Quantum fluctuations in quantum lattice-systems with continuous symmetry

TL;DR

This paper investigates how quantum fluctuations prevent spontaneous breaking of continuous symmetries in quantum lattice systems at zero temperature, providing conditions for invariance and decay of correlations across different dimensions.

Contribution

It establishes new theorems linking quantum fluctuations to symmetry preservation and correlation decay in quantum lattice systems with finite-range interactions.

Findings

Ground state invariance in 1D if susceptibility is finite

Correlation functions decay no faster than |r|^{-d+1} in higher dimensions when symmetry is broken

Quantum fluctuations prevent symmetry breaking in a broad class of systems

Abstract

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems at $T=0$. Our analysis is based on Pitaevskii and Stringari's idea that the uncertainty relation can be employed to show quantum fluctuations. For the one-dimensional systems, it is shown that the ground state is invariant under the continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than $|r|^{-d+1}$ whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice-systems having not-too-long-range interactions.