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Journal of Physics C Solid State Physics

Scaling stiffness and correlations in the spin-glass-ferromagnet transition: evidence for the mixed phase

Marek Cieplak, Marek Cieplak

Published Mar 10, 1985• 6 citations

Abstract

Results of numerical studies on the equilibrium scaling stiffness energy for 2D nearest-neighbour Heisenberg systems at T=0 are presented. The authors first consider the percolation problem and estimate the critical exponent for the spin wave stiffness to be close to 1. They then consider systems in which -J-bonds appear with probability x and +J-bonds with probability (1-x). The scaling stiffness energy, a measure of sensitivity to changes in the boundary conditions, disappears exponentially with x at x c approximately=0.13 (an estimate for large systems). This value of concentration is identified as an upper boundary of the usual ferromagnetic phase. Nevertheless the magnetisation persists above x c and disappears at X c approximately=0.30 when the system enters into the spin glass phase. The region between x c and X c is identified as corresponding to the Gabay-Toulouse mixed phase in which the system looks like a spin glass but has some non-zero magnetisation. This interpretation receives further support from studies of the two- and four-spin correlation functions in local energy minima. The correlations decay exponentially, at least on small distances. The correlation lengths of the two-spin correlations start to drop at x c . The spin components parallel to the magnetisation are actually less correlated than the transverse ones.

DOI: 10.1088/0022-3719/18/7/014

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Results of numerical studies on the equilibrium scaling stiffness energy for 2D nearest-neighbour Heisenberg systems at T=0 are presented. The authors first consider the percolation problem and estimate the critical exponent for the spin wave stiffness to be close to 1. They then consider systems in which -J-bonds appear with probability x and +J-bonds with probability (1-x). The scaling stiffness energy, a measure of sensitivity to changes in the boundary conditions, disappears exponentially with x at x c approximately=0.13 (an estimate for large systems). This value of concentration is identified as an upper boundary of the usual ferromagnetic phase. Nevertheless the magnetisation persists above x c and disappears at X c approximately=0.30 when the system enters into the spin glass phase. The region between x c and X c is identified as corresponding to the Gabay-Toulouse mixed phase in which the system looks like a spin glass but has some non-zero magnetisation. This interpretation receives further support from studies of the two- and four-spin correlation functions in local energy minima. The correlations decay exponentially, at least on small distances. The correlation lengths of the two-spin correlations start to drop at x c . The spin components parallel to the magnetisation are actually less correlated than the transverse ones.

For the Ising model with Gaussian random coupling of average

{J}_{0}

and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter

{J}_{0}

can be studied via the size-

L

-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension

d=2,3,4,5,6

. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.

The spin and the chirality orderings of the Heisenberg spin glass in two dimensions with the nearest-neighbour Gaussian coupling are investigated by equilibrium Monte Carlo simulations. Particular attention is paid to the behaviour of the spin and the chirality correlation lengths. In order to observe the true asymptotic behaviour, a fairly large system size L ⪆ 20 ( L the linear dimension of the system) appears to be necessary. It is found that both the spin and the chirality order only at zero temperature. At high temperatures, the chiral correlation length stays shorter than the spin correlation length, whereas at lower temperatures below the crossover temperature T × , the chiral correlation length exceeds the spin correlation length. The spin and the chirality correlation-length exponents are estimated above T × to be ν SG = 0.9 ± 0.2 and ν CG = 2.1 ± 0.3, respectively. These values are close to the previous estimates on the basis of the domain-wall-energy calculation. Discussion is given about the asymptotic critical behaviour realized below T × .

With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.

We examine three-dimensional ± J Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent θ s at zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers (Kawamura 1992 Phys. Rev. Lett. 68 3785, 1998 Phys. Rev. Lett. 80 5421, Hukushima and Kawamura 2000 Phys. Rev. E 61 R1008), a Gabay–Toulouse phase transition might occur when the anisotropies are absent, although the results, except for the correlation length of the chirality for larger sizes, suggest no phase transition when they are present.

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at

{T}_{c}

=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero

{T}_{c}

but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that

{T}_{c}

is nonzero with a long-range-ordered phase below

{T}_{c}

. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at

T=0

are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.