Hall conductivity as topological invariant in phase space

Abstract

It is well known that the quantum Hall conductivity in the presence of constant magnetic field is expressed through the topological TKNN invariant. The same invariant is responsible for the intrinsic anomalous quantum Hall effect (AQHE), which, in addition, may be represented as one in momentum space composed of the two point Green's functions. We propose the generalization of this expression to the QHE in the presence of non-uniform magnetic field. The proposed expression is the topological invariant in phase space composed of the Weyl symbols of the two-point Green's function. It is applicable to a wide range of non-uniform tight-binding models, including the interacting ones.

Wigner-Weyl calculus in description of non-dissipative transport phenomena

“…To conclude, in this paper we review the results obtained by the group working in Ariel University. We summarize here the results reported previously in a series of papers [26,[47][48][49][50][51][52][53] (see also references therein to the other papers of the group). In these works the non -dissipative transport phenomena have been investigated using the technique of Wigner -Weyl calculus.…”

“…In the present paper we review the recent works by the group working in Ariel University on the topological description of non -dissipative transport in non -homogeneous systems. We summarize here the results reported earlier in a series of papers (see [26,[47][48][49][50][51][52][53] and references therein). We used Wigner -Weyl formalism and start from the extension of Eq.…”

“…Weakness of inhomogeneity means that physical quantities almost do not vary at the distance of the order of the lattice spacing. Within this calculus we develop Wigner -Weyl field theory in equilibrium [26,48], and its extension to quantum kinetic theory [52]. In this theory currents and conductivities are expressed through the Wigner transformed fermion propagators and Weyl symbols of operators.…”

Hall conductivity of strained $Z_2$ crystals

“…For self-sufficiency of the paper, all the details of this approach are given in the Appendices A and B. Also see another paper of this volume [13].…”

“…As derived in Appendix B, or with somewhat more details in [13], the current density in the system can be given in terms of Weyl symbols of Q and Ĝ = Q−1 as…”