Equivalence between XY and dimerized models

TL;DR

This paper demonstrates the exact equivalence between XY and dimerized spin-1/2 chains in the bulk and finite systems, providing constructive mappings and extending the results to disordered couplings, with implications for disordered systems.

Contribution

It provides a constructive proof of the equivalence between XY and dimerized models and generalizes the mapping to fully site-dependent couplings.

Findings

Exact bulk equivalence of XY and dimerized models

Constructive unitary transformations mapping models onto each other

Extension of mapping to disordered XY chains

Abstract

The spin-1/2 chain with XY anisotropic coupling in the plane and the XX isotropic dimerized chain are shown to be equivalent in the bulk. For finite systems we prove that the equivalence is exact in given parity sectors, after taking care of the precise boundary conditions. The proof is given constructively by finding unitary transformations that map the models onto each other. Moreover, we considerably generalized our mapping and showed that even in case of fully site dependent couplings the XY chain can be mapped onto an XX model. This result has potential application in the study of disordered systems.