Fermionic symmetry-protected topological state in strained graphene
Ying-Hai Wu Tao Shi G. J. Sreejith Zheng-Xin Liu






Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany


Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany


Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany
Indian Institute for Science Education and Research, Pune 411008, India



Renmin University of China, Beijing 100876, China


The low-energy physics of graphene is described by relativistic Dirac fermions with spin and valley degrees of freedom. Mechanical strain can be used to create a pseudo magnetic field pointing to opposite directions in the two valleys. We study interacting electrons in graphene exposed to both an external real magnetic field and a strain-induced pseudo magnetic field. For a certain ratio between these two fields, it is proposed that a fermionic symmetry-protected topological state can be realized. The state is characterized in detail using model wave functions, Chern-Simons field theory, and numerical calculations. Our paper suggests that graphene with artificial gauge fields may host a rich set of topological states.


Fermionic symmetry-protected topological state in strained graphene
    Zheng-Xin Liu
    December 30, 2023
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§ INTRODUCTION


The study of topological phases has been a central topic of condensed matter physics since the observations of the quantum Hall effect <cit.>. The discovery of topological insulators signifies that the interplay between topology and symmetry can lead to a variety of exotic phenomena <cit.>. The concept of symmetry-protected topological (SPT) states has been introduced to describe gapped quantum states which are non-trivial only if certain symmetries are enforced. The SPT states are different from topologically ordered states in that they do not exhibit fractionalization or long-range entanglement, while they are distinguished from trivial product states by their non-trivial edge states if the protecting symmetries are not broken explicitly or spontaneously.

For non-interacting fermions, the possible SPT states have been fully classified for various symmetries <cit.>. It is natural to ask what happens if interactions are introduced. On one hand, distinct SPT states in free fermion systems may be adiabatically connected if interactions are allowed <cit.>. On the other hand, new SPT states with no free fermion counterparts may emerge because of interactions. The SPT states in one- and two-dimensional interacting bosonic systems can be constructed and classified by the group cohomology theory <cit.>, while SPT states beyond group cohomology in three dimensions have been reported <cit.>. For two-dimensional systems, the Chern-Simons field theory turns out to be very useful for classifying the SPT states and studying their physical properties <cit.>. As an example of bosonic SPT states, the integer quantum Hall (IQH) states of bosons have been identified in several microscopic models <cit.>. The theory of fermionic SPT states is less complete except for in one dimension <cit.>. One exotic possibility is that topological orders may be found on the surfaces of three-dimensional SPT states in some ways that are impossible in strictly two-dimensional systems <cit.>.

In this work, we construct a fermionic SPT state in two dimensions and analyze its properties in detail. This state depends essentially on interactions because its physical responses cannot appear in the same setup without interactions. The occurrence of this state requires two types of fermions coupled to two different magnetic fields. One possible platform fulfilling such a condition is strained graphene. The band structure of graphene contains two inequvialent valleys 𝐊^± in the Brillouin zone where the physics is described by Dirac fermions with linear dispersion. By applying suitably designed strain, a pseudo magnetic field pointing to opposite directions in the two valleys can be generated <cit.>. If an external magnetic field is also applied, the electrons in the two valleys would experience different total magnetic fields. This paper is organized as follows: the model of our interest is defined in Sec. <ref>, the fermionic SPT state is characterized using wave function and field theory in Sec. <ref>, the relevance of this state in strained graphene is corroborated by numerical calculations in Sec. <ref>, and we conclude in Sec. <ref>.



§ MODEL



We define the graphene sheet as the xy plane, the unit vector perpendicular to the xy plane as e_z, and the electron charge as -e. The system experience both a real magnetic field and a strain induced pseudo magnetic field as shown in Fig. <ref> (a). The real (pseudo) magnetic field is denoted as B^r (B^p) and the corresponding vector potential is 𝐀^r (𝐀^p). It is assumed that B^r>B^p and the total magnetic fields B^±=B^r±B^p in the two valleys point to the -e_z direction. The single-particle Hamiltonians in the two valleys are

    ℋ_𝐊^±   =    v_ F[
    [            0 π^±_x±iπ^±_y; π^±_x∓iπ^±_y            0 ]]

where v_ F is the Fermi velocity and π^±_i=p_i-e(A^r_i±A^p_i)/c are the canonical momentum operators. The solutions in each valley contain a set of zero energy states, whose spatial components are the same as the non-relativistic lowest Landau level (LL) wave functions.

For an infinite disk with symmetric gauge, the non-relativistic lowest LL wave functions have the simple forms ψ_m(x,y) ∼ z^mexp[-|z|^2/(4ℓ^2_B)], where ℓ_B=√(hc/(eB)) is the magnetic length, z=(x+iy) is the complex coordinate, and m is the angular momentum. If the direction of the magnetic field is reversed, the solutions are complex conjugates ψ^*_m(x,y). The electrons interact with each other via the Coulomb potential e^2/(ε|𝐫_j-𝐫_k|) (ε is the dielectric constant of the system). The magnetic field is taken to be sufficiently strong so the electrons are confined to the zero energy states. The many-body problem can be studied on compact surfaces such as sphere and torus <cit.> to avoid edge effects (see Appendix for some technical details). The low-energy states are assumed to be spin-polarized due to the Zeeman splitting and/or quantum Hall ferromagnetism <cit.> (this will be corroborated by numerical results later). The numbers of electrons in the two valleys are denoted as N^±_e. The system respects a U(1)_r×U(1)_p symmetry, where U(1)_r [U(1)_p] corresponds to the conservation of N^+_e+N^-_e (N^+_e-N^-_e). The numbers of magnetic fluxes through the surface of sphere or torus in the two valleys are denoted as N^±_ϕ. The magnetic length ℓ^+_B=√(hc/(eB^+)) associated with B^+ is used as the length scale and e^2/(εℓ^+_B) is used as the energy scale. For electrons on the sphere, an angular momentum quantum number L labels the many-body eigenstates. For electrons on the torus, special momentum quantum number Y labels the many-body eigenstates. 





§ TOPOLOGICAL STATE





 §.§ Wave Function


To construct a many-body state, we start from a non-interacting system with N^+_e=N^-_e and choose B^p appropriately such that the two valleys form two decoupled IQH states at filling factor 1 with opposite chiralities (B^r is zero at this stage). This system has the wave function

    ∏_j<k (z^+_j-z^+_k) ∏_j<k (z^-_j-z^-_k)^*

where the superscripts ± are used to label the two valleys. As we turn on the real magnetic field and the interaction between electrons, the composite fermion theory <cit.> suggests that the interaction energy can be efficiently minimized if the electrons each absorb two magnetic fluxes as shown in Fig. <ref> (b). This transformation is implemented by the factor ∏_j<k (z_j-z_k)^2, where the coordinates without superscripts are for all electrons. If we choose B^r=4B^p, the real magnetic field is completely absorbed by the electrons and the resulting composite fermions form two decoupled IQH states. The wave function for this system is

    𝒫_ LLL∏_j<k (z^+_j-z^+_k) ∏_j<k (z^-_j-z^-_k)^* ∏_j<k (z_j-z_k)^2

where 𝒫_ LLL projects wave functions to the lowest LLs (i.e. the zero energy single-particle states in the two valleys). The filling factors in the two valleys are ν^+=1/5 and ν^-=1/3 respectively.

The mapping to composite fermions also helps us to understand the excitations because the composite fermions can be taken as non-interacting objects and form their effective LLs. For the ground state, the two types of composite fermions completely fill their respective lowest effective LLs. A neutral excitation is created if we promote one composite fermion from its lowest LL to second LL. There are four types of charged minimal excitations: one type I (II) quasihole is present if the composite fermions occupy all orbitals of the lowest LLs except for one orbital in the 𝐊^+ (𝐊^-) valley; one type I (II) quasiparticle is present if the composite fermions occupy all orbitals of the lowest LLs and one orbital of the second LL in the 𝐊^+ (𝐊^-) valley. The local charges of the type I (II) quasihole/quasiparticle with respect to 𝐀^r are denoted as ±Q^r_ I (±Q^r_ II). The local charges with respect to 𝐀^p are denoted similarly with the superscript r replaced by p. Let us consider what happens if one electron is removed from the ground state [Fig. <ref> (c)]. The effective magnetic fluxes for the composite fermions in the 𝐊^+ (𝐊^-) valley increase (decrease) by two units. If the electron is from the 𝐊^+ valley, the real and pseudo charges increase by e and three type I quasiholes and two type II quasiparticles are created. If the electron is from the 𝐊^- valley, the real (pseudo) charge increases (decreases) by e and two type I quasiholes and one type II quasiparticles are created. This analysis yields the equations

    3Q^r_ I-2Q^r_ II=e     2Q^r_ I-Q^r_ II=e 
           3Q^p_ I-2Q^p_ II=e     2Q^p_ I-Q^p_ II=-e

so we have Q^r_ I=e, Q^r_ II=e, Q^p_ I=-3e, and Q^p_ II=-5e, which demonstrates that the charges of electron do not fractionalize in our system.

The Hall conductances can be obtained using the Laughlin flux insertion argument <cit.>. We place the system on a disk and insert one flux at the center. If the inserted flux is for the real magnetic field (which increases the flux values in both valleys), one type I quasihole and one type II quasiparticle are created at the center. The change of real charge at the center is Q^r_ I-Q^r_ II=0 so the real Hall conductance σ^r_xy=0. If the inserted flux is for the pseudo magnetic field [which increases (decreases) the flux value in the 𝐊^+ (𝐊^-) valley], one type I quasihole and one type II quasihole are created at the center. The change of pseudo charge at the center is Q^p_ I+Q^p_ II=-8e so the pseudo Hall conductance σ^p_xy=-8e^2/h. To measure the change of real (pseudo) charge due to the insertion of a pseudo (real) flux, we define a mutual Hall conductance σ^rp_xy whose value turns out to be 2e^2/h. It should be emphasized that these three Hall conductance values cannot appear simultaneously in a system of non-interacting two-component electrons, where two decoupled IQH states with the same or opposite chiralities at any filling factors can in principle be realized using real plus pseudo magnetic fields.

The system is expected to possess two counterpropagating edge modes corresponding to the edge modes of the IQH states of composite fermions. These edge modes will not be gapped out if there is no tunneling between the valleys. Another possibility is that the composite fermion edge states will remain gapless so long as an emergent time-reversal symmetry of composite fermions is preserved. This is intuitively plausible because the ν=±1 IQH states can be viewed as the simplest two-dimensional topological insulator. This will be formulated in a more precise way using effective field theory below. However, if electrons can tunnel between the valleys, N^±_e, σ^p_xy, and σ^rp_xy are no longer well-defined.



 §.§ Field Theory


The wave function Eq. <ref> can be described by the Chern-Simons theory with Lagrangian density <cit.>

    ℒ_1 = 1/4πħϵ^λμν K_IJ a_Iλ∂_μ a_Jν - j_Iλ a_Iλ

where a_Iλ (I=1,2, λ=0,x,y) are internal gauge fields, j_Iλ is the excitation current, and

    K = (
    [ 3 2; 2 1 ])

One important signature of topological phases is the number of degenerate ground states on torus. The existence of multiple degenerate ground states implies the presence of fractionalization. For the Chern-Simons Lagrangian ℒ_1, the ground state degeneracy on torus is |K|=1, which is consistent with our conclusion that there are no fractionally charged excitations.

An excitation in the Chern-Simons field theory can be labeled by an integer vector 𝐥, which is (±1,0)^T for type I quasiparticle/quasihole and (0,∓1)^T for type II quasiparticle/quasihole. The statistical angle of an excitation labeled by 𝐥 with itself is θ=π l^T K^-1 l, so all the minimal excitations have fermionic self braid statistics. The statistical angle of two excitations labeled by 𝐥_1 and 𝐥_2 is θ_12 = 2π l_1^T K^-1 l_2, so one type I quasihole/quasiparticle and one type II quasihole/quasiparticle have bosonic mutual braid statistics. If two probing U(1) gauge fields A^r_μ and A^p_μ coupled to the excitation current are introduced, we need to add an extra term

    ℒ_2 = e/2πħϵ^λμν ( t^r_I A^r_λ∂_μ a_Iν + t^p_I A^p_λ∂_μ a_Iν)

to ℒ_1, where the charge vectors 𝐭^r=(1,1)^T and 𝐭^p=(1,-1)^T. The U(1) charges of an excitation labeled by 𝐥 are -e[ t^r]^T K^-1𝐥 and -e[𝐭^p]^T K^-1𝐥, which yield the same results for the minimal excitations as our previous analysis. The Hall conductance with respect to the real gauge field is σ^r_xy = e^2 [𝐭^r]^TK^-1𝐭^r/h=0, the one with respect to the pseudo gauge field is σ^p_xy = e^2 [𝐭^p]^TK^-1𝐭^p/h = -8e^2/h, and the mutual Hall conductance is σ^rp_xy = e^2 [𝐭^r]^TK^-1𝐭^p/h = 2e^2/h, which reproduce the results derived using the Laughlin argument.

For a system described by ℒ_1+ℒ_2 in the bulk, it has gapless edge states on an open mainfold as captured by 

    ℒ_ edge = 1/4πħ( K_IJ∂_0ϕ_I∂_xϕ_J - V_IJ∂_xϕ_I∂_xϕ_J)

where ϕ_I are chiral boson fields that satisfy the Kac-Moody algebra

    [ ∂_xϕ_I(x) , ∂_yϕ_J(y) ] = 2πi K^-1_IJ∂_xδ(x-y)

and V_IJ depends on the microscopic details at the edge <cit.>. The electron annihilation operators for the two valleys are

    C_1 = e^-i(3ϕ_1+2ϕ_2)    C_2 = e^-i(2ϕ_1+ϕ_2)

The number of edge modes is the dimension of K and their chiralities are given by the sign of the eigenvalues of K. This means that our system has two counterpropagating edge modes and is consistent with our previous analysis based on wave functions. These edge modes can be gapped out by some perturbations, but one can rule out such perturbations by imposing certain symmetries on the system <cit.>. If the system has a symmetry group G and is acted upon by an element g of G, the K matrix transforms as

    K → W^T_g K W_g = s_g K

and the gauge fields ϕ transform as

    ϕ→ W^-1_gϕ + δ_gϕ

where W_g is an integer matrix, s_g is 1 for unitary symmetry and -1 for anti-unitary symmetry, and δ_gϕ is a constant. 

If we choose the symmetries to be U(1)_r×U(1)_p, the elements of the symmetry groups can be labeled by θ_r,θ_p∈[0,2π) and the symmetry operators are

    𝒰^r_θ_r=exp(-iθ_r∑_I t^r_I C^†_I C_I)

and

    𝒰^p_θ_p=exp(-iθ_p∑_I t^p_I C^†_I C_I)

The quantities W_g and δ_gϕ in Eqs. <ref> and <ref> can be written as

    W^r_θ_r = (
    [ 1 0; 0 1 ])    δ^r_θ_rϕ = θ_r∑_I t^r_I K^-1_IJ

for the U(1)_r group and

    W^p_θ_p = (
    [ 1 0; 0 1 ])    δ^p_θ_pϕ = θ_p∑_I t^p_I K^-1_IJ

for the U(1)_p group. The edge modes can also be protected by another combination of symmetries. To understand how this works, we convert the K matrix to the diagonal form K= Diag(1,-1)=X^TKX using

    X=(
    [  0 -1;  1  2 ])

The physics of our system would remain the same if the fields a_Iλ,ϕ_I and the vectors 𝐥, 𝐭 are also transformed properly (the transformed ones will be denoted by symbols with a tilde). In the transformed basis, the system is a quantum spin Hall insulator with edge modes that are protected by particle number conservation and time-reversal symmetry. Its symmetry group can be written as G^-_-[U(1)_r,𝒯], where 𝒯 is an emergent time-reversal symmetry, the subscript - means 𝒯^2=P_f (the fermion parity operator), and the superscript - means 𝒰^r_θ𝒯 = 𝒯𝒰^r_-θ. The time-reversal symmetry operates on K and ϕ_I as

    𝒯K𝒯^-1 = σ_xKσ_x = -K
          𝒯ϕ_1𝒯^-1 = ϕ_2
          𝒯ϕ_2𝒯^-1 = ϕ_1 + π

As one goes back to the original basis, an emergent time-reversal symmetry can be defined and it operates on K and ϕ_I as

    𝒯 K 𝒯^-1 = W^T_𝒯 K W_𝒯 = -K 
          𝒯ϕ_1 𝒯^-1 = -2ϕ_1 - ϕ_2 + π
          𝒯ϕ_2 𝒯^-1 = 3ϕ_1 + 2ϕ_2

The quantities W_g and δ_gϕ in Eqs. <ref> and <ref> can be written as

    W_𝒯 = ( 
    [ -2 -1;  3  2 ])    δ_𝒯ϕ = (
    [ π; 0 ])




§ NUMERICAL RESULTS









Exact diagonalization can be used to check the validity of our previous analysis. Let us consider the cases where the electrons are spin-polarized and N^+_e=N^-_e. For electrons on the sphere, the relation between N^±_e and N^±_ϕ is N^±_ϕ=N^±_e/ν^±-𝒮^±, where 𝒮^+=3 and 𝒮^-=1 are the shift quantum numbers. There is no shift quantum number on the torus. Fig. <ref> shows the energy spectra of the N^+_e=N^-_e=5 system on sphere and torus. The existence of a unique ground state on torus is confirmed in all the systems that can be studied. The energy gap Δ (the energy difference between the first excited state and the ground state) is found to be ≈0.018e^2/(εℓ^+_B) in the thermodynamic limit based on the finite size scaling analysis in Fig. <ref> (c). For ε=3 and B^r=20 T, the numerical value is about 17 K. Based on previous experiences <cit.>, we expect that disorder and other imperfections in experimental systems would reduce the actual value to 30%∼50% of the ideal value.

The spherical version of Eq. <ref> can be constructed explicitly and its high overlap with the exact ground state (0.9797 for N^+_e=N^-_e=5 and similar values for smaller systems) corroborates the accuracy of our ansatz. Besides the ground state, there are several low-energy neutral excitations in Fig. <ref> (a), which can be modeled by exciting one composite fermion in the 𝐊^+ valley to an originally unoccupied state. This is an interesting feature that distinguishes our state from fractional quantum Hall states in graphene, for which composite fermions in both valleys contribute to the low-energy neutral excitations <cit.>. Fig. <ref> shows the energy spectra of systems that contain one charged minimal excitation, where the trial wave functions also provide excellent approximations of the exact eigenstates. 

The edge states can be seen from entanglement spectrum <cit.>. For this calculation, the sphere is cut along its equator such that two hemispheres are separated by a virtual edge <cit.>. The ground state |Ψ⟩ is decomposed as |Ψ⟩ = ∑_ij F_ij |Ψ^S_i⟩⊗ |Ψ^R_j⟩, where S (R) is the southern hemisphere and its basis states are |Ψ^S_i⟩ (|Ψ^R_j⟩). The Schmidt decomposition of F_ij gives |Ψ⟩ = ∑_μ e^-ξ_μ/2 |Ψ^S_μ⟩⊗ |Ψ^R_μ⟩ and the ξ_μ levels comprise the entanglement spectrum. The entanglement levels can be labeled by the good quantum numbers N^S±_e (the numbers of electrons) and L^S_z (the z-component angular momentum) of the southern hemisphere. Fig. <ref> (d) presents the entanglement spectrum of the N^+_e=N^-_e=6 system in the N^S+_e=N^S-_e=3 subspace with the levels organized according to L^S_z. There is a foward-moving branch with counting 1,1,2,3,5 and a backward-moving branch with counting 1,1,2,3, which are consistent with the presence of two counterpropagating edge modes each described by a single boson field.

The assumption of spin polarization can also be tested numerically. To this end, we denote the numbers of electrons with spin-valley indices as N^↑+_e,N^↓+_e,N^↑-_e,N^↓-_e. The electrons in the same valley with different spin experience the same magnetic field. For the two valleys, we define total spin operators 𝐒^2_± and z-component spin opertors S^z_±. Because the Hamiltonian is SU(2) symmetric in the spin space, the energy eigenstates are also eigenstates of 𝐒^2_± [with eigenvalues S_±(S_±+1)] and S^z_± [with eigenvalues S^z_±=(N^↑±_e-N^↓±_e)/2]. To check for all possible spin polarizations, we should study the subspace with minimal z-component spin values S^z_±. The ground states for the two systems with N^+_e=N^-_e=3 and N^+_e=N^-_e=4 are found to be spin-polarized.

For a fixed N_e=N^+_e+N^-_e, there are many possible choices of N^±_e but only the special cases with N^+_e=N^-_e were studied above. It is useful to compare the ground state energy E_N^-_e at different N^-_e. Fig. <ref> shows that the lowest one appears at N^-_e=0 in the N_e=8 and 10 systems. If a pseudo Zeeman term α(N^+_e-N^-_e) is added to the Hamiltonian <cit.>, E_N^-_e increases (decreases) for N^-_e<N_e/2 (N^-_e>N_e/2) and E_N_e/2 will become the global minimum in a certain range of α (estimated to be 0.0344≲α≲0.0860). This can be seen easily from the spectra because the lines connecting all the pairs of data points at N^-_e and N_e-N^-_e lie above E_N_e/2 (one line is shown explicitly for each case).

The K matrix in Eq. <ref> suggests another possible trial wave function

    ∏_j<k (z^+_j-z^+_k)^3 ∏_j<k (z^-_j-z^-_k) ∏_j,k (z^+_j-z^-_k)^2

for the ground state. It is the zero energy ground state of the Hamiltonian g_1∑_j<k∇^2 δ^(2)(𝐫^+_j - 𝐫^+_k) + g_2∑_j<kδ^(2)(𝐫^+_j - 𝐫^-_k) + g_3∑_j<k∇^2 δ^(2)(𝐫^+_j - 𝐫^-_k), where the positive coefficients g_1,2,3 determine the interaction strength <cit.>. However, if there is no interaction within the 𝐊^- valley but repulsion between the two valleys as required by this parent Hamiltonian, the system would be unstable to phase separation. We have computed the overlap between Eq. <ref> and the exact ground state but find that it is much worse than Eq. <ref> (0.6972 for the N^+_e=N^-_e=5 system on sphere).



§ CONCLUSION



In conclusion, we have studied the properties of a fermionic SPT state in detail and demonstrated that it can be realized in strained graphene. The combination of real and pseudo magnetic fields proposed here allows us to explore a broad range of gauge field configurations for multi-component electrons in graphene. We expect that many other quantum phases in such systems will be revealed. It would also very interesting if one can find some methods to engineer non-Abelian gauge fields and study the quantum phases in such systems.



§ ACKNOWLEDGEMENT


Exact diagonalization calculations are performed using the DiagHam package for which we are grateful to all the authors. YHW and TS were supported by the DFG within the Cluster of Excellence NIM. ZXL was supported by the Research Funds of Remin University of China (No. 15XNFL19), NSF of China (No. 11574392), and the Major State Research Development Program of China (2016YFA0300500).

*





§ HAMILTONIAN MATRIX ELEMENTS


The two valleys are labeled using σ,τ=± and the creation (annihilation) operator for the single-particle state with quantum number m is denoted as C^†_σ,m (C_σ,m). The electrons interact via the Coulomb potential V(𝐫_1-𝐫_2)=e^2/(ε|𝐫_1-𝐫_2|). The second quantized form of the many-body Hamiltonian is

    1/2∑_στ∑_{m_i} F^σττσ_m_1m_2m_4m_3 C^†_σ,m_1 C^†_τ,m_2 C_τ,m_4 C_σ,m_3




 §.§ Sphere


The particles on a sphere experience a radial magnetic field generated by a magnetic monopole at the center. If the magnetic flux through the sphere is N^σ_ϕ, the LLL single-particle wave functions are <cit.>

    ψ^N^σ_ϕ_m(θ,ξ) = [ N^σ_ϕ+1/4πN^σ_ϕN^σ_ϕ-m]^1/2 u^N^σ_ϕ/2+m v^N^σ_ϕ/2-m

where u=cos(θ/2)e^iξ/2,v=sin(θ/2)e^-iξ/2 are spinor coordinates (θ and ξ are the azimuthal and radial angles in the spherical coordinate system) and m is the z component of the angular momentum. The magnetic length is related to the radius of the sphere as R=ℓ^+_B√(N^+_ϕ/2)=ℓ^-_B√(N^-_ϕ/2). The coefficients F^σττσ_m_1m_2m_4m_3 are

    ∫ dΩ_1 dΩ_2  [ ψ^N^σ_ϕ_m_1(Ω_1) ]^* [ ψ^N^τ_ϕ_m_2(Ω_2) ]^* V(𝐫_1-𝐫_2) ψ^N^τ_ϕ_m_4(Ω_2) ψ^N^σ_ϕ_m_3(Ω_1)

where 𝐫=R(sinθcosϕ,sinθsinϕ,cosθ) and Ω=𝐫/R. The product of two wave functions can be expressed as 

    ψ^N^σ_ϕ_m_1ψ^N^τ_ϕ_m_2 = (-1)^N^σ_ϕ-N^τ_ϕ[ (N^σ_ϕ+1)(N^τ_ϕ+1)/4π(N^σ_ϕ+N^τ_ϕ+1)]^1/2⟨N^σ_ϕ/2,-m_1;N^τ_ϕ/2,-m_2 | N^σ_ϕ/2+N^τ_ϕ/2,-m_1-m_2 ⟩ψ^N^σ_ϕ+N^τ_ϕ_m_1+m_2

The Coulomb potential can be expanded as

    e^2/ε|𝐫_1-𝐫_2| = 4πe^2/εR∑^∞_L=0∑^L_M=-L1/2L+1[ ψ^0_LM(Ω_1) ]^* ψ^0_LM(Ω_2)

These relations help us to obtain

    F^σττσ_m_1m_2m_4m_3 = δ_m_1+m_2,m_3+m_44πe^2/εR∑^ min(N^σ_ϕ,N^τ_ϕ)_L=01/2L+1 (-1)^(N^σ_ϕ+N^τ_ϕ)/2-m_1-m_4 S^1_L S^2_L

where the two coefficients S^1,2_L are defined by

    [ ψ^N^σ_ϕ_m_1(Ω_1) ]^* [ ψ^0_LM(Ω_1) ]^* ψ^N^σ_ϕ_m_3(Ω_1) = ∑^N^σ_ϕ_L_1=0 (-1)^N^σ_ϕ/2-m_1 S^1_L_1[ ψ^0_LM(Ω_1) ]^* ψ^0_L_1,m_3-m_1(Ω_1) 
          [ ψ^N^τ_ϕ_m_2(Ω_2) ]^* ψ^0_LM(Ω_2) ψ^N^τ_ϕ_m_4(Ω_2) = ∑^N^τ_ϕ_L_2=0 (-1)^N^τ_ϕ/2-m_4 S^2_L_2[ ψ^0_L_2,m_2-m_4(Ω_2) ]^* ψ^0_LM(Ω_2)




 §.§ Torus


The torus is spanned by the vectors 𝐋_1=L_1e_x,𝐋_2=L_2e_y and we choose the Landau gauge 𝐀^σ=(0,B^σx,0). If the magnetic flux through the torus is N^σ_ϕ, the LLL single-particle wave functions are <cit.>

    ψ^N^σ_ϕ_m(x,y) = 1/(√(π) L_2ℓ^σ_B)^1/2∑^ℤ_kexp{ - 1/2[ x/ℓ^σ_B - 2πℓ^σ_B/L_2( m+kN^σ_ϕ) ]^2 + i 2πy/L_2( m+kN^σ_ϕ) }

where the magnetic length ℓ^σ_B=√(L_1L_2/(2πN^σ_ϕ)). By defining the reciprocal lattice vectors 𝐆_1=2πe_x/L_1,𝐆_2=2πe_y/L_2, we transform the interaction potential to momentum space as 

    V(𝐫_1 - 𝐫_2)    =   1/L_1L_2∑_𝐪 V(𝐪) e^i𝐪·(𝐫_1 - 𝐫_2)

where 𝐪=q_1𝐆_1+q_2𝐆_2. The coefficients F^σττσ_m_1m_2m_4m_3 are

    ∫ d^2 𝐫_1 d^2 𝐫_2  [ ψ^N^σ_ϕ_m_1(𝐫_1) ]^* [ ψ^N^τ_ϕ_m_2(𝐫_2) ]^* V(𝐫_1-𝐫_2) ψ^N^τ_ϕ_m_4(𝐫_2) ψ^N^σ_ϕ_m_3(𝐫_1) 
    
    =       1/L_1L_2∑^N^σ_ϕ_m_1∑^N^τ_ϕ_m_2∑_q_1,q_2 V(𝐪) exp{ -𝐪^2/4 (ℓ^σ2_B+ℓ^τ2_B) + i 2πq_1 [ (m_1-q_2/2)/N^σ_ϕ - (m_2+q_2/2)/N^τ_ϕ] }δ^N^ G_ϕ_m_1+m_2,m_3+m_4

where N^ G_ϕ is the greatest common divisor of N^+_ϕ and N^-_ϕ and δ^N_ϕ_i,j is a generalized Kronecker delta defined as

    δ^N_ϕ_i,j=1    iff   i   mod  N_ϕ = j   mod  N_ϕ

The many-body eigenstates are labeled by a special momentum quantum number Y ≡ (∑_σ=±∑^N^σ_ϕ_i=1 m^σ_i)   mod  N^ G_ϕ. 





