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Physical Review Letters

Critical behavior of random spin systems at the percolation threshold

Jayanth R. Banavar, Alan J. Bray, Shechao Charles Feng

Published Apr 6, 1987• 13 citations
Publisher

Abstract

Ising and Heisenberg ferromagnets and spin-glasses are studied at the percolation threshold for power-law bond distributions, P(J)\ensuremath{\sim}\ensuremath{\Vert}J${\ensuremath{\Vert}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}1}$ for J\ensuremath{\rightarrow}0. Nonuniversal, \ensuremath{\alpha}-dependent, critical behavior is found. A hierarchical model for the ``backbone'' yields, via finite-size scaling, expressions for the exponent t in the relation \ensuremath{\Upsilon}\ensuremath{\propto}(p-${p}_{c}$${)}^{t}$ for the generalized stiffness coefficient, and for the percolation-to-thermal crossover exponent \ensuremath{\varphi}.

DOI
10.1103/physrevlett.58.1463

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Ising and Heisenberg ferromagnets and spin-glasses are studied at the percolation threshold for power-law bond distributions, P(J)\ensuremath{\sim}\ensuremath{\Vert}J\ensuremath∥\ensuremathα\ensuremath−1{\ensuremath{\Vert}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}1}\ensuremath∥\ensuremathα\ensuremath−1 for J\ensuremath{\rightarrow}0. Nonuniversal, \ensuremath{\alpha}-dependent, critical behavior is found. A hierarchical model for the ``backbone'' yields, via finite-size scaling, expressions for the exponent t in the relation \ensuremath{\Upsilon}\ensuremath{\propto}(p-{p}_{c}$${)}^{t} for the generalized stiffness coefficient, and for the percolation-to-thermal crossover exponent \ensuremath{\varphi}.
The thermal-to-percolative crossover exponent $\ensuremath{\phi}$, well known for ferromagnetic systems, is studied extensively for Edwards--Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold pc{p}_{c}pc​ using an improved reduction algorithm. Simulations extend to system sizes above N=108N={10}^{8}N=108 in dimensions d=2,…,7d=2,\dots{},7d=2,…,7. The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for Tg{T}_{g}Tg​ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.
The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$
Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-Hc{H}_{c}Hc​), where Hc{H}_{c}Hc​ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=kz{k}^{z}kzf(\ensuremath{\Delta}/kcphi{k}^{\mathrm{cphi}}kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: kz{k}^{z}kz for \ensuremath{\Delta}\ensuremath{\ll}kcphi{k}^{\mathrm{cphi}}kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}kcphi{k}^{\mathrm{cphi}}kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
We investigate random-cluster representations of the q=1 - and 2-state Potts models in three dimensions, i.e., the bond-percolation and the Ising model, respectively. Using a recently developed sampling technique, we determine the probabilities C1 (r) and C2 (r) that a pair of lattice sites at a distance r are connected by at least one and two mutually independent paths, respectively. The scaling behavior of C1 and C2 at criticality is governed by the magnetic and the backbone scaling dimension X(h) and X(b) , respectively. From a finite-size analysis of the numerical data, we determine X(h) =0.4768 (7) and X(b) =1.125 (3) for the percolation and X(h) =0.5178 (7) and X(b) =0.829 (4) for the Ising model.