The equivalence of the Bott index and the Chern number is established in the
thermodynamic limit for a gapped, short ranged and bounded Hamiltonian on a two
dimensional torus of linear size $ L $. A Kubo formula as an exact operatorial
identity is provided in real space and used to show the quantization of the
transverse conductance within corrections of order $L^-1$. In doing so the
physical foundations of the theory that introduces the Bott index in the realm
of condensed matter as proposed by Hastings and Loring in J. Math. Phys. (51),
015214, (2010) and Annals of Physics 326 (2011) 1699-1759 are recalled.
Description
On the equivalence of the Bott index and the Chern number on a torus,
and the quantization of the Hall conductivity with a real space Kubo formula
%0 Journal Article
%1 toniolo2017equivalence
%A Toniolo, Daniele
%D 2017
%K cond-mat diff-geom hall
%T On the equivalence of the Bott index and the Chern number on a torus,
and the quantization of the Hall conductivity with a real space Kubo formula
%U http://arxiv.org/abs/1708.05912
%X The equivalence of the Bott index and the Chern number is established in the
thermodynamic limit for a gapped, short ranged and bounded Hamiltonian on a two
dimensional torus of linear size $ L $. A Kubo formula as an exact operatorial
identity is provided in real space and used to show the quantization of the
transverse conductance within corrections of order $L^-1$. In doing so the
physical foundations of the theory that introduces the Bott index in the realm
of condensed matter as proposed by Hastings and Loring in J. Math. Phys. (51),
015214, (2010) and Annals of Physics 326 (2011) 1699-1759 are recalled.
@article{toniolo2017equivalence,
abstract = {The equivalence of the Bott index and the Chern number is established in the
thermodynamic limit for a gapped, short ranged and bounded Hamiltonian on a two
dimensional torus of linear size $ L $. A Kubo formula as an exact operatorial
identity is provided in real space and used to show the quantization of the
transverse conductance within corrections of order $L^{-1}$. In doing so the
physical foundations of the theory that introduces the Bott index in the realm
of condensed matter as proposed by Hastings and Loring in J. Math. Phys. (51),
015214, (2010) and Annals of Physics 326 (2011) 1699-1759 are recalled.},
added-at = {2017-08-22T09:26:28.000+0200},
author = {Toniolo, Daniele},
biburl = {https://www.bibsonomy.org/bibtex/2112d7005aa8d3a575b7df491c9cc3a35/vindex10},
description = {On the equivalence of the Bott index and the Chern number on a torus,
and the quantization of the Hall conductivity with a real space Kubo formula},
interhash = {1eb4ec4611fff8ae7a23f04da9ad860c},
intrahash = {112d7005aa8d3a575b7df491c9cc3a35},
keywords = {cond-mat diff-geom hall},
note = {cite arxiv:1708.05912},
timestamp = {2017-08-22T09:26:28.000+0200},
title = {On the equivalence of the Bott index and the Chern number on a torus,
and the quantization of the Hall conductivity with a real space Kubo formula},
url = {http://arxiv.org/abs/1708.05912},
year = 2017
}