Local conservation laws in spin-1/2 XY chains with open boundary conditions

TL;DR

This paper investigates local conservation laws in spin-1/2 XY chains with open boundaries, revealing parity-dependent properties and proposing a framework for generalized models, with implications for understanding conserved quantities in quantum spin chains.

Contribution

It identifies new families of local charges in XY chains, analyzes their dependence on chain parity, and introduces a framework for extending these results to models with transverse fields.

Findings

Half of the conservation laws depend on the chain's parity.

In even chains, charges form an abelian set; in odd chains, they are non-abelian.

Evidence of reduced quasilocal conservation laws in certain parameter regions.

Abstract

We revisit the conserved quantities of the spin-1/2 XY model with open boundary conditions. In the absence of a transverse field, we find new families of local charges and show that half of the seeming conservation laws are conserved only if the number of sites is odd. In even chains the set of noninteracting charges is abelian, like in the periodic case when the number of sites is odd. In odd chains the set is doubled and becomes non-abelian, like in even periodic chains. The dependence of the charges on the parity of the chain's size undermines the common belief that the thermodynamic limit of diagonal ensembles exists. We consider also the transverse-field Ising chain, where the situation is more ordinary. The generalization to the XY model in a transverse field is not straightforward and we propose a general framework to carry out similar calculations. We conjecture the form of the bulk part of the local charges and discuss the emergence of quasilocal conserved quantities. We provide evidence that in a region of the parameter space there is a reduction of the number of quasilocal conservation laws invariant under chain inversion. As a by-product, we study a class of block-Toeplitz-plus-Hankel operators and identify the conditions that their symbols satisfy in order to commute with a given block-Toeplitz.