Macroscopic loops in the Bose gas, Spin O(N) and related models

Abstract

We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in Z d , d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate Z d by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.

Reflection positivity and infrared bounds for quantum spin systems

“…Another extension of the method deals with "chessboard estimates", proposed by Fröhlich and Lieb [16] (they were partly motivated by [23]). Among many interesting works that use these ideas, let us mention the flux phase problem [30,35]; spin reflection positivity applied to Hubbard models [31,43,44]; itinerant electron models [32][33][34]; high spin systems whose classical limit has long-range order [6,7]; spin nematic phases [4,42]; Néel order in spin-1 model with biquadratic interactions [29]; hard-core bosons [1,26]; loop models associated with quantum spin systems [47] (motivated by [2,45]) and other loop models associated with classical spin systems [39]. Finally, let us mention an alternate extension of [17] to quantum systems by Albert, Ferrari, Fröhlich, and Schlein [3].…”