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Physical review. B, Solid state

Critical properties of a three-dimensional spherical model with random spin layers

Alexander Rauh

Published Mar 1, 1976• 8 citations
Publisher

Abstract

A ferromagnetic system with frozen-in defects is investigated within a modified spherical model: The three-dimensional lattice is formed out of homogeneous layers which are occupied at random either by $A$ or by $B$ spins. We assume three exchange constants ${J}_{A}$, ${J}_{B}$, ${J}_{\mathrm{AB}}$ which correspond to the coupling between $A\ensuremath{-}A$, $B\ensuremath{-}B$, and $A\ensuremath{-}B$ spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness $d$, then its influence on the free energy decreases exponentially with $d$; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to ${J}_{\mathrm{AB}}=0$. In case (i) we find a finite critical temperature ${T}_{c}$ which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.

DOI
10.1103/physrevb.13.2080

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A ferromagnetic system with frozen-in defects is investigated within a modified spherical model: The three-dimensional lattice is formed out of homogeneous layers which are occupied at random either by AAA or by BBB spins. We assume three exchange constants JA{J}_{A}JA​, JB{J}_{B}JB​, JAB{J}_{\mathrm{AB}}JAB​ which correspond to the coupling between A\ensuremath−AA\ensuremath{-}AA\ensuremath−A, B\ensuremath−BB\ensuremath{-}BB\ensuremath−B, and A\ensuremath−BA\ensuremath{-}BA\ensuremath−B spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness ddd, then its influence on the free energy decreases exponentially with ddd; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to JAB=0{J}_{\mathrm{AB}}=0JAB​=0. In case (i) we find a finite critical temperature Tc{T}_{c}Tc​ which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.
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Analyses the critical behaviour of a three-dimensional spherical model with nearest-neighbour ferromagnetic interactions (of strength J>0) in which there is a regular array of planes of bond defects (perpendicular to the planes) of strength J':0<or=J'<or=J spaced n lattice sites apart. For a fixed, non-zero defect strength J', the critical temperature has the form T c (J',n)=T c (pure)+c 1 (J')n -2 +c 2 (J')n -2 +o(n -2 ). For J'=0 the problem reduces to that of finite size effect, while at J'=J the system becomes homogeneous. Two asymptotically distinct scaling regions correspond to these limiting cases. In both scaling regions the spherical field scales with the variable x identical to n 2 phi where phi is proportional to the deviation in spherical field. In terms of the variable lambda identical to 1-(J('/J) 2 and its conjugate lambda =(J'/J) 2 , the impurity strength scales as y=n lambda in the region near J'=J( lambda =0) and as lambda for J' away from zero. These scaling relationships are valid for the spherical constraint equation and the thermodynamic quantities such as the specific heat and entropy.
The Privman-Fisher hypothesis on the singular part of the free-energy density of a finite system, near the bulk critical point T=Tc{T}_{c}Tc​, is examined in the context of the spherical model of ferromagnetism. A d-dimensional hypercubical lattice (of size N1{N}_{1}N1​a\ifmmode\times\else\texttimes\fi{}N2{N}_{2}N2​a\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Nd{N}_{d}Nd​a) is considered and, subject to periodic boundary conditions, explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of the system at temperatures close to Tc{T}_{c}Tc​. The relevant scaling functions governing the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis are verified. By implication, the passage of the given system towards standard bulk behavior, as Nj{N}_{j}Nj​\ensuremath{\rightarrow}\ensuremath{\infty}, is also elucidated.
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A convenient way to study phase transitions is by considering finite spin systems of linear size LLL with boundary conditions that produce a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension d=2d=2d=2 within Migdal--Kadanoff real-space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of nonboundary states. Our main conclusion is that all singularities involve the correlation length ${\ensuremath{\xi}}_{\mathit{av}}(T)\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\ensuremath{\nu}}$ appearing in the average free-energy $\overline{F}\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{av}}(T)]}^{{d}_{s}}$ of the interface of dimension ${d}_{s}=d\ensuremath{-}1$, except for the free-energy width $\ensuremath{\Delta}F\ensuremath{\sim}{[L∕{\ensuremath{\xi}}_{\mathit{var}}(T)]}^{\ensuremath{\theta}}$ that involves the droplet exponent $\ensuremath{\theta}$ and another correlation length ${\ensuremath{\xi}}_{\mathit{var}}(T)$ that diverges more rapidly than ${\ensuremath{\xi}}_{\mathit{av}}(T)$. We compare with the spin-glass transition in d=3d=3d=3, where ${\ensuremath{\xi}}_{\mathit{var}}(T)$ is the ``true'' correlation length and where the interface energy presents unconventional scaling with a chaos critical exponent ${\ensuremath{\zeta}}_{c}>1∕\ensuremath{\nu}$ [M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale Lch(T){L}_{\mathit{ch}}(T)Lch​(T) associated with the chaotic nature of the low-temperature phase diverges more slowly than the correlation length.