The free energy in a class of quantum spin systems and interchange processes

Abstract

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin $S=\frac{1}{2}$ the model is the Heisenberg ferromagnet, for general spin $S\in \frac{1}{2}\mathbb{N}$ 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 $S=\frac{1}{2}$). 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 $S\ge 1$.

22 pages

References in corpus (5)

• Vanishing critical magnetization in the quantum Ising model
• The random interchange process on the hypercube
• Decay of transverse correlations in quantum Heisenberg models
• Large cycles in random permutations related to the Heisenberg model
• Existence of Néel order in the S=1 bilinear-biquadratic Heisenberg model via random loops

Cited by in corpus (10)

• Quantum spins and random loops on the complete graph
• Phase transition for the interchange and quantum Heisenberg models on the Hamming graph
• Critical parameter of random loop model on trees
• Loop correlations in random wire models
• Bounded entanglement entropy in the quantum Ising model
• The mean-field quantum Heisenberg ferromagnet via representation theory
• Quantum group deformation of the Kittel--Shore model
• Critical Parameters for Loop and Bernoulli Percolation
• On a class of orthogonal-invariant quantum spin systems on the complete graph
• The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems