The breaking of Mermin-Wagner-Hohenberg's theorem for fractional systems

Abstract

The Mermin-Wagner-Hohenberg theorem forbids the existence of long range order in two dimensional systems, since continuous symmetries cannot be spontaneously broken in d≤2. However, one of the premises for the validity of the theorem is that the energy-momentum dispersion of the quanta responsible for the order parameter fluctuations has a quadratic form, ε(k)=Ak2. In this paper we address the emergence of a fractional order energy-momentum dispersion in d-dimensional systems, leading to important consequences for the existence of long-range order in the case d=2. A first order renormalization group analysis in the ϵ-expansion is also performed in the fractional regime. It is shown that for fractional systems the universality classes are characterized by the number of spatial dimensions d and the number of components n of the order parameter, as in the case of integer order systems, but the labeling by the fractional order α is also required.