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Physical review. B, Condensed matter

Spin-glass with long-range random exchange interaction

Mau-chung Chang, Josef Sak

Published Mar 1, 1984• 18 citations
Publisher

Abstract

We consider the spin-glass (SG) with long-range (LR) random exchange interaction varying with distance $R$ as ${R}^{\ensuremath{-}\frac{d}{2}\ensuremath{-}\frac{\ensuremath{\sigma}}{2}}$, where $d$ is the space dimensionality and $\ensuremath{\sigma}>0$. It is shown that the LR fixed point is stable if $\ensuremath{\sigma}<2$. The nonclassical ($d<3\ensuremath{\sigma}$) LR critical exponents in powers of $3\ensuremath{\sigma}\ensuremath{-}d$ are not continuous at $\ensuremath{\sigma}=2$ when the transition to short-range (SR) exchange is made.

DOI
10.1103/physrevb.29.2652

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Physical review. B, Condensed matter•Dec 1, 1983•20 citations
We consider the spin-glass (SG) with long-range (LR) random exchange interaction varying with distance RRR as R\ensuremath−d2\ensuremath−\ensuremathσ2{R}^{\ensuremath{-}\frac{d}{2}\ensuremath{-}\frac{\ensuremath{\sigma}}{2}}R\ensuremath−2d​\ensuremath−2\ensuremathσ​, where ddd is the space dimensionality and \ensuremathσ>0\ensuremath{\sigma}>0\ensuremathσ>0. It is shown that the LR fixed point is stable if \ensuremathσ<2\ensuremath{\sigma}<2\ensuremathσ<2. The nonclassical (d<3\ensuremathσd<3\ensuremath{\sigma}d<3\ensuremathσ) LR critical exponents in powers of 3\ensuremathσ\ensuremath−d3\ensuremath{\sigma}\ensuremath{-}d3\ensuremathσ\ensuremath−d are not continuous at \ensuremathσ=2\ensuremath{\sigma}=2\ensuremathσ=2 when the transition to short-range (SR) exchange is made.
A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance rij{r}_{\mathrm{ij}}rij​ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.
The q-state Potts spin-glass (SG) with short-ranged random-exchange interaction is formulated field theoretically. The one-loop calculation [M.-c. Chang, Phys. Rev. B 28, 6592 (1983)] shows that the Potts SG transition is of first order when q\ensuremath{\ge}4 and for q=3 it is undetermined because the Potts SG fixed point is only marginally stable. The calculations are carried out to two-loop order with use of renormalized perturbation theory by means of minimal subtraction procedure. The results show that the three-state Potts SG fixed point is infrared stable, and we conclude that it is second-order transition. The critical exponents to order ${\ensuremath{\epsilon}}^{2}$ (\ensuremath{\epsilon}=6-d) are given.
We consider an Ising chain with Hamiltonian $H=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}$, where the ${\ensuremath{\epsilon}}_{\mathrm{ij}}$ are independent random variables. We find a phase transition for $\frac{1}{2}<\ensuremath{\sigma}<1$. For $\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}$ the critical exponents exhibit mean-field classical behavior. Near $\ensuremath{\sigma}=1$ we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an $\ensuremath{\epsilon}$ expansion around $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part WMF{W}_{\text{MF}}WMF​ coincides with previous results. The short-range part WSR{W}_{\text{SR}}WSR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as dc=5{d}_{c}=5dc​=5 at very low temperatures due to the dimensionality shift Dc=dc+1=6{D}_{c}={d}_{c}+1=6Dc​=dc​+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power $\ensuremath{\sigma}$ of the distance. We show that there is a value of $\ensuremath{\sigma}$ of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of $\ensuremath{\sigma}$ for which we find the critical behavior to be compatible with that of the three-dimensional model, although we have much less precision than in the four-dimensional case.
The critical behavior of an nnn-component system in ddd dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.