Quantum spin correlations and random loops

We review the random loop representations of Toth and Aizenman-Nachtergaele for quantum Heisenberg models. They can be combined and extended so as to include the quantum XY model and certain SU(2)-invariant spin 1 systems. We explain the calculations of correlation functions.Comment: 9 pages, 4 figures. Proceeding for QMATH 12 (Berlin, 10-13 September 2013

Universal behaviour of 3D loop soup models

These notes describe several loop soup models and their {\it universal behaviour} in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, long-range order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a {\it Poisson-Dirichlet distribution}. This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.Comment: 31 pages, 11 figures. Notes prepared for the 6th Warsaw School of Statistical Physics, held from 25 June to 2 July 2016 in Sandomierz, Polan

Ferromagnetism, antiferromagnetism, and the curious nematic phase of S=1 quantum spin systems

We investigate the phase diagram of S=1 quantum spin systems with SU(2)-invariant interactions, at low temperatures and in three spatial dimensions. Symmetry breaking and the nature of pure states can be studied using random loop representations. The latter confirm the occurrence of ferro- and antiferromagnetic transitions and the breaking of SU(3) invariance. And they reveal the peculiar nature of the nematic pure states which MINIMIZE \sum_x (S_x^i)^2.Comment: 12 pages, 6 figure

A self-avoiding walk with attractive interactions

A self-avoiding walk with small attractive interactions is described here. The existence of the connective constant is established, and the diffusive behavior is proved using the method of the lace expansion.Comment: 13 pages, 4 figure

Analyticity in Hubbard models

The Hubbard model describes a lattice system of quantum particles with local (on-site) interactions. Its free energy is analytic when \beta t is small, or \beta t^2/U is small; here, \beta is the inverse temperature, U the on-site repulsion and t the hopping coefficient. For more general models with Hamiltonian H = V + T where V involves local terms only, the free energy is analytic when \beta ||T|| is small, irrespectively of V. The Gibbs state exists in the thermodynamic limit, is exponentially clustering and thermodynamically stable. These properties are rigorously established in this paper.Comment: 16 pages, LaTeX 2e, 7 figures. To appear in J. Stat. Phys. 95 (May 1999

Segregation in the asymmetric Hubbard model

We study the `asymmetric' Hubbard model, where hoppings of electrons depend on their spin. For strong interactions and sufficiently asymmetric hoppings, it is proved that the ground state displays phase separation away from half-filling. This extends a recent result obtained with Freericks and Lieb for the Falicov-Kimball model. It is based on estimates for the sum of lowest eigenvalues of the discrete Laplacian in arbitrary domains.Comment: 12 page