Interacting relativistic boson fields in the De Sitter universe with two space-time dimensions

The positive temperature Gibbs state of a scalar boson field with a relativistic local self-interaction in two space-time dimensional Minkowski universe as constructed in [1] is not relativistic invariant. We prove in this paper that the corresponding state in the De Sitter universe is actually relativistic invariant if the temperature is given byT=1/2πR whereR is the constant radius of curvature of the De Sitter universe. Moreover the construction gives that the Schwinger functions or imaginary time Wightman functions are the moments of a generalized Markoff process on the sphere of radiusR.     

The scaling limit and Osterwalder-Schrader axioms for the two-dimensional Ising model

From a Feynman-Kac formula in a Fermion Fock space for the Schwinger functions of the infinite lattice periodic two-dimensional Ising model, scaled and scaling limit Schwinger functions are defined and shown to admit an absolutely convergent series representation. As the critical temperature is attained, it is shown that the scaled Schwinger functions converge and that the resulting scaling limit Schwinger functions obey the Osterwalder-Schrader axioms.     

Non-integrable aspects of the multi-frequency sine-Gordon model

We consider the two-dimensional quantum field theory of a scalar field self-interacting via two periodic terms of frequencies α and β. Looking at the theory as a perturbed sine-Gordon model, we use form factor perturbation theory to analyse the evolution of the spectrum of particle excitations. We show how, within this formalism, the non-locality of the perturbation with respect to the solitons is responsible for their confinement in the perturbed theory. The effects of the frequency ratio α/β being a rational or irrational number and the occurrence of massless flows from the gaussian to the Ising fixed point are also discussed. A generalisation of the Ashkin-Teller model and the massive Schwinger model are presented as examples of application of the formalism.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding von Neumann Algebras

In the present paper a model with competing ternary (J ₂) and binary (J ₁) interactions with spin values ±1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J ₁,J ₂ (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation-invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the unordered phase is a factor of type III _λ , where λ=exp{?2βJ ₂}, β>0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III _δ , otherwise they have type III₁.     

One-dimensional quantum liquids with power-law interactions: the Luttinger staircase

We study one-dimensional fermionic and bosonic gases with repulsive power-law interactions 1/|x|(β), with β>1, in the framework of Tomonaga-Luttinger liquid (TLL) theory. We obtain an accurate analytical expression linking the TLL parameter to the microscopic Hamiltonian, for arbitrary β and strength of the interactions. In the presence of a small periodic potential, power-law interactions make the TLL unstable towards the formation of a cascade of lattice solids with fractional filling, a "Luttinger staircase." Several of these quantum phases and phase transitions are realized with ground state polar molecules and weakly bound magnetic Feshbach molecules.     

A Three State Hard-Core Model on a Cayley Tree

Abstract

We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ? λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure ^*. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.     

A microscopic model with quasicrystalline properties

A classical lattice gas model with two-body nearest neighbor interactions and without periodic ground-state configurations is presented. The main result is the existence of a decreasing sequence of temperatures for which the Gibbs states have arbitrarily long periods. It is possible that the sequence accumulates at nonzero temperature, giving rise to a quasiperiodic equilibrium state.     

A microscopic theory of the lambda transition,II

The previously proposed finite temperature field theory of the lambda transition based on the Schwinger functional method is investigated further. A systematic method for calculating the higher-order loop terms is presented by introducing the one-loop Green's functions, which are found to be a natural finite temperature extension of the Beliaev-Hugenholtz-Pines-Gavoret-Nozières zero-temperature Green's functions. The application of the finite temperature loop expansion to the dynamical properties is presented by calculating the retarded density correlation functions at the one-loop level. The result gives a microscopic basis for the form of the dynamical structure factor recently proposed by Woods and Svensson. From a general point of view, without using any approximations or model interactions, Goldstone's theorem for the lambda transition at finite temperature is presented.     

Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface

We present a rigorous discussion of the analyticity properties of the beta function and of the effective potential for the theory of the ground state of a one dimensional system of many spinless fermions. We show that their analyticity domain as a function of the running couplings is a polydisk with positive radius bounded below, uniformly in, all the cut offs (infrared and ultraviolet) necessary to give a meaning to the formal Schwinger functions. We also prove the vanishing of the scale independent part of the beta function showing that this implies the analyticity of the effective potential and of the Schwinger functions in terms of the bare coupling. Finally we show that the pair Schwinger function has an anomalous long distance behaviour.     

Quantum Phase Transition on a Spin-One Spatially Anisotropic Frustrated Zigzag Square Lattice

We study quantum phase transitions in the easy-plane spin-one Heisenberg model on a zigzag square lattice with next-nearest-neighbor interactions at zero temperature using the SU(3) Schwinger boson formalism in a mean field approximation. We present the phase diagram and the dispersion relation for several values of the parameters. A magnetically disordered region in the phase diagram is found, even when the anisotropy parameter vanishes.     

The one-dimensional long-range ferromagnetic Ising model with a periodic external field

We consider the one-dimensional ferromagnetic Ising model with very long-range interaction under a periodic, biased and weak external field and prove that at sufficiently low temperatures the model has a unique limiting Gibbs state.     

Breaking of periodicity at positive temperatures

We discuss a classical lattice gas model without periodic or quasiperiodic ground states. The only ground state configurations of our model are nonperiodic Thue-Morse sequences. We show that low temperature phases of such models can be ordered. In fact, we prove the existence of an ordered (nonmixing) low temperature translation invariant equilibrium state which has nonperiodic Gibbs states in its extremal decomposition.     

Anderson Localization for the Holstein Model

A one-dimensional system of electrons on a lattice, interacting with a periodic potential, with period incommensurate with the lattice spacing and satisfying a Diophantine condition, is considered in the case of strong interaction. The Schwinger functions are computed and their asymptotic behaviour is studied, proving Anderson localization. The decay of the Schwinger functions is shown to depend critically on the value of the chemical potential. Received: 20 October 1999 / Accepted: 18 May 2000     

Taming Griffiths' singularities: Infinite differentiability of quenched correlation functions

We prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of Griffiths' singularities. We also show uniqueness of the Gibbs state and exponential decay of truncated correlation functions with probability one. Our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments. Our results require no assumptions on the probability distributions of the random parameters, except for the obvious one of no percolation of infinite couplings, and, in the strong field situation, for the also obvious requirement that zero magnetic fields do not percolate.Partially supported by the CNPq and FAPESP.Partially supported by the NSF under grants DMS-9208029 and INT-9016926.Partially supported by the CNPq.     

Correlation inequalities and the mass gap inP(φ)2

For theP()₂ field theory, we prove that the falloff of the (vacuum subtracted) two point Schwinger function dominates the higher order (vacuum subtracted) Schwinger functions. As applications, we prove that for even polynomials, the first excited state is odd, and that when there is a one particle state in the infinite volume limit, it is coupled to the vacuum by a single power of the field. The main inputs are the theory of Markov fields and the F.K.G. inequalities.     

Ordered valence-bond states in symmetric two-dimensional spin-orbital systems

We consider a superexchange Hamiltonian, H = -SUM ()(2S(i) . S(j)-(1/2)) (2T(i) . T(j)-(1/2)), which describes systems with orbital degeneracy and strong electron-phonon coupling in the limit of large on-site repulsion. In an SU(4) Schwinger boson representation, a reduced spin-orbital interaction is derived exactly, and a mean field theory has been developed. In one dimension, a spin-orbital liquid state with a finite gap is obtained. On a two-dimensional square lattice a novel type of spin-orbital ferromagnetically ordered state appears, while spin and orbital are antiferromagnetic. An important relation has been found, relating the spin and orbital correlation functions to the combined spin-orbital ones.     

On aC*-algebra approach to phase transition in the two-dimensional Ising model. II

We investigate the statesφ _β on theC*-algebra of Pauli spins on a one-dimensional lattice (infinitely extended in both directions) which give rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction) at inverse temperature β. We show that if β _c is the known inverse critical temperature, then there exists a family {v _β :β≠β _c } of automorphisms of the Pauli algebra such that $$\phi _\beta = \left\{ {_{\phi _\infty \circ v_\beta ,}^{\phi _0 \circ v_\beta ,} } \right. _{\beta > \beta _c .}^{0 \leqslant \beta< \beta _c } $$ .     

Boltzmann Limit and Quasifreeness for a Homogenous Fermi Gas in a Weakly Disordered Random Medium

We discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under negligence of the particle pair interactions. We derive the kinetic scaling limit for the momentum distribution function with a translation invariant initial state and prove that it is determined by a linear Boltzmann equation. Moreover, we prove that if the initial state is quasifree, then the time evolved state, averaged over the randomness, has a quasifree kinetic limit. We show that the momentum distributions determined by the Gibbs states of a free fermion field are stationary solutions of the linear Boltzmann equation; this includes the limit of zero temperature.