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arXiv:cond-mat/0307699 · Physical Review E

Crossover behavior in three-dimensional dilute spin systems

Pasquale Calabrese, Pietro Parruccini, Andrea Pelissetto C, Ettore Vicari

Published Mar 31, 2004• 23 citations
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Abstract

We study the crossover behaviors that can be observed in the high-temperature phase of three-dimensional dilute spin systems, using a field-theoretical approach. In particular, for randomly dilute Ising systems we consider the Gaussian-to-random and the pure-Ising-to-random crossover, determining the corresponding crossover functions for the magnetic susceptibility and the correlation length. Moreover, for the physically interesting cases of dilute Ising, XY, and Heisenberg systems, we estimate several universal ratios of scaling-correction amplitudes entering the high-temperature Wegner expansion of the magnetic susceptibility, of the correlation length, and of the zero-momentum quartic couplings.

DOI
10.1103/physreve.69.036120

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We study the crossover behaviors that can be observed in the high-temperature phase of three-dimensional dilute spin systems, using a field-theoretical approach. In particular, for randomly dilute Ising systems we consider the Gaussian-to-random and the pure-Ising-to-random crossover, determining the corresponding crossover functions for the magnetic susceptibility and the correlation length. Moreover, for the physically interesting cases of dilute Ising, XY, and Heisenberg systems, we estimate several universal ratios of scaling-correction amplitudes entering the high-temperature Wegner expansion of the magnetic susceptibility, of the correlation length, and of the zero-momentum quartic couplings.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
We study the crossover from pure to random critical behavior in ddd-dimensional ($d=4\ensuremath{-}\ensuremath{\epsilon}$) classical spin systems which is induced by quenched random fields with spatial correlations such that ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}\frac{{h}_{0}^{2}}{{R}^{d+\ensuremath{\theta}}}$, where $\ensuremath{\theta}=2\ensuremath{-}{\ensuremath{\epsilon}}_{d}$. An effective critical exponent for the magnetic susceptibility ${\ensuremath{\gamma}}_{\mathrm{eff}}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}\ensuremath{\chi}}{\ensuremath{\partial}\mathrm{ln}t}$ is calculated with the use of a renormalization-group approcah and a double $\ensuremath{\epsilon}$,$\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}$ expansion to linear order in $\ensuremath{\epsilon}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}\ensuremath{\equiv}\ensuremath{\epsilon}+{\ensuremath{\epsilon}}_{d}$. The crossover scaling function for the susceptibility is obtained and the width of the crossover region is estimated.
Crossover functions are obtained to three-loop order in an expansion around dimension d=4 for the ${\ensuremath{\varphi}}^{4}$ theory. Expressions for the internal energy, susceptibility, and correlation length are given for the general n-component system in zero magnetic field and above the critical temperature. The full Helmholtz free energy, magnetization equation of state, and susceptibility are given for the Ising-like n=1 case. Corrected values for the critical amplitude ratios as well as for the correction-to-scaling amplitude ratios are given to three-loop order.
High-temperature series for the inverse spin-spin correlation function of the s=12s=\frac{1}{2}s=21​ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at Tc{T}_{c}Tc​ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.
In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in d=3−−5d=3--5d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In d=4d=4d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in d=4d=4d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.