A Note on Exponential Decay in the Random Field Ising Model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate correlations parallel to the layering in the diagonally layered model with periodv=2, the so-called “general square lattice” model (GS). If the model has a finite critical temperature,T _c>0, we have a spontaneous magnetization belowT _c vanishing atT _c with the Ising exponent β=1/8. AtT _c correlations decay algebraically with critical exponnet η=1/4 and exponentially forT>T _c. In the frustrated case we have oscillatory behaviour superposed on the exponential decay where the wavevector of the oscillations changes at some “disorder temperature”T _D(>T _c) from commensurate to temperature-dependent in commensurate periods. If the critical temperature vanishes,T _c=0 we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index η=1/2, i.e.T=0 is thus a critical point.     

Stochastic generalization for a hyperbolic model of spinodal decomposition

A model for diffusion and phase separation which takes into account exponential relaxation of the solute diffusion flux and its fluctuations is developed. The model describes a system undergoing phase separation governed by a partial differential equation of hyperbolic type. The analysis is done for the evolution of patterns in spinodal decomposition for the system supercooled below critical temperature. Analytical results show that relaxation processes of the solute diffusion flux lead to the selection of patterns with different wavenumbers. Considering spatial-temporal correlations of the flux fluctuations, we have found that the temporal correlations promote selecting large-period patterns, whereas the corresponding spatial correlations accelerate such processes.     

Mixing and its rate in ‘soft’ and ‘hard’ billiards motivated by the Lorentz process

Billiards corresponding to planar periodic Lorentz processes are considered in the usual (hard) sense and in the case when the hard core potential of the scatterers is replaced by some other circularly symmetric potential. A review on certain important aspects of the history of the subject is given and some new results on exponential decay of correlations are formulated. Both the results from the literature and those of our own mentioned are mathematically rigorous, nevertheless, proofs are only briefly sketched. For further details, see the preprint [Correlation decay in certain soft billiards, Commun. Math. Phys., in press].     

Decay of correlations in a chaotic measure-preserving transformation

For a chaotic, area-preserving map on the torus, we study the decay of correlations in detail. Taking as observables the square-integrable functions, we find examples of decay rates which are algebraic, exponential, and faster than exponential. For correlations that decay exponentially the rate is sensitive to the choice of function. The implications for numerical experiments of this nonuniformity in the decay are discussed.     

Effect of suddenly turning on interactions in the Luttinger model

The evolution of correlations in the exactly solvable Luttinger model (a model of interacting fermions in one dimension) after a suddenly switched-on interaction is analytically studied. When the model is defined on a finite-size ring, zero-temperature correlations are periodic in time. However, in the thermodynamic limit, the system relaxes algebraically towards a stationary state which is well described, at least for some simple correlation functions, by the generalized Gibbs ensemble recently introduced by Rigol et al. (cond-mat/0604476). The critical exponent that characterizes the decay of the one-particle correlation function is different from the known equilibrium exponents. Experiments for which these results can be relevant are also discussed.     

Critical behavior of three-state Potts model on decagonal covering quasilattice

The three-state Potts model on a 2D decagonal covering quasilattice is investigated by means of the Monte Carlo simulation. The periodic boundary conditions are realized on a rhombus-like covering pattern. By use of the finite-size scaling analysis, we obtain the critical temperature and the critical exponents. The critical temperature is higher than that of the square lattice mainly due to the larger mean coordination number of the covering model. The critical exponents are close to the corresponding values of the 2D periodic lattices, which means that the Potts model on the covering structure may belong to the same universal class as that of the periodic lattices.     

Decay of Correlations and Dispersing Billiards

We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points     

Correlations in inhomogeneous Ising models

We study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant either in horizontal or in diagonal direction, i.e. the models have a layered structure. By using transfer matrix techniques the spin-spin correlations are calculated parallel to the layering and are expressed as Toeplitz determinants. After working out the general methods we discuss two special examples in detail: the fully frustrated square lattice (FFS) and the chessboard model, both having no phase transition. At zero temperature correlations in the chessboard model decay exponentially, while in the FFS model one has algebraic decay with a critical index =1/2, i.e.T=0 is a critical point. At finite temperature we find exponential decay in both models with a correlation length determined by the excitation gap in the fermion spectrum. Due to frustration correlations may develop on oscillatory structure and spins separated by an odd diagonal distance are totally uncorrelated at all temperatures.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köla     

Local Statistics of Realizable Vertex Models

We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including the dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an n × n torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.     

Beyond Inverse Ising Model: Structure of the Analytical Solution

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.     

Ising models on the pentagon lattice

We solve inhomogeneous Ising models on the pentagon lattice using the transfer matrix formalism. As two special cases we study the ferromagnetic and the fully frustrated antiferromagnetic model on this lattice. The ferromagnet shows a phase transition at someT _c>0 with the usual Ising behaviour. In the frustrated case no transition occurs at any temperature due to frustration. Frustration also causes a nonvanishing rest entropy. We also calculate the spin-spin-correlation for large distance in both cases. In the ferromagnetic model we thus get the magnetization and the expected algebraic (exponential) decay of the correlations at (above)T _c. The correlations of the frustrated model decay exponentially for all temperatures, includingT=0, indicating that evenT=0 belongs to the disordered high temperature phase. Superimposed to the exponential decay the correlation shows an interesting oscillatory behaviour with temperature dependent wave number, i.e. an incommensurate structure.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

A simple way of understanding some exact results on critical phenomena in non-homogeneous and finite Ising systems II: Cooperativity in periodically layered systems

By means of a continuum approximation introduced in a preceding paper, we present a more transparent treatment of the critical phenomena in the specific heat of a two-dimensional Ising model with a “layer” type spatial inhomogeneity of its exchange coupling. This calculational procedure (which will be tested by comparison with an earlier exact treatment by Becker and Hahn) enables us to develop a physical picture of cooperative effects in real (spatially non-homogeneous) systems with quenched impurities. To be more concrete: By examination of the one-particle eigenvalues and eigenfunctions of the transfer matrix, it becomes apparent how the critical behaviour is dominated by i) “local” correlation lengths, describing the spatial decay of correlations within homogeneous subdomains of the system and becoming large near certain temperatures which indicate transitions from disorder to spin alignment within each of these domains, and ii), by the spatial average of the inverse of these local decay lengths, termed “global” reciprocal correlation length, which describes correlations between different subdomains on a scale on which these domains look like fluctuating “block spins”. This “global” decay length diverges at a true phase transition of the system as a whole. Correspondingly, for each sort of subdomain a sharp, but finite logarithmic peak in the specific heat and, as long as the layering of the inhomogeneous Ising system is periodic, a true logarithmic singularity are found.     

The Potts Model Built on Sand

We consider theq = 4 Potts model on the square lattice with an additional nonlocal interaction. That interaction arises from the choice of the reference measure taken to be the uniform measure on the recurrent configurations for the abelian sandpile model. In that reference measure some correlation functions have a power-law decay. We investigate the low-temperature phase diagram and we prove the existence of a single stable phase with exponential decay of correlations. For all boundary conditions the density of 4 in the infinite volume limit goes to one as the temperature tends to zero.     

Analytical properties of the plasmon decay profile in a periodic metal-nanoparticle chain

We show that the spatial decay of plasmons in a periodic metal-nanoparticle chain is composed of exponential and power-law decays. Our results show a high level of similarity between the absorptive and radiative decay channels. By analyzing the poles (and the corresponding residues) of the generating function for the lattice Green's function, we explain the details of the spatial decay profile. We also present an analytical formula for the decay profile.     

研究温度波的传播特性

根据振动与波的原理,把样品上某点温度随时间的周期性变化看作一种振动,把这种温度变化向外传播的过程看作波动,引进温度波,说明样品上各点温度随时间、距离的变化;采用一维模型,写出温度波的传播方程.考虑到样品(铜棒)散热,引进衰减系数,描述温度幅度随频率及传播距离的变化关系.利用傅里叶变换分析实验数据,得到温度波幅度与角频率、位置的对应关系.根据温度幅度衰减公式拟合数据,算出基频及倍频对应的衰减系数.结果说明:衰减系数与温度波的频率相关,温度波的频率越高,衰减系数越大,温度幅度衰减得越快.     

The Abelian Sandpile Model on a Random Binary Tree

We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar and Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution; (2) quenched and annealed estimates of the eigenvalues of a product of n random transfer matrices.     

Coulomb and liquid dimer models in three dimensions

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.     

Eigenfunctions of Unbounded Support for Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

It is known that if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported—that is, if the Fermi surface is irreducible, which occurs generically in dimension two or higher. This article constructs a class of operators whose Fermi surface is reducible for all energies by coupling several periodic systems. The components of the Fermi surface correspond to decoupled spaces of hybrid states, and in certain frequency bands, some components contribute oscillatory hybrid states (corresponding to spectrum) and other components contribute only exponential ones. This separation allows a localized defect to suppress the oscillatory (radiation) modes and retain the exponential ones, thereby leading to embedded eigenvalues whose associated eigenfunctions decay exponentially but are not compactly supported.