The Frustration of being Odd: How Boundary Conditions can destroy Local Order

TL;DR

This paper demonstrates that boundary conditions can influence local order parameters in quantum spin chains, challenging the traditional assumption that bulk properties are boundary-condition independent, especially in antiferromagnetic regimes.

Contribution

It provides direct calculations showing boundary effects on local magnetization in XYZ chains with odd sites, revealing boundary influence in antiferromagnetic cases.

Findings

Boundary conditions affect local magnetization in antiferromagnetic regimes.

Magnetizations decay algebraically with system size in AFM case.

Standard bulk property assumptions are violated in certain boundary conditions.

Abstract

A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the AFM coupling. We term this behavior {\it ferromagnetic mesoscopic magnetization}. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one--spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.