On the thermodynamics of the random one-dimensional Ising chain in a transverse field (original) (raw)

Dynamics of the random one-dimensional transverse Ising model

Physical Review B, 1999

We study the dynamics of the spin-1/2 random transverse Ising model in the high-temperature limit by means of the method of recurrence relations. We analyze two types of disorder: a disorder on the transverse field, and a disorder on the exchange coupling. We find that the dynamics undergoes a crossover from a central peak behavior onto a collective mode behavior as a function of the disorder. ͓S0163-1829͑99͒04937-1͔ PHYSICAL REVIEW B 1 OCTOBER 1999-I VOLUME 60, NUMBER 13 PRB 60 0163-1829/99/60͑13͒/9555͑6͒/$15.00 9555

Transverse-field Ising spin chain with inhomogeneous disorder

We consider the critical and off-critical properties at the boundary of the random transversefield Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance l from the surface, deviates from its uniform bulk value by terms of order l −κ with an amplitude A. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, κ < 1/2, the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of A. In the marginal situation, κ = 1/2, the average surface magnetization decays as a power law with a continuously varying, A-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions. 05.50.+q, 64.60.Fr, 68.35.Rh

Spectral Analysis of the Disordered Stochastic¶1-D Ising Model

Communications in Mathematical Physics, 1999

We consider the generator of the Glauber dynamics for a 1-D Ising model with random bounded potential at any temperature. We prove that for any realization of the potential the spectrum of the generator is the union of separate branches (so-called k-particle branches, k = 0; 1; 2; :::), and with probability one it is a nonrandom set. We nd the location of the spectrum and prove the localization for the one-particle branch of the spectrum.

Strongly coupled Ising chain under a weak random field

Physica A: Statistical Mechanics and its Applications, 1993

We present here an analytical method based on the transfer matrix M to study quenched disorder in one-dimensional spin systems in the limit of strong couplings and weak disorder. The procedure is formulated for the random-field Ising chain of finite length L, and its properties, represented as functions of M, satisfy a differential equation of the Fokker-Planck type. This equation describes "evolution" with L, and a central-limit theorem of a novel kind provides the equation and its solutions with universal character. We obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit. We find that the random-field free energy fn and the Edwards-Anderson order parameter m 2 satisfy a simple relation. We discuss our results in connection to previous work by Luck and Nieuwenhuizen.

Dynamical properties of random-field Ising model

Physical Review E, 2013

Extensive Monte Carlo simulations are performed on a two-dimensional random field Ising model. The purpose of the present work is to study the disorder-induced changes in the properties of disordered spin systems. The time evolution of the domain growth, the order parameter and the spin-spin correlation functions are studied in the non equilibrium regime. The dynamical evolution of the order parameter and the domain growth shows a power law scaling with disorder-dependent exponents. It is observed that for weak random fields, the two dimensional random field Ising model possesses long range order. Except for weak disorder, exchange interaction never wins over pinning interaction to establish long range order in the system.

The 3d random field Ising model at zero temperature

Europhysics Letters (EPL), 1997

We study numerically the zero temperature Random Field Ising Model on cubic lattices of various linear sizes L in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength J. We find that in the infinite volume limit the magnetization is discontinuous in J. The energy and its first J derivative are continuous. The approch to the thermodynamic limit is slow, behaving like L −p with p ∼ .8 for the gaussian distribution of the random field. We also study the bimodal distribution hi = ±h, and we find similar results for the magnetization but with a different value of the exponent p ∼ .6. This raises the question of the validity of universality for the random field problem.

Localization in the ground state of the ising model with a random transverse field

Communications in Mathematical Physics, 1991

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by H=-J Σ σ,(x)σ,(y) <*,y> where J > 0, x, yeZ d , σ 1? σ 3 are the usual Pauli spin ^matrices, and h = (/z(x), are independent identically distributed random variables. We consider the ground state correlation function <σ 3 (x)σ 3 (-y)> and prove: 1. Let d be arbitrary. For any m > 0 and J sufficiently small we have, for almost every choice of the random transverse field h and every xeZ d , that for all yeZ d with C xh < oo. 2. Let d ^ 2. If J is sufficiently large, then, for almost every choice of the random transverse field h, the model exhibits long range order, i.e., Iim o <σ 3 (x)σ 3 ();)>>0 for any xeZ d .

The Lifshitz Tail and Relaxation to Equilibrium in the One-Dimensional Disordered Ising Model

2000

We study spectral properties of the generator of the Glauber dynamics for a 1D disordered stochastic Ising model with random bounded couplings. By an explicit representation for the upper branch of the generator we get an asymptotic formula for the integrated density of states of the generator near the upper edge of the spectrum. This asymptotic behavior appears to have the form of the Lifshitz tail, which is typical for random operators near fluctuation boundaries. As a consequence we find the asymptotics for the average over the disorder of the time-autocorrelation function to be ((_ | 0 (t), _ 0 (0)) P(|)) | =exp[& gt&kt 1Â3 (1+o(1))] as t Ä with constants g, k depending on the distribution of the random couplings.

Temperature-dependent criticality in random 2D Ising models

The European Physical Journal Plus, 2021

We consider 2D random Ising ferromagnetic models, where quenched disorder is represented either by random local magnetic fields (random-field Ising model) or by a random distribution of interaction couplings (random-bond Ising model). In both cases, we first perform zero- and finite-temperature Monte Carlo simulations to determine how the critical temperature depends on the disorder parameter. We then focus on the reversal transition triggered by an external field and study the associated Barkhausen noise. Our main result is that the critical exponents characterizing the power law associated with the Barkhausen noise exhibit a temperature dependence in line with existing experimental observations.

The random field Ising model in one and two dimensions: A renormalization group approach

Physica A: Statistical Mechanics and its Applications, 1990

We study tile one-and two-dimensional random field [sing models, using a real space renormalization group approach. We consider a bimodai distribution such that the random field assumes the values of + H or -H with probabilities p and 1 -p, respectively (instead of the usual case p = t_,). We obtain the phase diagrams and exponents associated with the uniform (p = 0, 1) and the random field ( p = t ) problems. Our results are consistent with the absence of a spontaneous magnetization for H ~0 and p #0, 1 in d = 1.2 even at zero temperature. We finally discuss the nature of the singularities in the thermodynamics quantities occurring at T= 0 for discrete values of the random field intensity. Wc compare these results with tho~e obtained previously for the dalutc anttferromagnet in a umtorm field using the same approach.