}q_{y}\qquad\mbox{\kern
0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{2}=-\frac{\sqrt{3}}{2}aq_{x}+\frac{a}{2}q_{y},$
$\displaystyle\mbox{\kern 0.0pt${\bm{q}}$}\cdot\mbox{\kern
0.0pt${\bm{\delta}}$}_{3}=-aq_{y},$

which may be used to evaluate Eq. (35); to this order Goerbig (2011)

$\gamma_{\bm{q}}^{\pm}\simeq\mp\frac{3}{2}a(q_{x}\pm iq_{y}).$ (36)

To deal with the $\pm$ and $\mp$ symbols that follow from the two-fold
degeneracy of solutions in Eq. (27), it is convenient to employ the valley
isospin quantum number $\xi$ introduced earlier, with $\xi=+1$ corresponding
to the ${\rm K}$ point at $\bm{K}$ and $\xi=-1$ to the K′ point at $-\bm{K}$
[modulo translations by reciprocal lattice vectors; see Eq. (27) and Fig. 5].
Thus, Eq. (36) becomes

$\gamma_{\bm{q}}^{\xi}=-\frac{3}{2}\xi a(q_{x}+i\xi q_{y}).$ (37)

Since we are evaluating only to first order, the $t_{\mbox{\scriptsize NNN}}$
term in Eq. (28) may be dropped and inserting the preceding expression for
$\gamma_{k}$ in Eq. (28) gives

$H_{k}^{\xi}\simeq t\begin{pmatrix}0&\gamma_{k}^{*}\\\
\gamma_{k}&0\end{pmatrix}\simeq-\frac{3}{2}\xi at\begin{pmatrix}0&q_{x}-i\xi
q_{y}\\\ q_{x}+i\xi q_{y}&0\end{pmatrix}$

as the effective low-energy Hamiltonian. Defining a Fermi velocity

$v_{\mbox{\scriptsize F}}\equiv\frac{-3at}{2\hbar}=\frac{3a|t|}{2\hbar}$ (38)

(where $t$ is generally negative), gives for the Hamiltonian

$H_{k}^{\xi}=\hbar\xi v_{\mbox{\scriptsize F}}\begin{pmatrix}0&q_{x}-i\xi
q_{y}\\\ q_{x}+i\xi q_{y}&0\end{pmatrix},$ (39)

which may be expressed as

$H_{q}^{\xi}=\hbar\xi v_{\mbox{\scriptsize F}}(q_{x}\sigma^{x}+\xi
q_{y}\sigma^{y}),$ (40)

upon utilizing the standard $2\times 2$ representations of the Pauli matrices
$\sigma^{x}$ and $\sigma^{y}$. The dispersion at this level of approximation
is obtained by dropping the term proportional to $t_{\mbox{\scriptsize NNN}}$
from Eq. (23) and inserting $\gamma_{\bm{k}}$ from Eq. (37), giving

$\epsilon_{\bm{k}}=\lambda t|\gamma_{\bm{q}}|=\frac{3}{2}a\lambda
t|\bm{q}|=\hbar v_{\mbox{\scriptsize F}}\lambda|\bm{q}|,$ (41)

where Eq. (38) was used. The energy $\epsilon_{\bm{k}}$ depends on the band
index $\lambda$ but not the valley isospin $\xi$.

###  IV.5  Chirality

The sublattice symmetry of graphene (two interlocking triangular sublattices)
should be approximately valid for low energy excitations. Since it corresponds
to two degrees of freedom, it is convenient to describe it in terms of a
pseudospin ${\bm{\sigma}}$ that takes two values. To the degree that the
sublattice symmetry is respected, there is an associated conserved quantity.
For actual relativistic electrons it is convenient to express the spin degree
of freedom in terms of helicity, which is the projection of the spin on the
direction of motion. Since the solution near the Dirac points (low-energy
states) behaves effectively as that for massless relativistic electrons, it is
convenient to define the helicity $\eta_{q}$ operator associated with
projection of the sub-lattice pseudospin on the wavevector by Goerbig (2011)

$\eta_{q}\equiv\frac{\mbox{\kern 0.0pt${\bm{\sigma}}$}\cdot\mbox{\kern
0.0pt${\bm{q}}$}}{|\mbox{\kern 0.0pt${\bm{q}}$}|},$ (42)

which has eigenvalues $\eta_{q}\left|\eta=\pm 1\right\rangle=\pm\left|\eta=\pm
1\right\rangle$. The helicity for massless particles commutes with the Dirac
Hamiltonian, so it is a conserved quantum number.

For massless particles the helicity is the same as the chirality
(“handedness”), which is the eigenvalue of the Dirac $\gamma_{5}$ operator. We
shall assume electrons near the Dirac points to be exactly massless [see Fig.
4(d)], so that their helicity and chirality may be identified. It is common to
refer to these fermions as massless chiral fermions, and to the sublattice
symmetry as conservation of chirality. (Remember that this is a “pseudo-
chirality” associated with the sublattice pseudospin, not the actual spin $s$
of the electron.) The band index $\lambda$ is the product of the sublattice
chirality and the valley isospin, $\lambda=\eta\xi$. Conservation of chirality
implies that there is no backscattering of electrons upon encountering a
slowly-varying barrier, since to rotate the wavevector by $\pi$ would flip the
sign of the chirality. From the tight-binding model, the effective low-energy
Hamiltonian at the corners of the Brillouin zone may be written

$H=\hbar v_{\mbox{\scriptsize F}}\,\mbox{\kern
0.0pt${\bm{\sigma}}$}\cdot\mbox{\kern 0.0pt${\bm{p}}$},$ (43)

which is equivalent to the Dirac equation for massless chiral fermions (also
called the Weyl equation), but with the speed of light $c$ replaced by the
Fermi velocity $v_{\mbox{\scriptsize F}}$.

##  V Classical and Quantum Hall Effects

The integer and fractional quantum Hall effects represent remarkable physics
that appears when a strong magnetic field is applied to a low-density electron
gas confined to two dimensions at very low temperature. These effects were
first observed in the early 1980s for 2D electron gases created in
semiconductor devices. To understand the quantum Hall effect in graphene it is
important to understand the basics of this extensive earlier work. This
section gives a general introduction to the quantization of non-relativistic
electrons in a magnetic field that is the basis of the quantum Hall effect,
and the following two sections give a general introduction to the integer and
fractional quantum Hall effects, respectively. Then we shall be prepared to
address the issue of quantum Hall effects in graphene.

###  V.1 The classical Hall effect

Let’s begin by recalling the basics of the classical Hall effect.

Figure 7:  The classical Hall effect. (a) An electric field $E_{x}$ causes a
current $j_{x}$ to flow through a thin rectangular sample in the $x$
direction. (b) A uniform magnetic field $B_{z}$ is placed on the sample in the
positive $z$ direction. The curved arrows indicate the Lorentz-force response
of the electrons to the magnetic field, causing a deflection in the $y$
direction. (c) This causes electrons to accumulate on one edge and a positive
ion excess on the opposite edge, producing a transverse electric field $E_{y}$
(the Hall field) that just cancels the force produced by the magnetic field.
Adapted from Ref. Guidry and Sun (2022)

In Fig. 7(a), an electric field $E_{x}$ causes a current $j_{x}$ to flow
through a thin rectangular sample in the $x$ direction. In Fig. 7(b) a uniform
magnetic field $B_{z}$ is placed on the sample in the positive $z$ direction,
which causes a deflection of the electrons in the $y$ direction (Lorentz
force). As indicated in Fig. 7(c), electrons accumulate on one edge and a
positive ion excess accumulates on the opposite edge, producing a transverse
electric field $E_{y}$ (the Hall field) that in steady state just cancels the
Lorentz force produced by the magnetic field. As a result, the current is
entirely in the $x$ direction and for uniform samples the Hall field is
perpendicular to the current, in the direction $\bm{j}\times\bm{B}$. Typically
the transverse voltage $V_{\mbox{\scriptsize H}}$ and the longitudinal voltage
$V_{\mbox{\scriptsize L}}$ are measured in the experiment, as indicated in
Fig. 7(d).

The situation may be analyzed quantitatively in terms of the classical Lorentz
force $\bm{F}$ acting on the electrons in the magnetic field,

$\bm{F}=-e\left(\bm{E}+\frac{1}{c}\,\bm{v}\times\bm{B}\right),$ (44)

where $\bm{B}$ is a magnetic field oriented in the $+z$ direction, $\bm{E}$ is
the electric field, and $\bm{v}$ is the electron velocity (taken to be in the
$x$ direction). Requiring forces in the $y$ direction to cancel,

$F_{y}=-e\left(E_{y}-\frac{1}{c}v_{x}B_{z}\right)=0,$

implies that the transverse electric field and the perpendicular magnetic
field are related by

$E_{y}=\frac{1}{c}v_{x}B_{z}.$ (45)

The velocity $\bm{v}$ can be estimated using a force
$\bm{$}derivedfromtheDrudemodel\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{kitt2004,quii2009,phil2003}{\@@citephrase{(}}{\@@citephrase{)}}},\begin{equation}\bm{F}=m\left(\frac{d\kern
0.0pt\bm{v}}{d\kern
0.0ptt}+\frac{\bm{v}}{\tau}\right)=-e\left(\bm{E}+\frac{1}{c}\,\bm{v}\times\bm{B}\right),\end{equation}where$m$istheeffectivemassofanelectronand$τ$isthemeantimebetweenelectroncollisions.Insteadystate,$dv/dt
=
0$andtheequationsofmotionare\begin{equation}\begin{gathered}v_{x}=-\frac{e\tau}{m}\,E_{x}-\omega_{\mbox{\scriptsize
c}}\tau v_{y}\qquad v_{y}=-\frac{e\tau}{m}\,E_{y}+\omega_{\mbox{\scriptsize
c}}\tau v_{x},\\\
v_{z}=-\frac{e\tau}{m}\,E_{z},\end{gathered}\end{equation}where$B ≡B_z =
|B|$andthe{\em
cyclotronfrequency}$ω_c$isdefinedby\begin{equation}\omega_{\mbox{\scriptsize
c}}\equiv\frac{eB}{mc}.\end{equation}InsertingthefirstofEqs.\
(\ref{eq:classicalHall1.4})intoEq.\
(\ref{eq:classicalHall1.2})givesarelationshipbetweenthetransverseelectricfield$E_y$andlongitudinalelectricfield$E_x$,\@@eqnarray
E_{y}=\frac{B}{c}\,v_{x}=\frac{B}{c}\left(-\frac{e\tau}{m}\,E_{x}-\omega_{\mbox{\scriptsize
c}}\tau v_{y}\right)=-\frac{eB\tau}{mc}\,E_{x},\cr whereatequilibrium$v_y =
0$hasbeenrequired.If$τ$isthemeancollisiontimeforanelectron,thecurrentdensitymaybeapproximatedintheDrudemodelas\begin{equation}\bm{j}=\frac{e^{2}n_{e}}{m}\,\tau\bm{E}=\sigma\bm{E}\qquad\sigma\equiv\frac{n_{e}e^{2}\tau}{m},\end{equation}where$n_e$istheelectronnumberdensityand$σ$istheconductance.\par
InaHalleffectexperiment,onetypicallymeasuresthetransversevoltage$V_H$andthelongitudinalvoltage$V_L
= E_x
ℓ$,where$ℓ$isthedistanceoverwhichthevoltagechangesby$V_L$.Ifthesampleisapproximatedas2Dwithtransversewidth$w$[seeFig.~{}\ref{fg:classicalHallEffect}(a)],therelationshipbetweentheHallvoltage$V_H$,Hallfield$E_y$,andtotalcurrent$I$isgivenby\begin{equation}E_{y}=\frac{V_{\mbox{\scriptsize
H}}}{w}\qquad I=wj_{x}.\end{equation}The{\em
Hallresistance}$R_H$isthendefinedby[seeAppendix\ref{ResistanceAndStuff})]\begin{equation}R_{\mbox{\scriptsize
H}}=\frac{V_{\mbox{\scriptsize
H}}}{I}=\frac{wE_{y}}{wj_{x}}=\frac{E_{y}}{j_{x}}=-\frac{B}{ecn_{e}},\end{equation}whereEqs.\
(\ref{eq:classicalHall1.5})--(\ref{eq:classicalHall1.7})wereused.Likewise,thelongitudinalresistance$R_L$isgivenby\begin{equation}R_{\mbox{\scriptsize
L}}=\frac{V_{\mbox{\scriptsize
L}}}{I}=\frac{LE_{x}}{wj_{x}}=\frac{\ell}{w\sigma},\end{equation}where(\ref{eq:classicalHall1.6})wasused.Nowletusseehowthisclassicalpictureismodifiedbyquantumeffects,guidedbythepresentationinRef.\
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{guid2022}{\@@citephrase{(}}{\@@citephrase{)}}}.\par\par\par$

###  V.2 Quantum Hall effects

A much richer set of possibilities is found when Hall effect experiments are
performed at low temperatures and high magnetic fields. These quantum Hall
effects (QHE) can be separated into two sets of phenomena:

  1. 1.

the integer quantum Hall effect (IQHE), and

  2. 2.

the fractional quantum Hall effect (FQHE).

To understand these quantum Hall effects we must first examine the
quantization of electrons confined to two dimensions in a strong magnetic
field.

##  VI  Non-Relativistic Landau Levels

Consider the quantum description of non-relativistic electrons in a 2D gas
with a strong magnetic field transverse to the 2D plane. The magnetic field
will lead to substantial spin polarization, so we assume states to have a
single electron spin polarization and neglect the constant Zeeman energy for
each polarized spin. This is not strictly justified since some spin effects
are observable even at high magnetic field, but it should not change
substantially the general features that we shall emphasize. Initially the
Coulomb interaction also will be neglected relative to the effect of the
strong magnetic field, but we will return to its effect later.

###  VI.1 Schrödinger equation and solution

This section follows the presentation by Phillips Phillips (2003). By the
usual minimal prescription to ensure gauge invariance, the effect of an
electromagnetic field may be included by adding to the momentum operator
$\bm{p}=(p_{x},p_{y})$ a term depending on the vector potential and the
Hamiltonian may be written

$H=\frac{1}{2m}\left(\frac{\hbar}{i}\,\mbox{\kern
0.0pt${\bm{\nabla}}$}+\frac{e}{c}\,\,\bm{A}\right)^{2},$ (54)

where $\bm{A}$ is a 3-vector potential having a curl equal to the magnetic
field $\bm{B}$:

$\bm{B}=\bm{\nabla}\times\bm{A}\qquad\bm{B}\equiv(B_{x},\,B_{y},\,B_{z})=(0,0,B).$
(55)

Various gauge choices give this magnetic field. Two common ones are the Landau
gauge and the symmetric gauge, which are defined by

$\displaystyle\bm{A}=(0,\,Bx,\,0)\quad\mbox{\,(Landau gauge)},$ (56a)
$\displaystyle\bm{A}=\frac{B}{2}(-y,\,x,\,0)\quad\mbox{\,(symmetric gauge)}.$
(56b)

The electromagnetic field is unchanged by gauge transformations, so no
physical quantities depend on the gauge choice, but the forms of the vector
potential $\bm{A}$ and wavefunction $\psi$ do. That is not an ambiguity
because $\bm{A}$ and $\psi$ are not observables  in this context.

It will prove convenient for the initial discussion to work in the Landau
gauge, $\bm{A}=(0,\,Bx,\,0)$, so that $A_{y}=Bx$ and $A_{x}=0$ (with the $z$
dimension neglected, since we are concerned with 2D electron gases). The
Schrödinger equation is then

$-\frac{\hbar^{2}}{2m}\left[\partial_{x}^{2}+\left(\partial_{y}-\frac{ieBx}{\hbar
c}\right)^{2}\right]\psi(x,y)=E\psi(x,y),$ (57)

with $m$ the effective mass, $\partial_{y}\equiv\partial/\partial y$, and
$\partial_{x}^{2}\equiv\partial^{2}/\partial x^{2}$. Since the Hamiltonian
does not depend on $y$ in the chosen gauge, it is convenient to write the
wavefunction in the separable form $\psi(x,y)=\phi(x)e^{iky}$. Inserting this
into Eq. (57) indicates that $\phi(x)$ obeys

$\frac{\hbar\omega_{\mbox{\scriptsize
c}}}{2}\left[-\Lambda^{2}\partial_{x}^{2}+\left(\frac{x}{\Lambda}-\Lambda\,k\right)^{2}\right]\phi(x)=E\phi(x),$
(58)

where $\omega_{\mbox{\scriptsize c}}=eB/mc$ is the cyclotron frequency of Eq.
(48) and the magnetic length $\Lambda$ is

$\Lambda\equiv\sqrt{\frac{\hbar c}{eB}}.$ (59)

But Eq. (58) is just the harmonic-oscillator Schrödinger equation, so the
wavefunction is of harmonic oscillator form

$\psi_{nk}(x,y)=H_{n}\left(\frac{x}{\Lambda}-\Lambda
k\right)e^{-(x-x_{k})^{2}/2\Lambda^{2}}e^{iky},$ (60)

where $H_{n}$ is a Hermite polynomial and $x_{k}=\Lambda^{2}k$. The
corresponding energy is

$\epsilon_{n}=\hbar\omega_{\mbox{\scriptsize c}}\left(n+\frac{1}{2}\right),$
(61)

which depends on the principle Landau quantum number $n=0,1,2,\ldots,$ but not
on $k$.

###  VI.2 Properties of Landau levels

From Eq. (60) the wavefunction is extended in $y$ but localized in $x$ near
$x_{k}=\Lambda^{2}k$. This is specific to our gauge choice: under a local
gauge transformation the vector potential $A(\bm{r},t)$ and the wavefunction
$\psi(\bm{r},t)$ are changed simultaneously according to

$\displaystyle A(\bm{r},t)\ \rightarrow\ A(\bm{r},t)+\mbox{\kern
0.0pt${\bm{\nabla}}$}\chi(\bm{r},t),$ (62a) $\displaystyle\psi(\bm{r},t)\
\rightarrow\ \psi(\bm{r},t)e^{i\hbar c\chi(\bm{r},t)/e},$ (62b)

where $\bm{r}\equiv(x,y)$ and $\chi(\bm{r},t)$ is some scalar function, so the
forms of both $\bm{A}$ and $\psi$ (but no observable quantities) depend on the
gauge chosen. If $L$ is the spatial extent in the $y$ direction, $k_{m}=2\pi
m/L$, with $m$ an integer. Therefore the spacing in $k$ is given by

$k_{m+1}-k_{m}=\frac{2\pi}{L}.$ (63)

Energy levels labeled by $n$ in Eq. (61) are termed Landau levels.
Semiclassically, they correspond to electrons moving in circles of quantized
radius specified by the quantum number $n$, with quantization arising from
requiring an integer number of electron de Broglie wavelengths to fit around
the cyclotron orbit. For a given Landau orbit the radius of the circle is
termed the cyclotron radius and the center of the circle is termed the guiding
center [see Fig. 16(b)]. Landau levels are highly degenerate in strong
magnetic fields because there are many possible locations for the center of
the circle of some radius defining a cyclotron orbit.

###  VI.3 Degeneracy and level densities

The degeneracy for a Landau level labeled by $n$ is equal to the number of $k$
values associated with that $n$. If electronic interactions are neglected, the
level density $g(\epsilon)$ is a series of $\delta$-functions at energies
corresponding to discrete values of $n$, weighted by a degeneracy factor $N$
equal to the number of electrons that can occupy the Landau level,

$g(\epsilon)=N\sum_{n}\delta\left[\epsilon-\hbar\omega_{\mbox{\scriptsize
c}}\left(n+\frac{1}{2}\right)\right].$ (64)

For a 2D sample of monolayer graphene having width $w$, length $\ell$, and
magnetic length $\Lambda$,

$N=\frac{B\ell w}{hc/e}=\frac{\ell w}{2\pi\Lambda^{2}}.$ (65)

Equation (65) has a simple physical interpretation. The magnetic flux through
a 2D sample of area $A=\ell w$ is $B\ell w$, so $N$ is the magnetic flux in
units of the quantum of magnetic flux, $hc/e$, where $h$ is Planck’s constant,
$e$ is the electronic charge, and $c$ is the speed of light. From Eq. (65),
the number of states per unit area in each Landau level is

$n_{B}=\frac{B}{hc/e}=\frac{1}{2\pi\Lambda^{2}},$ (66)

which does not depend on the Landau level but scales linearly with the
strength of the magnetic field. Thus, large magnetic fields imply large
degeneracies. If the actual number density of electrons is $n_{e}$, a filling
factor $\nu$ may be defined by by

$\nu\equiv\frac{n_{e}}{n_{B}}=\frac{n_{e}hc}{eB}.$ (67)

If $\nu$ is an integer, the lowest $\nu$ Landau levels will be filled
completely and all other Landau levels will be empty, while if $\nu$ is not
integer a Landau level will be partially filled. For $\nu$ equal to an integer
$m$ there will be $m$ completely-filled Landau levels and there will be an
energy gap between the $n=m$ and $n=m+1$ Landau levels. Thus, at $T=0$ the
flow of charge is suppressed because electronic excitation is inhibited by the
gap.

##  VII  The Integer Quantum Hall Effect

What happens to the classical Hall effect for a dilute 2D electron gas at very
low temperature and very strong magnetic field? If the filling factor $\nu$
given by Eq. (67) is an integer the Hall conductance
$\sigma_{\mbox{\scriptsize H}}=1/R_{\mbox{\scriptsize H}}$ becomes quantized,
since

$\sigma_{\mbox{\scriptsize H}}=\frac{1}{R_{\mbox{\scriptsize
H}}}=-\frac{ecn_{e}}{B}=-\frac{ec\nu n_{B}}{B}=\nu\frac{-e^{2}}{h}$ (68)

As electrons are added beyond $\nu$ completely filled Landau levels, they will
go into the next level. One might expect that the longitudinal conductance
$\sigma_{\mbox{\scriptsize L}}$ would at first increase with added electrons
until the level becomes half full, and then decrease with addition of further
electrons until the level becomes completely full, as illustrated by the
dashed curves in Fig. 8(a).

Figure 8:  Conductance as a function of Landau-level filling factors $\nu$.
(a) Longitudinal conductance $\sigma_{\mbox{\scriptsize L}}$. (b) Hall
conductance $\sigma_{\mbox{\scriptsize H}}$. Dashed green curves indicate the
expectation from simple considerations. Solid blue curves suggest what is
actually observed (compare experimental data in Fig. 9).

However, that is not what is found. Instead,

  1. 1.

the longitudinal conductance is finite only in narrow ranges [solid blue peaks
in Fig. 8(a)], and

  2. 2.

the Hall conductance $\sigma_{\mbox{\scriptsize H}}$ has very flat plateaus in
the regions around integer filling factor $\nu$ that coincide with regions of
very small longitudinal conductance, as illustrated in Fig. 8(b).

The next section will describe these quantum Hall experiments and attempt to
understand these properties.

###  VII.1 Discovery of the integer quantum Hall effect

The integer quantum Hall effect (IQHE) was discovered in 1980 by von Klitzing,
Dorda, and Pepper von Klitzing et al. (1980). Experimental data exhibiting the
IQHE are shown in Fig. 9

Figure 9:  Data showing the integer quantum Hall effect von Klitzing et al.
(1980). The Hall resistance $R_{\mbox{\scriptsize H}}$ varies stepwise between
plateaus with changes in magnetic field B (upper curve). Step height is given
by the physical constant $h/e^{2}$ divided by an integer $i$. The figure shows
steps for $i=2,3,4,5,6,8$ and 10. The lower curve with multiple peaks
represents the ohmic (longitudinal) resistance $R_{\mbox{\scriptsize L}}$,
which is finite at transitions between plateaus, but drops to almost zero over
each plateau.

and a schematic experimental apparatus is illustrated in Fig. 10.

Figure 10:  Schematic diagram of a Hall bar experimental apparatus employing a
gallium arsenide heterostructure. The dotted red line indicates the 2D
electron gas at the interface between the gallium arsenide (GaAs) and aluminum
gallium arsenide (AlGaAs) layers. The magnetic field is $B$, the electrical
current is $I$, the Hall voltage is $V_{\mbox{\scriptsize H}}$, and the
longitudinal voltage is $V_{\mbox{\scriptsize L}}$. Adapted from Ref.
Eisenstein and Stormer (1990).

In the classical Hall effect the Hall resistance $R_{\mbox{\scriptsize H}}$
should increase linearly with field strength, according to Eq. (52), while the
longitudinal resistance $R_{\mbox{\scriptsize L}}$ should be independent of
the magnetic field, according to Eq. (53). This is approximately true for
magnetic fields with strength less than about 1 tesla in Fig. 9, but for
stronger fields the Hall resistance $R_{\mbox{\scriptsize H}}$ develops
plateaus where it remains constant with increasing magnetic field at values

$R_{\mbox{\scriptsize H}}=\rho_{yx}=\frac{V_{y}}{I_{x}}=\frac{h}{ne^{2}}$ (69)

(see Appendix A for the definition of the resistivity tensor $\rho_{ij}$),
where $n$ is an integer, and in the center of regions where
$R_{\mbox{\scriptsize H}}$ is constant the longitudinal resistance
$R_{\mbox{\scriptsize L}}$ behaves as $R_{\mbox{\scriptsize
L}}\simeq\exp(-\Delta/2k_{\mbox{\scriptsize B}}T)$, which tends to zero as
$T\rightarrow 0$, indicating the onset of charge transport without
dissipation. This is suggestive of an energy gap of magnitude $\Delta$
suppressing the scattering at low temperature. The plateaus in
$R_{\mbox{\scriptsize H}}$ are associated with quantization of the Hall
conductance, with the precision of the quantization being remarkably high (of
order one part in a billion). This behavior constitutes the integer quantum
Hall effect (IQHE). The IQHE diverges from classical behavior at larger
magnetic fields; let’s see if we can understand this using the quantum Landau-
level theory of Section VI.

###  VII.2 Understanding the integer quantum Hall effect

The integer quantum Hall effect does not require electron–electron or
electron–phonon correlations for its explanation. It related solely to the
filling of Landau levels with non-interacting electrons and our discussion of
quantized Landau levels in Section VI, supplemented by impurity scattering
effects and topological constraints to be discussed below, provides an
understanding of its essential features. As implied by Eq. (65), the number of
electrons that can be accommodated by each Landau level is governed by the
magnetic field strength $B$. At certain special values of the field strength
$B$ a match between the number of electrons and the capacity of the Landau
levels causes an integer number of Landau levels to become exactly filled,
producing (in outline) the integer quantum Hall effect. At these special
values of $B$ the ratio of the number of electrons per unit area to the number
of units of flux $h/e^{2}$, which is the filling factor $\nu$ defined in Eq.
(67), takes integer values. However, this simple Landau-level picture is not
sufficient to understand all features of Fig. 9. Level filling and the
associated “shell closures” suggest the broad outlines of the IQHE, but there
are two essential details of Fig. 9 that remain unexplained.

  1. 1.

The longitudinal resistance is very near zero over the entire filling-factor
range of a plateau in the Hall resistance.

  2. 2.

The individual plateaus of Fig. 9 have heights quantized with a precision as
high as one part in $10^{9}$ for broad ranges of filling factors.

The first suggests that only some states in a Landau level contribute to
longitudinal conductance. The second hints at a fundamental principle
responsible for quantizing the Hall resistance that renders it insensitive to
details. As will now be shown, the first is explained by the effect of
impurities on the transport properties of the electron gas, and the second by
a conserved topological quantum number that protects quantization of the Hall
resistance.

Figure 11:  Level densities $g(\epsilon)$ for a 2D electron gas in a strong
magnetic field. (a) In the clean limit, Landau level densities correspond to
$\delta$-functions, as in Eq. (64). (b) Interactions with impurities lead to a
broadening of the $\delta$-function distributions. (c) The broadened peaks
consist of extended states near the centroids and localized states in the
wings. Adapted from a figure in Ref. Guidry and Sun (2022).  Figure 12:
Density of states (a) for a band in the clean limit and (b) for a band in a
disordered system. The mobility edges mark the boundary between localized and
extended states.

####  VII.2.1 Impurity scattering and mobility gaps

The integer quantum Hall effect is observed for impure samples and the effect
increases up to a point with increased impurity concentration. Let us try to
understand that. Impurities break translational invariance and increase
resistance by scattering charge carriers, so we must consider the influence of
disorder and impurity scattering on the results of Fig. 9. First note that the
electronic states in a sample may be separated broadly into two categories.

  1. 1.

Some states are extended (metallic) states, with wavefunctions that fall off
slowly with distance; such states facilitate charge transport and increase the
conductance.

  2. 2.

Some states are localized (insulating) states, with wavefunctions that are
finite only in a small region; such states suppress charge transport and
increase the resistance.

The effect of impurities on charge transport is often framed in terms of
Anderson localization, which is described briefly in Appendix B. The Anderson
model of localization is difficult to solve exactly but qualitative insight
from it will suffice for our discussion.

Figure 11(a) illustrates the density of states implied by Eq. (64) for a clean
system, which corresponds to $\delta$-functions for each Landau level $n$,
weighted by a degeneracy factor $N$ depending on the magnetic field strength.
For disordered samples the $\delta$-functions are broadened into peaks of
finite width by impurity scattering, as illustrated in Fig. 11(b).
Intuitively, impurity scattering tends to localize a state and impede charge
transport, with stronger impurity scattering implying greater deviation of the
energy from the clean limit. Thus, as suggested in Fig. 11(c), only states
near the centers of the broadened peaks in Fig. 11(b) are extended spatially
and can carry a current, with the states in the wings of the peaks localized
by impurity scattering so that they are insulating.

The general idea is illustrated in Fig. 11(c) and Fig. 12. In Fig. 12(a) the
density of states expected as a function of energy for a band in a clean
system is illustrated. Figure 12(b) indicates that the effect of impurity
scattering is to spread those states in energy, with states near the center of
the resulting distribution being spatially extended and those on the wings
being spatially localized, with a mobility edge characterizing the boundary
between the two regions. This leads to mobility gaps, corresponding to ranges
of energies having localized states that do not support charge transport.

The influence of the magnetic field is crucial for the present argument.
Normally in a 2D system all current-carrying states are destroyed by even a
tiny amount of impurity scattering. However, this conclusion can be
invalidated by the presence of a magnetic field, which breaks time-reversal
symmetry and interferes with scattering processes that promote localization.
Thus, 2D systems subject to a magnetic field can carry current, even in the
presence of impurities. More detailed studies than our simple discussion here
confirm that in 2D systems current-carrying states near the energy of the
unperturbed Landau level survive in the presence of a magnetic field, as
illustrated schematically in Fig. 11(c).

Figure 11(c) explains the Hall plateaus of Fig. 9. Only states near the
unperturbed Landau-level energy are delocalized and carry current, which
causes $R_{\mbox{\scriptsize H}}$ to jump discontinuously as the chemical
potential is tuned through those states, but $R_{\mbox{\scriptsize H}}$
remains constant as the chemical potential is tuned through localized states
in the wings of the peak, as illustrated in Fig. 13.

Figure 13:  Origin of Hall plateaus. Extended states (near the centroid of the
peak) can carry current; localized states (in the wings of the peak) cannot.
Thus conductance $\sigma_{\mbox{\scriptsize H}}$ changes only when the energy
$\epsilon$ is near the centroid of a Landau peak.

Thus the Hall plateaus of Fig. 9 correspond to ranges of magnetic field
strength where the population of extended states is constant because the
chemical potential is not near the center of the broadened Landau level. In
effect, when the Fermi level of a disordered sample lies in a mobility gap
between Landau levels $n$ and $n+1$, the transport characteristics mimic those
of a pure sample with $\nu=n$. The plateaus in Fig. 9 are broad so the ranges
of extended states must be narrow. Note that thermal excitation invalidates
the preceding argument by scattering electrons between localized and extended
states. Thus, quantum Hall experiments require very low temperatures,
typically less than several kelvin.

####  VII.2.2 Edge states and conduction

For an electron gas confined to two dimensions with a magnetic field
orthogonal to the 2D plane, the classical Lorentz force causes electrons to
move in circular (cyclotron) orbits. But as suggested by Fig. 14(a),

Figure 14:  (a) Conduction by edge states for a 2D electron gas in a
perpendicular magnetic field. (b) First three Landau levels with a smooth
confining potential on the right Wen (2004). States in each level are labeled
by $k$ and indicated by circles. Filled states are shaded, empty states are
unshaded, and the chemical potential is indicated by the dashed horizontal
line. Bulk excitations are inhibited by a gap $\Delta
E=\hbar\omega_{\mbox{\scriptsize c}}$ (where $\omega_{\mbox{\scriptsize c}}$
is the cyclotron frequency), but the gap for edge excitations tends to zero in
the thermodynamic limit, implying gapless edge excitations.

no net current flows in the central region because the motion in one cyclotron
orbit is offset by the opposite motion in the adjacent cyclotron orbit. Thus,
for the 2D electron gas in a magnetic field the current vanishes in the bulk.
However, because of the confinement provided by the edge of the sample, states
near the edge do not suffer this cancellation. Thus boundary electrons can
“skip” along the edges and in confined 2D electron gases subject to a
transverse magnetic field the current is carried entirely by edge states. This
edge current is chiral: it is right-going along the upper edge and left-going
along the lower edge in Fig. 14(a).

The preceding argument is basically correct, even though it is classical and
Fig. 14 is just a cartoon. Solution of the Schrödinger equation with an edge
confining potential leads to a delocalization of the wavefunction parallel to
the edge and currents that are chiral because they flow in a single direction
along each edge. Energy levels in the first few Landau levels are illustrated
in Fig. 14(b) for a confining potential along one edge Wen (2004). States are
gapped by $\Delta E=\hbar\omega_{\mbox{\scriptsize c}}$ in the bulk, but are
gapless near the edge, where energy between states tends to zero in the
thermodynamic limit of many particles.

The chiral nature of the edge states protects their extended character, even
if there is impurity scattering (Figs. 14 and 11). Normally, scattering
between time-reversed states of opposite momentum in low-dimensional systems
leads to Anderson localization and insulating character (Appendix B). But if
all states are of a single chirality, scattering of a particle of definite
chirality into a time-reversed state would change the chirality, which is
forbidden if no states of opposite chirality are accessible. Thus, if
tunneling across the device is negligible, impurity scattering is ineffective
in localizing edge states.

####  VII.2.3  Topology of the 2D Brillouin Zone

Let us now understand why quantization of conductance in integer multiples of
$e^{2}/h$ is so precise across varying experimental conditions, with samples
having different geometries, electron densities, and impurity concentrations.
The independent variables are momenta $\bm{k}$ in the Brillouin zone (BZ), but
these are turned into angular variables by the constraint $k(0)=k(2\pi)$
arising from the periodicity of Bloch waves on the lattice. Taking the
Brillouin zone for a 1D crystal to range from $k=-\pi/a$ to $+\pi/a$ for
lattice spacing $a$, the momentum dependence of a band energy can be displayed
as in Fig. 15(a).

Figure 15:  Topology of the Brillouin zone. (a) Momentum dependence of a
single band in a 1D crystal. (b) The periodicity of the crystal requires that
$k=-\pi/a$ and $k=+\pi/a$ be identified, which converts the Brillouin zone to
a circle and the band-energy plot to a cylinder. (c) The 2D Brillouin zone.
(d) Because of crystal periodicity in the $x$ and $y$ directions, the top and
bottom edges, as well as the left and right edges, of the 2D BZ must be
identified, giving the geometry of a (closed) 2-torus for the Brillouin zone.
(e) Different closed paths in the toroidal 2D Brillouin zone, implying non-
trivial topology. Adapted from Refs. Guidry and Sun (2022); Vanderbilt (2018).

But this doesn’t indicate clearly the equivalence of the right and left sides
of the BZ at $k=\pm\pi/a$. We can improve things by identifying $k=+\pi/a$ and
$k=-\pi/a$, which maps the 1D BZ from an interval on the real number line to a
circle and the band-energy plot to a cylindrical, as in Fig. 15(b).

The Brillouin zone for a 2D crystal is displayed in Fig. 15(c). Periodicity in
$k_{y}$ identifies top and bottom edges, converting the square to a cylinder,
and periodicity in $k_{x}$ identifies left and right edges, joining the ends
of the cylinder to form the 2-torus of Fig. 15(d). The toroidal geometry of
the 2D Brillouin zone implies different classes of closed paths on the
manifold, as illustrated in Fig. 15(e). Loops C and D can be shrunk
continuously to a point (D is closed because of the periodic boundary
condition); loops A and B cannot. Thus the loops A, B, and C lie in distinct
topological sectors that cannot be deformed continuously into one another. The
non-trivial connectivity of the 2D Brillouin zone in Fig. 15 implies states
characterized by topological quantum numbers that cannot be deformed smoothly
into each other. Thus the Hilbert space for the integer quantum Hall effect is
defined on a momentum-space torus and this leads to inequivalent topological
sectors labeled by Chern numbers, as we now discuss.

####  VII.2.4  Topological Protection of Hall Plateaus

Topological protection accounts for the remarkable flatness of the plateaus in
Fig. 9, because it can be shown that, as a consequence of the non-trivial
topology of the Brillouin zone and the Chern theorem (Appendix C), the Hall
conductance $\sigma_{\mbox{\scriptsize H}}=1/R_{\mbox{\scriptsize H}}$
contributed by a single non-degenerate band is given by Tong

$\sigma_{\mbox{\scriptsize H}}=\frac{e^{2}}{2\pi h}\int_{S}\Omega
dS=C_{1}\frac{e^{2}}{h},$ (70)

where $e$ is electrical charge, $h$ is Planck’s constant, $S$ is a closed
surface in the manifold of quantum states, $\Omega$ is local Berry curvature
in that manifold, and $C_{1}$ is a topological Chern number that takes integer
values. Thus the exquisite flatness of the plateaus for $R_{\mbox{\scriptsize
H}}$ in Fig. 9 follows from topological protection within a particular quantum
topological phase ensured by Eq. (70), and the vertical jumps between plateaus
correspond to quantum phase transitions between topological states
characterized by different Chern numbers (see Appendix C).

####  VII.2.5 Bulk–Boundary Correspondence

In the quantum Hall effect, and more generally in topological matter (material
characterized by topological quantum numbers), it is important to distinguish
the bulk (interior) of a sample and the boundary (surface in 3D, edges in 2D,
and ends in 1D). In the quantum Hall effect the strong magnetic field
localizes electrons in the bulk, while forcing boundary electrons into
delocalized chiral edge states [Fig. 14(a)]. Thus the bulk states are
insulating but the boundary states are conducting (metallic). This
relationship between the bulk states and the boundary states for the IQHE is
an example of bulk–boundary correspondence, which is a characteristic feature
of topological matter and implies that topological quantum numbers
characterizing the bulk also predict boundary properties.

####  VII.2.6  Incompressible States

When $N$ electrons exactly fill an integer number $\nu$ of Landau levels there
is an energy gap $\hbar\omega_{\mbox{\scriptsize c}}$ between filled and empty
states. Now a decrease in the area $A$ of the system at constant magnetic
field and electron density will decrease the number of flux quanta $N_{\phi}$
piercing the sample. By Eq. (66), this reduces the capacity of the Landau
levels and requires that electrons be promoted to higher Landau levels. There
is an energy cost $\hbar\omega_{\mbox{\scriptsize c}}$ for each promoted
electron, so the system resists compression and is said to be incompressible.
Thus, a quantum Hall system is an incompressible quantum fluid.

###  VII.3 Summary: Integer Quantum Hall Effect

In summary, for the integer quantum Hall effect the crucial points are (1)
Landau-level quantization in the magnetic field, (2) a random scattering
potential caused by impurities, and that (3) each Landau level contributes a
fixed value to the Hall conductance as a consequence of mobility gaps created
by the impurity scattering. Thus conductance is quantized because effectively
it counts the number of filled Landau levels. The robustness of this
quantization derives from the non-trivial Brillouin zone topology and the
Chern theorem, as described in Sections VII.2.3 and VII.2.4, and Appendix C.

##  VIII  The Fractional Quantum Hall Effect

Tsui, Stormer, and Gossard Tsui et al. (1982) discovered the fractional
quantum Hall effect (FQHE) in 1982 when they found that at higher magnetic
fields and lower temperatures the quantum Hall effect can occur for the
fractional value of $\tfrac{1}{3}(e^{2}/h)$, corresponding to a filling
fraction $\nu=1/3$, unlike the integer quantum Hall effect (IQHE), which
occurs only at integer multiples of $e^{2}/h$. Laughlin Laughlin (1983)
proposed that the FQHE corresponds to a new state of matter formed by strong
electron–electron correlations within a single Landau level. It was shown in
later work that the FQHE could occur for whole series of fractional values
$\tfrac{1}{m}(e^{2}/h)$, with $m$ an even or odd integer.

###  VIII.1  Properties of the fractional quantum Hall state

Let us consider a theoretical understanding of this fractional quantum Hall
effect. First, note that fractional filling factors in themselves are not
unusual. The conductance is expected to be $\nu e^{2}/h$ if we add non-
interacting electrons to the lowest Landau level up to a fractional occupation
$\nu$. But in the absence of interactions there will be no gap at the Fermi
surface and adding electrons will cost almost no energy, destroying the
plateau structure of the Hall resistance in Fig. 9. If we assume interacting
electrons instead, they will scatter into empty Landau states, leading to a
finite longitudinal resistance, which contradicts experiments. Thus,
explanation of the fractional quantum Hall effect requires a ground state that

  1. 1.

has a fractionally-filled Landau level and

  2. 2.

has an energy gap with respect to excitation at that fractional filling.

As noted above, a state with an energy gap resists compression, so we may also
term such a state incompressible. Ground states with a gap and small
resistance were known when the fractional quantum Hall effect was discovered,
but none had the right properties to explain the FQHE. A fundamentally new
kind of state is required for which strong electronic correlations in a
partially-filled Landau level produce a ground state corresponding to an
incompressible electronic liquid, with a fractional filling factor and an
energy gap for all excitations.

Both the IQHE and the FQHE result from quantizing electronic motion for 2D
electron gases in strong magnetic fields, but their mechanisms are
fundamentally different. The IQHE involves weakly-interacting electrons and
depends on impurity scattering for its plateau structure indicating
quantization of conductance, but the FQHE requires very pure samples because
it is a consequence of strong Coulomb interactions between the degenerate
electrons in Landau levels. Strongly-correlated electrons in a magnetic field
represent an inherently difficult problem; it was solved only by an educated
guess for the form of the wavefunction. The difficulty of solving this problem
derives from some features of the FQHE that distinguish it from most other
problems in many-body physics Jain (2007).

(1) It is generally assumed that the FQHE corresponds to partial filling of
the lowest Landau level (LLL). Absent interactions there are many ways to fill
the LLL partially that give the same energy. What principle should guide us to
choose the linear combination of these states that will define the physical
ground state when the interactions are turned on?

(2) In the very high magnetic fields that characterize FQHE experiments the
problem reduces approximately to Coulomb interactions between electrons in the
lowest Landau level. The strength of the Coulomb interaction just sets an
energy scale, so the fractional quantum Hall problem contains no obvious
parameters to adjust in order to gain intuition about the physics responsible
for the solution.

(3) In many-body physics it is common to view an emergent state as resulting
from an instability of a parent “normal state” that would be favored in the
absence of interactions. But turning off the interactions for the FQHE state
does not lead to a normal state that is the obvious parent of the FQHE state.

For these reasons the solution of the FQHE problem cannot be obtained by
successive small steps or other standard approximations and we are reduced to
making an educated guess for a wavefunction that leads to the observed FQHE.
The best such educated guess to date is the Laughlin wavefunction.

###  VIII.2  The Laughlin wavefunction

Let us consider a wavefunction to describe the fractional quantum Hall state,
guided by the discussion in Phillips Phillips (2003). We assume the magnetic
field to be large and as a first approximation ignore the Coulomb interactions
between electrons, and assume the electron spins to be completely polarized by
the field and drop the corresponding (constant) spin term. This gives again
the Hamiltonian (54), but now let’s work in the symmetric gauge defined in Eq.
(56b). It is convenient to introduce the complex coordinate
$z_{i}=x_{i}+iy_{i}=re^{i\phi}$ for the $i$th particle, in which case the
solution of the Shrödinger equation in symmetric gauge gives the $N$-body
wavefunctions

$\psi_{N}=\prod^{N}_{j<k}f(z_{j}-z_{k})e^{-\sum_{j=1}^{N}|z_{j}|^{2}/4\Lambda^{2}},$
(71)

where $f(z_{j},z_{k})$ is a polynomial in the electron coordinates $z_{i}$ and
$\Lambda$ is the magnetic length defined in Eq. (59). Analysis of this
wavefunction for two and three electrons led Laughlin to propose what is now
termed the Laughlin wavefunction,

$\psi_{1/m}=\prod^{N}_{j<k}(z_{j}-z_{k})^{m}\,e^{-\sum_{j=1}^{N}|z_{j}|^{2}/4\Lambda^{2}},$
(72)

which specifies the FQHE states at filling factors $\nu=1/m$, with the integer
$m$ required to be odd to ensure antisymmetry of the fermionic wavefunction.
Thus, the first FQHE state discovered with filling factor $\nu=\tfrac{1}{3}$
corresponds to the wavefunction (72) with $m=3$.

The exponent $m$ was introduced originally as a variational parameter but
$\psi_{1/m}$ is an eigenstate of the angular momentum operator with angular
momentum $M$ given by

$M=\frac{1}{2}N(N-1)m$ (73)

for $N$ particles. Thus $m$ is related directly to the angular momentum
content of the Laughlin wavefunction (72), which may be viewed as a
superposition of states from the lowest Landau level having the same angular
momentum.

The Laughlin wavefunction given by Eq. (72) has several features that make it
a plausible candidate for a FQHE wavefunction.

  1. 1.

It minimizes the kinetic energy by placing all electrons in the lowest Landau
level.

  2. 2.

It vanishes if the coordinates of any two electrons are the same because of
the $(z_{k}-z_{\ell})^{m}$ factor, which is desirable for a wavefunction in a
strongly-correlated system with repulsive interactions because it reduces the
total potential energy.

  3. 3.

It is antisymmetric under the exchange of any pair of particles if $m$ is odd.

  4. 4.

It can be shown to describe a circular droplet with $\nu=\tfrac{1}{3}$ and an
approximately uniform density.

  5. 5.

Simple models and numerical simulations suggest that it describes an
incompressible state that has an energy gap to all excitations.

The wavefunction (72) is now accepted broadly as a valid description of the
first $\nu=\tfrac{1}{3}$ FQHE state, and by inference of other fractional
quantum Hall states discovered later.

The Laughlin wavefunction provides only limited fundamental insight into the
underlying microscopic physics responsible for the FQHE state since it is in
essence a highly-educated guess at the many-body wavefunction (similar in
spirit to the “guess” by Bardeen, Cooper, and Schrieffer for the form of the
BCS wavefunction that led to comprehensive understanding of the
superconducting problem). However, the influence of the Laughlin wavefunction
on understanding the fractional quantum Hall effect has been seminal.

##  IX Graphene in Strong Magnetic Fields

As we have seen, rather remarkable quantum Hall physics is observed when a
cold 2D electron gas is subject to a strong magnetic field. Since a single
layer of graphene is the ideal 2D material, it is of interest whether similar
quantum Hall effects could be observed in graphene. To begin addressing that
question we consider in this section the quantum states of electrons confined
in 2D and governed by a massless Dirac single-particle Hamiltonian, with a
strong magnetic field placed perpendicular to the 2D plane. Our discussion
will have some overlap with the earlier treatment of a similar system with
electrons governed by a Schrödinger equation, but there will be substantial
differences associated with the Dirac rather than Schrödinger physics
appropriate for graphene. We shall use the review by Goerbig Goerbig (2011)
for guidance, and often use $c=1$ and $\hbar=1$ units.

###  IX.1 Quantization of Dirac Landau levels

By the usual minimal substitution, electromagnetism may be included in a
Hamiltonian by replacing the canonical momentum $\bm{p}$ with the gauge-
invariant kinetic momentum $\bm{\Pi}$,

$\bm{\Pi}=\bm{p}+e\bm{A}(\bm{r}),$ (74)

where the magnetic field $\bm{B}$ is related to the vector potential $\bm{A}$
by

$\bm{B}=\bm{\nabla}\times\bm{A}.$ (75)

From Eq. (40), to lowest order in $|\bm{q}|a$ the Hamiltonian with the minimal
substitution is

$H=\xi v_{\mbox{\scriptsize F}}(\Pi_{x}\sigma^{x}+\Pi_{y}\sigma^{y}).$ (76)

The electron energies will also split into two branches according to the
Zeeman energy in the magnetic field $\Delta_{z}=g\mu_{\mbox{\scriptsize B}}B$,
where $g\sim 2$ for graphene. In this initial discussion we ignore
$\Delta_{z}$, but will return to the role of spin later.

The canonical method may be used to quantize this system. In the absence of
the vector potential the coordinates and momenta would obey the commutation
relations

$\begin{gathered}[\,x,p_{x}\,]=[\,y,p_{y}\,]=i\hbar\\\
[\,x,y\,]=[\,p_{x},p_{y}\,]=[\,x,p_{y}\,]=[\,y,p_{x}\,]=0.\end{gathered}$ (77)

From Eqs. (74) and (77), the commutator of $\Pi_{x}$ and $\Pi_{y}$ is Goerbig
(2011)

$\displaystyle[\,\Pi_{x},\Pi_{y}\,]=-ie\hbar\left(\frac{\partial{A_{y}}}{\partial{x}}-\frac{\partial{A_{x}}}{\partial{y}}\right).$
(78)

But the magnetic field is assumed aligned along the $z$-axis,
$\bm{B}(B_{x},B_{y},B_{z})=(0,0,B)$ and from Eq. (75),

$\frac{\partial{A_{y}}}{\partial{x}}-\frac{\partial{A_{x}}}{\partial{y}}=B_{z}\equiv
B,$

and therefore Eq. (78) becomes

$[\,\Pi_{x},\Pi_{y}\,]=-ie\hbar
B=-i\hbar^{2}\frac{eB}{\hbar}=-i\,\frac{\hbar^{2}}{\Lambda^{2}},$ (79)

with the magnetic length $\Lambda$ introduced in Eq. (59) given by
$\Lambda=\sqrt{\hbar/eB}$ in $c=1$ units. It will be convenient to introduce
the raising and lowering operators

$a^{\dagger}=\frac{\Lambda}{\sqrt{2}\,\hbar}\left(\Pi_{x}+i\,\Pi_{y}\right)\qquad
a=\frac{\Lambda}{\sqrt{2}\,\hbar}\left(\Pi_{x}-i\,\Pi_{y}\right),\qquad$ (80)

where the normalization factor is chosen so that

$[\,a,a^{\dagger}\,]=1.$ (81)

Adding and subtracting Eqs. (80) gives the inverse relations

$\Pi_{x}=\frac{\hbar}{\sqrt{2}\Lambda}(a^{\dagger}+a)\qquad\Pi_{y}=\frac{\hbar}{\sqrt{2}\Lambda
i}(a^{\dagger}-a),$ (82)

and inserting this result into the Hamiltonian (76) gives

$\displaystyle H$ $\displaystyle=\xi v_{\mbox{\scriptsize
F}}\left(\Pi_{x}\,\sigma^{x}+\Pi_{y}\,\sigma^{y}\right)$ $\displaystyle=\xi
v_{\mbox{\scriptsize F}}\left[\Pi_{x}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}+\Pi_{y}\begin{pmatrix}0&-i\\\
i&\phantom{-}0\end{pmatrix}\right]$ $\displaystyle=\xi v_{\mbox{\scriptsize
F}}\begin{pmatrix}0&\Pi_{x}-i\Pi_{y}\\\\[0.0pt]
\Pi_{x}+i\Pi_{y}&0\end{pmatrix}$
$\displaystyle=\frac{\sqrt{2}\,\hbar}{\Lambda}\,\xi v_{\mbox{\scriptsize
F}}\begin{pmatrix}0&a\\\\[0.0pt] a^{\dagger}&0\end{pmatrix}$
$\displaystyle=\hbar\,\xi\omega\begin{pmatrix}0&a\\\\[0.0pt]
a^{\dagger}&0\end{pmatrix},$ (83)

where Eqs. (80) were used and $\omega\equiv\sqrt{2}v_{\mbox{\scriptsize
F}}/\Lambda$ is the analog of the cyclotron frequency (48) for the present
relativistic problem. Introducing the 2-component spinor

$\psi_{n}=\begin{pmatrix}u_{n}\\\ v_{n}\end{pmatrix},$ (84)

we must then solve the eigenvalue equation $H\psi_{n}=\epsilon_{n}\psi_{n}$,
which takes the form

$\hbar\xi\omega\begin{pmatrix}0&a\\\\[0.0pt]
a^{\dagger}&0\end{pmatrix}\begin{pmatrix}u_{n}\\\
v_{n}\end{pmatrix}=\epsilon_{n}\begin{pmatrix}u_{n}\\\ v_{n}\end{pmatrix},$
(85)

implying the simultaneous equations

$\displaystyle\hbar\xi\omega av_{n}$ $\displaystyle=\epsilon_{n}u_{n},$ (86a)
$\displaystyle\hbar\xi\omega a^{\dagger}u_{n}$
$\displaystyle=\epsilon_{n}v_{n}.$ (86b)

Solving Eq. (86a) for $u_{n}$ and substituting into Eq. (86b) gives
$a^{\dagger}av_{n}=(\epsilon_{n}/\hbar\omega)^{2}v_{n}$. But because of Eq.
(81) this is just the usual harmonic oscillator number equation
$a^{\dagger}a\left|\psi\right\rangle=n\left|\psi\right\rangle$ with
$n=(\epsilon_{n}/\hbar\omega)^{2}$, and up to numerical factors the state
$\left|v_{n}\right\rangle$ may be identified with a number eigenstate
$\left|n\right\rangle$. Solving for $\epsilon_{n}$ in terms of $n$ gives
$\epsilon_{n}=\pm\hbar\omega\sqrt{n}$, which may be written

$\displaystyle\epsilon_{n}=\lambda\hbar\omega\sqrt{n}=\lambda\frac{\hbar
v_{\mbox{\scriptsize F}}}{\Lambda}\sqrt{2n}\propto\lambda\sqrt{nB},$ (87)

where the quantum number $\lambda=\pm 1$ has been introduced to distinguish
the positive and negative square roots; it plays the same role as the band
index in the zero-field case [see Eq. (12)], corresponding to the graphene
two-fold valley degeneracy for $B=0$. Thus, the energy disperses as
$\epsilon_{n}\sim\pm(nB)^{1/2}$ with magnetic field strength, as illustrated
in Fig. 16(a).

Figure 16:  (a) Relativistic Landau energy levels as a function of magnetic
field strength $B$. (b) Guiding center coordinates $\bm{R}$ and cyclotron
coordinates $\bm{r}$. Classically the guiding center coordinates define the
center of the cyclotron orbit and the cyclotron coordinates describe the
motion of the electron relative to the guiding center. The shaded area in the
center of the circle represents the uncertainty in the position of the guiding
center implied by Eq. (95). Figure adapted from Ref. Goerbig (2011).

From Eq. (87) and Fig. 16(a) the $n=0$ state is not split into positive and
negative branches, as is the case for all other values of $n$. From Eq. (86a)
we have $u_{n}\propto av_{n}\sim
a\left|n\right\rangle\sim\left|n-1\right\rangle$, because of the usual
oscillator operator requirements that

$\displaystyle
a^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle\qquad
a\left|n>0\right\rangle=\sqrt{n}\left|n-1\right\rangle$ $\displaystyle
a\left|n=0\right\rangle=0.$

Thus, for $n=0$ the eigenspinor (84) has only a single component,

$\psi_{n}=\begin{pmatrix}u_{n}\\\ v_{n}\end{pmatrix}=\begin{pmatrix}0\\\
\left|n=0\right\rangle\end{pmatrix},$ (88)

with the non-zero component corresponding to the B sublattice in the $K$
valley ($\xi=+1$) and the A sublattice in the $K^{\prime}$ valley ($\xi=-1$).
This implies that the valley pseudospin and the sublattice pseudospin coincide
for $n=0$. For all other values of $n$, there are two spinor components and
positive and negative energy solutions.

###  IX.2 Degeneracy of Landau levels

Just as for the non-relativistic case, Landau levels for relativistic fermions
in a strong magnetic field become highly degenerate. In both the non-
relativistic and relativistic cases the degeneracy $N_{B}$ is equal to the
number of flux quanta threading the 2D electron gas,

$N_{B}=\frac{B}{(h/e)}\,S,$ (89)

where $S$ is the area of the 2D sample, $BS$ is the total flux, and $h/e$ is
the flux quantum (in $c=1$ units); see Eq. (65) and the discussion following
it. Let’s investigate this degeneracy in more detail for the relativistic case
appropriate for graphene.

It is convenient to separate the relativistic cyclotron motion of the
electrons into guiding center coordinates $\bm{R}=(X,Y)$ that commute with the
Hamiltonian, and relative coordinates termed cyclotron variables
$\bm{\eta}=(\eta_{1},\eta_{2})$, according to

$\bm{r}=\bm{R}+\bm{\eta},$ (90)

as illustrated in Fig. 16(b). Classically, the guiding center coordinates
represent the center of the cyclotron motion for an electron and the cyclotron
coordinates describe the time-dependent (dynamical) component of motion. From
Fig. 16(b) the cyclotron variable $\bm{\eta}$ is orthogonal to the electron
velocity and hence is related to the kinetic momentum $\bm{\Pi}$ defined in
Eq. (74) by

$\eta_{x}=\frac{\Pi_{y}}{eB}\qquad\eta_{y}=-\frac{\Pi_{x}}{eB}.$ (91)

From this relation and Eq. (79), the commutator of the cyclotron variables is

$\displaystyle[\,\eta_{x},\eta_{y}\,]=\frac{[\,\Pi_{x},\Pi_{y}\,]}{(eB)^{2}}=-i\Lambda^{2}.$
(92)

This can be used to find the commutation relation for the guiding center
coordinates $(X,Y)$. From Eq. (90) with $r=(x,y)$ we have $x=X+\eta_{x}$ and
$y=Y+\eta_{y},$ and since $[\,x,y\,]=0$,

$\displaystyle[\,x,y\,]$ $\displaystyle=[\,X+\eta_{x},Y+\eta_{y}\,]$
$\displaystyle=[\,X,Y\,]+[\,X,\eta_{y}\,]+[\,\eta_{x},Y\,]+[\,\eta_{x},\eta_{y}\,]=0.$

Assuming the guiding center and cyclotron coordinates to commute, this yields

$[\,X,Y\,]=-[\,\eta_{x},\eta_{y}\,]=i\Lambda^{2},$ (93)

where Eq. (92) was used. Thus $X$ and $Y$ are conjugate variables and Eq. (93)
is an uncertainty relation associated with simultaneous specification of $X$
and $Y$. Introduce

$b=\frac{1}{\sqrt{2}\Lambda}(X+iY)\qquad
b^{\dagger}=\frac{1}{\sqrt{2}\Lambda}(X-iY),$ (94)

which satisfy the commutation relation $[\,b,b^{\dagger}\,]=1$ and commute
with the Hamiltonian (since $X$ and $Y$ do). Because of this commutation
relation this is a second set of harmonic-oscillator operators, so we expect
$b^{\dagger}b$ to be a number operator with
$b^{\dagger}b\left|m_{k}\right\rangle=m_{k}\left|m_{k}\right\rangle$, where
the integer $m_{k}\geq 0$ (separate from $n$) counts the states created by
operating repeatedly on the vacuum with $b^{\dagger}$. Since $b$ and
$b^{\dagger}$ commute with the Hamiltonian, $m_{k}$ is an additional quantum
number that labels electronic states in the magnetic field and we conclude
that the states on a given branch of Fig. 16(a) may be labeled by the quantum
numbers $n$ and $m_{k}$.

As noted above, Eq. (93) is an uncertainty relation indicating that we cannot
measure the $X$ and $Y$ coordinates of the guiding center for an electron
simultaneously with arbitrary precision. The location of the guiding center is
found to be uncertain according to Goerbig (2011)

$\Delta X\Delta Y=2\pi\Lambda^{2},$ (95)

which is represented by the shaded area in the center of Fig. 16(b). Let the
total area of the 2D gas be $S$ and introduce the flux density $n_{B}$
measured in units of the flux quantum $h/e$ through

$n_{B}=\frac{N_{B}}{S}=\frac{B}{(h/e)},$ (96)

where Eq. (89) was used. The number of quantum electronic states found in the
area $S$ is

$N_{B}=\frac{S}{\Delta X\Delta Y}=\frac{S}{2\pi\Lambda^{2}}=n_{B}S,$ (97)

and so

$n_{B}=\frac{B}{(h/e)}=\frac{1}{2\pi\Lambda^{2}}.$ (98)

Thus, the flux density and the density of possible electronic states in a
magnetic field coincide for the 2D gas. The filling factor $\nu$ can then be
defined by the ratio of the electron number density $n_{e}$ and the flux
density $n_{B}$,

$\nu\equiv\frac{n_{e}}{n_{B}}=\frac{hn_{e}}{eB}.$ (99)

This treatment assumes the only degeneracies to be those of the Landau levels
for massless Dirac electrons. In the following sections we shall see how this
is modified by additional degeneracies associated with sublattice and valley
degrees of freedom in graphene.

##  X Quantum Hall Effects in Graphene

The quantum Hall effect occurs for 2D electron gases in strong magnetic
fields. Because of degrees of freedom associated with the honeycomb lattice
structure, we may expect that quantum Hall effects in graphene could be richer
than those in the usual 2D electron gas. From the preceding discussion we may
expect that analogs of the integer quantum Hall states might occur for weakly-
interacting electrons in graphene, but the fractional quantum Hall state can
be produced only by strongly-correlated electrons. Most of the low-energy
properties of graphene are described well assuming weakly interacting massless
Dirac fermions in two dimensions, but we may ask whether there are particular
situations in which monolayer graphene might exhibit strong electron–electron
correlations. The relative effectiveness of correlations are expected to
depend on (1) the intrinsic strength of the electronic interactions, and (2)
the density of states at the Fermi surface.

###  X.1 Strength of correlations in graphene

For the Coulomb gas of electrons characterizing a layer of graphene, the
average interaction energy at a characteristic length scale $k_{\rm F}^{-1}$
is $E_{\mbox{\scriptsize int}}\sim e^{2}k_{\mbox{\scriptsize F}}/\epsilon$,
where $e$ is the electronic charge, $\epsilon$ is the effective dielectric
constant of the environment, and $k_{\mbox{\scriptsize F}}$ is the Fermi
momentum. The average kinetic energy at the same length scale is
$E_{\mbox{\scriptsize kin}}\sim\hbar\epsilon v_{\mbox{\scriptsize F}}$, where
$v_{\mbox{\scriptsize F}}$ is the Fermi velocity. The ratio of interaction
energy and kinetic energy at this characteristic length scale defines a
coupling constant for grapheneGoerbig (2011); Goerbig and Regnault (2012),

$\alpha_{\mbox{\scriptsize G}}\equiv\frac{E_{\mbox{\scriptsize
int}}}{E_{\mbox{\scriptsize kin}}}=\frac{e^{2}k_{\mbox{\scriptsize
F}}/\epsilon}{\hbar v_{\mbox{\scriptsize F}}k_{\mbox{\scriptsize
F}}}=\frac{e^{2}}{\hbar\epsilon v_{\mbox{\scriptsize F}}}.$ (100)

This has the same form as the fine-structure constant $\alpha$ of quantum
electrodynamics (QED) if the speed of light $c$ is exchanged for the Fermi
velocity $v_{\mbox{\scriptsize F}}$. The “fine structure constant”
$\alpha_{\mbox{\scriptsize G}}$ of graphene is $c/v_{\mbox{\scriptsize F}}\sim
300$ times that for QED ($\alpha\sim 1/137$) at comparable energies, implying
a relatively large Coulomb interaction. However, the low-energy excitations in
graphene obey an approximately linear dispersion relationship and in the
undoped compound the density of states vanishes as the Fermi surface [at the
apex of the Dirac cones in Fig. 4(b)] is approached. This low density of
states near the Fermi surface suppresses electron–electron correlations and
much of low-energy physics for monolayer graphene is dominated by weakly-
correlated particles. If we wish to search for strongly-correlated physics in
graphene, it is necessary to find circumstances that correspond to increased
level density near the Fermi surface Goerbig and Regnault (2012).

One way to attain higher local level density is to place a strong magnetic
field $B$ perpendicular to the sample. Then the electronic levels become
quantized into Landau levels that are highly degenerate, with a strongly-
peaked density of states

$\rho(E)=gn_{B}\sum_{n}f(E-E_{n}),$ (101)

where $g$ is a degeneracy factor for internal degrees of freedom (in graphene,
$g=4$ because of the four-fold spin–valley degeneracy), $n_{B}=eB/h$ is the
flux density in units of the flux quantum $h/e$, and $f(E-E_{n})$ is strongly
peaked, tending to a $\delta$-function in the clean limit. In this limit, each
LL may then be approximated as an infinitely flat (dispersionless, because
kinetic energy is negligible) energy band in which the state density grows
linearly with the magnetic field strength Goerbig and Regnault (2012). The
effect of Fermi surface position relative to the Landau levels may be studied
by fixing the magnetic field and sweeping a gate voltage applied to the
sample, or by fixing the gate voltage and sweeping the magnetic field.

###  X.2 The graphene integer quantum Hall effect

Electron spin has largely been ignored in the discussion of the quantum Hall
effect to this point, but if it is not neglected IQHE conditions are satisfied
when both spin branches (split by the Zeeman term) of the last Landau level
(LLL) are completely filled. Thus $\nu_{\mbox{\scriptsize IQHE}}=2n$, where
$n$ is an integer and filling factors are even integers in the usual IQHE. The
graphene IQHE is expected to resemble the usual IQHE, except for two key
points.

  1. 1.

In addition to the 2-fold spin degeneracy (if Zeeman splitting is neglected),
there is a 2-fold $K$ and $K^{\prime}$ valley degeneracy for graphene. Thus
the filling factor changes in steps of four between plateaus in the Hall
resistance for graphene.

  2. 2.

For graphene the filling factor

$\nu=\frac{n_{e}}{n_{B}}=\frac{hn_{e}}{eB}$ (102)

vanishes at the Dirac point for particle–hole symmetric half filling of the
graphene lattice, because the electron density $n_{e}$ tends to zero at the
Dirac point. Therefore, absent a Zeeman effect or electron correlations, one
expects no integer quantum Hall effect effect in graphene for $\nu=0$.

Thus the signature of a graphene IQHE is Hall resistance quantization for
filling factors

$\nu=\pm 4\left(n+\frac{1}{2}\right)=\pm(4n+2)=\pm 2,\pm 6,\pm 10,\ldots$
(103)

where $n$ is an integer, the factor of 4 is because of the 4-fold valley and
spin degeneracy, and the added $\frac{1}{2}$ (not present in non-relativistic
2D systems) is because of the special status of the $n=0$ state for massless
Dirac fermions in graphene.

Evidence for an integer quantum Hall effect in graphene was reported in Refs.
Novoselov et al. (2005); Zhang et al. (2005), and is illustrated in Fig.
17(a).

Figure 17:  Evidence for integer and fractional quantum Hall effects in
monolayer graphene Novoselov et al. (2005); Bolotin et al. (2009); Feldman et
al. (2012); Zhang et al. (2005). (a) Integer quantum Hall effect Novoselov et
al. (2005). The filling factors are anomalous, as described in the text. (b)
Fractional quantum Hall effect; lower fields Bolotin et al. (2009). (c)
Fractional quantum Hall effect; higher fields Bolotin et al. (2009). The
plateau labeled A has features expected for a $\nu=1/3$ fractional quantum
Hall state. (d) Incompressible states with fractional filling numbers
(marked), as a function of magnetic field $B$ and carrier density $n$ Feldman
et al. (2012); contours of $d\mu/dn$ in units of
$10^{-10}\mbox{\,meV\,cm}^{2}$ are displayed.

These experiments confirmed the predicted anomalous spacing of Eq. (103) for a
graphene integer quantum Hall effect. Later experiments observed fragile IQHE
states at filling factors $0,\,\pm 1,\pm 4$, that are thought to be caused by
Zeeman splitting and electron interactions breaking degeneracies of the $n=0$
Landau level.

###  X.3 The effect of strong correlations in graphene

Graphene is a unique environment because of two aspects not found in non-
relativistic quantum Hall experiments: (1) its charge carriers are effectively
massless chiral fermions, and (2) the symmetries associated with the four-fold
spin and valley degrees of freedom introduce new features for states and
correlations. As we have seen in Section X.1, low-energy excitations in normal
monolayer graphene occur in regions of reduced electron density, which
disfavors electron correlations. But by applying a strong magnetic field the
resulting Landau quantization leads to a bunching of levels into regions of
high degeneracy, giving conditions more amenable to the development of strong
electronic correlations if the Fermi surface lies in one of those regions.

Landau levels (LL) become strongly correlated when inter-LL excitations may be
neglected and the low-energy excitations involve only the same level. Then the
kinetic energy is a constant that can be ignored. This limit of strong
electronic correlations leads to two important physical effects Goerbig and
Regnault (2012).

  1. 1.

The 4-fold approximate spin–valley degeneracy of the graphene Landau levels
leads to quantum Hall ferromagnetic states.

  2. 2.

The strong correlations can lead to a fractional quantum Hall effects (FQHE).

Because of the approximate 4-fold degeneracy, these effects can be discussed
within the framework of an SU(4) symmetry that we shall elaborate in Section
XI. Let us now consider the fractional quantum Hall effect in graphene, which
is a clear signal of a strongly-correlated state.

###  X.4  The graphene fractional quantum Hall effect

The fractional quantum Hall effect in graphene can be observed only in samples
extremely free of impurities. Strong electron–electron correlations in
general, and the FQHE in particular, were initially very difficult to observe
in graphene because of samples that were not sufficiently clean to suppress
impurity scattering. The fractional quantum Hall effect in graphene was
discovered in 2009 Bolotin et al. (2009); Du et al. (2009), when it became
possible to make measurements on suspended graphene sheets, thus avoiding
impurities and scattering associated with substrate contact. Evidence for a
fractional quantum Hall effect in graphene from those experiments is presented
in Fig. 17(b)-(c). A plateau labeled A at a density dependent on the magnetic
field emerges for $B>11$ T. The inset to Fig. 17(c) plots this feature as a
function of filling factor. The different traces for different magnetic field
strength collapse into a single universal feature, suggesting that A be
identified with a $\nu=\tfrac{1}{3}$ fractional quantum Hall state.

In Fig. 17(d), further evidence for incompressible states at fractional
filling in graphene is shown Feldman et al. (2012). The inverse
compressibility is defined by $\kappa^{-1}=d\mu/dn$, where $n$ is the local
carrier density and $\mu$ is the chemical potential. In Fig. 17(d) a scanning
single-electron transistor was used to measure the local incompressibility of
electrons in a suspended graphene sample and the derivative $d\mu/dn$ was
plotted versus density and magnetic field strength. High incompressibility is
indicated by vertical red bands. Localized states are broad and curve as the
magnetic field is varied but the fractional quantum Hall states are narrow and
vertical, and are labeled by fractional filling numbers. The states that are
observed follow the standard sequence for filling factors $\nu=0-1$, but only
even-numerator fractions are seen for $\nu=1-2$. It is thought that these
sequences and the corresponding energy gaps for the incompressible states
reflect the interplay of strong electron correlations and the characteristic
symmetries of graphene that will be discussed further in Section XI Feldman et
al. (2012).

##  XI SU(4) Quantum Hall Ferromagnetism

Graphene FQHE states can exhibit various symmetries because of possible
degeneracies associated with the pseudospin and valley degrees of freedom. In
the normal two-dimensional electron gas (2DEG) produced in semiconductor
devices the Landau levels (LL) can contain $eB/h$ states, where $e$ is the
electronic charge, $B$ is the magnetic field strength, and $h$ is Planck’s
constant. In graphene, there is an additional 4-fold degeneracy associated
with the spin and valley degrees of freedom. It is common to unite these four
degrees of freedom (often termed flavors) in terms of an SU($4$) symmetry
called quantum Hall ferromagnetism (QHF).

###  XI.1 Degeneracies and filling of Landau levels

Single-particle states within a Landau level may be labeled by quantum numbers
$(n,m_{k})$, with $n$ indicating the Landau level and $m_{k}$ labeling
degenerate states associated with that level. These Landau states $(n,m_{k})$
hold a maximum of $2\Omega_{k}=BS/(h/e)$ electrons if spin and valley degrees
of freedom are neglected, where $B$ is the strength of the magnetic field, $S$
is the area of the two-dimensional sample, and $h/e=4.136\times
10^{-15}\mbox{\,Wb}$ is the magnetic flux quantum. But the 4-fold degenerate
internal $\left|\rm spin\right\rangle\otimes\left|\rm isospin\right\rangle$
space for graphene implies that there are four copies of each Landau level
$(n,m_{k})$ and the total electron degeneracy $2\Omega$ for graphene is given
by

$2\Omega=4(2\Omega_{k})=\frac{4BS}{(h/e)}.$ (104)

For a single Landau level the fractional occupation $f$ of the single Landau
level is

$f\equiv\frac{n}{2\Omega}=\frac{N}{\Omega},$ (105)

where $n$ is the electron number and $N=\tfrac{1}{2}n$ is the electron pair
number. Note that the fractional occupation $f$ and the quantum Hall filling
factor $\nu$ defined in Eq. (67) are related by Wu et al. (2017)

$\nu=4\left(f-\frac{1}{2}\right).$ (106)

For the ground state of undoped graphene the $n=0$ Landau level located at the
Fermi surface is half filled and the electron number $n_{\mbox{\scriptsize
gs}}$ is then

$n_{\mbox{\scriptsize gs}}=\Omega=\frac{2BS}{(h/e)}.$ (107)

These degeneracies and occupation numbers are standard results for
relativistic Landau electrons in a 2D electron gas, but now modified by the
intrinsic graphene degrees of freedom.

###  XI.2 SU(4) quantum Hall states in graphene

The graphene honeycomb lattice is bipartite, corresponding to the interlocking
A and B sublattices shown in Fig. 18.

Figure 18:  The full bipartite lattice for graphene and the A (solid red
circles) and B (open blue circles) sublattices.

The A and B sublattices are related by inversion, as illustrated in Fig. 5(a).
The $n=0$ LL is located exactly at the Dirac point corresponding to
$\epsilon=0$. For low-energy excitations in each $K$ or $K^{\prime}$ valley
the inter-valley tunneling may be ignored and the wavefunctions in the valley
reside entirely on either the A or B sublattice, as in Fig. 19.

Figure 19:  Localization of valley wavefunctions in the $n=0$ LL. In each
valley $K$ or $K^{\prime}$, the wavefunction resides on only one sublattice,
either A (solid red circles on left) or B (open black circles on right).

For the $n=0$ LL, the valley isospin (labeling whether the electron is in a
$K$ or $K^{\prime}$ valley) is identical to the sublattice pseudospin
(labeling whether the electron is on the A or B sublattice). Thus, this is
analogous to a Néel antiferromagnetic state with spins on two different
sublattices, with a Néel order defined by the difference in spins on the A and
B sublattices.

####  XI.2.1 Effective Hamiltonian

The two largest energy scales for monolayer graphene in a strong magnetic
field are

  1. 1.

the Landau-level separation, $H_{\Delta}=\sqrt{2}\hbar v_{\mbox{\scriptsize
F}}/\Lambda$, and

  2. 2.

the Coulomb energy $H_{\mbox{\scriptsize C}}$.

The separation of Landau levels is typically several times larger than the
Coulomb energy, which is in turn considerably larger than any other terms in
the interaction. Therefore, we adopt a strategy of ignoring excitations
between Landau levels and projecting onto the $n=0$ LL. Such an approximation
gives the correct qualitative physics, which will be sufficient for our
discussion. Within this single LL we assume that the Hamiltonian is dominated
by an SU($4$)-symmetric, long-range Coulomb interaction, with shorter-range
spin and valley isospin interactions originating in electron–electron
interactions and electron–phonon interactions that break SU($4$) symmetry. We
shall use an approximate graphene Hamiltonian for the $n=0$ Landau level that
was proposed in Ref. Kharitonov (2012) and employed in Ref. Wu et al. (2014),

$H=H_{\mbox{\scriptsize C}}+H_{\mbox{\scriptsize v}}+H_{\mbox{\scriptsize
Z}},$ (108)

with the valley-independent Coulomb interaction $H_{\mbox{\scriptsize C}}$,
the Zeeman energy $H_{\mbox{\scriptsize Z}}$, and the short-range, valley-
dependent interaction $H_{\mbox{\scriptsize v}}$ given respectively by

$\begin{gathered}H_{\mbox{\scriptsize C}}=\frac{1}{2}\,\sum_{i\neq
j}\,\frac{e^{2}}{\epsilon|\bm{r}_{i}-\bm{r}_{j}|}\qquad H_{\mbox{\scriptsize
Z}}=-\mu_{\mbox{\scriptsize B}}B\sum_{i}\sigma_{z}^{i},\\\
H_{\mbox{\scriptsize v}}=\frac{1}{2}\sum_{i\neq
j}\left[g_{z}\tau_{z}^{i}\tau_{z}^{j}+g_{\perp}(\tau_{x}^{i}\tau_{x}^{j}+\tau_{y}^{i}\tau_{y}^{j})\right]\delta(\bm{r}_{i}-\bm{r}_{j}),\end{gathered}$
(109)

where $g_{z}$ and $g_{\perp}$ are coupling constants, $\mu_{\mbox{\scriptsize
B}}$ is the Bohr magneton, $B$ is magnetic field strength, the Pauli matrices
$\tau_{\alpha}$ and $\sigma_{\alpha}$ operate on valley isospin and spin,
respectively, and the $z$ direction for the spin space is assumed to be
aligned with the magnetic field.

####  XI.2.2  Symmetry and explicit symmetry breaking

Labeling states in a single Landau level by $m_{k}$ and defining composite
indices $\alpha\equiv\\{x,y,z\\}$ and $\beta\equiv\\{x,y\\}$, we may introduce
the 15 operators

$\displaystyle\mathscr{S}_{\alpha}$
$\displaystyle=\sum_{m_{k}}\sum_{\tau\sigma\sigma^{\prime}}\left\langle\sigma^{\prime}\right|\sigma_{\alpha}\left|\sigma\right\rangle
c^{\dagger}_{\tau\sigma^{\prime}m_{k}}c_{\tau\sigma m_{k}},$ (110)
$\displaystyle T_{\alpha}$
$\displaystyle=\sum_{m_{k}}\sum_{\sigma\tau\tau^{\prime}}\left\langle\tau^{\prime}\right|\tau_{\alpha}\left|\tau\right\rangle
c^{\dagger}_{\tau^{\prime}\sigma m_{k}}c_{\tau\sigma m_{k}},$ $\displaystyle
N_{\alpha}$
$\displaystyle=\frac{1}{2}\sum_{m_{k}}\sum_{\sigma\sigma^{\prime}\tau}\left\langle\tau\right|\tau_{z}\left|\tau\right\rangle\left\langle\sigma^{\prime}\right|\sigma_{\alpha}\left|\sigma\right\rangle
c^{\dagger}_{\tau\sigma^{\prime}m_{k}}c_{\tau\sigma m_{k}},$
$\displaystyle\Pi_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\sum_{m_{k}}\sum_{\sigma\sigma^{\prime}\tau\tau^{\prime}}\left\langle\tau^{\prime}\right|\tau_{\beta}\left|\tau\right\rangle\left\langle\sigma^{\prime}\right|\sigma_{\alpha}\left|\sigma\right\rangle
c^{\dagger}_{\tau^{\prime}\sigma^{\prime}m_{k}}c_{\tau\sigma m_{k}}.$

The operator $\mathscr{S}_{\alpha}$ represents the total spin and the operator
$T_{\alpha}$ represents the total valley pseudospin. In the $n=0$ Landau level
for graphene there is an equivalence between valley and sublattice degrees of
freedom, so $N_{\alpha}$ can be identified as a Néel vector in the $n=0$
Landau level. The operators $\Pi_{\alpha\beta}$ couple the spin and valley
isospin and will be discussed further below.

Under commutation the operators of Eq. (110) satisfy an SU($4$) Lie algebra
that commutes with the Coulomb interaction $H_{\mbox{\scriptsize C}}$ Wu et
al. (2014). Thus, if $H_{\mbox{\scriptsize v}}$ and $H_{\mbox{\scriptsize Z}}$
are much smaller than $H_{\mbox{\scriptsize C}}$, the Hamiltonian (108) has an
approximate SU($4$) symmetry. If Zeeman splitting is ignored, SU($4$) symmetry
is broken explicitly by $H_{\mbox{\scriptsize v}}$ in Eq. (109), with the
degree of symmetry breaking depending on the coupling parameters $g_{z}$ and
$g_{\perp}$. For characteristic graphene lattice spacings, terms involving
$g_{z}$ and $g_{\perp}$ are much smaller than the SU($4$)-symmetric Coulomb
term and explicit breaking of SU($4$) should be small in realistic systems.
Four basic symmetry-breaking patterns have been studied, as summarized in Fig.
20 Kharitonov (2012); Wu et al. (2014).

Figure 20:  Symmetry-breaking pattern for SU($4$) quantum Hall ferromagnetism
generated by the operators in Eq. (110) Kharitonov (2012); Wu et al. (2014).
The SU($4$) symmetry is broken by explicit terms $H_{\mbox{\scriptsize v}}$
and $H_{\mbox{\scriptsize C}}$ in the Hamiltonian (108) that depend on the
parameters $g_{z}$, $g_{\perp}$, and $H_{z}$.

1\. For $g_{z}\neq 0$ and $g_{\perp}\neq 0$, the symmetry is broken to

${\rm SU(4)}\supset{\rm SU(2)}_{\mbox{\scriptsize s}}\times{\rm
U(1)}_{\mbox{\scriptsize v}}\supset{\rm U(1)}_{\mbox{\scriptsize s}}\times{\rm
U(1)}_{\mbox{\scriptsize v}},$ (111)

where SU($2$)${}_{\mbox{\scriptsize s}}$ is associated with spin conservation,
U($1$)${}_{\mbox{\scriptsize s}}$ with conservation of its $z$ component, and
U($1$)${}_{\mbox{\scriptsize v}}$ with conservation of $T_{z}$. If Zeeman
splitting is ignored, spin is conserved but only the $z$ component of valley
isospin is conserved. The full Hamiltonian (108) with the Zeeman term
conserves only the $z$ components of spin and valley isospin.

2\. If $g_{\perp}=0$ but $g_{z}\neq 0$, the symmetry is broken in the manner

$\displaystyle{\rm SU(4)}$ $\displaystyle\supset{\rm SU(2)}_{\rm\scriptstyle
s}^{K}\times{\rm SU(2)}_{\rm\scriptstyle s}^{K^{\prime}}\times{\rm
U}(1)_{\mbox{\scriptsize v}}$ $\displaystyle\supset{\rm U(1)}_{\rm\scriptstyle
s}^{K}\times{\rm U(1)}_{\rm\scriptstyle s}^{K^{\prime}}\times{\rm
U(1)}_{\mbox{\scriptsize v}}.$ (112)

If Zeeman splitting is neglected this implies conservation of spins
independently in the $K$ and $K^{\prime}$ valleys, but that valley isospin is
broken to U(1)${}_{\mbox{\scriptsize v}}$. If the Zeeman term is included, the
full Hamiltonian (108) conserves only the $z$ components of spin in each
valley, and the $z$ component of valley isospin.

3\. If $g_{\mbox{\scriptsize z}}=g_{\perp}\neq 0$, the symmetry-breaking
pattern is

${\rm SU(4)}\supset{\rm SU(2)}_{\mbox{\scriptsize s}}\times{\rm
SU(2)}_{\mbox{\scriptsize v}}\supset{\rm U(1)}_{\mbox{\scriptsize
s}}\times{\rm SU(2)}_{\mbox{\scriptsize v}}.$ (113)

In the absence of Zeeman splitting, this corresponds to full rotational
symmetry in the spin and valley isospin spaces. If the Zeeman term is included
in the full Hamiltonian (108), the SU($2$) isospin symmetry is conserved but
only the $z$ component of the spin is conserved.

4\. If $g_{\perp}=-g_{z}\neq 0$, the 10 generators
$\\{\Pi_{\alpha}^{\beta},\vec{S},T_{z}\\}$ commute with the Hamiltonian,
forming the Lie group SO($5$). Thus

${\rm SU(4)}\supset{\rm SO(5)}\supset{\rm U(1)}_{\mbox{\scriptsize
s}}\times{\rm SU(2)_{z}},$ (114)

where $\\{T_{z},\Pi_{z}^{x},\Pi^{y}_{z}\\}$ generates the SU($2$)z symmetry Wu
et al. (2014). If Zeeman splitting is neglected, the system exhibits an
SO($5$) symmetry generated by spin and valley isospin. The full Hamiltonian
(108) with the Zeeman term included conserves SU($2$)z but only the $z$
component of spin. At a mean-field level, the 10 generators of the SO($5$)
symmetry in Eq. (114) produce rotations on a vector space defined by order
parameters associated with the remaining five generators of SU(4):
$\\{T_{x},T_{y},N_{x},N_{y},N_{z}\\}$ Wu et al. (2014). The vector space and
the rotations generated by SO($5$) are illustrated schematically in Fig.
21(b), which generalizes the Bloch sphere of Fig. 21(a).

Figure 21:  (a) Bloch sphere representing a system with two basis states
$\left|0\right\rangle$ and $\left|1\right\rangle$. (b) The 5D space operated
on by the SU(4) $\supset$ SO(5) subgroup and the actions of the generators on
this space. This represents a generalization of the Bloch sphere in (a).
Adapted from Ref. Wu et al. (2014).

The spin operators $\bm{S}$ generate rotations in the Néel vector space
$\bm{N}$, the operator $T_{z}$ generates rotations in the valley vector space
defined by $T_{x}$ and $T_{y}$, and the $\Pi$ operators rotate between the
Néel and valley spaces. Thus SO($5$) is a symmetry interpolating between Néel
states associated with $\bm{N}_{\alpha}$ and states associated with valley
isospin degrees of freedom $T_{x}$ and $T_{y}$.

##  XII  Emergent States in Magnetic Fields

Little evidence exists for emergent states in neutral monolayer graphene, as
expected because of the low density of states near the Fermi surface. However,
in a strong magnetic field this situation changes since electrons in the
resulting highly degenerate Landau levels can undergo strong electron–electron
and electron–phonon interactions that can break symmetries spontaneously
rather than explicitly. The spontaneous breaking of symmetry can produce
highly collective states that differ qualitatively from low-lying states of
the weakly interacting system with explicit symmetry breaking, since the
emergent and weakly-interacting states are typically separated by quantum
phase transitions.

As we have discussed in Ref. Wu and Guidry (2016), there is evidence that the
ground state of monolayer graphene in a strong magnetic field is such a
(spontaneously) broken-symmetry state. In particular the ground state for
monolayer graphene in a magnetic field is known to be very strongly
insulating, exhibiting a rapid divergence of the longitudinal resistance
$R_{\mbox{\scriptsize L}}$ at a critical magnetic field $B_{\mbox{\scriptsize
c}}$, with $B_{\mbox{\scriptsize c}}$ smaller for cleaner samples. This
behavior suggests that the resistance in this state is a consequence of
emergent intrinsic properties of the state itself and not of impurity
scattering.

With this motivation, let us turn to a discussion of emergent states in
graphene created through strong electron–electron and electron–phonon
correlations that break symmetries spontaneously rather than explicitly. We
give first a more conventional view. Then we shall reformulate the discussion
of graphene emergent states in terms of powerful new approaches based on
fermion dynamical symmetries. In all of this discussion “graphene” will be
shorthand for monolayer graphene in a strong magnetic field.

###  XII.1  The role of correlations

Some general basis states for collective states that could be realized by
placing two electron spins in $n=0$ graphene valleys are illustrated in Fig.
22.

Figure 22:  Some possible basis states for collective states in the graphene
$n=0$ Landau level. Electron occupation for adjacent valleys A and B are
shown, with up and down arrows indicating spin-up electrons and spin-down
electrons, respectively. In the absence of Landau level mixing valley indices
and the sublattice indices A and B coincide for $n=0$. The corresponding
lattice occupation is illustrated on the right. (a) A charge density wave
(CDW). (b) A spin density wave (SDW). (c) A ferromagnetic state (F). Figure
adapted from Ref. Jung and MacDonald (2009).

(1) In Fig. 22(a), both electrons are in the same valley (with opposite spins)
and the adjacent valley is unoccupied, so valleys alternate in charge
$+2e,\,0,\,+2e,\,0\ldots,$ around the carbon ring. The right side of Fig.
22(a) illustrates the general lattice occupation, with adjacent sites
alternating between being occupied by two spin-singlet electrons and by no
electrons. This is a charge density wave (CDW).

(2) In Fig. 22(b), the electrons are in adjacent valleys with opposite spin
projections. The right side of Fig. 22(b) illustrates the lattice occupation,
with adjacent sites alternating between spin-up and spin-down electrons. This
is an antiferromagnetic (AF) spin-density wave (SDW).

(3) In Fig. 22(c), the electrons are in adjacent valleys with the same spin
projections. The right side of Fig. 22(c) illustrates the lattice occupation,
with a spin-up electron at each site. This is a ferromagnetically polarized
state (F).

(4) A fourth possible collective mode on the lattice is Kekulé distortion
(KD), which induces alternating shorter and longer bonds around the ring in
the graphene structure (a “bond density wave” Chamon (2000)). The terminology
references August Kekulé, who proposed the alternating single and double bond
structure for the benzene ring in Fig. 1(b). A Kekulé distortion state will be
illustrated in Fig. 23(d) below.

The collective modes described above are illustrated schematically in Fig. 23.

Figure 23:  Some collective states in the monolayer graphene, adapted from
Ref. Kharitonov (2012). (a) Charge density wave (CDW). (b) Ferromagnetic state
(F). (c) Spin density wave (SDW), corresponding to an antiferromagnetic Néel
state. (d) Kekulé distortion (KD) mode (“bond density wave”).

Assuming the effective Hamiltonian of Eq. (109) to capture the essential
physics, we expect in the simplest picture that the system could be in one of
four possible phases exhibiting different forms of long-range order Kharitonov
(2012).

  1. 1.

A spin-singlet charge density wave phase CDW illustrated schematically in Fig.
23(a) that is not affected by the Zeeman term $H_{\mbox{\scriptsize Z}}$ in
the Hamiltonian.

  2. 2.

A ferromagnetic phase F, illustrated schematically in Fig. 23(b). The symmetry
is broken to U($1$) by $H_{\mbox{\scriptsize Z}}$, corresponding to
conservation of spin projection.

  3. 3.

An antiferromagnetic or Néel phase AF, illustrated schematically in Fig.
23(c). The symmetry is broken to U(1) by $H_{\mbox{\scriptsize Z}}$,
corresponding to conservation of spin projection.

  4. 4.

A spin-singlet Kekulé distortion phase KD illustrated schematically in Fig.
23(d) that is not affected by the Zeeman term in the Hamiltonian.

These competing emergent modes are a result of correlations. Which, if any, of
these modes are realized depends on which correlations dominate, which will
now be discussed.

###  XII.2 Symmetry-breaking interactions

Symmetries may be broken spontaneously by short-range electron–electron (e–e)
interactions, and short-range electron–phonon (e–ph) interactions between the
lattice and the electrons, with the electron–electron interactions generally
being repulsive while the electron–phonon interactions are attractive. The
most important e–ph interaction is thought to be Kekulé distortion, which
perturbs the valley locations and contributes to the $g_{\perp}$ coefficient
in Eq. (109). Hence the valley-isospin $T_{x}$ and $T_{y}$ degrees of freedom
should couple to Kekulé distortion modes. A systematic analysis by Kharitonov
Kharitonov (2012) suggests that the e–e interactions could favor any of the
states described above: F, AF, CDW, or KD, depending on details of the
symmetry-breaking interactions, but the leading terms in the attractive e–ph
interactions always favor the KD mode.

Thus, we deal with a system having multiple possible emergent ground states,
with the actual ground state determined by a delicate balance among the
different e–e and e–ph interactions. This is the hallmark of complexity: a