On the Swendsen-Wang dynamics. II. Critical droplets and homogeneous nucleation at low temperature for the two-dimensional Ising model

We consider the Swendsen-Wang dynamics for the two-dimensional Ising model at low temperature in the presence of a small negative magnetic fieldh and with plus boundary conditions at the boundary of an arbitrarily large square. We analyze in detail the tunneling from the metastable phase to the stable one. In particular, we obtain an upper bound on the tunneling timet by explicitly constructing paths in the space of spin configurations that drive the system from the metastable phase to the stable one. In our analysis the transition takes place through the formation of droplets of the right phase inside the wrong one with side greater than a certain critical valuel _c . The values of the tunneling time and ofl _c coincide with those found for a single-spin-flip dynamics in finite volume by Jordao-Neves and Schonmann.     

Probability distribution of random paths in the Ising model at low temperature

Communications in Mathematical Physics - The random walk representation of then-dimensional Ising model exhibits the 2-point correlation function 〈σ(x) σ(y)〉 as a sum of...     

Approximate calculations for the two-dimensional Ising model

A self-consistent molecular field approximation for the two-dimensional, square-lattice Ising model is used to calculate the energy and magnetization. Agreement with the exact calculations is good except near the critical temperature, which differs from the exact critical temperature by 4%. The specific heat has no anomalous behavior asT approachesT _c from above, and the magnetization follows the incorrect Weiss (T _c-T)^1/2 law asT approachesT _c from below.     

Boundary conditions and correlation functions in the ν-dimensional Ising model at low temperature

The boundary condition dependence of the correlation functions in a phase transition region of the thermodynamic parameters is of great importance to understand the character and properties of the phase transition itself. In this paper we study the boundary condition dependence of certain correlation functions in the Ising model at low temperature.     

New results for the susceptibility of the two-dimensional Ising model at criticality

We report several new results for the susceptibility of the two-dimensional Ising model in the critical region including the exact evaluation of the constant (first regular term) in the asymptotic expansion near T_c for the ferromagnetic case.     

Debye screening for two-dimensional Coulomb systems at high temperatures

The grand canonical ensemble of a two-dimensional Coulomb system with±1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region is assumed to be (–)^–1, where is the Laplacian with zero boundary conditions on. The hard-core condition is not assumed. The model is set up by separating (–)^–1 into a shortrange part and a long-range part depending on a parameter. The self-energies are subtracted only for the short-range part and therefore a choice of is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.     

Partition function zeros for the two-dimensional Ising model

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation derived from a generalized scaling theory. Corresponding results for the isotropic square, triangular and honeycomb lattices are also presented.     

Ising model on mixed two-dimensional lattices

We inform results on physical and topological magnitudes related to the ground level of Ising model on mixed two-dimensional lattices of coordination numbers 4 (Kagomé lattices) and 5 (five-point star lattices). We consider little clusters of size N, where N represents the total number of spins, subject to periodic boundary conditions. On these systems we randomly distribute ±J nearest-neighbor interactions (+J: antiferromagnetic, −J: ferromagnetic (F)). Concentration x of F interactions is varied in the interval (0,1). Two different methods are used to obtain results reported here. First, a numerical method related to multiple replicas. Second, an analytical method based on probabilistic analysis of flat and curved plaquettes. Both methods are complementary to each other. Initially, this study is restricted to calculate frustration of plaquettes and bonds, energy and bond order parameter at T=0. The results of magnitudes informed here are compared with the similar ones obtained for honeycomb, square and triangular lattices.     

A new rigorous upper bound for the inverse critical temperature of the two-dimensional Coulomb gas

In this paper we show how to improve the recent result _c 17.2 on the inverse critical temperature for the two-dimensional Coulomb gas at low density to get the following upper bound: _c 16.     

Analyticity of correlation functions for the two-dimensional Ising model

Analyticity of correlation functions for the two-dimensional Ising model as a function of the inverse temperature except for the singularity at the critical temperature is proved. A crucial step is the establishment of the correspondence between extremal equilibrium states of the model and pure ground states of a one-dimensional spin system below the critical temperatureT _c . An exact decay rate of the clustering property along axes is also determined for allTT _c .     

An upper bound on the critical temperature for a continuous system with short-range interaction

A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(, z) denotes the thermodynamic pressure at inverse temperature and activityz, then it follows from the Mayer expansion thatp(, z) is infinitely differentiable provided andz are sufficiently small. Here it is shown that there exists ₀>0 such thatp(, z) is infinitely differentiable if< ₀ andz>0. One can interpret this result as saying that ( ₀)^–1 is an upper bound on the critical temperature for the system.     

Heat conduction in a two-dimensional Ising model

A cylindrical Ising model between thermostats is used to explore the heat conduction for any temperature interval. The standard Q2R and Creutz dynamics, previously used by Saito, Takesue and Miyashita, fail below the critical temperature, limiting the analysis to high temperatures intervals. We introduce improved dynamics by removing limitations due to the chessboard-like refresh, and by supplementing the Q2R rule with Kadanoff-Swift moves. These new dynamics not only prove highly efficient in recovering old results in their domains of validity, but also allow exploration of steady heat transport between two arbitrary temperatures, i.e. very far from equilibrium. From an ansatz avoiding references to quasi equilibrium or to local temperature, and from comparison with numerical simulations, we can consistently define a generalized diffusivity. Its dependence on the energy density may be evaluated without any recourse to the Green-Kubo formula.     

Magnetization at corners in two-dimensional Ising models

We investigate the corner spin magnetization of two-dimensional ferromagnetic Ising models in various wedge geometries. Results are obtained for triangular and square lattices by numerical studies on finite wedges using the star-triangle transformation, as well as analytic calculations using corner transfer matrices and a fermionic representation of the row-to-row transfer matrix. The corner magnetizations vanish at the bulk critical temperature T_c with an exponent _c differing from the bulk exponent. For isotropic systems with free edges we find that _c is given simply by _c =/2, where is the angle at the corner. For apex magnetizations of conical lattices we obtain the strikingly similar result _a=/4. These formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle, ^eff, determined by the relative strengths of the interactions.     

A Remark on High Temperature Polymer Expansion for Lattice Systems with Infinite Range Pair Interactions

We consider the standard high temperature-small activity polymer expansion for lattice spin systems and show how, in many cases using a tree graph equality, various classical results may be improved in some cases or at least proved in a much simpler way.     

Single-cluster Monte Carlo dynamics for the Ising model

We present an extensive study of a new Monte Carlo acceleration algorithm introduced by Wolff for the Ising model. It differs from the Swendsen-Wang algorithm by growing and flipping single clusters at a random seed. In general, it is more efficient than Swendsen-Wang dynamics ford>2, giving zero critical slowing down in the upper critical dimension. Monte Carlo simulations give dynamical critical exponentsz _w=0.33±0.05 and 0.44+0.10 ind=2 and 3, respectively, and numbers consistent withz _w=0 ind=4 and mean-field theory. We present scaling arguments which indicate that the Wolff mechanism for decorrelation differs substantially from Swendsen-Wang despite the apparent similarities of the two methods.     

On the analyticity of the pressure in the hierarchical dipole gas

The convergence of the Mayer expansion is proved by estimating directly the convergence radius.     

A cryostat design for radiation damage experiments at low temperature

Abstract

A cryostat for electron irradiations at liquid hydrogen or helium temperatures is described. The cryostat uses two concentric liquid coolant containers, of which the inner reservoir is sealed and acts as a heat exchanger through ebullition and recondensation of the coolant gas. This allows irradiations with a power input from the beam of up to a few watts, and when the sample is thermally isolated allows it to be annealed up to room temperature.