Theory of broken symmetry quantum Hall states in the N1Landau level of Graphene Nikolaos Stefanidis1and Inti Sodemann Villadiego2 1

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Theory of broken symmetry quantum Hall states in the N=1Landau level of

Graphene

Nikolaos Stefanidis1, ∗and Inti Sodemann Villadiego2, 1, †

1Max-Planck-Institut f¨ur Physik komplexer Systeme, Dresden 01187, Germany

2Institut f¨ur Theoretische Physik, Universit¨at Leipzig, D-04103, Leipzig, Germany

We study many-body ground states for the partial integer ﬁllings of the N=1 Landau level in

graphene, by constructing a model that accounts for the lattice scale corrections to the Coulomb

interactions. Interestingly, in contrast to the N=0 Landau level, this model contains not only pure

delta function interactions but also some of its derivatives. Due to this we ﬁnd several important

diﬀerences with respect to the N=0 Landau level. For example at quarter ﬁlling when only a single

component is ﬁlled, there is a degeneracy lifting of the quantum hall ferromagnets and ground states

with entangled spin and valley degrees of freedom can become favourable. Moreover at half-ﬁlling

of the N=1 Landau level, we have found a new phase that is absent in the N=0 Landau level, that

combines characteristics of the Kekul´e state and an antiferromagnet. We also ﬁnd that according to

the parameters extracted in a recent experiment, at half-ﬁlling of the N=1 Landau level graphene

is expected to be in a delicate competition between an AF and a CDW state, but we also discuss

why the models for these recent experiments might be missing some important terms.

Introduction. The quantum Hall regime in graphene

realizes a rich landscape of broken symmetry and topo-

logical states, stemming in part from the near four-fold

degeneracy of its Landau levels (LLs) associated with its

valley and spin degrees of freedom [1]. Most studies to

date have focused on the states in the N=0 LL, with

transport and magnon transmission experiments favor-

ing an anti-ferromagnetic (AF) state at neutrality [2–6],

while STM experiments reporting evidence for Kekul´e-

type valence-bond-solids and charge density wave states

(CDW) [7–9].

While the projected Coulomb interaction is typically

the dominant term in the Hamiltonian, it possesses a

large symmetry that leaves the quantum Hall ground

states undetermined. Therefore, it is crucial to account

for the corrections that reﬂect the lower symmetry of

the underlying graphene lattice to select the ground

states [1, 10–15]. A convenient model to capture these

symmetry breaking interactions in the N=0 LL was in-

troduced by Kharitonov in Ref. [15]. This model can be

viewed as a projection into the N=0 LL of a more gen-

eral model introduced by Aleiner, Kharzeev and Tsve-

lik [15, 16] that includes all possible delta-function in-

teractions allowed by symmetries. There is no study to

this date that has constructed an analogous model in the

N=1 LL that includes all possible short-distance interac-

tions allowed by symmetry, although a related model con-

taining some of these terms was introduced in Ref. [17].

The purpose of our study is therefore to construct this

model of symmetry breaking interactions in the N=1 LL

and to determine its ground states at partial integer ﬁll-

ings. Intrestingly, we will see that in contrast to the N=0

LL [15], the N=1 LL model contains interactions that

are not pure delta functions [17]. Therefore, in contrast

to the N=0 LL, a unique ground state is selected even

when a single component is ﬁlled (to be denoted by ˜ν=1),

and some of the possible ground states are spin-valley

entangled, in the sense discussed in Ref. [18]. Moreover,

when two components are ﬁlled (to be denoted by ˜ν=2),

we ﬁnd a new type of Kekul´e-Antiferromagnetic state in

addition to those found in the N=0 LL. Based on the pa-

rameters estimated in Ref. [17], graphene is expected to

be in a delicate competition between an AF and a CDW

state. However, as we will discuss, these parameters are

possibly missing some important terms.

Model and Symmetries. We begin by reviewing the

continuum model of short-range symmetry breaking in-

teractions of Aleiner, Kharzeev and Tsvelik [16] in the

absence of a magnetic ﬁeld. This is described by the

following Hamiltonian:

H=HD+HC+HA,(1)

with:

HD=vF∑

i(τi

zpi

xσi

x+pi

yσi

y),(2)

being the linearized single particle hamiltonian around

the Dirac points,

HC=∑

i<j

ri−rj,

the Coulomb interaction, and

HA=∑

i<j{∑

α,β

Vαβ Ti

αβ Tj

αβ }δ(ri−rj),(3)

the sublattice-valley dependent interactions. We have

deﬁned Ti(j)

αβ =τi(j)

α⊗σi(j)

β⊗si(j)

0, and τi(j)

α, σi(j)

β, si(j)

α, β =0, x, y, z to be the Pauli matrices acting on valley,

sublattice and spin respectively.

By denoting the valley (sublattice) states as τ(σ),

with τ(σ)=±1 corresponding to the K, K′(A, B)val-

leys (sublattices), then the action of lattice symmetry on

arXiv:2210.03752v1 [cond-mat.mes-hall] 7 Oct 2022

these states is given by:

C6τ, σ=Z−τ σ −τ, −σ,

Mxτ, σ=τ, −σ,

Myτ, σ=−τ, σ,

TR1,2τ, σ=Z±ττ, σ,

(4)

with Z=ei2π

3.C6is the rotation by π3, Mx, Mythe

two mirrors and TRithe translations by the two basis

vectors of graphene (see Fig. 1(a) for the illustration of

these symmetries). These symmetries reduce the cou-

plings of Eq. (3) to nine independent couplings satisfying

the following relations [16]:

F⊥z≡Vxx =Vyx,

Fz⊥≡V0x=Vzy,

F⊥⊥ ≡Vxz =Vy0=Vyz =Vx0,

F0⊥≡Vzx =V0y,

F⊥0≡Vyy =Vxy,

Fzz ≡V0z,

Fz0≡Vz0,

F0z≡Vzz.

(5)

Projected Model in the N=1Landau level. By projecting

HAfrom Eq. (1), with the constraints in Eq. (5), one

obtains the following Hamiltonian of symmetry breaking

interactions in the Nth LL:

A=∑

i<j{VN

z(rij )τi

zτj

z+VN

⊥(rij )τi

⊥τj

⊥},(6)

with τi

⊥τj

⊥=τi

xτj

x+τi

yτj

y(see S-II for further details). As we

see there is an eﬀective U(1)valley conservation arising

from the underlying lattice symmetries. Speciﬁcally for

the N=1 LL we have:

Vz,⊥(rij )=2

∑

n=0

gz,⊥

n∇2nδ(rij ).(7)

Here gz,⊥

nare independent constants that parametrize

the projected interactions that are linear combinations

of those in Eq. (5) (see Eq.(S-28) for their explicit rela-

tions). Therefore we have a model with 6 independent

parameters characterizing the interactions in the N=1

LL, in contrast to the more restricted model of Ref. [17]

with only 2 parameters. The model of Ref. [17] is a spe-

cial case of our Eq. (7), in which gz

0,1=g⊥

0,2=0. Notice,

in particular, that in our model the n=0 terms in Eq. (7)

are pure delta function interactions, which are absent in

Ref. [17] (see S-IV for further details).

On the other hand, if we project HAonto the N=0

LL we obtain the model from Ref. [15] for which the

interactions would include only pure delta functions (see

Eq.(S-29) for the deﬁnition of gz,⊥):

Vz,⊥(rij )=gz,⊥δ(rij ).(8)

Therefore, the main diﬀerence between the model of

Eq. (7) for the N=1 LL and the model of Ref. [15]

for the N=0 LL is the existence of interactions which

are not pure delta functions. As we will show, this leads

to several important diﬀerences in the physics of these

two Landau levels.

Mean-ﬁeld ground states. We will now derive the

Hartree-Fock (HF) functional for the Hamiltonian of

Eqs. (6), (7) and obtain the phase diagram in the integer

ﬁllings of the N=1 LL, ˜ν=1 (˜ν=2) when one (two)

out of the four valley-spin degenerate LL are ﬁlled 1. We

consider the competition of translational invariant inte-

ger quantum Hall ferromagnets that can be described by

a particle-hole condensate order parameter of the form,

c†

X1τ1s1cX2τ2s2=Ps1s2

τ1τ2δX1,X2, with Xilabeling intra-

LL guiding center coordinates. Here c†

Xτ s denotes the

electron creation operator with valley τand spin s, and

Pis the projector in spin-valley space into either a one-

dimensional subspace (for ˜ν=1) or a two-dimensional

subspace (for ˜ν=2). The general form of the Hartree-

Fock functional is then (EHF [P]≡2A

EHF [P]):

EHF [P]=∑

i=x,y,z uH

i(T r{TiP})2−uX

iT r{(TiP)2},(9)

with uH,X

⊥=uH,X

x=uH,X

y. Therefore the possible ground

states depend only on 4 eﬀective Hartree and exchange

constants, uH

z, uX

z, uH

⊥, uX

⊥, which are linear combinations

of the constants gz,⊥

nthat appear in Eq. (7) (see Eq.(S-35)

for explicit relations). Moreover, while in the N=0 LL

(see Eq.(S-33)) the Hartree and the exchange constants

are forced to be equal, uH

z,⊥=uX

z,⊥[15], in the N=1

LL they are independent due to the appearance of non-

delta interactions (see S-III for further details). Similar

functionals have been proposed, however phenomenolog-

ically, for the N=0 LL to capture the physics beyond

the delta-functions in Refs. [18, 19].

We will consider general spin-valley entangled [18, 19]

variational states. The following two orthonormal spinors

can be used to uniquely parametrize the state character-

ized by Pin Eq. (9),

F1=cos a1

2ηs+eiβ1sin a1

2−η−s,

F2=cos a2

2η−s+eiβ2sin a2

2−ηs.

(10)

Here ηand sare states parametrized by unit vec-

tors ηand sin the spin and valley Bloch spheres re-

spectively and a1,2and β1,2are real constants. No-

tice that in general these states might not be separable

into a tensor product of spin and valley components and

1The partial ﬁlling of ˜ν=3(˜ν=2)is equivalent to ˜ν=1(˜ν=4)by

a particle-hole conjugation.

a) b)

Figure 1. a) Graphene unit cell and its lattice symmetries

(top) and its reciprocal unit cell (bottom). b) Phase diagram

at ˜ν=1. It contains four phases: charge density wave (CDW),

Kekul´e distortion (KD) and the two entangled phases, the

antiferrimagnetic phase (AFI) and the canted antiferromagnet

(CAF).

therefore can account for spin-valley entanglement [18].

For ˜ν=1, we take P=F1>< F1, and for ˜ν=2

P=F1><F1+F2><F2.

Ground states for ˜ν=1.As discussed in Ref. [18], the

energy functional in this case reduces to:

E˜ν=1

HF =cos2a1{∆zη2

z+∆⊥η2

⊥},(11)

with ∆z=uH

z−uX

z, ∆⊥=uH

⊥−uX

⊥and η2

⊥=η2

x+η2

y(see

S-V-A) for further details). The resulting phase diagram

is shown in Fig.1(b) and contains four phases. These

are a charge density wave (CDW) with η=ˆz, s=ˆz

and a1=0, and a Kekul´e distortion (KD) state with

η=η⊥,s=ˆzand a1=0. Interestingly, we see that also

spin-valley entangled phases with a1=π2, appear when

∆z>0,∆⊥>0. These entangled phases are degenerate in

the absence of Zeeman ﬁelds, but in their presence they

split antiferrimagnetic phase (AFI) with η=ˆz, s=ˆz

and a1=π

2and the canted antiferromagnet (CAF) with

η=η⊥,s=ˆzand a1=π

2, as discussed in Ref. [18].

Notice that in the N=0 LL, ∆z=∆⊥=0, and therefore

all of the above states would be degenerate and with a

vanishing HF energy.

Ground states for ˜ν=2.The HF functional for ν=2 is

more diﬃcult to minimize analytically. To make progress,

we ﬁrst consider the subset of states from Eq. (10) with-

out spin-valley entanglement. These can be classiﬁed into

the valley active states [20] :

F1=η1s,F2=η2−s,(12)

in which the valley degree of freedom varies, and the spin

active states :

F1=ηs1,F2=−ηs2,(13)

in which the spin degree of freedom varies. We ﬁrst

minimize the energy functional within this subspace and

States appearing at ˜ν=2

States Wavefunctions {∣F⟩1,∣F⟩2}

CDW (Charge density wave) {∣ˆz⟩ ∣s⟩,∣ˆz⟩ ∣−s⟩}

KD (Kekul´e distortion) {∣η⊥⟩ ∣s⟩,∣η⊥⟩ ∣−s⟩}

FM (Ferromagnet) {∣ˆz⟩ ∣s⟩,∣−ˆz⟩ ∣s⟩}

AF (Antiferromagnet) {∣ˆz⟩ ∣s⟩,∣−ˆz⟩ ∣−s⟩

KD-AF (Kekul´e

antiferromagnet)

{∣η⊥⟩ ∣s⟩,∣−η⊥⟩ ∣−s⟩}

Table I. Competing states at ˜ν=2 and their wavefunctions.

then perform a quadratic expansion of all possible devia-

tions of parameters that account for spin-valley entangled

states (see S-V-B), VI, VII for further details). For sim-

plicity we will also neglect the Zeeman term that is typi-

cally weak compared to the interaction terms [2, 21, 22].

In contrast to ˜ν=1, for ˜ν=2 we ﬁnd that whenever a

spin-valley disentangled state is energetically favorable it

is also an exact local minima of the energy with respect to

all possible quadratic deviations that include spin-valley

entanglement. This indicates that these spin-valley dis-

entangled states are also possibly exact global minima of

the energy.

Following this procedure, we ﬁnd a total of ﬁve pos-

sible ground states for ˜ν=2 that are realized as a

function of the four Hartree and exchange parameters

z, uX

z, uH

⊥, uX

⊥. These possible ﬁve states are listed in

Table I (see S-V-B) for more details on these states).

To visualize the energetic competition among these ﬁve

phases, we have chosen to draw two-dimensional phase

diagrams as functions of the two Hartree parameters

˜uH

z,⊥=uH

z,⊥∆z−∆⊥for ﬁxed values of ∆z=uH

z−uX

∆⊥=uH

⊥−uX

⊥. We ﬁnd that there are a total of six

diﬀerent kinds of phase diagrams depending on the val-

ues and signs of ∆z,⊥. Two of these representative phase

diagrams are depicted in Fig. 2, and the remainder are

presented in S-V-B).

Interestingly, according to the model and the estimates

of Ref. [17], uH

z=uH

⊥=0 and uX

z>0, uX

⊥<0 (see S-IV

for further details). This means that graphene in the

N=1 LL would have a phase diagram like the one in

Fig. 2(a), and it would be located exactly at the origin

of this phase diagram, which we indicate by a black dot

in Fig. 2(a). Therefore, we see that the model and the

parameter estimates of Ref. [17] place graphene right at

the boundary between the CDW and the AF states. We

note that even at this boundary, these phases remain

stable against spin-valley entangled rotations (see S-VII

for further details).

One of the interesting qualitative diﬀerences that we

have found in the N=1 LL is the existence of a new

phase that features a combination of Kekul´e state and

antiferromagnet, that we term the Kekul´e- antiferromag-

net (KD-AF). In this phase one set of electrons has an

XY vector in the valley sphere with spin up while the oth-

ers occupy the opposite valley vector with spin down, as

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摘要：

TheoryofbrokensymmetryquantumHallstatesintheN1LandaulevelofGrapheneNikolaosStefanidis1,andIntiSodemannVilladiego2,1,1Max-Planck-InstitutfurPhysikkomplexerSysteme,Dresden01187,Germany2InstitutfurTheoretischePhysik,UniversitatLeipzig,D-04103,Leipzig,GermanyWestudymany-bodygroundstatesforthepartia...

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Theory of broken symmetry quantum Hall states in the N1Landau level of Graphene Nikolaos Stefanidis1and Inti Sodemann Villadiego2 1

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