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Topological invariant in terms of the Green functions for the Quantum Hall Effect in the presence of varying magnetic field
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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.
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Cited by 2 Pith papers
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Hall conductivity as the topological invariant in phase space in the presence of interactions and non-uniform magnetic field
Interactions do not renormalize the phase-space topological expression for the Hall conductivity in a 2+1D tight-binding model with non-uniform magnetic field, to all orders of perturbation theory.
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Topological invariant responsible for the integer QHE and non-commutative geometry
The integer quantum Hall invariant N3 is expressed as a K-theory/cyclic-cohomology pairing; it vanishes on finite lattices and is only conditionally integer in the infinite-lattice limit.
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