Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians H=∑(ψ†iχijψj+H.c.) and H′=∑(ψ†iχ′ijψj+H.c.), we say that H and H′ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces. And the Wilson loop W(C) can be defined as the trace of matrix-product P(C)(see the notations here ). Now my questions are:
(1)"H and H′ are gauge equivalent" if and only if "W(C)=W′(C) for all loops C on the 2D lattice ". Is this true? How to prove or disprove it?
(2)If the system is on a 2D torus, is W(L) always a positive real number ? Which means that the 'total flux'(the phase of W(L)) through the torus is quantized as 2π×integer, where L is the boundary of the 2D lattice.
(3)If the Hamiltonian contains extra terms, say H=∑(ψ†iχijψj+ψTiηijψj+H.c.+ψ†ihiψi), is the Wilson loop still defined as W(C)=tr(P(C))?
Thanks a lot.
This post imported from StackExchange Physics at 2014-03-09 08:40 (UCT), posted by SE-user K-boy