The Spectral Gap of the 2-D Stochastic Ising Model with Mixed Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l×l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by l, where <1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on ), the spectral gap decreases exponentially in l.     

The Spectral Gap of the 2-D Stochastic Ising Model with Nearly Single-Spin Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N×N box, with boundary conditions which are plus except for small regions at the corners which are either free or minus. The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order logN. This means that removing as few as O(logN) plus spins from the corners produces a spectral gap far smaller than the order N ^–2 gap believed to hold under the all-plus boundary condition. Our results are valid at all subcritical temperatures.     

Absence of first-order phase transitions for antiferromagnetic systems

We consider a spin system with nearest-neighbor antiferromagnetic pair interactions in a two-dimensional lattice. We prove that the free energy of this system is differentiable with respect to the uniform external fieldh, for all temperatures and allh. This implies the absence of a first-order phase transition in this system.     

Restrictions on the phase diagrams for a large class of multisite interaction spin systems

Through the proof of two very general theorems involving Ising spin systems with multisite interactions, specific regions of the complexh plane, whereh is the external magnetic field, are shown to be free of zeros of the partition function. Hence in these regions the partition function is analytic and phase transitions are absent. As an example: for systems with ferromagnetic multisite interactions involving even numbers of sites, no phase transition occurs outside of an interval centered on the origin of the realh axis and of the form (–C(T),C(T)), whereT is the temperature. For T0,C(T)0 and phase transitions can occur only ath=0.     

A Constructive Description of Ground States and Gibbs Measures for Ising Model with Two-Step Interactions on Cayley Tree

We consider the Ising model with (competing) two-step interactions and spin values ± 1, on a Cayley tree of order k ≥ 1. We constructively describe ground states and verify the Peierls condition for the model. We define notion of a contour for the model on the Cayley tree. Using a contour argument we show the existence of two different Gibbs measures.     

Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems

We consider a two-dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice. The configuration space of the spin system is a direct product of mixing subshifts of finite type. The potential is defined on the set of all squares in Z² and decays exponentially with the linear size of the square. Via the polymer expansion technique we prove that for sufficiently high temperatures the limit Gibbs distribution is unique and has an exponential decay of correlations.     

Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model

In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube ((B/h) ash0 for variousB. We find a critical balance at a valueB ₀ ofB.     

The spin dynamics of the random transverse Ising chain with a double-Gaussian disorder

The dynamical properties of one-dimensional random transverse Ising model （RTIM） with a double-Gaussian disorder is investigated by the recursion method. Based on the first twelve recurrences derived analytically, the spin autocorrelation function （SAF） and associated spectral density at high temperature were obtained numerically. Our results indicate that when the standard deviation σg （or OrB） of the exchange couplings Ji （or the random transverse fields Bi） is small, no long-time tail appears in the SAE The spin system undergoes a crossover from a central-peak behavior to a collectivemode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when σJ （or σB） is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large σJ or a disordered behavior for large σB. In this instance, SAFs exhibit a similar long-time tail, i.e., C（t） ~ t ^-2 for large t. Similar properties are obtained when Ji （or Bi） satisfy the double-exponential distribution or the double-uniform distribution. Besides, when both the standard deviations and the mean values of the exchange couplings are small, the effects of the Gaussian random bonds may drive the system undergo two crossovers from a triplet state to a doublet state, and then to a collective-mode state.     

Partition function zeros for aperiodic systems

The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.     

Transfer matrix spectrum for the finite-width Ising model with adjustable boundary conditions: Exact solution

Using the spinor approach, we calculate exactly the complete spectrum of the transfer matrix for the finite-width, planar Ising model with adjustable boundary conditions. Specifically, in order to control the boundary conditions, we consider an Ising model wrapped around the cylinder, and introduce along the axis a seam of defect bonds of variable strength. Depending on the boundary conditions used, the mass gap is found to vanish algebraically or exponentially with the size of the system. These results are compared with recent numerical simulations, and with random-walk and capillary-wave arguments.     

Existence of the thermodynamic limit in nonequilibrium systems

The existence of a thermodynamic limit in nonequilibrium stochastic and quantal systems is proven for finite-range interactions and macrovariables which are bounded in the sense of norm. This condition is easily confirmed to be satisfied for specific models, such as the kinetic Ising model and quantal spin systems.Partially financed by Japanese Scientific Research Fund of the Ministry of Education.     

Thermodynamic calculation of spin scaling functions

Critical phenomena theory centers on the scaled thermodynamic potential per spin $?(\beta ,h)=|t{|}^{p}Y(h|t{|}^{?q})$, with inverse temperature $\beta =1/T$, $h=?\beta H$, ordering field H, reduced temperature $t=t(\beta )$, critical exponents p and q, and function $Y(z)$ of $z=h|t{|}^{?q}$. I discuss calculating $Y(z)$ with the information geometry of thermodynamics. Scaled solutions are found to obtain with three admissible functions $t(\beta )$: 1) $t=e^{?J\beta }$, 2) $t={\beta }^{?1}$, and 3) $t={\beta }_C?\beta$, where J and ${\beta }_C$ are constants. For $p=q$, information geometry yields $Y(z)=\sqrt{1+z^2}$, consistent with the one-dimensional (1D) ferromagnetic Ising model.     

Spectral Analysis of a Stochastic Ising Model in Continuum

We consider an equilibrium stochastic dynamics of spatial spin systems in ℝ ^d involving both a birth-and-death dynamics and a spin flip dynamics as well. Using a general approach to the spectral analysis of corresponding Markov generator, we estimate the spectral gap and construct one-particle invariant subspaces for the generator. Dedicated to our admired teacher and friend Robert Minlos on occasion of his 75th birthday. The financial support of SFB-701, Bielefeld University, is gratefully acknowledged. The work is partially supported by RFBR grant 05-01-00449, Scientific School grant 934.2003.1, CRDF grant RUM1-2693-MO-05.     

A measure of statistical complexity based on predictive information with application to finite spin systems

We propose the binding information as an information theoretic measure of complexity between multiple random variables, such as those found in the Ising or Potts models of interacting spins, and compare it with several previously proposed measures of statistical complexity, including excess entropy, Bialek et al.?s predictive information, and the multi-information. We discuss and prove some of the properties of binding information, particularly in relation to multi-information and entropy, and show that, in the case of binary random variables, the processes which maximise binding information are the ‘parity’ processes. The computation of binding information is demonstrated on Ising models of finite spin systems, showing that various upper and lower bounds are respected and also that there is a strong relationship between the introduction of high-order interactions and an increase of binding-information. Finally we discuss some of the implications this has for the use of the binding information as a measure of complexity.     

A heuristic theory of the spin glass phase

We study the low-temperature phase of the nearest-neighbor Ising spin glass. Our analysis of gauge-invariant ground state Peierls contours suggests the existence of infinitely many disjoint Gibbs states at low temperatures, provided the dimension,d, is sufficiently large (presumablyd> 3 or 4), while ford=2 the Gibbs state is unique for all temperatures. Ind 3 we present arguments supporting the existence of a massless phase with broken spin-flip symmetry at low temperatures.     

Critical indices for systems of different space dimensionality

Equations for calculation of correlation length critical indices and the other critical indices for systems of different space dimensionality have been derived on the basis of the fluctuation theory of phase transition (FTPT) and inequalities for values of the critical indices. The values of the critical indices for two-, three-, four-spatial dimensionalities have been calculated by using these equations. The results obtained have a good agreement with classical theory of critical phenomena for d=4, three-dimensional Ising model (d=3), and satisfy the Onsager solution for two-dimensional Ising model (d=2).     

A stochastic model for the dynamics of a classical spin

A stochastic model for the dynamics of a macroscopic or classical spin based on a classical generalized Lagrangian formalism is proposed. The model can be used to describe the evolution of the magnetic moment of superparamagnetic particles. In this sense, it is a generalization of the model proposed by Brown, allowing for fluctuations on the magnitudes of the magnetic moments of the particles. The corresponding covariant Fokker-Planck equation is also obtained.     

On the purity of the limiting gibbs state for the Ising model on the Bethe lattice

We give a proof that for the Ising model on the Bethe lattice, the limiting Gibbs state with zero effective field (disordered state) persists to be pure for temperature below the ferromagnetic critical temperatureT _c ^F until the critical temperatureT _c ^SG of the corresponding spin-glass model. This new proof revises the one proposed earlier.     

Simulation of the Ising model,memory for Bell states and generation of four-atom entangled states in cavity QED

A scheme is proposed to simulate the Ising model and preserve the maximum entangled states (Bell states) in cavity quantum electrodynamics (QED) driven by a classical field with large detuning. In the strong driving and large-detuning regime, the effective Hamiltonian of the system is the same as the standard Ising model, and the scheme can also make the initial four Bell states of two atoms at the maximum entanglement all the time. So it is a simple memory for the maximal entangled states. The system is insensitive to the cavity decay and the thermal field and more immune to decoherence. These advantages can warrant the experimental feasibility of the current scheme. Furthermore, the genuine four-atom entanglement may be acquired via two Bell states through one-step implementation on four two-level atoms in the strong-driven model, and when two Greenberger-Horne-Zeilinger (GHZ) states are prepared in our scheme, the entangled cluster state may be acquired easily. The success probability for the scheme is 1. Supported by the National Natural Science Foundation of China (Grant No. 10774088) and the Key Program of the National Natural Science Foundation of China (Grant No. 10534030)