Quantum fluctuations inhibit symmetry breaking in the HMF model

TL;DR

Quantum fluctuations in the HMF model prevent symmetry breaking predicted by mean-field theory, restoring symmetry in the ground state for any finite number of particles, challenging the classical understanding of phase transitions.

Contribution

This work demonstrates that quantum fluctuations inhibit symmetry breaking in the HMF model, showing that the mean-field predicted ordered phase is not realized at finite N.

Findings

Quantum fluctuations restore $O(2)$ symmetry in the ground state.

The energetic cost of symmetry-breaking gradients scales as $\mathcal{O}(1/N^2)$.

Symmetry breaking is suppressed for any finite N, regardless of size.

Abstract

It is widely believed that mean-field theory is exact for a wide-range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian Mean Field (HMF) model for a system of indistinguishable bosons which is predicted (according to mean-field theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken $O(2)$ symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work, we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry breaking mean-field states. We find that the energetic cost of gradients in the center of mass wavefunction inhibit the breaking of the $O(2)$ symmetry, but that the energetic cost is very small --- scaling as $\mathcal{O}(1/N^2)$. Nevertheless, for any finite $N$, no matter how large, this implies that the ground state has a restored $O(2)$ symmetry. Implications for the finite temperature phases, and classical limit, of the HMF model are discussed.