Breakdown of the thermodynamic limit in quantum spin and dimer models

TL;DR

This paper demonstrates that the thermodynamic limit in quantum spin and dimer models can be geometry-dependent, showing boundary shape influences phase behavior contrary to traditional assumptions in statistical mechanics.

Contribution

It constructs quantum Hamiltonians whose ground states violate the thermodynamic limit independence, revealing geometry-dependent phases in quantum spin and dimer models.

Findings

Boundary shape affects quantum phase behavior.

Square-octagon lattice supports a single gapped phase.

Diamond-shaped domains exhibit additional ordered phases.

Abstract

The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system's boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, an additional ordered phase emerges near the corners, where the dimers are in a staggered pattern.