Spectral Gap and Decay of Correlations in U(1)-Symmetric Lattice Systems in Dimensions D<2

TL;DR

This paper proves that in U(1)-symmetric lattice systems with fractal dimensions less than 2, a spectral gap ensures correlations decay stretched exponentially, ruling out long-range order.

Contribution

It extends the McBryan-Spencer method to zero temperature for systems with U(1) symmetry and fractal dimensions, allowing for ground state degeneracy.

Findings

Spectral gap implies stretched exponential decay of correlations.

Results apply to quantum spin and electron models on fractal lattices.

Bounds exclude magnetic and electric long-range order.

Abstract

We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimensions D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range order.