Infinite-range Ising Spin-glass Model Research Articles

Code-division multiple-access multiuser demodulator by using quantum fluctuations.

We examine the average-case performance of a code-division multiple-access (CDMA) multiuser demodulator in which quantum fluctuations are utilized to demodulate the original message within the context of Bayesian inference. The quantum fluctuations are built into the system as a transverse field in the infinite-range Ising spin glass model. We evaluate the performance measurements by using statistical mechanics. We confirm that the CDMA multiuser modulator using quantum fluctuations achieve roughly the same performance as the conventional CDMA multiuser modulator through thermal fluctuations on average. We also find that the relationship between the quality of the original information retrieval and the amplitude of the transverse field is somehow a "universal feature" in typical probabilistic information processing, viz., in image restoration, error-correcting codes, and CDMA multiuser demodulation.

Infinite-range Ising spin glass with a transverse field under the static approximation

In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

Nature of ergodicity breaking in ising spin glasses as revealed by correlation function spectral properties

In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with<S(i)S(j)>. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.

Da Costa-Nobre-Yokoi Model with Spin S

An infinite-ranged Ising spin glass model with spin S, including S=∞, with uniform biquadratic interactions is investigated using replica symmetry (RS) solutions, de Almeida and Thouless (AT) stability analyses, and one-step replica symmetry breaking (RSB) solutions. We find that the model with integral spin S and the model with half-integral spin S exhibit different thermodynamic behavior. Namely, we show that for the model with integral spin S, certain physical quantities display two-sublattice structure, but that for the model with half-integral spin S, including S=∞, this is not the case.

Effects of random fields in an antiferromagnetic Ising spin glass

The effects of random fields on the two-sublattice infinite-ranged Ising spin-glass model are investigated. This model is expected to be appropriate as a mean-field description of antiferromagnetic spin glasses such as FexMn1-xTiO3. Within replica-symmetric calculations, we study the influence of Gaussian and bimodal random fields on the phase transitions and phase diagrams. It is shown that, in the presence of random fields, the first-order transitions are weakened and may become continuous. Also, the antiferromagnetic phases are always destroyed by sufficiently strong random fields. A qualitative comparison with existing experimental results and the limitations of the present calculations are discussed.

Dynamics of the Infinite-Range Ising Spin-Glass Model in a Transverse Field

We use quantum Monte Carlo methods and various analytic approximations to study the infinite-range Ising spin-glass model in a transverse field in the disordered phase. We focus on the behavior of the frequency dependent susceptibility of the system above and below the critical field. We establish that, at the quantum critical point, there exists an equivalence between the long-time behavior of this model and that of the single-impurity Kondo model. Our predictions for the long-time dynamics of the model are in good agreement with experimental results on ${\mathrm{LiHo}}_{0.167}{\mathrm{Y}}_{0.833}{\mathrm{F}}_{4}$.

Spin-glass model with uniform biquadratic interactions

We study a spin-1 infinite-ranged Ising spin-glass model with uniform biquadratic exchange interactions. The phase diagram of the model is obtained in the framework of the replica-symmetric solution. For the case of attractive biquadratic interactions the usual spin-glass phase is stabilized at low temperatures, but for the repulsive case, the antiquadrupolar and antiquadrupolar-glass phases with two-sublattice structure are also possible. We investigate the stability of the replica-symmetric solution and show that the paramagnetic and antiquadrupolar phases are stable, whereas the spin-glass and antiquadrupolar-glass phases are unstable. Parisi's replica-symmetry-breaking procedure is implemented in the neighbourhood of the spin-glass - paramagnetic critical frontier.

Ordering temperature of the infinite-range +/-J Ising spin glass.

The spin glass susceptibility of infinite-range±J ising spin glass samples has been measured as a function of sample size and temperature using numerical simulations. A slow-cooling annealing schedule was used. The data indicate that spin glass order appears to set in only at a temperature well below T=1, the accepted theoretical ordering temperature T g for the infinite-range ising spin glass model

Cavity-field approach to quantum spin glasses: The Ising spin glass in a transverse field.

The cavity-field method, recently used for classical spin glasses, is extended to study the infinite-range Ising spin-glass model in a transverse field, by means of the Trotter-Suzuki transformation. Within this approach, which relies on well-known mathematics and deals with quantities that have transparent physical significance by avoiding the replica method, the quantum version of the classical Thouless-Anderson-Palmer equation is obtained. The phase diagram of the model is also obtained and full agreement with previous replica-method results is found.

The nature of attractors in an asymmetric spin glass with deterministic dynamics

The authors study the attractors in an infinite-range Ising spin-glass model with deterministic dynamics where the interactions have asymmetry, specified by a parameter k. They find a duality relation between the attractors for models with asymmetry parameters k and 1/k. The attractors are fixed points or limit cycles of short length, except for k=1, at which the average cycle length diverges, reminiscent of a phase transition, and the model has many similarities to the random map model as well as to the infinite-range symmetric spin glass in thermal equilibrium, including the fact that a few attractors dominate the weight. The extent of this dominance varies from sample to sample and so is given by a non-trivial probability distribution, Pi (Y), which they compute numerically.

Metastable states of the SK spin glass model

The properties of metastable states at zero temperature are examined numerically for the infinite-ranged Ising spin-glass model. It is shown that the energy levels of the metastable states behave like a random energy model. Furthermore, it is found that the barrier energy between them is an increasing function of the Hamming distance. The result is consistent with the prediction that the metastable states have an ultrametric organisation in the phase space.

Reply to "Comment on 'Temperature dependence of the response time of dilute metallic spin glasses' "

In this response we reply to the criticism of Nordblad et al. on our measurements of the dynamic repsonse of dilute metallic spin glasses. We argue that we have, in our experiments on dilute metallic spin glasses, separated the thermalization dynamics from the dynamic response to a change in applied field. This is contrary to experiments with observation times that exceed the characteristic thermalization time. On the basis of this observation we show that our interpretation of our experimental results is completely consistent with a recent model calculation of the dynamic response of the infinite-range Ising spin-glass model.

Temperature dependence of the response time of dilute metallic spin glasses.

The temperature dependence of the time decay of the thermoremanent magnetization has been measured for four dilute metallic spin glasses: Ag:${\mathrm{Mn}}_{2.6\phantom{\rule{0ex}{0ex}}\mathrm{at}.\phantom{\rule{0ex}{0ex}}\mathrm{%}}$, Ag:${\mathrm{Mn}}_{4.1}$ at. $_{\mathrm{%}}$, Ag:${\mathrm{Mn}}_{2.6}$ at. % +${\mathrm{Sb}}_{0.46}$ at. %, and Cu:${\mathrm{Mn}}_{4.0}$ at. %. After cooling in a constant applied field, the field was cut to zero and the time decay of the thermoremanent magnetization was observed over a period of 0.2--500 sec. The time dependence of the thermoremanent magnetization can be described well by a stretched exponential: ${\ensuremath{\sigma}}_{\mathrm{TRM}}$(t)=${\ensuremath{\sigma}}_{0}$exp[-(t/${\ensuremath{\tau}}_{p}$ ${)}^{1\mathrm{\ensuremath{-}}n}$], with an apparent rate, 1/${\ensuremath{\tau}}_{p}$, which varies exponentially with the inverse reduced temperature over nearly the entire temperature range of measurement below ${T}_{g}$, the glass temperature. Moreover, the temperature dependence of 1/${\ensuremath{\tau}}_{p}$ is given by a universal function, 1/${\ensuremath{\tau}}_{p}$=A exp(-2.5${T}_{g}$/T), with A${=10}^{\mathrm{\ensuremath{-}}3}$ ${\mathrm{sec}}^{\mathrm{\ensuremath{-}}1}$. Very near to ${T}_{g}$, the apparent rate 1/${\ensuremath{\tau}}_{p}$ decreases much more rapidly as T diminishes, and the scaling with ${T}_{g}$ breaks down. All of these results can be mapped onto a recent calculation of the dynamics of the infinite-range Ising spin-glass model by De Dominicis et al. We are able to fit the observed temperature dependence of 1/${\ensuremath{\tau}}_{p}$ over the full temperature range (T\ensuremath{\le}${T}_{g}$) using either the experimentally determined values for the quantities in the theory, or those values extracted from the Sherrington-Kirkpatrick model.

Phase diagrams for dilute spin glasses

A generalised, dilute, infinite-ranged Ising spin-glass model is introduced and studied as a function of the concentration p and temperature T. The phase diagram is investigated and paramagnetic (P), ferromagnetic (F), spin glass (SG) and mixed (M) phases, meeting at a multicritical point (p*,T*), are identified. The P/F and P/SG phase boundaries are derived, and the F/M and M/SG boundaries are calculated close to (p*,T*). The condition for having a re-entrant spin-glass transition is derived. In non-zero magnetic field a p-dependent A-T instability line is obtained. The authors apply their results to the insulator EuxSr1-xS, it is predicted to exhibit re-entrant behaviour.

Statics and dynamics of the infinite-range Ising spin glass model

The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below Tc there is a spectrum of relaxation times which diverge like exp(cN14/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg, which is the time to turn over all the spins. Averaging over samples (ln tau eg)j(=ln tau eg) varies as N12/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N12/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.

Bounds for the entropy of the infinite-ranged Ising spin-glass model

Abstract It is shown that the TAP expression for the entropy of the SK model is non-negative for all possible distributions of the local magnetizations. In addition an upper bound for the entropy is presented.

Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model

It is shown that the power expansion of the Gibbs potential of the SK model up to second order in the exchange couplings leads to the TAP equation. This result remains valid for the general (including a ferromagnetic exchange) SK model. Theorems of power expansions and resolvent techniques are employed to solve the convergence problem. The convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations.

Ordering in the infinite-ranged Ising spin glass—Exact solutions for small samples

A new order parameter q(2) for the infinite-ranged Ising spin glass model is introduced. q(2) can be defined without specifying any symmetry breaking field, and is a direct measure of the effect of frustration on the internal energy below the spin glass transition temperature, Tsg. An exact evaluation of q(2)(T) for small systems (with up to 20 spins) establishes the existence of Edwards-Anderson ordering, and sheds light on several of the controversies which surround this model.