On the upper critical dimensions of random spin systems

A set of critical exponent inequalities is proved for a large class of classical random spin systems. The inequalities imply rigorous (and probably the optimal) lower bounds for the upper critical dimensions, i.e.,d _u4 for regular and random ferromagnets,d _u6 for spin glasses and random field systems.     

Comparison and disjoint-occurrence inequalities for random-cluster models

A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed elsewhere in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical pointp _c (q) of the random-cluster model with cluster-weighting factorq (1) is strictly monotone inq. Our second result is a BK inequality for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.     

Characterizations of classical and quantum logics

In classical logic (Boolean algebras) probability systems involving correlations are fully characterized by the system of generalized Bell inequalities. On the other hand, probability systems with pairwise correlations on orthomodular lattices (OML) representing quantum logics are so general that the only inequalities that hold universally are the trivial inequalities 0p _i1, 0p _ijmin {p _i,p _j}. In this paper it is shown that every correlation sequence p=(p₁,...,p _n,...,P _ij,...) satisfying the above inequalities can be represented by a probability measure on an orthomodular latticeL admitting a full set of {0,1}-valued probability measures with the additional property that isL ortho-Arguesian.     

Correlation length and its critical exponent for percolation processes

Some critical exponent inequalities are given involving the correlation length of site percolation processes on ^d. In particular, it is shown thatv2/d, which implies that the critical exponentv cannot take its mean-field value for the three-dimensional percolation processes.     

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.     

On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT _c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x–y| couplings, for which 2/(–1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).     

Generalized quantum spins,coherent states,and lieb inequalities

A mathematical generalization of the concept of quantum spin is constructed in which the role of the symmetry groupO ₃ is replaced byO _v (=2,3,4, ...). The notion of spin direction is replaced by a point on the manifold of oriented planes in ^v . The theory of coherent states is developed, and it is shown that the natural generalizations of Lieb's formulae connecting quantum spins and classical configuration space hold true. This leads to the Lieb inequalities [1] and with it to the limit theorems as the quantum spinl approaches infinity. The critical step in the proofs is the validity of the appropriate generalization of the Wigner-Eckart theorem.This paper is based largely on the Indiana University Ph. D. thesis of the first named author     

Critical temperature for two-dimensional ising ferromagnets with quenched bond disorder

The configuration-averaged free energy of a quenched, random bond Ising model on a square lattice which contains an equal mixture of two types of ferromagnetic bonds J₁ and J₂ is shown to obey the same duality relation as the ordered rectangular model with the same two bond strengths. If the random.system has a single, sharp critical point, the critical temperature T_c must be identical to that of the ordered system, i.e., sinh(2J ₁/kT _c) sinh(2J ₂/kT _c) = 1. Since _c ^(B) = 1/2, we can takeJ ₂ 0 and use Bergstresser-type inequalities to obtain(/dp) exp(–2J ₁/kT_c¦_p=p_c + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.Work supported in part by National Science Foundation Grant No. DMR 76-21703, Office of Naval Research Grant No. N00014-76-C-0106, and National Science Foundation MRL program Grant No. DMR 76-00678.Paper presented at the 37th Yeshiva University Statistical Mechanics Meeting, May 10, 1977.     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

Correlation inequalities are used to show that the two component (²)² model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant.For then-componentP(||²)₂+₁ model, absence of continuous symmetry breaking, as goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN _loc and higher order estimates.     

Almost Sure Weak Convergence for the Generalized Orthogonal Ensemble

The generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the Gaussian isoperimetric inequality, and an analogue of Wigner's law. Let v be a continuous and even real function such that V(X)=tracev(X)/n defines a uniformly p-convex function on the real symmetric n×n matrices X for some p2. Then (dX)=e ^–V(X) dX/Z satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure^{(6, 27)}, but for the Schatten c ^p norm. The map, that associates to each XM ^s _n () its ordered eigenvalue sequence, induces from a measure which satisfies similar inequalities. It follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non-random compactly supported probability distribution as n.     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Correlation inequalities for two-component hypercubicϕ 4 models

A collection of new and already known correlation inequalities is found for a family of two-component hypercubic ⁴ models, using techniques of duplicated variables, rotated correlation inequalities, and random walk representation. Among the interesting new inequalities are: rotated very special Dunlop-Newman inequality _1,x ² ; _1,z ² + _2g ² 0, rotated Griffiths I inequality _{1,x 1,y} ; _1z ² 0, and anti-Lebowitz inequalityu ₄ ¹¹¹¹ >-0.     

Critical exponents and elementary particles

Particles are shown to exist for a.e. value of the mass in single phase ⁴ lattice and continuum field theories and nearest neighbor Ising models. The particles occur in the form of poles at imaginary (Minkowski) momenta of the Fourier transformed two point function. The new inequalitydm ²/dZ, where =m ₀ ² is a bare mass² andZ is the strength of the particle pole, is basic to our method. This inequality implies inequalities for critical exponents.Supported in part by the National Science Foundation under grant PHY 76-17191Supported in part by the National Science Foundation under grant MPS 75-21212     

On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.     

New inequalities from local realism

The probabilistic formulation of local realism is shown to imply the existence of physically meaningful limits for arbitrary linear combinations of joint probabilities. The set of the so generated inequalities (setA) is wider than the previously known set of inequalities for linear combinations of correlation functions (setB). One particular inequality of the setA is shown to be violated by the probabilities of the Garg-Mermin model. The same model satisfies instead all the inequalities of the setB. As a consequence, the Garg-Mermin model is nonlocal and the setA provides physical restrictions not contained in the setB. 1. In the adopted formalism it is implicitly assumed that physical properties of the type are not created in the act of measurement. IfB(b) is measured on the systems, the setT is split into two parts,T(b _±), corresponding to the resultsB(b) = ±1, respectively. AlsoS is split intoS(b _±) from the existing correlation between and systems. If it is possible to predict that a measurement ofA(a) on the's of, say,S(b ₊) will give the results ±1 with respective probabilitiesP _±, then, on the basis of the probabilistic criterion of reality, we can attribute a physical property ₊ toS(b ₊) such that p(a ₊, ₊) is the probability ofA(a) = +1 inS(b ₊), p(a _–, ₊) is the probability ofA(a) = –1 inS(b ₊).It is natural to assume that ₊ belongs toS(b ₊) also ifA(a) isnot measured. In so doing, we exclude that future measurements create, with a retroaction in time, the physical properties of the statistical ensembles on which these measurements are performed.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

On the equivalence of different order parameters and coexistence of phases for Ising ferromagnet. II

We investigate Ising spin systems with general ferromagnetic, translation invariant interactions,H=–J _BB,J _B0. We show that the critical temperatureT _i for the order parameterp _i defined as the temperature below whichp _i>0, is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the equivalent correlation functions have the same critical exponents asT T_i from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperatureT at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R.J. Baxter et al. concerning the equality of some critical exponents.Supported by NSF Grant PHY 77-22302     

Specific heat of Eu x Sr1−x Te

In this paper, we report on measurements of the specific heatC of single-crystalline Eu _x Sr_1–x Te at temperatures between 60 mK and 15 K and in magnetic fields up to 6 T. Pure antiferromagnetic EuTe shows unusual critical behavior in the vicinity of the Néel temperatureT _N=9.8 K with a positive critical exponent instead of the 3d-Heisenberg exponent =–0.12. Possible reasons for this discrepancy between theory and experiment include magnetic anisotropy effects due to magnetic dipole-dipole interactions, which may give rise to a cross-over of the critical behavior very close toT _N. This anisotropy is also seen in the specific heat below 1 K where an exponential decay ofC is observed, and in the dependence of the magnetic susceptibility on the direction of the applied field. With increasing dilution of EuTe with nonmagnetic Sr, the critical behavior changes: becomes negative and decreases continuously towards –1 atxx _c. This concentration dependence of was previously observed in the diluted ferromagnetic system Eu _x Sr_1–x S. Our data thus support that the apparent change in the critical behavior depends on the degree of disorder. Samples with concentrationx lower than the critical concentrationx _c reveal spin-glass behavior in the specific heat. In addition, the dependence ofT _N on magnetic fields is discussed. The data yield a normalized magnetic phase boundaryB _c(T)/B_c(T=0) vs.T _N(B)/T_N(B=0) which is independent of concentration.     

High-temperature series expansion for the relaxation times of the two dimensional Ising model

We derive the high temperature series expansions for the two relaxation times of the single spin-flip kinetic Ising model on the square lattice. The series for the linear relaxation time _l is obtained with 20 non-trivial terms, and the analysis yields 2.183±0.005 as the value of the critical exponent _l , which is equal to the dynamical critical exponentz in the two-dimensional case. For the non-linear relaxation time we obtain 15 non-trivial terms, and the analysis leads to the results _nl = 2.08 ± 0.07. The scaling relation _l – _nl = ( being the exponent of the order parameter) seems to be fulfilled, though the error bars of _nl are still quite substantial. In addition, we obtain the series expansion of the linear relaxation time on the honeycomb lattice with 22 non-trivial terms. The result for the critical exponent is close to the value obtained on the square lattice, which is expected from universality.