Spherical Spin-glass Model Research Articles

Ultrametricity between states at different temperatures in spin-glasses

We prove the existence of correlations between the equilibrium states at different temperatures of the multi-$p$-spin spherical spin-glass models with continuous replica symmetry breaking: there is no chaos in temperature in these models. Furthermore, the overlaps satisfy ultrametric relations. As a consequence the Parisi tree is essentially the same at all temperatures with lower branches developing when lowering the temperature. We conjecture that the reference free energies of the clusters are also fixed at all temperatures as in the generalized random-energy model.

A conjectured scenario for order-parameter fluctuations in spin glasses

We study order-parameter fluctuations (OPF) in disordered systems by considering the behavior of some recently introduced paramaters $G,G_c$ which have proven very useful to locate phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and $G_c$ (for connected disorder averages) take the respective universal values 1/3 and 13/31 in the $T\to 0$ limit for any {\em finite} volume provided the ground state is {\em unique} and there is no gap in the ground state local-field distributions, conditions which are met in generic spin-glass models with continuous couplings and no gap at zero coupling. This makes $G,G_c$ ideal parameters to locate phase transitions in disordered systems much alike the Binder cumulant is for ordered systems. We check our results by exactly computing OPF in a simple example of uncoupled spins in the presence of random fields and the one-dimensional Ising spin glass. At finite temperatures, we discuss in which conditions the value 1/3 for G may be recovered by conjecturing different scenarios depending on whether OPF are finite or vanish in the infinite-volume limit. In particular, we discuss replica equivalence and its natural consequence $\lim_{V\to\infty}G(V,T)=1/3$ when OPF are finite. As an example of a model where OPF vanish and replica equivalence does not give information about G we study the Sherrington-Kirkpatrick spherical spin-glass model by doing numerical simulations for small sizes. Again we find results compatible with G=1/3 in the spin-glass phase.

Model glasses coupled to two different heat baths

In a $p$-spin interaction spherical spin-glass model both the spins and the couplings are allowed to change in the course of time. The spins are coupled to a heat bath with temperature $T$, while the coupling constants are coupled to a bath having temperature $T_{J}$. In an adiabatic limit (where relaxation time of the couplings is much larger that of the spins) we construct a generalized two-temperature thermodynamics. It involves entropies of the spins and the coupling constants. The application for spin-glass systems leads to a standard replica theory with a non-vanishing number of replicas, $n=T/T_J$. For $p>2$ there occur at low temperatures two different glassy phases, depending on the value of $n$. The obtained first-order transitions have positive latent heat, and positive discontinuity of the total entropy. This is the essentially non-equilibrium effect. The predictions of longtime dynamics and infinite-time statics differ only for $n<1$ and $p>2$. For $p=2$ correlation of the disorder (leading to a non-zero $n$) removes the known marginal stability of the spin glass phase. If the observation time is very large there occurs no finite-temperature spin glass phase. In this case there are analogies with the broken-ergodicity dynamics in the usual spin-glass models and non-equilibrium (aging) dynamics. A generalized fluctuation-dissipation relation is derived.

Quenched complexity of the mean-fieldp-spin spherical model with external magnetic field

We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one-step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what the general connection is between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. A dynamical analysis of the model confirms the validity of our results.

Spin-glass model with partially annealed asymmetric bonds

We have considered the two-spin interaction spherical spin-glass model with asymmetric bonds (coupling constants). Besides the usual interactions between spins and bonds and between the spins and a thermostat with temperature ${T}_{\ensuremath{\sigma}}$ there is also an additional factor: the bonds are not assumed random a priori but interact with some other thermostat at the temperature ${T}_{J}.$ We show that when the bonds are frozen with respect to the spins a first order phase transition to a spin-glass phase occurs, and the temperature of this transition tends to zero if ${T}_{J}$ is large. Our analytical results show that a spin-glass phase can exist in mean-field models with nonrelaxational dynamics.

Dynamical fluctuations in an exactly solvable model of spin glasses

In this work we calculate the dynamical fluctuations at O(1/N) in the low temperature phase of the $p=2$ spherical spin glass model. We study the large-times asymptotic regimes and we find, in a short time-differences regime, a fluctuation dissipation relation for the four-point correlation functions. This relation can be extended to the out of equilibrium regimes introducing a function $X_{t}$ which, for large time $t$, we find scales as $t^{-1/2}$ as in the case of the two-point functions.

Series expansion of the off-equilibrium mode coupling equations

We show that computing the coefficients of the Taylor expansion of the solution of the off-equilibrium dynamical equations characterizing models with quenched disorder is a very effective way to understand the long time asymptotic behavior. We study the $p=3$ spherical spin glass model, and we compute the asymptotic energy (in the critical region and down to $T=0$) and the coefficients of the time decay of the energy.

Inergodicity and dynamical phase transitions in the soft spherical spin-glass model.

The appearance of infinite relaxation times indicating the breaking of ergodicity in the spin-glass phase is demonstrated in the analytical study of finite-field dynamics of the soft spherical spin-glass model. The finite-field inergodic effects such as the difference between field-cooled and zero-field-cooled (ZFC) magnetizations and history dependence of a hysteresis loop were described for the case of the infinite-range random exchange. The existence of the dynamical phase transitions resulting from the jumps over finite microscopic barriers and trapping in the ZFC subspace were established in this case. The hysteresis loop form was analyzed for various temperatures, frequencies, and amplitudes of the external ac field.

Full dynamical solution for a spherical spin-glass model

We present a detailed analysis for the Langevin dynamics of a spherical spin-glass model (the spherical Sherrington-Kirkpatrick model). All the spins in the system are coupled by pairs via a random interaction matrix taken from the Gaussian ensemble. One finds that for a general initial configuration the system never reaches an equilibrium state and the theorems associated to `equilibrium dynamics' are violated. Only very particular initial conditions drive the system to equilibrium. The weak ergodicity breaking scenario is demonstrated for general `non-equilibrium' initial conditions. Two-time quantities such as the auto-correlation function explicitly depend on both times. When the time difference is short compared to the smaller time one finds stationary dynamics with time translation invariance and the fluctuation-dissipation theorem satisfied. Instead, when the time difference is of the order of the smaller time one finds non-stationary dynamics with aging phenomena, the system {\em remembers} the time spent after the initial time (the quench below the critical temperature). Interestingly enough the short time-difference dynamics (FDT regime) for non-equilibrium initial conditions is identical to the relaxation within the equilibrium states obtained for the particular `equilibrium' initial conditions. This points to a self-similaririty of the energy landascape. In addition we analyse the effect of temperature variations on the behaviour of the correlation function and energy density. In this way we are able to make some comparisons between experimentally observed results and exact calculations for the model. Somewhat surprisingly, this simple model captures some of the effects seen in laboratory spin-glasses.

Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model

We analyze the Thouless-Anderson-Palmer (TAP) approach to the spherical p-spin spin glass model in zero external field. The TAP free energy is derived by summing up all the relevant diagrams for N → ∞ of a diagrammatic expansion of the free energy. We find that if the multiplicity of the TAP solutions is taken into account, there is a first order transition in the order parameter at the critical temperature T c higher than that predicted by the replica solution T RSB , but in agreement with the results of dynamics. The transition is of geometrical nature since the new state has larger free energy but occupies the largest volume in phase space. The transition predicted by the replica calculation is also of geometrical nature since it corresponds to the states with smallest free energy with positive complexity.

Exactly solvable quantum magnets and spin glasses

The spherical model for a ferromagnet is exactly solvable, but the result is unrealistic at low temperatures. Here a quantum version is discussed, which has a fully acceptable low temperature behavior. Next a spherical spin glass model is discussed, that exhibits the full replica symmetry breaking properties of the Ising spin glass. It can be solved exactly in the whole low temperature phase.

Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model.

We study the non-equilibrium relaxation of the spherical spin-glass model with p-spin interactions in the $N \rightarrow \infty$ limit. We analytically solve the asymptotics of the magnetization and the correlation and response functions for long but finite times. Even in the thermodynamic limit the system exhibits `weak' (as well as `true') ergodicity breaking and aging effects. We determine a functional Parisi-like order parameter $P_d(q)$ which plays a similar role for the dynamics to that played by the usual function for the statics.

Spherical spin-glass models with short-range ferromagnetic interaction: Thermodynamics and correlation functions.

The free energy is solved for spherical spin-glass models with nearest-neighbor ferromagnetic and infinite-range random interactions, of either random-bond or random-site type. The long-range random-bond interaction destroys the ferromagnetic phase for 2&lt;d\ensuremath{\le}4, whereas the random-site interaction does not. The spin-glass--ferromagnetic phase-transition scaling laws, satisfied in the random-site model, are not valid in the random-bond model. Correlation functions of the order parameters are evaluated.

Free energies of the spherical model of a spin glass

Free energies g ( m, m s ) and f ( m, q ) of the spherical spin glass (SG) model due to Kosterlitz et al. are calculated explicitly as functions of the uniform magnetization m , and SG order parameter m s and the Edwards-Anderson order parameter q . It is shown that g (0, m s ) and f (0, q ) below the transition temperature T g are constant in the ranges 0 ≦ m s ≦ m s 0 and 0 ≦ q ≦ q 0 respectively, where q 0 = (1 -− T T g ) = m 2 s 0 . The proper equilibrium values of m s ( = m s 0 ) and q ( d = q 0 ) are then fixed from the inspection of their behaviors under infinitesimal uniform field proproportional to N -a ( a ≧ 0 ).

A quantum spherical model of spin glass

A quantum spherical model of spin glass is solved. The solution exhibits the usual spin glass phase transition, but with-out any unphysical features at low temperatures. In particular, the entropy is always positive and vanishes like T 3 as T → 0.

Mean-field theory of spin glasses: a spinodal analogy

The author compares known results of the Sherrington-Kirkpatrick model (1975) of a spin glass with the spinodal behaviour of a ferromagnet. This analogy suggests that the massless modes of the Ising spin glass are similar to the spinodal modes of the homogeneous case. Moreover, it seems possible to understand qualitatively the differences of the Ising and spherical spin glass models. A short dynamical study is also given.