Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang–Swendsen–Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length 5000; the data are consistent with ()=Ae ^{2 p} (1+a ₁ e ^– + ...) as , with p1. The staggered susceptibility behaves as _stagg ^5/3. For q=4 the model is disordered (2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.     

Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperature _t , where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities at _t of the magnetization and of the mass gap, with the magnetization vanishing below _t and the mass gap vanishing above _t . We also show that the surface tensions between ordered stable phases are strictly positive up to _t , and the surface tension between an ordered phase and the disordered one is strictly positive at _t . For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).On leave from ENS Rabat, Morocco.     

Calculation of the force-dipole tensor for hydrogen in palladium and platinum

The force-dipole tensorP , also known as the elastic dipole and the double-force tensor, is calculated for hydrogen dissolved in palladium and platinum, using a microscopic model for the interaction potentials; the so-called effective medium theory. The force-dipole tensor describes the long range displacement field induced by hydrogen dissolved in the host metal lattice in the dilute limit. It can be related to the mean elastic hydrogen-hydrogen interaction energy and the critical temperatureT _c for the gas-liquid phase transition observed in systems such as PdH _x and NbH _x . Comparison show a fair agreement between theoretical and experimental values for the force-dipole tensor.     

Calculation of characteristics of benzene derivatives

The C₆H₅X compounds are considered as regards the energy and wave function _x as functions of within limits of –3 and +3, and also as functions of _cx within limits of 0. 5 and 1. 5. Convenient numerical tables are compiled.We are indebted to N.A. Prilezhaev and V.I. Danilov for extensive collaboration in this work.     

Gentle perturbations

By introducing a specific type of perturbation,A, in the Hamiltonian, we define a class of gently perturbed states, ,A, of a canonical ensemble, . The perturbations are chosen so as to preserve a relationship of the form ,A constant ×. Applications in ergodic theory and phase transitions are described.     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

Spatial fluctuations in reaction-limited aggregation

A general method is used for describing reaction-diffusion systems, namely van Kampen's method of compounding moments, to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areK _ij=ij (i, j=1, 2,). For thediffusion constants D _j of clusters of sizej the following class of models is considered:D _j=D if 1Js andD _j=0 ifj>s. The casess= ands< are studied separately. For the models= the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs< the density fluctuations remain finite att _c, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isd _c=2.     

The lanthanideβ″ aluminas

The entire sodium ion content of sodium alumina (Na_1.67Mg_0.67Al_10.33O₁₇) can be replaced with a variety of lanthanide ions by simple diffusion reactions at moderate temperatures (500–700°C). Lanthanide alumina crystals are hard, clear, chemically stable, and have well-defined crystal structures. The fluorescence spectrum of Nd³⁺ in alumina is similar to that in YAG. The lifetime of the⁴ F _3/2 state of Nd³⁺ in completely-exchanged alumina (350s at 10²¹ Nd³⁺ cm^–3) is about 45% longer than in YAG (240s at 10²⁰Nd³⁺ cm^–3). The lanthanide aluminas may be of considerable interest as new phosphor and laser host materials.     

Microstructure of fiber-like SiC/Si3N4 composite particulate

The microstructure of fiber-like SiC/Si₃N₄ composite particulate was investigated using high-resolution transmission electron microscopy techniques. The SiC/Si₃N₄ composite particulate consisted of a-SiC core and a -Si₃N₄ outer shell. Two kinds of composite particulate were distinguished when the observed orientation of the SiC core was <110>. In one type of the SiC/Si₃N₄ composite particulate, a crystal relationship of (111)-SiC | (102) -Si₃N₄ and (111)-SiC (114) -Si₃N₄ was identified; in the other type of the SiC/Si₃N₄ composite particulate, a crystal relationship of (111)-SiC (001) -Si₃N₄, and (111)-SiC (101) -Si₃N₄ was observed.     

Finite-Volume Glauber Dynamics in a Small Magnetic Field

We consider Glauber dynamics on a finite cube in d-dimensional lattice (d2), which is associated with basic Ising model at temperature T=1/1 under a magnetic field h > 0. We prove that if the effective magnetic field is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (, h) is so near the phase transition line, (, h); _c < &h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.     

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ _x,y const/¦x–y¦². For translation-invariant systems with well-defined lim _x x ² J _x =J ⁺ (possibly 0 or ) we establish: (1) There is no long-range order at inverse temperatures withJ ⁺1. (2) IfJ ⁺>q, then by sufficiently increasingJ ₁ the spontaneous magnetizationM is made positive. (3) In models with 0<J ⁺< the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM( _c )1/( _c J ⁺)^1/2. (4) For Ising (q=2) models withJ ⁺<, it is noted that the correlation function decays as xy()c()/|x–y|² whenever< _c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.     

Mixing properties,differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature

For the Ising model with nearest neighbour interaction it is shown that the spin correlations _{A B} - _{A B} decrease exponentially asd(A, B) in a pure phase when the temperature is well belowT _c. This is used to prove that the free energyF(,h) is infinitely differentiable in and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(,h).     

Quadratic Lagrangians and the Reissner-Nordström-de Sitter metric

The purpose of this note is to point out that the Einstein-Maxwell equations with cosmological constant can be derived from the quadratic Lagrangians R² and F F . The linear combination R R + ²+k ₂F F leads to field equations not satisfied by the Reissner-Nordström-de Sitter metric.     

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.     

Lattice vibrations in ordered and disordered thiourea

The critical dynamics of the Syozi model for dilute ferromagnetism is considered by the use of master equations. The dynamics is soluble as it is assumed that the time scale of motion on the sublattice on which the impurities move is so much faster than on the other sublattice that fast relaxing variables may be adiabatically eliminated, leaving a new soluble master equation. It is found that the linear and non-linear relaxation of magnetization exponents ^(l) and ^(nl) increase on dilution to ^(l)/(1–) and ^(nl)/(1–) respectively ( is the specific heat exponent for the pure system, which itself changes on dilution to –/(1–)). Thus if the exponents for the pure system obey the scaling law of Rácz and Fisher ^(nl)= ^(l)– ( is the magnetization exponent which changes on dilution to /(1–)) then so do the exponents for the diluted system. Similarly the exponent for spin diffusion changes on dilution to /(1–).     

Experimental investigation of the nonlinear properties of the surface helical instability in germanium

The results of an experimental investigation of the parameters of the surface helical instability (SHI) in germanium are presented. The dependence of the amplitude of the current oscillations of the helical wave and frequency on the electric E and magnetic B fields with E and B significantly exceeding the threshold values E_t and B_t are presented for the case of nonequilibrium (for temperatures T < 343°K) and equilibrium (T 343°K) plasma. It is shown that for small supercriticality (_E = (E-E_t)/Et 1, = ( -_t)/_t 1) > the square of the amplitude of the current oscillations and the frequency shift are proportional to the supercriticality, i.e., the regime for excitation of SHI is soft.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 18–21, May, 1981.     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941     

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.     

Additivity of the entropy and definition of the temperature for quantum systems

A closed quantum systemL is considered which is described by a microcanonical ensemble.L consists of two rather weakly interacting subsystemsL ₁,L ₂. In a rigorous way, the additivity of the entropy is proved by deriving an expression for the entropy density ofL in terms of the entropy densities ofL ₁ andL ₂. Rigorous implies that the thermodynamic limit is taken. In the second part, it is shown how a microcanonical state of the composite system — provided this limit exists — gives rise to a canonical state , when restricted toL ₁, providedL ₁ is very small as compared toL ₂; is defined as a limit of Gibbs states. This yields a definition of the equilibrium temperature ^–1.On leave of absence from the Institut für Theoretische Physik, Universität Göttingen.