Full reduction of large finite random Ising systems by real space renormalization group

Abstract

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain Gamma(r)=<sigma(0)sigma(r)>-<sigma(0)><sigma(r)>, the "connected" correlation function, and S(r)=<sigma(0)sigma(r)>, the "disconnected" correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function Gamma(r) along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such nu or eta). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.

Lack of self-averaging in random systems—Liability or asset?

“…[2] and [19], the signal to noise of the susceptibility remains small anywhere in the 6 ordered phase and is very large in the disordered phase. For general singular thermodynamic quantities, x, at criticality, 7 Aharony & Harris [5] have obtained later, that when the disorder is relevant, the corresponding signal-to-noise ratio, γ x , 8 tends to a constant with increasing system size.…”

“…As long as higher derivatives are avoided, this procedure is limited, however, to thermal averages of spin 36 products which already appear in the original Hamiltonian (such as nearest-neighbor pair products). In the following we 37 will see how averages involving non-surviving spins can also be calculated [19]. 38 We start with a set of N = L 3 Ising spins, with L = 2 n , situated on a three dimensional cubic lattice with periodic boundary 39 conditions.…”

“…In the random field case the situation is quite different. The correlation length is of the order of the 19 linear size of the system everywhere in the ordered phase and not just at the boundary of the phase [2]. 20 We describe next our procedure for obtaining numerically the relevant physical quantities, which is based on 21 Casher-Schwartz RSRG [10].…”

Noise-to-noise ratios in correlation length calculations near criticality

“…Take, for instance, the Random Field Ising Model (RFIM) and the various techniques used over the years for calculating the critical exponent, 𝜈, related to the correlation length, yielding results wideranging from 0.62 to 2.26. (The list of methods includes: Exact Ground State [1][2][3] Domain Wall Renormalization Group simulations [4], Monte Carlo simulations [5][6][7][8][9], Migdal Kadanoff Renormalization Group [7,10,11], Casher Schwartz Renormalization Group [12,13], Modified Dimensional Reduction [14] and experiment [15][16][17][18]).…”

“…In this paper, an effort is made to correct the situation by presenting a new method for calculating the correlation length as a function of temperature, for quenched infinite and rather general, random Ising systems. The correlation length can also be obtained by measuring the correlation function [13], but that requires much larger systems and many more calculations per each realization than the present technique. The reason is that it requires the calculation of the correlation function as a function of distance within each realization.…”