Fractional Quantum Hall Effect in n=0 Landau Band of Graphene with Chern Number Matrix
Koji Kudo Yasuhiro Hatsugai


Interplay between speed and fidelity in off-resonant quantum-state transfer protocols
    Guilherme M. A. Almeida
    December 30, 2023
=====================================================================================

The concept of topological order
<cit.> substantially expands our understanding of phases of
matter that cannot be described by the conventional order parameters
associated with symmetry breaking. The quantum Hall effect
<cit.> is a prominent example
of topologically non-trivial quantum phases. The quantized Hall conductance is
expressed as the Chern number
<cit.>
associated with the Berry connection<cit.>. Although the integer 
quantum
Hall state can be described by non-interacting electrons, the
electron-electron interaction plays a crucial role in the fractional quantum
Hall (FQH) phase. The characteristics of these correlated quantum states are
well captured by the Laughlin wave function<cit.>. In 
addition,
its excitations are quasiparticles with fractional
charges and fractional statistics<cit.>. This FQH effect
is understood as an integer quantum Hall effect of composite fermions as flux
charge composites <cit.>. It also provides a consistent 
picture
at an even-denominator Landau level (LL) filling <cit.>.

The internal degrees of freedom, such as spin or layer index, bring further
diversity to the FQH phases. The ground state at ν=5/2
<cit.> is described by the Moore-Read
Pfaffian state <cit.> with the excitation obeying
non-Abelian statistics <cit.> when the interaction is 
short-range. As for two-component Abelian FQH systems, the Halperin 
lmn
state <cit.> is a typical example. This is realized in a
bilayer FQH system at ν=1/2 <cit.>, for example, but its
appearance depends strongly on the system parameters. For 
multi-component systems, the symmetric integer matrix K
<cit.>
provides classification of the FQH phases, which is discussed in relation to
the Chern number matrix <cit.>.

The FQH effect of graphene <cit.> is also an example of the
multi-component FQH systems. The low-energy behavior of electrons in graphene 
is
described by the doubled massless Dirac fermions at K and K' points in the
Brillouin zone. These characteristics give rise to the FQH
phases peculiar to graphene. 
<cit.>.
Since the n=0 LL is a standard lowest LL of the valley polarized Dirac
fermions, the FQH effect of the n=0 LL has been discussed similarly with the
SU(2) invariance arising from the valley degree of freedom. The ground states
at the n=0 LL filling factor ν=1/3 and 1/2 are described by the
pseudospin (valley) polarized Laughlin state and pseudospin singlet composite
fermion Fermi sea, respectively.
<cit.>

In this study, the FQH system for the n=0 Landau band is investigated
by fully taking into account the honeycomb lattice structure of the
interaction. Short-range
electron-electron interaction of the nearest neighbor (NN) and next-nearest 
neighbor (NNN) is
discussed in this paper by constructing the pseudopotential 
<cit.>
projected into the n=0 Landau band. The chiral symmetry of the honeycomb 
lattice plays an important role in the many-body problems as well
<cit.>.
The quantum phase transitions associated with the chirality ferromagnetism
occur by changing the interaction range. Since the total pseudospin is not
conserved because of the lattice effects, the SU(2) symmetry discussed by the
continuous approximation is absent. In order to characterize 
the quantum phases topologically, the Chern number matrices specified by the 
chiral basis are constructed numerically. The results are also discussed in
relation to the conventional bilayer quantum Hall system.

Let us begin by introducing the projected fermion operators into
the n=0 Landau band. Here, we assume that the system is always
spin-polarized. The kinetic Hamiltonian is written as

    H_kin
     =t∑_⟨ ij⟩e^iϕ_ijc_i^† c_j
     =c^† h_kinc,

which describes hopping between the NN pairs of sites with
strength t. Here, c^†=(c^†_∙,c^†
_∘), c^†_∙(∘)=(c^†_1∙(∘),⋯,
c^†_N_cell∙(∘)) and
c^†_i∙(∘) creates a fermion at the sublattice ∙
(∘) for unit cell i.
The Peierls phase ϕ_ij is determined such that the sum of the phases
around an elementary hexagon is equal to the magnetic flux 2πϕ in units 
of the flux quantum ϕ_0=h/e. In the calculation, the string gauge
<cit.> is employed,
which enables us to realize the minimum magnetic fluxes that are consistent 
with the lattice periodicity. When the system is put on the N_x× N_y
unit cells with a periodic boundary condition, the magnetic field can be
provided as ϕ=N_ϕ/N_cell (N_ϕ=1,2,⋯,N_cell),
where N_cell=N_xN_y. Here, N_ϕ corresponds to the total magnetic
flux. The lattice model with ϕ=p/q(p,q: relatively prime) has 2q
single-electron bands, where 2 comes from the sublattice degree of freedom.
The number of states per band is obtained as N_xN_y/q. For the weak magnetic
field (ϕ≪1), 2p bands flow into each other around the zero energy, which
form the n=0 LL in the large q limit. Thus, in this paper, “the n=0
Landau band” is defined as a group of these bands, where there are
(N_xN_y/q)×2p=2N_ϕ one-body states. 

Since the honeycomb lattice is bipartite, the Hamiltonian H_kin has
chiral symmetry. The 2N_cell× 2N_cell matrix
Γ=diag(I_N_cell,-I_N_cell) anticommutes with
the Hamiltonian as {h_kin,Γ}=0 and Γ^2=I_2N_
cell. If ψ_k is the eigenvector of h_kin with the energy
ϵ_k, the chiral symmetry guarantees that Γψ_k is identical
to the one with -ϵ_k. Thus, the chiral operator Γ can be
diagonalized within the one-body states of the n=0 Landau band. Then, the 
chiral basis can be defined as ψ=(ψ_+,ψ_-),
ψ_±=(ψ_1,±,⋯,ψ_N_ϕ,±), and 
Γψ_k,±=±ψ_k,±. Note that ψ_k,+(-) is localized on
the sublattice ∙(∘) so that the multiplet can be expressed as 
ψ=(
[ ψ_∙   0;   0 ψ_∘ ]),
where ψ_∙(∘) is a proper N_cell× N_ϕ matrix
<cit.>.

Next, let us consider the two-body interactions written as

    H_int=
     ∑_i<j∑_σ=∙,∘
     V_ij^σσn_iσn_jσ
     +∑_i,j
     V_ij^∙∘n_i∙n_j∘,

where n_i∙(∘)=c_i∙(∘)^† c_i∙(∘) and
V_ij^σσ' is the strength of the electron-electron interaction.
In order to construct the pseudopotential, the projected creation-annihilation
operators are defined as c̃^†=c^† P
<cit.>, where P=ψψ^†. This expression is
simplified by writing c̃^†_∙(∘)
=c^†_∙(∘)P_∙(∘) and 
P_∙(∘)=ψ_∙(∘)ψ^†_∙(∘).
Note that these projected operators no longer satisfy the canonical
anticommutation relations ({c̃_i,c̃^†_j}=
P_ij≠δ_ij). By taking into account the ordering of fermions,
the replacement of c^†, c with c̃^†,
c̃ causes the Hamiltonian to be projected into the n=0 Landau
band. Then, the projected Hamiltonian can be defined as

    H̃_int   =∑_i<j∑_σ=∙,∘
     V_ij^σσc̃^†_iσc̃^†_jσc̃_jσc̃_iσ
     +∑_i,j
     V_ij^∙∘c̃^†_i∙c̃^†_j∘c̃_j∘c̃_i∙
       =∑_klmn
     (∑_σ=∙,∘A^σσ_klmn
     d^†_k, χ_σd^†_l, χ_σ
     d_m, χ_σd_n, χ_σ
                           +A^∙∘_klmn
     d^†_k, + d^†_l, - d_m, -d_n, +),

where
A^σσ'_klmn
=∑_i<jV_ij^σσ'
(ψ_σ)^∗_ik(ψ_σ')^∗_jl
(ψ_σ')_jm(ψ_σ)_in,
d_+(-)^†=(d^†_1, +(-),⋯,d^†_N_ϕ, +(-))
=c_∙(∘)^†ψ_∙(∘), and χ_
∙(∘)=+(-). Here, we choose the strength of the interaction such
that its energy scale is much larger than the energy width of the n=0 Landau
band, so that only the interaction term is considered. The projected
Hamiltonian H̃_int commutes with the total chirality operator
written as

    𝒢=c̃^†Γc̃=d^†_+d_+-d^†_-d_-,

which enables us to classify the N_e-body states by the total 
chirality
χ_tot=-N_e,-N_e+2,⋯,N_e. Now, the filling
factor is defined as ν=N_e/N_ϕ.

Hereafter, we consider the electron-electron interaction between
NN and NNN pairs, 

    H̃_int
     =V_1∑_⟨ ij⟩c̃^†_ic̃^†_jc̃_jc̃_i
     +V_2∑_⟨⟨ ij⟩⟩c̃^†_ic̃^†_jc̃_jc̃_i,

where V_1 and V_2 are the strength of the interaction. Note that the
interaction V_1 (V_2) contributes V^∙∘_ij (V^∙∙_ij and V^∘∘_ij). Figures <ref> (a) and (b)
show the V_2/V_1 dependence of the four-electron energy obtained by the 
exact diagonalization at ν=1/3 and 1/2.
Since the exact topological degeneracy due to the 
center-of-mass translation <cit.> is given by the Landau 
gauge in a finite system, we choose a system of 12×12 size for 
ϕ=1/12 as shown in Fig. <ref> (a) at ν=1/3. For example, the
chirality-polarized ground state multiplet is exactly three-fold degenerated. 
(See the inset in Fig. <ref> (a).) As for the system at ν=1/2, to
be compatible to the Dirac cones at K and K' points of graphene, the system
size should be 3n×3n (n=1,2,⋯). Then, we choose the magnetic flux
as ϕ=8/(12×12)=1/18 using the string gauge. In this case, the 
topological degeneracy is not exact. However, the energy separation of the 
entangled states within the ground state multiplet is very small. For example, 
it is less than 0.003 times the energy gap for the six-fold ground state 
multiplet with χ_tot=N_e. 
(See the inset in Fig. <ref> (b).)
Note that the magnetic length l_B=√(ħ/eB)=√(S/2πϕ), 
where S=3√(3)a^2/2 is the area of the elementary hexagon with lattice
constant a, is approximately 2.2a and 2.7a for ϕ=1/12 and 1/18,
respectively.

In the case of V_1≠ V_2=0, that is, only the bipartite interactions, 
many-body states with |χ_tot|=N_e should be eigenstates with
zero eigenvalues. Generally, the system at ν<1 
provides 2×_N_ϕC_N_e-fold degenerate ground states 
(2 comes from the sign of chirality), which are lifted by the infinitesimal 
V_2. Further, an increase in V_2 leads the phase transition from |χ
_tot|=N_e to 0 since the NNN interactions act between the same
sublattices. In Fig. <ref> (c) and (d), the ratio
V_2/V_1 at the phase transition points is plotted against the magnetic flux
ϕ at ν=1/3 and 1/2. 
The results suggest that the
strong short-range interaction compared with the magnetic flux (ϕ≪1)
favors the unpolarized chirality unless V_2 is not vanishing.

Since the pseudopotential is constructed on the basis of the honeycomb lattice 
model,
only the z-component of the pseudospin, χ_tot, is conserved. The
SU(2) symmetry arising from the chirality is absent in contrast to the cases
in the continuum limit <cit.>. Then,
in order to investigate the internal topological structure of
the many-body states, we evaluate the Chern number matrices associated with the
chirality. First, we investigate the twisted boundary condition
<cit.>,
c^†_n_x+N_x,n_y=e^iθ^xc^†_n_x,n_y and
c^†_n_x,n_y+N_y=e^iθ^yc^†_n_x,n_y, where n_x and
n_y are the labels of the unit cell for the x and y directions. Since the
chiral symmetry remains in the kinetic Hamiltonian H_kin(θ^x,
θ^y), the connection between chirality and sublattice is preserved. 

Let us consider each projected fermion operator c̃_∙
and c̃_∘ with different twisted boundary conditions,

    c̃_∙(∘)
     =c̃_∙(∘)(θ_∙(∘)),

where θ_∙(∘)=(θ_∙(∘)^x,θ_∙
(∘)^y). The projected Hamiltonian H̃_int(θ_∙,
θ_∘) can be defined by replacing c̃_∙(∘)
with c̃_∙(∘)(θ_∙(∘)) in
Eq. (<ref>). Note that this Hamiltonian can be written by the fermion
operators d^†_+(-)(θ_∙(∘))=
c^†_∙(∘)ψ_∙(∘)(θ_∙(∘)),
which satisfy the canonical anticommutation relations 
{d_i, +(θ_∙),d_j, -(θ_∘)}
={d^†_i, +(θ_∙),d_j, -(θ_∘)}=0
for any θ_∙(∘).

Now, let us further define the non-Abelian Berry connection and curvature
<cit.>
of the m-fold ground state multiplet Φ=(|G_1⟩,⋯,
|G_m⟩) by selecting the sublattices in each direction x,y as

    A_σ_xσ_y
     =Φ^† dΦ, 
     d=∑_μ=x,ydθ_σ_μ^μ∂/∂θ_σ_μ^μ,
       F_σ_xσ_y
     =dA_σ_xσ_y+A_σ_xσ_y^2,

where σ_μ=∙,∘, and the two parameters except for
θ^x_σ_x and θ^y_σ_y are fixed to 0. Then, the Chern
number matrix is defined as

    C
     =(
     [ C_∙∙ C_∙∘; C_∘∙ C_∘∘;      ]), 
     C_σ_xσ_y
     =1/2π i∫_T^2Tr F_σ_xσ_y.

The element is evaluated as 
<cit.>
C_σ_xσ_y
=1/2π i∑_θ_σF̃_σ_xσ_y
(θ_σ) numerically, where 
F̃_σ_xσ_y(θ_σ)
 =Log 
 [U_σ_x^x(θ_σ)
 U_σ_y^y(θ_σ +Δ_σ_x^x)
 U_σ_x^x(θ_σ+Δ_σ_y^y)^-1
 U_σ_y^y(θ_σ)^-1],
U_∙(∘)^μ(θ_σ)
 =[Φ^†(θ_σ)
 Φ(θ_σ+Δ_∙(∘)^μ)]
 /|[Φ(θ_σ)
 Φ(θ_σ+Δ_∙(∘)^μ)]|,
θ_σ=(θ_σ_x^x,θ_σ_y^y), and
Δ_∙(∘)^μ represents the lattice displacement in the
direction μ=x,y for the sublattice ∙(∘).

In order to construct the ground state multiplet Φ(θ_σ)
numerically, a basis of N_e-electron states classified by the total
chirality χ_tot is defined as
Ψ(θ_σ)
=(|Ψ_1(θ_σ)⟩,⋯, |Ψ_N_D(θ_σ)⟩).
Here, N_D is the dimension of the Hilbert space, and

    |Ψ_i(θ_σ)⟩
     =(∏_n∈ P_i, +d_n, +^†(θ_σ))
     (∏_n∈ P_i, -d_n, -^†(θ_σ))
     |0⟩,

where P_i, ± is one of the possible ways to occupy N_ϕ states by
N_± electrons, and N_±=(N_e±χ_tot)/2. Then, the
ground state multiplet is expressed as Φ=Ψ u_G, where u_G=
(u_G_1,⋯,u_G_m), and u_G_i is the eigenvector of
Ψ^†H̃_intΨ. We have
Φ^†(θ_σ)Φ(θ_σ
+Δ_∙(∘)^μ)=u_G^†(θ_σ)O(θ_σ,
Δ_∙(∘)^μ)u_G(θ_σ+Δ_∙(∘)^μ) and

    O_ij(θ_σ,Δ_∙(∘)^μ)
     =   ⟨Ψ_i(θ_σ)|Ψ_j(θ_σ
     +Δ_∙(∘)^μ)⟩
    
     =   δ_P_i, -(+)P_j, -(+)⟨0|
     (
     ∏_n∈ P_i, +(-)d_n, +(-)^†(θ_σ)
     )^†
                   (∏_n∈ P_j, +(-)d_n, +(-)^†(θ_σ
     +Δ_∙(∘)^μ))|0⟩
    
     =   δ_P_i, -(+)P_j, -(+)[ψ̃_P_i, +(-)^†
     (θ_σ)
     ψ̃_P_j, +(-)
     (θ_σ+Δ_∙(∘)^μ)],

where
ψ̃_P_i, +(-)
=(ψ_α_1, ∙(∘),⋯,
ψ_α_N_e, ∙(∘)),
ψ_i, ∙(∘) is the i-th column vector of ψ_∙
(∘), and P_i, +(-)={α_1⋯,α_N_e}.

We first focus on the chirality-polarized states with χ_tot=
N_e for V_2≠0. Since the polarized many-body states occupy only
the sublattice ∙, the staggered sublattice potential written as 
H̃_site
=-M∑_ic̃_i∙^†c̃_i∙
+M∑_ic̃_i∘^†c̃_i∘
=-M𝒢 
(M>0) stabilizes the polarized many-body states. Now, the ground
state multiplet is defined as a group of lowest-energy many-body states that is
separated from the other excited states. Numerically obtained ground
states at ν=1/3 and 1/2 are similar to those
provided by the pseudopotentials projected into the lowest Landau band
<cit.>.
At ν=1/3, the ground state is always three-fold degenerated 
irrespective of the number of electrons. This three-fold ground state multiplet
is gapped in the thermodynamic limit similar to the
Ref. doi:10.7566/JPSJ.86.103701. In addition, its Chern number
C_∙∙ is 1. 
This implies that the ground state is the lattice analogue of the Laughlin 
state.
On the other hand, for 
the ν=1/2 case, the degeneracy of the ground state has no such universal 
feature, and there is no sign of a finite energy gap from its scaling. This 
is also consistent with the composite fermion Fermi sea
<cit.>.

Next, let us consider the many-body states with χ_tot=0, which
occupy the same number of sublattices ∙ and ∘. Here, we  
focus on the ν=1/2 state. Figure <ref> (a) and (b) shows the
θ_x^∙ dependence of the many-body spectrum at V_2/V_1=0.3 and 
1.0, respectively. In Fig. <ref> (a), the ground state mixes 
with
higher states with the change in boundary conditions, and the ground state
multiplet is not well-defined. On the other hand, in Fig. <ref> 
(b), eight low-energy states are entangled and do not mix with excited states. 
In Fig. <ref> (c), the many-body spectrum with 
χ_tot=0 for N_e=4 and ϕ=1/18 is plotted as a function
of V_2/V_1. 

The behavior of the ground state in the spectral flow changes at 
V_2/V_1≃0.53. 
Note that the eight-fold ground state multiplet is always gapped for V_2/V_1
≳0.53 as long as V_1≠0 (For V_1=0, the energy gap of the 
eight-fold
ground state multiplet vanishes, and the two decoupled ν/2+ν/2 states
should be the ground states). In the inset of the Fig. <ref> (b),
the energy gaps  Δ E=E_9-E_1 at V_2/V_1=1 with periodic boundary
conditions are plotted as a function of the magnetic flux ϕ, where E_i
is the i-th eigenvalue of H̃_int with χ_tot=0. 
The result indicates that the scaling low Δ E∝ϕ^2 is roughly
valid in the wide range of ϕ. Since the electron density is obtained as
ρ=N_e/(2N_cell)=ϕν/2, we have Δ E∝ρ^2,
which means that the excitations are local.

Following the argument of the Ref. PhysRevB.95.125134, the obtained 
Chern number matrix C suggests the K-matrix as K=(C/m)
^-1, where m is the degeneracy.
The matrices defined by the eight-fold ground state at ν=1/2 are 
numerically observed as

    C
     =(
     [  3 -1; -1  3 ]), K
     =(
     [ 3 1; 1 3 ]).


As for a conventional bilayer FQH system with finite 
interlayer separation d, the pseudopotential does not have the SU(2) symmetry
arising from the pseudospin. However, in the limit d→∞ or 
d=0, the system is exceptionally SU(2) invariant. Therefore, the ground 
states of the ν=1/2 case in these two limits are described by the 
composite 
fermion Fermi sea. The former is the two decoupled states and the latter is the
pseudospin-singlet state.
<cit.>
On the other hand, the Halperin 331 state, which does not have the SU(2) 
symmetry, is realized as the ground state in the intermediate separation.
<cit.>
The FQH system for the n=0 LL of graphene in the continuum limit 
corresponds to the conventional bilayer FQH system with d=0, where the 
Halperin 331 state should not be observed because there is SU(2) symmetry 
in the system.

In contrast, the NN and NNN interactions in 
the honeycomb lattice act between the different and same sublattices 
(chiralities). 
Therefore, by connecting the parameter V_1 and V_2 with the interlayer 
and intralayer interactions, the FQH system of the n=0 Landau band 
corresponds to the conventional bilayer quantum Hall system 
with d≠0; the 
chirality ±1 plays the role of the two-layer index. Note that the only 
z-component of the pseudospin is conserved in both systems.
Thus, it is natural to expect that the analogue of the Halperin
331 state is realized for the n=0 Landau band when the ratio V_2/V_1 is on
the order of unity. We confirmed this scenario for the four-electron system 
in terms of the Chern number matrix as Eq. (<ref>).

The chirality-unpolarized ground state at ν=1/3 is also considered in terms
of the Chern number matrix. In the conventional bilayer system at ν=1/3,
for example, the Halperin 551 state is one of the candidates of the gapped 
ground state<cit.>. 
However, the results at ν=1/3 are not systematic, and no clear picture is 
obtained in contrast to the ν=1/2 case. They will be discussed later
elsewhere.

To summarize, we have constructed the Chern number matrix in association with
the chiral basis by using the Hamiltonian projected into the n=0 
Landau band of the honeycomb lattice. Modifying the interaction range induces 
the quantum phase transitions associated with the chirality ferromagnetism.
When the NN interactions are sufficiently strong, the ground 
states at ν=1/3 and 1/2 are chirality-polarized, and consistent with the 
Laughlin state and the composite fermion Fermi sea, respectively. On the 
other hand, an increase in the strength of the NNN interaction
leads the ground state to chirality-unpolarize. The obtained Chern 
number matrix indicates that the unpolarized ground state for large V_2 at 
ν=1/2 is consistent with the Halperin 331 state.

This work is partly supported by Grants-in-Aid for Scientific Research, 
(KAKENHI), Grant numbers 17H06138, 16K13845, and 25107005.





















































































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