Giant number-parity effect leading to spontaneous symmetry breaking in finite-size quantum spin models

TL;DR

This paper demonstrates that spontaneous symmetry breaking can occur in finite-size quantum spin systems under specific conditions, challenging the traditional view that it only occurs in the thermodynamic limit, and shows how to prepare such states with spin squeezing.

Contribution

It identifies conditions for finite-size SSB in quantum spins and introduces a quasi-adiabatic method to achieve symmetry-broken states with Heisenberg scaling.

Findings

SSB can occur in finite systems with odd N under certain conditions.

Symmetry-broken states exhibit spin squeezing with Heisenberg scaling.

A quasi-adiabatic protocol enables preparation of symmetry-broken states in finite systems.

Abstract

Spontaneous symmetry breaking (SSB) occurs when a many-body system governed by a symmetric Hamiltonian, and prepared in a symmetry-broken state by the application of a field coupling to its order parameter $O$, retains a finite $O$ value even after the field is switched off. SSB is generally thought to occur only in the thermodynamic limit $N\to \infty$ (for $N$ degrees of freedom). In this limit, the time to restore the symmetry once the field is turned off, either via thermal or quantum fluctuations, is expected to diverge. Here we show that SSB can also be observed in \emph{finite-size} quantum spin systems, provided that three conditions are met: 1) the ground state of the system has long-range correlations; 2) the Hamiltonian conserves the (spin) parity of the order parameter; and 3) $N$ is odd. Using a combination of analytical arguments and numerical results (based on time-dependent variational Monte Carlo and rotor+spin-wave theory), we show that SSB on finite-size systems can be achieved via a quasi-adiabatic preparation of the ground state -- which, in U(1)-symmetric systems, is shown to require a symmetry breaking field vanishing over time scales $\tau \sim O(N)$. In these systems, the symmetry-broken state exhibits spin squeezing with Heisenberg scaling.