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Journal of Physics C Solid State Physics

Zero-temperature critical behaviour of vector spin glasses

B W Morris, S G W Colborne, M A Moore, Alan J. Bray, J. Canisius

Published Mar 20, 1986• 134 citations

Abstract

The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1<or=d<or=4 (m= infinity ). Scaling towards weak coupling, J'(L,m) approximately L -1 nu /, L>>L c (m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T - nu , for T to 0 but no finite-temperature transition. The characteristic length L c (m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.

DOI: 10.1088/0022-3719/19/8/014

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The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1<or=d<or=4 (m= infinity ). Scaling towards weak coupling, J'(L,m) approximately L -1 nu /, L>>L c (m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T - nu , for T to 0 but no finite-temperature transition. The characteristic length L c (m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.

A transfer matrix method is used to study the variation with length scale L of the distribution P L (J eff ) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod J eff mod ) varies as L -1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L -1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.

Vector spin glasses with random short-range anisotropic couplings are studied using the defect-wall renormalization-group method. By calculating the correlation length exponent \ensuremath{\nu} and the fractal dimension of the defect wall,

{\mathit{d}}_{\mathit{s}}

, we obtain the first direct evidence that in two dimensions these systems have a zero-temperature transition in the same universality class as the Ising spin glass. We also present results which strongly suggest that these systems have a transition at nonzero temperature in three dimensions.

The transverse dynamics of a vector spin glass chain at zero temperature is studied by a numerical scaling method which provides a new physical insight into the nature of the spin excitations. In the range of low energies considered we find that the states are all localized with the coupling between spins V decaying asymptotically for large lengths separating the sites l , like V ∼ exp[- l /ξ], where ξ measures the localization length. A dispersion relation is found between the energies and the associated inverse localization lengths of the form ω ∝ (1/ξ) z with the dynamic exponent z = 1.501 ± 0.044. This implies that the system at zero temperature has nontrivial dynamic critical behaviour, in agreement with our recent analytical results; it moreover gives for the first time a clear interpretation for the characteristic length of the spin fluctuations as the localization length.

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 × .

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.

Critical point scaling in a field

H

applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function

f(R)

defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically

f(R)

for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension

d=3

, we have determined the exponents and the critical scaling function

f(R)

within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for

f(R)

in

d=3

in reasonable agreement with those of the Monte Carlo simulations at the values of

R

for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function

f(R)

at some negative value of

R

, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.