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arXiv:1308.2037 · Physical Review B

Large deviations of correlation functions in random magnets

Flaviano Morone, Giorgio Parisi A, Federico Ricci‐Tersenghi

Published Jun 23, 2014• 14 citations
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Abstract

We present a large deviations theory of the spin-spin correlation functions in the random field Ising model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: The phase transition is driven by a few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero-temperature critical point the number of spin pairs correlated over a distance $\ensuremath{\ell}$ is shown to be no longer exponential, but only linear in the spins separation.

DOI
10.1103/physrevb.89.214202

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We present a large deviations theory of the spin-spin correlation functions in the random field Ising model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: The phase transition is driven by a few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero-temperature critical point the number of spin pairs correlated over a distance \ensuremathℓ\ensuremath{\ell}\ensuremathℓ is shown to be no longer exponential, but only linear in the spins separation.
An exact expression for the spin-spin correlation function is derived for the zero-temperature random-field Ising model defined on a Bethe lattice of arbitrary coordination number.The correlation length describing dynamic spinspin correlations and separated from the intrinsic topological length scale of the Bethe lattice is shown to diverge as a power law at the critical point.The critical exponents governing the behaviour of the correlation length are consistent with the mean-field values found for a hypercubic lattice with dimension greater than the upper critical dimension.
We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra{r}^{a}ra, where rrr is the distance between two lattice sites and a<0a<0a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.
We develop a transfer matrix method to compute exactly the spin-spin correlation functions 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 of Bethe lattice spin models in the external magnetic field hhh and for any temperature T.T.T. We first compute 〈s0sn〉〈{s}_{0}{s}_{n}〉〈s0​sn​〉 for the most general spin-SSS Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-SSS Ising model includes the spin-12\frac{1}{2}21​ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and TTT approaches the critical temperature Tc.{T}_{c}.Tc​. Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d≤2d\le 2d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).