We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange- and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur–Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase transition, the ground-state phase diagram, and the expected structure of extremal states.
We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S=12, the model is the Heisenberg ferromagnet, and for general spin S∈12N, it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S=12). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.
This thesis focuses on some results about quantum and classical lattice spin systems. We study a wide class of two-dimensional quantum models which enjoy a U(1) symmetry. Using the so called complex rotation method we show that the decay of the relevant correlation functions is at least algebraically fast. We provide some examples of relevant models which belong to our class. We review some results present in the literature concerning the so called GriffthsGinibre inequalities for the classical XY model and propose a generalisation to its quantum counterpart. Correlation inequalities indeed hold for the quantum XY model with spin- 1 2 and for the ground state of the spin-1 system. We propose some applications of these results, namely that the infinite volume limit of some correlation functions exists and that it is possible to compare quenched and annealed averages for a quantum XY model with random couplings. We investigate loop representations for O(n) classical spin systems. We propose a generalised random current representation and show its relationship with the Brydges-Fröhlich-Spencer one. We review some conjectures regarding the expected behaviour of these loop models { namely that macroscopic loops appear and their lengths are distributed according to a Poisson-Dirichlet distribution. We propose some arguments in favour of these conjectures for O(n) loop models, using a mix of exact results and heuristic considerations. In order to do so we de ne a stochastic process which is an effective split-merge process for macroscopic loops and we reformulate some correlation functions for the O(2) spin system in terms of loop properties.
The Stochastic Series Expansion (SSE) method along with resummation over the spin or flavor values maps the partition function of a quantum antiferromagnet to a closely-packed loop gas model in one higher dimension. Earlier work by Nahum et al. has shown that certain closely-packed three-dimensional loop gas models exhibit phases dominated by macroscopic loops, wherein the corresponding joint distribution of loop lengths is Poisson-Dirichlet. On grounds of universality, the same is expected of the ensemble of loops obtained in (2+1)-dimensional quantum antiferromagnets, albeit the loops emerge from a different microscopic origin. We sample the SSE loop ensemble for SU(N) antiferromagnets on a square lattice using Monte Carlo and study how the joint distribution varies with the degree of representation N and inverse temperature
β. We observe that, for low temperatures and small N(
≤ 4), the distribution indeed shows characteristics of Poisson-Dirichlet behaviour when antiferromagnetic correlations dominate the system.
We review random loop representations for the spin-1/2 quantum Heisenberg models, that are due to Toth (ferromagnet) and Aizenman-Nachtergaele (antiferromagnet). These representations can be extended to models that interpolate between the two Heisenberg models, such as the quantum XY model. We discuss the relations between long-range order of the quantum spins and the size of the loops. Finally, we describe conjectures about the joint distribution of the lengths of macroscopic loops, and of symmetry breaking.
Abstract We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin
S=21 it is equivalent to the XXZ model, and in spin
S=1 to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin
S=21 and
S=1 we give an explicit formula for the free energy for all values of the two parameters, and for spin
S>1 for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin
S=21 and
S=1, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.