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Physical review. B, Condensed matter

Critical behavior in systems with a long-range correlated frozen-in random field

Demetris Nicolaides

Published Jan 1, 2000• 1 citation
Publisher

Abstract

The critical behavior of inhomogeneous systems with a frozen-in random field having a nonlocal correlation function decaying according to the power law $|\mathbf{x}\ensuremath{-}{\mathbf{x}}^{\ensuremath{'}}{|}^{\ensuremath{-}b}$ is considered. The problem is studied within the context of a model which partially considers interactions of fluctuations and belongs in the same universality class as the spherical model. Depending on the relationship between the power b and the space dimensionality d, a new critical behavior arises.

DOI
10.1103/physrevb.61.246

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The critical behavior of inhomogeneous systems with a frozen-in random field having a nonlocal correlation function decaying according to the power law ∣x\ensuremath−x\ensuremath′∣\ensuremath−b|\mathbf{x}\ensuremath{-}{\mathbf{x}}^{\ensuremath{'}}{|}^{\ensuremath{-}b}∣x\ensuremath−x\ensuremath′∣\ensuremath−b is considered. The problem is studied within the context of a model which partially considers interactions of fluctuations and belongs in the same universality class as the spherical model. Depending on the relationship between the power b and the space dimensionality d, a new critical behavior arises.
Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.
We investigate the behavior of the correlation function and the susceptibility of the random-field spherical model at the critical point. In particular we calculate the critical exponents \ensuremath{\eta} and \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{} describing the divergences of the susceptibility and its disconnected part, respectively. In the case of short-range interactions we obtain \ensuremath{\eta}=\ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=0. For power-law interactions Uij{\mathit{U}}_{\mathit{i}\mathit{j}}Uij​\ensuremath{\sim}${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ we find \ensuremath{\eta}=1/2\ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=D+2-\ensuremath{\sigma} (for DD+2), where D is the spatial dimension. The Schwartz-Soffer exponent inequality \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\le}2\ensuremath{\eta} is satisfied and becomes an equality independent of the functional form of the interaction.
We present the results of the first analytical study of finite-size effects in a random-field model. In particular we consider the random-field spherical model in partially infinite geometries (infinite in ${d}^{\ensuremath{'}}$ dimensions and finite in $d\ensuremath{-}{d}^{\ensuremath{'}}$ dimensions) under periodic and antiperiodic boundary conditions in the finite directions. We find a simultaneous dimensional reduction by 2 in ddd and ${d}^{\ensuremath{'}}$ for 4<d<64<d<64<d<6 and $2<~{d}^{\ensuremath{'}}<~4$. For $0<~{d}^{\ensuremath{'}}<2$, we find new power laws describing the approach to bulk in different geometries.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.