Continuous symmetry breaking in 1D spin chains and 1+1D field theory

TL;DR

This paper demonstrates that 1D spin chains can spontaneously break continuous U(1) symmetry at a critical point, challenging traditional beliefs, through a field theory approach considering perturbations and renormalization-group flows.

Contribution

It introduces a new understanding of symmetry breaking in 1D systems via a continuum field theory and RG analysis, showing easier symmetry breaking than previously thought.

Findings

Ground states can break U(1) symmetry via long-range order in (S^x, S^y).

Critical points can be reached without fine-tuning, via RG flow.

Goldstone modes can be interacting even in the infrared.

Abstract

We argue that ground states of 1D spin chains can spontaneously break U(1) ``easy-plane'' spin rotation symmetry, via true long-range order of $(S^x, S^y)$, at the phase transition between two quasi-long-range-ordered phases. The critical point can be reached by tuning a single parameter in a Hamiltonian with the same symmetry as the XXZ model, without further fine-tuning. Equivalently, it can arise in systems of bosons with particle-hole symmetry, as a long-range-ordered transition point between two quasi-long-range-ordered superfluids. Our approach is to start with the continuum field theory of the isotropic Heisenberg ferromagnet and consider generic perturbations that respect easy-plane symmetry. We argue for a renormalization-group flow to a critical point where long-range order in $(S^x, S^y)$ is enabled by coexisting critical fluctuations of $S^z$. (We also discuss multicritical points where further parameters are tuned to zero.) These results show that it is much easier to break continuous symmetries in 1D than standard lore would suggest. The failure of standard intuition for 1D chains (based on the quantum--classical correspondence) can be attributed to Berry phases, which prevent the 1+1D system from mapping to a classical 2D spin model. The present theory also gives an example of an ordered state whose Goldstone mode is interacting even in the infra-red, rather than becoming a free field.