Low-temperature solution of the Sherrington-Kirkpatrick model

I propose a simple scaling ansatz for the full replica symmetry breaking solution of the Sherrington-Kirkpatrick model in the low energy sector. This solution is shown to become exact in the limit x --> 0, Bx --> infinity of the Parisi replica symmetry breaking scheme parameter . The distribution function of the frozen fields has been known to develop a linear gap at zero temperature. The scaling equations are integrated to find an exact numerical value for the slope of the gap thetaP(x,y)/delta|(y --> 0) = 0.301 046.... I also use the scaling solution to devise an inexpensive numerical procedure for computing finite time scale (x =1) quantities. The entropy, the zero field cooled susceptibility, and the local field distribution function are computed in the low-temperature limit with high precision, barely achievable by currently available methods.     

Free-energy fluctuations and chaos in the Sherrington-Kirkpatrick model

The sample-to-sample fluctuations Delta FN of the free-energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent mu governing the growth of the fluctuations with system size N, Delta FN approximately Nmu, is bounded by mu< or =1/4.     

The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter, thus formulating the thermodynamic formalism in terms of random processes. The well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments, a generalization of a result proved by Aizenman, Lebowitz and Ruelle [1].     

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

We prove that if _N is the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity converges in the variance to a nonrandom limit asN, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the non-self-averaging of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN, i.e., the free energy has the self-averaging property.See the Remarks after the proof of Theorem 1 on the validity of our results for more general distributions ofJ _ij .     

Some rigorous results on the Sherrington-Kirkpatrick spin glass model

We prove that in the high temperature regime (T/J>1) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed log Av(Z _N ({J})) converges in distribution, asN , to a (shifted) Gaussian variable. Some weak results about the low temperature regime are also obtained.Dedicated to Walter Thirring on his 60^th birthdayResearch supported in part by the NSF grant PHY-8605164. Present address: Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USAResearch supported in part by the NSF grant DMR 86-12369On leave from Institut des Hautes Etudes Scientifiques, Bures-Sur-Yvette, France     

Some comments on the Sherrington-Kirkpatrick model of spin glasses

In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated explicitly for the SK model, and uniform bounds on quenched susceptibilities are established. It is also shown that, at high temperatures, mean-field spin glasses are limits of short-range spin glasses, as the range of the interactions tends to infinity.     

Dynamics in the Sherrington-Kirkpatrick model. I. The first step

We study properties of the random configuration {s _j (1)} _j=1 ^N produced by the first step of the parallel dynamics in the Sherrington-Kirkpatrick model. We show that the law of large numbers holds for the sequence of overlaps between the initial (nonrandom) configuration {s _j (0)} _j=1 ^N and {s _j (1)} _j=1 ^N , and obtain the distribution of the fluctuations around the limiting value. As a by-product we derive the average number of the fixed points {s _j (1)} _j=1 ^N with a given value of the magnetization .     

Scaling and renormalization group in replica-symmetry-breaking space: evidence for a simple analytical solution of the Sherrington-Kirkpatrick model at zero temperature

Using numerical self-consistent solutions of a sequence of finite replica symmetry breakings (RSB) and Wilson's renormalization group but with the number of RSB steps playing a role of decimation scales, we report evidence for a nontrivial T-->0 limit of the Parisi order function q(x) for the Sherrington-Kirkpatrick spin glass. Supported by scaling in RSB space, the fixed point order function is conjectured to be q*(a)=sqrt[pi]/2 a/xi erf(xi/a) on 0 a at T =0 and xi approximately 1.13+/-0.01. Xi plays the role of a correlation length in a-space. q*(a) may be viewed as the solution of an effective 1D field theory.     

Double criticality of the Sherrington-Kirkpatrick model at T=0

Numerical results up to the 42nd order of replica-symmetry breaking (RSB) are used to predict the singular structure of the Sherrington-Kirkpatrick spin glass at T=0. We confirm predominant single parameter scaling and derive corrections for the T=0 order function q(a), related to a Langevin equation with pseudotime 1/a. a=0 and a=infinity are shown to be two critical points for infinity-RSB, associated with two discrete spectra of Parisi block size ratios, attached to a continuous spectrum. Finite-RSB-size scaling, associated exponents, and T=0-energy are obtained with unprecedented accuracy.     

The Sherrington-Kirkpatrick spin glass model: Results of a new theory

A new solution of the Sherrington-Kirkpatrick spin glass model is discussed which is valid below the temperature where the replica method becomes unstable. The free energy is lowered, the entropy has no unphysical properties forT 0. The specific heat approaches zero exponentially and has a second maximum near a lower critical temperature for given external field. Near the critical point the difference of the free energy to that of Sherrington and Kirkpatrick is extremely small. Thus the Sherrington-Kirkpatrick phase may be undercooled.     

Chaos in generalized Jaynes-Cummings model

The possibility of chaos formation is studied in terms of a generalized Jaynes-Cummings model which is a key model in the quantum electrodynamics of resonators. In particular, the dynamics of a three-level optical atom which is under the action of the resonator field is considered. The specific feature of the considered problem consists in that not all transitions between the atom levels are permitted. This asymmetry of the system accounts for the complexity of the problem and makes it different from the three-level systems studied previously. We consider the most general case, where the interaction of the system with the resonator depends on the system coordinate inside the resonator. It is shown that, contrary to the commonly accepted opinion, the absence of resonance detuning does not guarantee the system state controllability. In the course of evolution the system performs an irreversible transition from the purely quantum-mechanical state to the mixed state. It is shown that the asymmetry of the system levels accounts for the fact that the upper excited level turns out to be the most populated one.     

Percolation model, Sherrington-Kirkpatrick model and localization-delocalization model of spin glass transition: A comparative study

We recast the original results of our percolation model of the spin glass transition into alternative forms so as to reveal some apparent “similarities” with the Sherrington-Kirkpatrick model. However, because of the basic difference in the nature of the interactions in the two models, the reason for these “similarities” remains unclear. A close relation between the percolation model of the spin glass transition and the localization-decolization transition is also discussed qualitatively.     

Chaos and limit cycles in the Lorenz model

The Lorenz model is interpreted as a damping motion under a time-dependent force. The range of the Rayleigh number r in which limit cycles exist is studied by numerical simulation. The shape of the limit cycle is given.