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Physical review. B, Condensed matter

Quantum spin glass with long-range random interactions

Amit Dutta

Published Jun 10, 2002• 9 citations
Publisher

Abstract

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 ${r}_{\mathrm{ij}}$ 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.

DOI
10.1103/physrevb.65.224427

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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/rij(d+\ensuremathσ)/2.{1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.1/rij(d+\ensuremathσ)/2​. Here, apart from the strength of the quantum fluctuations, the interaction range \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
The one-dimensional long-range Ising spin glass provides useful insights into the properties of finite-dimensional spin glasses with short-range interactions. The defect energy renormalization-group equations derived for it by Kotliar, Anderson, and Stein have been reexamined and a new fixed point has been found. The fixed points which have previously been studied are found to be inappropriate. It is shown that the renormalization-group equations themselves directly imply that that the spin-glass phase is replica symmetric, in agreement with droplet model expectations, when the range exponent $\ensuremath{\sigma}$ lies between the upper and lower critical values.
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 examine the ground state of the random quantum Ising model in a transverse field using a generalization of the Ma-Dasgupta-Hu renormalization group (RG) scheme. For spatial dimensionality d=2,d=2,d=2, we find that at strong randomness the RG flow for the quantum critical point is towards an infinite-randomness fixed point, as in one dimension. This is consistent with the results of a recent quantum Monte Carlo study by Pich et al. [Phys. Rev. Lett. 81, 5916 (1998)], including estimates of the critical exponents from our RG that agree well with those from the quantum Monte Carlo. The same qualitative behavior appears to occur for three dimensions; we have not yet been able to determine whether or not it persists to arbitrarily high d. Some consequences of the infinite-randomness fixed point for the quantum critical scaling behavior are discussed. Because frustration is irrelevant in the infinite-randomness limit, the same fixed point should govern both ferromagnetic and spin-glass quantum critical points. This RG maps the random quantum Ising model with strong disorder onto a novel type of percolation/aggregation process.
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.
We study the effect of long-range connections on the infinite-randomness fixed point associated with the quantum phase transitions in a transverse Ising model (TIM). The TIM resides on a long-range connected lattice where any two sites at a distance rrr are connected with a nonrandom ferromagnetic bond with a probability that falls algebraically with the distance between the sites as $1∕{r}^{d+\ensuremath{\sigma}}$. The interplay of the fluctuations due to dilutions together with the quantum fluctuations due to the transverse field leads to an interesting critical behavior. The exponents at the critical fixed point (which is an infinite randomness fixed point) are related to the classical ``long-range'' percolation exponents. The most interesting observation is that the gap exponent $\ensuremath{\psi}$ is exactly obtained for all values of $\ensuremath{\sigma}$ and ddd. Exponents depend on the range parameter $\ensuremath{\sigma}$ and show a crossover to short-range values when $\ensuremath{\sigma}\ensuremath{\ge}2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}$ where ${\ensuremath{\eta}}_{\mathrm{SR}}$ is the anomalous dimension for the conventional percolation problem. Long-range connections are also found to tune the strength of the Griffiths phase.