Topological invariant in terms of the Green functions for the Quantum Hall Effect in the presence of varying magnetic field

Abstract

Recently the Wigner -Weyl formalism has been applied to the lattice models of solid state physics and to the lattice regularized quantum field theory. This allows to demonstrate that the electric current of intrinsic Anomalous Quantum Hall effect is expressed through the momentum space topological invariant composed of the Green functions both for the two -and the three -dimensional systems. Here we extend this consideration to the case of the Quantum Hall Effect existing in the presence of arbitrarily varying external magnetic field. The corresponding electric current appears to be proportional to the topological invariant in phase space composed of the Wigner transformed Green function that depends both on coordinates and momenta.

Precise Wigner-Weyl calculus for lattice models

“…Our Weyl symbols differ from those proposed in the other papers, and allow us to derive the topological expression for the Hall conductivity presented in Sect. II C. This expression remains valid for arbitrary magnetic field unlike our previous considerations [35][36][37], which were limited to magnetic fields much smaller than several thousands Tesla and of wavelengths much larger than 1 Angstrom. Thus, our present construction can be also used for consideration of artificial crystal lattices [84][85][86], in which external magnetic field of order of several Tesla may be sufficient to observe the Hofstadter butterfly.…”

“…Following [35,38,78,79] we start with the lattice tight-binding fermionic model with the partition function of the following form…”

“…The case of varying magnetic field has been considered recently in the series of papers written with the participation of the authors on the present paper. In [35] a new expression for the Hall conductivity has been proposed. It was given in a topologically invariant form, composed, however, of approximate Wigner transformation of the two-point Green's functions.…”

“…It was given in a topologically invariant form, composed, however, of approximate Wigner transformation of the two-point Green's functions. The adaptation of Weyl symbol used in [35] is described in Appendix B of the present paper. Graphene in the presence of inhomogeneities invoked by elastic deformations was considered in [36].…”

“…Graphene in the presence of inhomogeneities invoked by elastic deformations was considered in [36]. In [37] the proof was presented that in the presence of interactions the Hall conductivity is still given by the expression proposed in [35] with the complete two-point Green's function. The mentioned version of the Wigner-Weyl calculus has been also discussed in [38].…”