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Physical review. B, Condensed matter

One-dimensional spin-glass model with long-range random interactions

Gabriel Kotliar, Philip Warren Anderson, D. L. Stein

Published Jan 1, 1983• 151 citations
Publisher

Abstract

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 $\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}$.

DOI
10.1103/physrevb.27.602

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Physical Review B•Mar 11, 2009•2 citations
We consider an Ising chain with Hamiltonian H=\ensuremathΣi>j(\ensuremathϵijSiSj)(a∣i\ensuremath−j∣)\ensuremathσH=\ensuremath{\Sigma}{i>j}^{}\frac{({\ensuremath{\epsilon}}_{\mathrm{ij}}{S}_{i}{S}_{j})}{{(a|i\ensuremath{-}j|)}^{\ensuremath{\sigma}}}H=\ensuremathΣi>j(a∣i\ensuremath−j∣)\ensuremathσ(\ensuremathϵij​Si​Sj​)​, where the \ensuremathϵij{\ensuremath{\epsilon}}_{\mathrm{ij}}\ensuremathϵij​ are independent random variables. We find a phase transition for 12<\ensuremathσ<1\frac{1}{2}<\ensuremath{\sigma}<121​<\ensuremathσ<1. For 12<\ensuremathσ<23\frac{1}{2}<\ensuremath{\sigma}<\frac{2}{3}21​<\ensuremathσ<32​ the critical exponents exhibit mean-field classical behavior. Near \ensuremathσ=1\ensuremath{\sigma}=1\ensuremathσ=1 we find a smoothly varying specific heat. We investigate the critical behavior near the upper and lower critical range by means of an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion around \ensuremathσ=1and23\ensuremath{\sigma}=1 \mathrm{and} \frac{2}{3}\ensuremathσ=1and32​.
An Ising chain with Hamiltonian H=-J Sigma i<j epsilon ij S i S j / mod i-j mod (1+ sigma )/2 is considered where the epsilon ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when 1 / 3 < sigma <1 and mean-field exponents beyond the upper critical value of sigma = 1 / 3 . By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 / 3 < sigma <1.
The spin-glass transition in a magnetic field is studied both in and out of the limit of validity of mean-field theory on a diluted one dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to changing the dimension in spin-glass short-range models. Evidence for a spin-glass transition in a magnetic field is found also for systems whose equivalent dimension is below the upper critical dimension in a zero magnetic field.
We consider the spin-glass (SG) with long-range (LR) random exchange interaction varying with distance RRR as ${R}^{\ensuremath{-}\frac{d}{2}\ensuremath{-}\frac{\ensuremath{\sigma}}{2}}$, where ddd 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.
The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.
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 analyze the critical properties of a quantum spherical spin-glass model with random inverse power-law interactions ${r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$. It was shown in a previous publication that the effective partition function calculated with 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 the ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0,\ensuremath{\omega},\ensuremath{-}\ensuremath{\omega})$ and a strictly long-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 long-range part WLR{W}_{\text{LR}}WLR​ describes a phase transition in a Q3{Q}^{3}Q3-field theory that 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 upper critical dimensionality as ${d}_{c}=\frac{5\ensuremath{\sigma}}{2}$ at very low temperatures due to the dimensionality shift ${D}_{c}={d}_{c}+\frac{\ensuremath{\sigma}}{2}=3\ensuremath{\sigma}$. We then perform an ${ϵ}^{\ensuremath{'}}={d}_{c}\ensuremath{-}d$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.