Infinite differentiability for one-dimensional spin system with long range random interaction

We consider one-dimensional spin systems with Hamiltonian: $$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$ , where ? _tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α>3/2 andE(? _tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ^∞ both inh and β. Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ? _tt′ 's, where the quenched magnetization isC ^∞ but fails to be analytic inh for suitableh and β.     

Exact renormalization transformation for the one-dimensional ising model with finite range interaction

The transfer matrix formalism is employed in order to construct the exact renormalization transformation for the one-dimensional Ising model with finite range interaction. The method is applied to the nearest-neighbour model and to the next nearest-neighbour model.     

Classical one-dimensional Heisenberg model with an interaction of finite range

It is shown that the thermodynamic quantities and spin correlation functions of the classical Heisenberg model on a linear chain are expressed in terms of the eigenvalue with the smallest absolute value and the corresponding eigenfunction of a homogeneous linear integral equation, where the range of the interaction is assumed to be finite. The magnetization and susceptibility at nonzero external magnetic fields are given as a function of temperature, for the case of the nearest neighbour ferromagnetic and antiferromagnetic interaction. Efforts are paid to determine the properties near zero temperature.     

Strong cluster properties for classical systems with finite range interaction

In a previous paper, “strong” decrease properties of the truncated correlation functions, taking into account the separation of all particles with respect to each other, have been presented and discussed. In this paper, we prove these properties for finite range interactions in various situations, in particular

1. at low activity for lattice and continuous systems,
2. at arbitrary activity and high temperature for lattice systems,
3. at ReH≠0, β arbitrary and atH=0 for appropriate temperatures in the case of ferromagnets.

We also give some general results, in particular an equivalence, on the links between analyticity and strong cluster properties of the truncated correlation functions.     

Dynamics of the one-dimensional random transverse Ising model with next-nearest-neighbor interactions

The dynamics of the one-dimensional random transverse Ising model with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions is studied in the high-temperature limit by the method of recurrence relations. Both the time-dependent transverse correlation function and the corresponding spectral density are calculated for two typical disordered states. We find that for the case of bimodal disorder the dynamics of the system undergoes a crossover from a collective-mode behavior to a central-peak one and for the case of Gaussian disorder the dynamics is complex. For both cases, it is found that the central-peak behavior becomes more obvious and the collective-mode behavior becomes weaker as K_i increase, especially when K_i>J_i/2 (J_i and K_i are the exchange couplings of the NN and NNN interactions, respectively). However, the effects are small when the NNN interactions are weak (K_i<J_i/2).     

Low-temperature behavior of a one-dimensional random Ising model

The random Ising chain is a very simple model with a large number of metastable states. Simple analytical calculation of the relaxation of energy and magnetization is presented. The effect of a nonzero magnetic field is discussed qualitatively. The slow relaxation in this simple model resembles that observed in spin glasses. A weak magnetic field can produce rather strong effects. The magnetization is shown to be a nonanalytic function of the field. The field also greatly alters the metastability characteristics.     

Hydrodynamic limit for a system with finite range interactions

We study a system of interacting diffusions. The variables present the amount of charge at various sites of a periodic multidimensional lattice. The equilibrium states of the diffusion are canonical Gibbs measures of a given finite range interaction. Under an appropriate scaling of lattice spacing and time, we derive the hydrodynamic limit for the evolution of the macroscopic charge density.     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Metastability in a long range one-dimensional Ising model

Decay of the metastable state of the one-dimensional Ising model with an x^-α potential is investigated using instanton (critical droplet) techniques. Calculations indicating that the analytic structure of the free energy is modified by droplet-droplet interactions for 1 < α ? 1.2 are presented.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Spectral self-similarity,one-dimensional Ising chains and random matrices

Partition functions of one-dimensional Ising chains with specific long distance exchange between N spins are connected to the N-soliton τ-functions of the Korteweg-de Vries (KdV) and B-type Kadomtsev-Petviashvili (BKP) integrable equations. The condition of translational invariance of the spin lattice selects infinite-soliton solutions with soliton amplitudes forming a number of geometric progressions. The KdV equation generates a spin chain with exponentially decaying antiferromagnetic exchange. The BKP case is richer. It comprises both ferromagnets and anti ferromagnets and, as a special case, includes an exchange decaying as 1/(i − j)² for large |i − j|. The corresponding partition functions are calculated exactly for a homogeneous magnetic field and some fixed values of the temperature. The connection between these Ising chains and random matrix models is considered as well. A short account of the basic ideas underlying the present work has been published in JETP Lett. 66 (1997) 789.     

Strong-coupling limit for one-dimensional polarons in a finite box

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n ² discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.     

Relaxation times in a finite Ising system with random impurities

Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM ²(t). In addition, universal cumulant ratios are calculated to second order in ^1/4, where =4–d andd<4 denotes the spatial dimension.     

Rigorous bounds of the Lyapunov exponents of the one-dimensional random Ising model

We find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations.     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

Diffusion in random one-dimensional systems

Diffusion on the one-dimensional lattice is described by a master equation with nearest-neighbor transfer rates (symmetric or asymmetric). The transfer rates associated with bonds are assumed to be independent, equally distributed random variables. Under various conditions on their common distribution the large time behavior of averaged site probabilities and/or related quantities is exhibited.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Infinite-range Ising ferromagnet: Thermodynamic limit within Generalized Statistical Mechanics

We first discuss, for a variety of similar systems, the physical need for departure from Boltzmann-Gibbs statistical mechanics and thermodynamics. Then, we numerically discuss the infinite-range spin-1/2 Ising ferromagnet within the recently generalized statistical mechanics (canonical ensemble). Through the specific heat, we exhibit (for the first time, as far as we know, for an interacting system) that the thermodynamic limit is well defined.     

Velocity correlation functions for finite one-dimensional systems

Certain systems consisting of a one-dimensional gas of a finite number of point particles interacting with a “hard-core” potential are considered.We use the technique developed by Jepsen to calculate exactly the velocity correlation functions for these systems. We discover that after a slow decay for times of the order of the relaxation time, there is a “fast” decay to the equilibrium value on a macroscopic time scale characterized by L/v_TH (L is the length of the container and v_TH the thermal velocity).The dependence of the velocity correlation functions on the initial position of the specified particle is also considered. In particular, the behaviour approaching the boundary of the container is analyzed. These considerations are generalized to systems of higher spatial dimension.