Random loop representations for quantum spin systems

Abstract

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)-invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.

42 pages, 8 figures

References in corpus (6)

• Duality and hidden symmetries in interacting particle systems
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• Competition between two- and three-sublattice ordering for S=1 spins on the square lattice
• Rigorous analysis of discontinuous phase transitions via mean-field bounds
• On gapped phases with a continuous symmetry and boundary operators
• Macroscopic diffusion from a Hamilton-like dynamics

Cited by in corpus (26)

• Some Properties of Correlations of Quantum Lattice Systems in Thermal Equilibrium
• On gapped phases with a continuous symmetry and boundary operators
• The free energy in a class of quantum spin systems and interchange processes
• The interchange process with reversals on the complete graph
• Ferromagnetism, antiferromagnetism, and the curious nematic phase of S=1 quantum spin systems
• A numerical study of the 3D random interchange and random loop models
• Vanishing critical magnetization in the quantum Ising model
• Decay of correlations in 2D quantum systems with continuous symmetry
• Decay of transverse correlations in quantum Heisenberg models
• Macroscopic loops in the Bose gas, Spin O(N) and related models
• Dimerization and Néel order in different quantum spin chains through a shared loop representation
• Phase transition for loop representations of Quantum spin systems on trees
• Critical temperature of Heisenberg models on regular trees, via random loops
• Critical parameter of random loop model on trees
• Correlation inequalities for classical and quantum XY models
• Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction
• A direct proof of dimerization in a family of SU(n)-invariant quantum spin chains
• Fick's law in a random lattice Lorentz gas
• Existence of Néel order in the S=1 bilinear-biquadratic Heisenberg model via random loops
• Split-and-Merge in Stationary Random Stirring on Lattice Torus
• Poisson-Dirichlet distributions and weakly first-order spin-nematic phase transitions
• Characterising random partitions by random colouring
• Thermal stability of the Nagaoka-Thouless theorems
• On a class of orthogonal-invariant quantum spin systems on the complete graph
• Extension of the Lieb-Schupp theorem to the Heisenberg models with higher order interactions
• Site-monotonicity properties for reflection positive measures with applications to quantum spin systems