On the Truncation of Systems with Non-Summable Interactions

In this note we consider long-range q-states Potts models on Z ^d , d≥ 2. For various families of non-summable ferromagnetic pair potentials φ(x)≥ 0, we show that there exists, for all inverse temperature β > 0, an integer N such that the truncated model, in which all interactions between spins at distance larger than N are suppressed, has at least q distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type φ(x)∼ ‖x‖^−α, 0≤α≤ d. These results are obtained using simple percolation arguments. Work supported by Swiss National Foundation for Science, Conselho Nacional de Desenvolvimento Cientìfico e Tecnològico, and Programa de Auxìlio para Recèm Doutores PRPq-UFMG.     

The rotation number for almost periodic potentials

We define and analyze the rotation number for the almost periodic Schrödinger operatorL= –d ²/dx ²+q(x). We use the rotation number to discuss (i) the spectrum ofL; (ii) its relation to the Korteweg-de Vries equation.Partially supported by the National Science Foundation under Grant NSF-MCS 77-01986     

The low-temperature behavior of disordered magnets

We map out the low-temperature phase diagrams of dilute Ising ferromagnets and predominantly ferromagnetic ferrites, obtaining nonperturbative and essentially optimal conditions on the density of ferromagnetic couplings required to maintain long-range order. We also study mappings of dilute antiferromagnets in a uniform field onto random field ferromagnets.For the randomly dilute systems, we prove that ferromagnetically ordered states exist at low temperature if the density of ferromagnetic couplings exceeds the (appropriately defined) percolation threshold, thereby extending the result of Georgii to three or more dimensions. We also show that, for these systems, as the temperature tends to zero, the magnetization approaches the percolation probability of the corresponding Bernoulli system. In two dimensions, we prove that low-temperature ordering persists in the presence of antiferromagnetic impurities if the ferromagnetic couplings percolate and if the density of antiferromagnetic couplings is bounded above by the order of the inverse square of the corresponding percolation correlation length. For these systems, we rigorously compute the first order decrease in the zero-temperature nominal spontaneous magnetization, in terms of derivatives of the percolation probability, thereby establishing the existence of ferrimagnetically ordered states. Finally, we introduce a model of a random ferrite which exhibits spontaneous magnetization anticorrelated with the boundary conditions.National Science Foundation Postdoctoral Research Fellows. Work supported in part by the National Science Foundation under Grant No. PHY-8203669Work supported in part by the National Science Foundation under Grant No. MCS-8108814 (A03)     

Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces

LetZ(s, R) be the Selberg zeta function of a compact Riemann surfaceR. We study the behavior ofZ(s, R) asR tends to infinity in the moduli space of stable curves. The main result is an estimate forZ(s, R) valid fors in a neighborhood, depending only on the genus, ofs=1. Our analysis gives an alternate proof of the Belavin-Knizhnik double pole result, [5].Partially supported by the National Science Foundation and the Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA     

Tree graph inequalities and critical behavior in percolation models

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent associated with the expected cluster sizex and the structure of then-site connection probabilities =_n(x₁,..., x_n). It is shown that quite generally 1. The upper critical dimension, above which attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneousd-dimensional lattices with (x, y)=O(¦x -y¦^–(d–2+), atp=p _c, our criterion shows that =1 if > (6-d)/3. The connectivity functions _n are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of _n, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function ₂ (x, y). A. P. Sloan Foundation Research Fellow. Research supported in part by the National Science Foundation Grant No. PHY-8301493.Research supported in part by the National Science Foundation Grant No. MCS80-19384.     

A proof of part of Haldane's conjecture on spin chains

It has been argued that the spectra of infinite length, translation and U(1) invariant, anisotropic, antiferromagnetic spin s chains differ according to whether s is integral or 1/2 integral: There is a range of parameters for which there is a unique ground state with a gap above it in the integral case, but no such range exists for the 1/2 integral case. We prove the above statement for 1/2 integral spin. We also prove that for all s, finite length chains have a unique ground state for a wide range of parameters. The argument was extended to SU(n) chains, and we prove analogous results in that case as well.Work partially supported by U.S. National Science Foundation grant PHY80-19754 and by the A.P. Sloan Foundation.Work partially supported by U.S. National Science Foundation grant PHY-85-15288.     

Charge order and spin order in Nd0.5Sr0.5Mn1? x (Ga x , Ti x )O3 system

Magnetic properties of Nd0.5Sr0.5Mn1-x(Gax,Tix)O3 system (0.04≤x≤0.4) were investigated through magnetization and electron spin resonance (ESR) measurements. It was observed that a small amount of Ti substitution for Mn will destroy the charge-ordering (CO) phase completely and induce the cluster-spin-glass phase in the system, which displays a procedure of collapse of CO and of an enhancement of spin ordering (SO) phase. In contrast, the Ga substitution for Mn induces a melting of CO phase in the system. It was observed that with substitution the CO phase is suppressed gradually and the remanent CO phase is retained all the while, and withal, there is a co-existence of AFM CO phase and FM SO at low temperature. In addition, an abrupt rise of magnetization was observed in M-Tcurves. We attributed this abnormal phenomenon to a transition from canted AFM SO to FM SO in CO region.     

Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups

The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.Work supported in part by the National Science Foundation.Work supported in part by the U.S. Atomic Energy Commission.     

Localized excitation transport on substitutionally disordered lattices

Average-T-matrix and coherent medium theories are used to study the motion of localized excitations on Substitutionally disordered lattices. We derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer. Numerical results forP ₀(t), the probability of remaining at the origin for two-dimensional nearest-neighbor lattices are presented.Supported in part by the National Science Foundation. (CHE81-00407).     

Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN ₀, the number of distinct infinite clusters, is either 0, 1, or and the caseN ₀= (which might occur in sufficiently high dimension) is analyzed.Alfred P. Sloan Research Fellow, Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.Research supported in part by the U.S.Israel Binational Science Foundation.     

Stability of coulomb systems with magnetic fields

The ground state energy of an atom in the presence of an external magnetic filedB (with the electron spin-field interaction included) can be arbitrarity negative whenB is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, B ². For a hydrogenic like atom we prove that there is a critical nuclear charge,z _c , such that the atom is stable forz<z _c and unstable forz>z _c .Work partically supported by U.S. National Scinece Foundation grant DMS-8405264 during the author's stay at the Institute for Advanced Study, Princeton, NJ, USAWork partially supported by U.S. National Science Foundation grant PHY-8116101-A03Work partially supported by U.S. and Swiss National Science Foundation Cooperative Science Program INT-8503858. Current address: Institut f. Mathematik, FU Berlin, Arnimallee 3, D-1000 Berlin 33     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

Fractal magnetic microstructure in the (Co<Subscript>41</Subscript>Fe<Subscript>39</Subscript>B<Subscript>20</Subscript>)<Subscript>x</Subscript>(SiO<Subscript>2</Subscript>)<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript> nanocomposite films

Magnetostructural methods are applied to determine the exchange bond percolation limit in (Co₄₁Fe₃₉B₂₀)_x(SiO₂)_1?x nanocomposites (x _c = 0.30 ± 0.02), which separates the phase plane along the metal concentration axis into a superparamagnetic region and a ferromagnetic region. It is shown that, with respect to the singularities of the magnetization up to the magnetization saturation curves, the ferromagnetic region is further subdivided into three regions differing in the character of the spatial propagation of the magnetization ripples or in the magnetic correlation function characteristics. The fractal dimension of the nanocomposite magnetic microstructure near the percolation threshold is determined.     

Disordered system withn orbitals per site: 1/n expansion

Averaged Green's functions for a disordered electronic system withn orbitals per site are expanded in powers of 1/n. These expansions should be valid in the region of extended states. The expansion coefficients for the d.c. conductivity are finite for dimensionalityd>2 and diverge asd approaches 2. Similarities of two types of two-particle Green's functions with the transverse and longitudinal susceptibilities of a ferromagnet with broken continuous symmetry are pointed out. Arguments for two being the lower critical dimensionality for the hydrodynamics and the mobility edge are given. Provided our series can be exponentiated we find that no metallic conductivity exists for finiten andd=2 in one of our models. Critical exponents ford infinitesimal above two are given. In this limitv diverges like 1/(d–2) and the conductivity vanishes linearly at the mobility edge.The diagrams of the Green's functions are given in terms of vertices of short-range order and of the two-particle propagators of then= limit. Diagrams withs loops contribute in ordern ^–s . The diagrams can be rearranged so that a number of vertices vanishes like the square of the wavevector. This feature prevents infrared divergencies for the d.c. conductivity ford>2.Work supported in part by a DFG fellowship (R.O.), by the Material Research Laboratory of the National Science Foundation at the University of Chicago (F.W.), and by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 123 (Stochastic Mathematical Models) at the Universität Heidelberg     

Local Ward identities and the decay of correlations in ferromagnets

Using local Ward identities we prove a number of correlation inequalities forN-component, isotropically coupled, pair interacting ferromagnets; some for allN2 and some forN=2, 3, 4. These are used to prove a mass gap above the mean field temperature, for allN2. ForN=2, 3, 4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges asm0.Research partially supported by US National Science Foundation under Grant PHY-7825390 A01.Research partially supported by US National Science Foundation under Grant MCS-78-01885.     

Fixed points of Feigenbaum's type for the equationf p (λx)≡λf(x)

Existence and hyperbolicity of fixed points for the mapN _p :f(x) ^–1 f ^p (x), withf ^p p-fold iteration and =f ^p (0) are given forp large. These fixed points come close to being quadratic functions, and our proof consists in controlling perturbation theory about quadratic functions.Supported in part by the Swiss National Science Foundation     

A nontrivial Renormalization Group fixed point for the Dyson-Baker hierarchical model

We prove the existence of a nontrivial Renormalization Group (RG) fixed point for the Dyson-Baker hierarchical model ind=3 dimensions. The single spin distribution of the fixed point is shown to be entire analytic, and bounded by exp(–const×t ⁶) for large real values of the spint. Our proof is based on estimates for the zeros of a RG fixed point for Gallavotti's hierarchical model. We also present some general results for the heat flow on a space of entire functions, including an order preserving property for zeros, which is used in the RG analysis.Supported in Part by the National Science Foundation under Grant No. DMS-9103590.Supported in Part by the Swiss National Science Foundation.     

A model with persistent vacuum

It is shown that there exists a selfadjoint Hamilton operator corresponding to the interactionH ₀ ^(a) +H ₀ ^(b) + _b ⁺ (x) _a (x) _b ^– (x)d ³ x, wherea andb denote two types of scalar particles. We discuss the scattering theory of this operator.Work partially supported by the Swiss National Science Foundation.This paper contains results from the author's Ph. D. Thesis [3].     

Phase diagram for three-dimensional correlated site-bond percolation

The Coniglio-Stanley-Klein model is a random bond percolation process between the occupied sites of a lattice gas in thermal equilibrium. Our Monte Carlo simulation for 40³ and 60³ simple cubic lattices determines at which bond thresholdp _Bc , as a function of temperatureT and concentrationx of occupied sites, an infinite network of active bonds connects occupied sites. The curvesp _Bc (x, T) depend only slightly onT whereas they cross over if plotted as a function of the field conjugate tox. Except close toT _c we find 1/p _Bc to be approximated well by a linear function ofx, in the whole interval between the thresholdx _c (T) of interacting site percolation atp _Bc =1 and the random bond percolation limitx=1 atp _Bc =0.248±0.001. Thisx _c (T) varied between 0.22 forT=0.96 (coexistence curve) and 0.3117±0.0003 forT= (random site percolation). At the critical point (T=T andx=1/2) we confirmed quite accurately the predictionp _Bc =1-exp(–2J/k _B T _c ) of Coniglio and Klein. As a byproduct we found 0.89±0.01 for the critical exponent of the correlation length in random percolation.     

Incorporation of Effects of Diffusion into Advection-Mediated Dispersion in Porous Media

The distribution of solute arrival times, W(t;x), at position x in disordered porous media does not generally follow Gaussian statistics. A previous publication determined W(t;x) in the absence of diffusion from a synthesis of critical path, percolation scaling, and cluster statistics of percolation. In that publication, W(t;x) as obtained from theory, was compared with simulations in the particular case of advective solute transport through a two-dimensional model porous medium at the percolation threshold for various lengths x. The simulations also did not include the effects of diffusion. Our prediction was apparently verified. In the current work we present numerical results related to moments of W(x;t), the spatial solute distribution at arbitrary time, and extend the theory to consider effects of molecular diffusion in an asymptotic sense for large Peclet numbers, P_e. However, results for the scaling of the dispersion coefficient in the range 1<P_e<100 agree with those of other authors, while results for the dispersivity as a function of spatial scale also appear to explain experiment.