Suppose that we use the Schwinger-fermion (Si=12f†iσfi) mean-field theory to study the Heisenberg model on 2D lattices, and now we arrive at the mean-field Hamiltonian of the form HMF=∑<ij>(ψ†iuijψj+H.c.) with uij=tσz(t>0), where ψi=(fi↑,f†i↓)T, and σz is the third Pauli matrix.
Now let's find the IGG of HMF, by definition, the pure gauge transformations in IGG should satisfy GiuijG†j=uij⇒Gj=σzGiσz on link <ij>—(1) . Specifically, consider the IGGs on the following different 2D lattices:
(a)Square and honeycomb lattices(unfrustrated): These two lattices can be both viewed as constituted by 2 sublattices denoted as A and B. Due to Eq.(1), it's easy to show that for both of these two lattices the gauge transformations Gi in the same sublattice are site-independent while those in different sublattices differ by GA=σzGBσz and IGG=SU(2).
(b)Triangular and Kagome lattices(frustrated):Due to Eq.(1), it's easy to show that for both of these two lattices the gauge transformations Gi are global (site-independent) and Gi=(eiθ00e−iθ) which means that IGG=U(1).
So my question is: The same form mean-field Hamiltonian HMF may has different IGGs on different lattices?
This post imported from StackExchange Physics at 2014-03-09 08:43 (UCT), posted by SE-user K-boy