Department of Basic Science

The cumulant generating function φ(n) and rate function Σ(f ) of free energy is evaluated in the p-body Sherrington-Kirkpatrick model by the replica method with the finite replica number n. From a perturbational argument, we show that the cumulant generating function is constant in the vicinity of n = 0. On the other hand, with the help of two analytic properties of φ(n), the behavior of φ(n) is derived again. However, this is also shown to be broken at a finite value of n, which gives a characteristic value in the rate function near the thermodynamic value of the free energy. Through the continuation of φ(n) as a function of n, we find a way to derive the 1RSB solution at least in this model, which is to fix the RS solution to be a monotonically increasing function.

We study phase transition of a nonequilibrium statistical-mechanical model, in which two degrees of freedom with different time scales separated from each other touch to their own heat bath. A general condition for finding anomalous negative latent heat recently discovered is derived a from thermodynamic argument. As a specific example, phase diagram of a spin-lattice gas model is studied based on a mean-field analysis with replica method. While configurational variables are spin and particle in this model, it is found that the negative latent heat appears in a parameter region of the model, irrespective of the order of their time scale. Qualitative differences in the phase diagram are also discussed.

The Néel temperature T(N) of quasi-one- and quasi-two-dimensional antiferromagnetic Heisenberg models on a cubic lattice is calculated by Monte Carlo simulations as a function of interchain (interlayer) to intrachain (intralayer) coupling J(')/J down to J(')/J approximately = 10(-3). We find that T(N) obeys a modified random-phase approximationlike relation for small J(')/J with an effective universal renormalized coordination number, independent of the size of the spin. Empirical formulas describing T(N) for a wide range of J(') and useful for the analysis of experimental measurements are presented.

We study response to a twist in the two-dimensional p-state clock model, which has the discrete Zp symmetry. The response is measured in terms of helicity modulus, which is usually defined with respect to an infinitesimal twist. However, we demonstrate that such a definition is inappropriate for the clock model. The helicity modulus must be defined with respect to a finite, quantized twist which matches the discrete Zp symmetry of the model. Recent numerical results, which casted doubt on the standard picture that two Berezinskii-Kosterlitz-Thouless transitions occur for p > 4, are rather a consequence of use of the inappropriate quantity. Numerical calculation of the appropriately defined helicity modulus indeed supports the standard picture.

The presence (or absence) of spin glass phase in uniform magnetic field is studied in the three and four dimensional Edwards-Anderson Ising spin glass. Effective coupling J eff and effective field H eff of each sample are measured by a numerical renormalization-group method, and their sample-to-sample fluctuations s J and s H are estimated for various sizes L and fields H. As found in the Migdal-Kadanoff spin glass, s J and s H are scaled as a function of L='ðHÞ, where 'ðHÞ ¼ H À1=z is the so-called overlap length. Furthermore, our data show that the scaling function of s J drops to zero in the limit L='ðHÞ ! 1. These results indicate the absence of spin glass phase even in an infinitesimal field. r

Recently we proposed a simple statistical-mechanical model of interacting spins for studying evolutional process in biological system. In our model, a configuration of spins and their interactions J represent a phenotype and genotype, respectively. The phenotype dynamics are given by a stochastic process with temperature TS under a fixed Hamiltonian with J . Meanwhile, the evolution of J is also stochastic with temperature TJ and follows mutations introduced into J and selection based on a fitness given by equilibrium spin configurations of a given set of target spins. Using Monte Carlo simulations, it is found that frustration around the target spins is strongly suppressed for the interactions J evolved in the intermediate TS temperature region. The evolved J 's give the funnel-like dynamics and show robustness to mutations. This model can be regarded as an extension of partial annealing model, in which the interactions between spins are dynamical variables with characteristic time scale widely separated from that of the spins. We also discuss the partially annealed Sherrington-Kirkpatrick model by particularly paying attention to the frustration.

We develop a recently proposed importance-sampling Monte Carlo algorithm for sampling rare events and quenched variables in random disordered systems. We apply it to a two dimensional bond-diluted Ising model and study the Griffiths singularity which is considered to be due to the existence of rare large clusters. It is found that the distribution of the inverse susceptibility has an exponential tail down to the origin which is considered the consequence of the Griffiths singularity.

We investigate the distribution of zeros of the partition function of the two-and three-dimensional symmetric ±J Ising spin glasses on the complex field plane. We use the method to analytically implement the idea of numerical transfer matrix which provides us with the exact expression of the partition function as a polynomial of fugacity. The results show that zeros are distributed in a wide region in the complex field plane. Nevertheless we observe that zeros on the imaginary axis play dominant roles in the critical behaviour since zeros on the imaginary axis are in closer proximity to the real axis. We estimate the density of zeros on the imaginary axis by an importance-sampling Monte Carlo algorithm, which enables us to sample very rare events. Our result suggests that the density has an essential singularity at the origin. This observation is consistent with the existence of Griffiths singularities in the present systems. This is the first evidence for Griffiths singularities in spin glass systems in equilibrium.

The temperature (T )-shift protcol of aging in the 3 dimensional (3D) Edwards-Anderson (EA) spin-glass (SG) model is studied through the out-of-phase component of the ac susceptibility simulated by the Monte Carlo method. For processes with a small magnitude of the T -shift, ∆T , the memory imprinted before the T -shift is preserved under the T -change and the SG short-range order continuously grows after the T -shift, which we call the cumulative memory scenario. For a negative T -shift process with a large ∆T the deviation from the cumulative memory scenario has been observed for the first time in the numerical simulation. We attribute the phenomenon to the 'chaos effect' which, we argue, is qualitatively different from the so-called rejuvenation effect observed just after the T -shift.

We propose an efficient Monte Carlo algorithm for simulating a "hardlyrelaxing" system, in which many replicas with different temperatures are simultaneously simulated and a virtual process exchanging configurations of these replica is introduced. This exchange process is expected to let the system at low temperatures escape from a local minimum. By using this algorithm the three-dimensional ±J Ising spin glass model is studied. The ergodicity time in this method is found much smaller than that of the multicanonical method. In particular the time correlation function almost follows an exponential decay whose relaxation time is comparable to the ergodicity time at low temperatures. It suggests that the system relaxes very rapidly through the exchange process even in the low temperature phase.

Spin-glass (SG) states of the 3-dimensional Ising Edwards-Anderson model under a static magnetic field h are examined by means of the standard Monte Carlo simulation on the fieldshift aging protocol at temperature T . For each process with (T ; tw, h), tw being the waiting time before the field is switched on, we extract the dynamical crossover time, tcr(T ; tw, h). We have found a nice scaling relation between the two characteristic length scales which are properly determined from tcr and tw and then are normalized by the static field crossover length introduced in the SG droplet theory. This scaling behavior implies the instability of the SG phase in the equilibrium limit even under an infinitesimal h. In comparison with this numerical result the field effect on real spin glasses is also discussed.

An one-step replica-symmetry-breaking solution for finite connectivity spin-glass models with K body interaction is constructed at finite temperature using the replica method and thermodynamic constraints. In the absence of external fields, this construction provides a general extension of replica symmetric solution at finite replica number to one-step replica-symmetry-breaking solution. It is found that this result is formally equivalent to that of the one-step replica-symmetry-breaking cavity method. To confirm the validity of the obtained solution, Monte Carlo simulations are performed for K = 2 and 3. The thermodynamic quantities of the Monte Carlo results extrapolated to a large-size limit are consistent with those estimated by our solution for K = 2 at all simulated temperatures and for K = 3 except near the transition temperature.

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble (GOE), sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of ∼ 10 −200 , a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.

A new protocol of the zero-field-cooled (ZFC) magnetization process is studied experimentally on an Ising spin-glass (SG) Fe0.50Mn0.50TiO3 and numerically on the Edwards-Anderson Ising SG model. Although the time scales differ very much between the experiment and the simulation, the behavior of the ZFC magnetization observed in the two systems can be interpreted by means of a common scaling expression based on the droplet picture. The results strongly suggest that the SG coherence length, or the mean size of droplet excitations, involved even in the experimental ZFC process, is about a hundred lattice distances or less.

Domain-wall free-energy δF , entropy δS, and the correlation function, Ctemp, of δF are measured independently in the four-dimensional ±J Edwards-Anderson (EA) Ising spin glass. The stiffness exponent θ, the fractal dimension of domain walls ds and the chaos exponent ζ are extracted from the finite-size scaling analysis of δF , δS and Ctemp respectively well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation ζ = ds/2 − θ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.

From a consideration of high temperature series expansions in ferromagnets and in spin glasses, we propose an extended scaling scaling scheme involving a set of scaling formulae which express to leading order the temperature (T ) and the system size (L) dependences of thermodynamic observables over a much wider range of T than the corresponding one in the conventional scaling scheme. The extended scaling, illustrated by data on the canonical 2d ferromagnet and on the 3d bimodal Ising spin glass, leads to consistency for the estimates of critical parameters obtained from scaling analyses for different observables.

The stability of the spin-glass phase against a magnetic field is studied in the three and four dimensional Edwards-Anderson Ising spin glasses. Effective couplings J eff and effective fields H eff associated with length scale L are measured by a numerical domain-wall renormalization group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H → 0 but remains finite for any non-zero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.

Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. typeset using PTPT E X.sty <ver.1.0>

The relaxational modes and aging of the Glauber dynamics of the mean-field spin-glass model, SK model, are studied by a numerical diagonalization technique and Monte Carlo simulations. We found that the aging process of the model is understood as hierarchical growth of quasi-equilibrium domain in the phase-space.