ZQZ_Q Topological Invariants for Polyacetylene, Kagome and Pyrochlore lattices

Adiabatic $Z_Q$ invariants by quantized Berry phases are defined for gapped electronic systems in $d$-dimensions ($Q=d+1$). This series includes Polyacetylene, Kagome and Pyrochlore lattice respectively for $d=1,2$ and 3. The invariants are quantum $Q$-multimer order parameters to characterize the topological phase transitions by the multimerization. This fractional quantization is protected by the global $Z_Q$ equivalence. As for the chiral symmetric case, a topological form of the $Z_2$-invariant is explicitly given as well.Comment: 4 pgages, 4 figure

Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz

We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1/2 Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on ${\bar {\mathscr H}}$, the tensor product of auxiliary spaces. By changing the basis in ${\bar {\mathscr H}}$, we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo [Alcaraz F C and Lazo M J 2006 {\it J. Phys. A: Math. Gen.} \textbf{39} 11335.] apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions [Korepin V E 1982 {\it Commun. Math. Phys.}, \textbf{86} 391.] and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.Comment: 24 pages; minor typos are correcte

Entanglement entropy of two disjoint blocks in XY chains

We study the Renyi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking properly into account the Jordan-Wigner string between the two blocks. From this we can evaluate any Renyi entropy of finite integer order. We study in details critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results in the gapped phase and after a quantum quench.Comment: 34 pages, 11 figure