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arXiv:1106.3424 · Journal of Statistical Mechanics Theory and Experiment

Exact spin–spin correlation function for the zero-temperature random-field Ising model

T P Handford, F-J Perez-Reche, S N Taraskin

Published Jan 10, 2012• 7 citations
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Abstract

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.

DOI
10.1088/1742-5468/2012/01/p01001

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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 general behaviour of the correlation length for the two-point correlation function of the local fields in an Ising chain with binary distributed fields. At zero field it is shown that is the same as the zero-field correlation length for the spin - spin correlation function. For the field-dominated behaviour of we find an exponent for the power-law divergence which is smaller than the exponent for the spin - spin correlation length. The entire behaviour of the correlation length can be described by a single crossover scaling function involving the new critical exponent.
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.
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.
A high-temperature expansion of the partition function for a lattice of N spins with Hamiltonian H=−J ∑ (ij)σizσjz−mHz ∑ iσiz−mHx ∑ iσixis derived and thence an expansion for the zero-field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm2/q | J |, where q is the coordination number of the lattice. The perpendicular susceptibility is re-expressed in terms of near neighbor pair and higher-order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.
We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where ttt is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-ttt behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.