Quantum Hall conductivity in the presence of interactions

We discuss quantum Hall effect in the presence of arbitrary pair interactions between electrons. It is shown that irrespective of the interaction strength the Hall conductivity is given by the filling fraction of Landau levels averaged over the ground state of the system. This conclusion remains valid for both integer and fractional quantum Hall effect.


II. FIXED NUMBERS OF ELECTRONS
In this section we consider the interacting system with the constant homogeneous external magnetic field, and fixed number of electrons. The multi -particle Hamiltonian with the interaction term has the form: In the following, we take a shorthand notation a for N a=1 . The Kubo formula for Hall conductance is where S is the area of the system. The total electric current may be written aŝ Thus the Hall conductance of this quantum -mechanical system may be written as Written in a compact way, it becomes where |k labels a many-body state and |0 is the ground state, and i, j = x, y.
For the minimally coupled single-electron HamiltonianĤ 0 (π) =Ĥ 0 (p + eA) =Ĥ 0 (−i∂ x + eA(x)) we decompose the coordinates x 1 = x, x 2 = y as follows:x =π y eB +X =ξ x +X, The gauge-independent commutation relations follow: And for the interacting Hamiltonian the commutation relations Eq. (5), Eq. (6) are generalized into [Ĥ, Substituting Eq. (7) and Eq.(8) into Eq.(4), we get: If we denote by M the number of electron states in a fully occupied Landau level, then where Φ 0 is flux quanta. Then we have where ν is the filling fraction. As a result, we see that only the filling fraction is relevant for the Hall conductivity for both integer and fractional quantum Hall effect, which has been observed in experiment. Coulomb interaction does not affect the steps up to Eq. (9), and that is why (at least, in the absence of impurities) the Hall conductivity of integer quantum Hall effect can be calculated in the free electron system.

III. THE SYSTEM DESCRIBED BY A CHEMICAL POTENTIAL
Now we consider the situation, when the number of electrons is allowed to fluctuate, but the chemical potential is introduced. The Hamiltonian will be written in the second-quantized form: whereĤ 0 =Ĥ 0 (p+eA) is a single-electron Hamiltonian minimally coupled to the background gauge field, while a † (x) and a(x) are the fermionic creation and annihilation operators in coordinate space. Let us define the two single-body operators:F Here operatorsF andĜ act on a considered as a function of x. We omit possible internal symmetry indices for brevity. Formally the further expressions are valid for the spinless electrons, but in fact we can extend easily our consideration to the case, when the operators a have indices, and operatorsF ,Ĝ act on those indices as well. In a similar manner we construct the two-body operator: We have (see the derivation in Appendix A) and Next, we generalize the quantum-mechanical coordinate-sum operators into the second-quantized coordinate-sum operators: With Eq. (15) and Eq. (16), we can generalize expressions for the commutators from the last section to the present case (with constant chemical potential instead of the fixed number of particles). [ With the help of these commutators, we can express Hall conductivity as If we denote by M the number of electron states in a fully occupied Landau level, then where Φ 0 is the magnetic flux quantum. Then we have where 0|ν|0 is the expectation value of the filling fraction for the ground state. Again, the Coulomb interactions do not affect the steps up to Eq. (22). From Eq. (25) we see that the Hall conductivity is proportional to the number of electrons in the ground state. In the absence of interactions this is just the number of electrons with the energies below the chemical potential. In the presence of interactions the total energy of the given state is already not given by the sum of the occupied one -electron stares. Therefore, the meaning of the chemical potential is not so transparent. Notice, that the above derivation does not depend on the details of the ground state. Therefore, it is valid for both integer quantum Hall effect and the fractional quantum Hall effect. Above in Eq. (22) it is assumed that the energies of the excited states differ from the ground state energy. Otherwise, the singularities are present. Therefore, we assumed that the considered system is gapped. We cannot say definitely how our expressions are changed for the gapless systems. It is worth mentioning, however that in the 3D systems the (non -topological) QHE may appear in the gapless systems. The example is given by Weyl semimetals. In any case the more detailed analysis is needed to extend Eq. (22) to the gapless systems, which has to involve the precise expressions for the ground state wave function as well as the wave functions of excited states, and their energies.
We also notice that the above derivation fits the known phenomenological schemes of the FQHE. In particular, the one with the Laughlins wave function as the ground state gives the correct filling fraction. This wave function is known to reproduce many features of the real ground state that minimizes energy of the system of interacting fermions. Also the pattern of composite fermions does not exclude the application of our results. The composite fermion theory proposes a heuristical description of the interacting fermion systems. Within this pattern the appearance of the fractional filling fraction ratio is explained. This appearance is enough to apply Eq. (25).

IV. CONCLUSIONS
To conclude, in the present paper we consider the quantum Hall effect in the systems with pair interactions between the electrons. We do not consider the influence of disorder on Hall conductivity (for the discussion of this issue see, for example, [7,8]). The key tool used in our consideration is the standard operator formalism of equilibrium quantum field theory. Several useful identities of this formalism are accumulated in Appendix. Those identities were used while dealing with the commutation relations between particle momenta and coordinates. The coordinates are separated to those responsible for the "center of orbit" motion and the local motion within the orbits. This representation is an extension to many -body systems of an old approach of [44] (see also references therein).
The main advantage of our approach is that arbitrary pair interactions between the electrons are taken into account. We observe that they do not affect the basic commutation relations used to derive the final expression for the Hall conductivity. This expression is given by Eq. (25). It is the filling fraction operator averaged over the ground state. In the present paper we do not discuss the possible values of this average, and the nature of the ground state. The effective theories of the FQHE [12][13][14][15][16][17][18] prompt that the expectation value ofν may take the standard filling fraction values given by certain rational numbers. The microscopic theory that explains the appearance of these numbers remains out of the scope of the present paper. Notice, that at the given value of chemical potential modifications of interactions may change the value of the filling fraction via modification of the ground state. We also do not prove robustness of the obtained result with respect to the smooth modification of the system, which is to be the subject of a separate investigation.
The authors are grateful to I.Fialkovsky, C.Zhang, and M.Suleymanov for useful discussions.

Appendix A: Commutators in second quantization
Let us first check the relation between the commutators of operators in second quantization and commutators in quantum mechanics. We deal with the single-body operatorŝ and the two-body operatorsK In both cases the KernelsĜ,F ,K act as the operators on a(x) considered as functions of x. First we prove Eq. (15): We need the following commutators of creation and annihilation operators: Next we prove Eq. (16): Then the proof goes as −(a † (x ′ )a † (x ′′ )K(x ′ , x ′′ )[a(x ′′ )a(x ′ ), a † (x)]F (x)a(x)) .
Then substituting Eq. (A7) into the last step, we get