Lattice instantons in the large dimension limit

We give a lattice construction of the discretizations of the topologically nontrivial maps S ^2n–1S ⁿ . For n=1, 2, 4, 8, these are the Hopf maps. The construction, based on Barnes-Wall lattices, Reed-Muller error-correcting codes, and Hadamard matrices, generalizes to n=2 ⁱ for i any integer. Manton's result for the cases n=2 and 4 (i.e., the monopole and instanton) are included. We argue that discrete harmonic analysis will be exact in the infinite dimension limit.Work supported in part by the DOE contract #DE-ACO2-87ER-40325.B.Department of Energy Outstanding Junior Investigator supported in part by DOE contract number DE-FGO5-85ER-40226.     

On stationary topological solitons in a two-dimensional anisotropic Heisenberg model

We study stationary two-dimensional solitons in an easy-axis Heisenberg magnet with the Hamiltonian density wherei=1, 2,a=1, 2, 3, and (x _i ) is the angle between unit vector s(x _i ) and the easy axis, 0<p<. Stable solitons with a topological chargeQ=1 and localized distributionss ^a (x _i ) withQ=2 are found. The existence of the bound states of two solitons withQ=1 is shown numerically for 0<p<.     

Long-range order in a simple model of interacting fermions

We consider a model of spinless fermions on a lattice, interacting through a nearest neighbor repulsion. In the half-filled band case and for dimensionsd 2, we rigorously prove that there is long-range order in some domain of the parameters=(k _B T)^–1 andt/U, wheret is the hopping amplitude of the particles,U the strength of their repulsion, and the inverse temperature. Our technique is based on the usual Peierls argument of classical statistical mechanics but fails for the groundstate. We discuss the specific difficulties introduced by the Fermi statistics.Work supported in part by U.S. NSF grant PHY 90-19433-A02.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

A non-commutative version of the Arnold cat map

We provide a treatment of the ergodic properties of a noncommutative algebraic analogue of the dynamical system known as the Arnold cat map of the two-dimensional torus. Here, the algebra of functions on the torus is replaced by its noncommutative analogue, formulated by Connes and Rieffel, which arises in the quantum Hall effect. Our main results are that (a) the system is mixing and, as in the classical case, the unitary operator, representing its dynamical map, has countable Lebesgue spectrum; (b) for rational values of the noncommutativity parameter, , the model is a K-system, in the algebraic sense of Emch, Narnhofer, and Thirring, though not in the entropic sense of Narnhofer and Thirring; (c) for irrational values of , except possibly for a set of zero Lebesgue measures, it is neither an algebraic nor an entropic K-system.Supported in part by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P7101-PHY.     

An example of an infinite system with zero quantum dynamical entropy

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained new improved tree graph inequalities that have led to the extra 1/N! factors. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory.This work was supported in part by the National Science Foundation under grant no. PHY-87-01329.     

Localization for random potentials supported on a subspace

We consider the lattice Schrödinger operator acting onl ² ( ^d ) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.     

Correlation function formulas for some infrared asymptotic-free scalar field lattice models via the block renormalization group

We consider a class of scalar field lattice models with action 1/2()+V(), V small. After n block renormalization group transformations, new formulas are obtained for the finite lattice generating and correlation functions. For some infrared asymptotic-free models in the thermodynamic and n limits, the formulas for correlation functions are especially simple, isolate the correct dominant long-distance behavior, and can be used to control the subdominant contributions.     

Completely ℤ symmetric R matrix

An infinite-dimensional R matrix related to the limiting case n of the completely _n symmetric R matrix is discovered. This R matrix is expressed as an operator on C (S ¹×S ¹). Moreover, the fusion procedure of the R-operator is investigated and the finite-dimensional R matrices are constructed from the R operator.     

Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet

We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so-called random stirrings on ^d . By use of this representation, we improve the lower bound on the pressure given by Conlon and Solovej inLett. Math. Phys. 23, 223–231 (1991).Work supported by the Hungarian National Foundation for Scientific Research, grant No. 1902.     

Thomas-Fermi theory for matter in a magnetic field as a limit of quantum mechanics

The Thomas-Fermi theory for electrons and fixed nuclei in a homogeneous magnetic field is shown to be a limit of quantum mechanics in the following sense: If the nuclear charges and the number of eletrons are multiplied by a, and the magnetic field by a ^4/3, then the ground-state energy, divided by a ^7/3, converges to the TF energy when a . A similar result holds also for the electronic density. This generalizes corresponding theorems for zero magnetic field due to Lieb, Simon, and Baumgartner.     

On laplace integrals and transfer operators in large dimension: Examples in the nonconvex case

This Letter gives detailed proofs concerning the analysis of the pair correlations for a nonconvex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operator. Although the problem is similar to the semiclassical study of the Kac operator presented in a paper with M. Brunaud, which was devoted to the study of exp-(v/2) exp h ² exp-(v/2) for h small, new features appear for the model exp-(v/2h) exp h exp-(v/2h). Our principal results concern the splitting of this operator between the two largest eigenvalues. We give an upper and a lower bound for this splitting in the semi-classical regime. As a corollary, we get good control of the decay of the pair correlation. Some of the results were announced in a previous paper. Related WKB constructions will be developed in a later paper.Inspired by papers by M. Kac [15, 16].     

On log-Sobolev inequalities for infinite lattice systems

For a system on an infinite lattice, we show that a Gibbs measure for a smooth local specification ={E } satisfying the Dobrushin uniqueness theorem also satisfies log-Sobolev inequality, provided it is satisfied for one-dimensional measures E _l .     

Multicomponent composites,electrical networks and new types of continued fraction II

The outstanding problem of systematically developing rigorous bounds on the complex effective conductivity tensor * ofd-dimensional,n-component composites withn>2 is solved. The bounds incorporate information contained in successively higher order correlation functions which reflect the composite geometry. Explicit expressions are given for many of the bounds and some, but not all of them, are represented by nested sequences of circles in the complex plane that enclose, and in fact converge to, each diagonal element of *. They are derived from the fractional linear matrix transformations found in Part I that recursively link * with a hierarchy of complex effective tensors ^(j),j=0, 1, 2, ..., of increasing dimension,d(n–1) ^j . Elementary bounds on ^(j) confining the diagonal elements of ^(j) or its inverse to half-plane, wedge or open polygon regions of the complex plane, imply narrow bounds on * which converge to the exact value of * in the limit asj . When the component conductivities are real these bounds are more restrictive than the corresponding variational bounds. Besides applying to the effective conductivity *, the bounds extend to a wide class of matrix-valued multivariate functions called -functions, and thereby to conduction in polycrystalline media, viscoelasticity in composites, and conduction in multi-component, multiterminal, linear electrical networks. The analytic and invariance properties of -functions are explored and within this class of function most of the bounds are found to be optimal or at least attainable. The bounds obtained here are essentially a generalization to matrix-valued, multivariate functions of the nested sequence of lens-shaped bounds in the complex plane derived by Gragg and Baker for single variable Stieltjes functions.     

Monotonicity and concavity in Coulomb systems

The eigenvalues ofH()=H ₀+H ^*, whereH ^* is an arbitrary Coulomb potential, decrease with increasing 0. Linear and parabolic bounds for the ground-state energy are presented. These bounds are applied to the biexciton and the exciton at a neutral donor.     

Time-asymptotic boson states from infinite mean field quantum systems coupled to the boson gas

By means of cocycle techniques in a recent paper, the global dynamics of mean field-boson couplings has been studied. Here, by restricting to the bosonic system the infinite time limit (t ) for very general initial states, one obtains time-asymptotic states on the bosonicC ^*-Weyl algebra, in which one partially rediscovers the collective ordering of the infinite mean field lattice.     

TrigonometricR matrices related to ‘dilute’ Birman-Wenzl-Murakami algebra

Explicit expressions for three series ofR matrices which are related to a dilute generalisation of the Birman-Wenzl-Murakami algebra are presented. Of those, one series is equivalent to the quantumR matrices of theD _n+1 ⁽²⁾ generalised Toda systems, whereas the remaining two series appear to be new.     

The dynamical entropy of the quantum Arnold cat map

We present a rigorous computation of the dynamical entropyh of the quantum Arnold cat map. This map, which describes a flow on the noncommutative two-dimensional torus, is a simple example of a quantum dynamical system with optimal mixing properties, characterized by Lyapunov exponents ± 1n ⁺, ⁺ > 1. We show that, for all values of the quantum deformation parameter,h coincides with the positive Lyapunov exponent of the dynamics.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Absence of pair binding in the U=∞ Hubbard model

We consider the standard Hubbard model in the U= limit. We show that, for any finite lattice with all positive hopping matrix elements, t _i,j >0, the ground state energy of the system containing two particles in excess of half filling plus the energy of the system at half filling is never lower than twice the energy of the system with a single extra particle. Similar results are obtained for holes when the lattice is bipartite. As a byproduct, we obtain a simple alternative proof of Tasaki's generalization of the Nagaoka theorem for non-bipartite lattices (but without the uniqueness clause).