Session Y52: Focus Session: Spin Glasses: Advances, Algorithms, and Applications

How the Edwards-Anderson Model reaches its Mean-Field Limit; Simulations in d=3,...,7

Extensive computations of ground state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions $d=3,\ldots,7$. Results are presented for bond-densities exactly at the percolation threshold, $p=p_{c}$, and deep within the glassy regime, $p>p_{c}$, where finding ground-states becomes a hard combinatorial problem. The ``stiffness'' exponent $y$ that controls the formation of domain wall excitations at low temperatures is determined in all dimensions. Finite-size corrections of the form $1/N^{\omega}$ are shown to be consistent throughout with the prediction $\omega=1-y/d$. At $p=p_{c}$, an extrapolation for $d\to\infty$ appears to match our mean-field results for these corrections. In the glassy phase, $\omega$ does not approach the value of $2/3$ for large $d$ predicted from simulations of the Sherrington-Kirkpatrick spin glass. However, the value of $\omega$ reached at the upper critical dimension \emph{does} match certain mean-field spin glass models on sparse random networks of regular degree called Bethe lattices.\\[4pt] [1] S. Boettcher and S. Falkner, arXiv:1110.6242;\hfil\break [2] S. Boettcher and E. Marchetti, PRB77, 100405 (2008);\hfil\break [3] S. Boettcher, PRL95, 197205 (2005).   

Ensemble Inequivalence in Spin Glasses

We report on the ensemble inequivalence in many-body spin-glass models with Ising and integer spins. In the Ising case, for many-body interactions the transition between the ferromagnetic and paramagnetic phases is of first order, and the microcanonical and canonical ensembles give different results. The spin-glass transition is of first order for certain values of the crystal field strength in the integer-spin model and is dependent whether it was derived in the microcanonical or the canonical ensemble. We also discuss the ensemble inequivalence of random energy models, corresponding to the limit infinitely many-body interactions. This is the first systematic treatment of spin glasses with long-range interactions in the microcanonical ensemble using the replica approach, which shows how the two ensembles give different results.   

Algorithms and long-range order in the two-dimensional +/-J spin glass

Numerical methods and results of their application to the two-dimensional Ising spin glasses will be described. For a random mix of ferromagnetic and antiferromagnetic bonds of equal strength, long range correlations at zero-temperature are derived from scaling relations between computed exponents and are confirmed in numerical simulations. This long range order is stabilized by large entropy differences, as large domain walls often have zero energy cost. The order resembles that in higher-dimensional models at finite temperature. A publically distributed implementation of the algorithms has been developed for computing partition functions and exactly sampling configurations according to their Boltzmann weight for the general spin-glass and related two-dimensional models.   

Replica theory of partition-function zeros in spin-glass systems

We study the phase transitions in spin-glass systems by analysing the partition-function zeros (Lee-Yang zeros) with respect to the complex temperature/field. For several models as the random energy and spherical models with many-body interactions, we extend the replica method and the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex plane and calculate the density of zeros in each phase. We find that there is a replica symmetric phase having a large density near the imaginary axis away from the origin. In the spin-glass phase, the density is finite only when the chaos effect is present. This result indicates that the density of zeros is more closely connected to the chaos effect than the replica symmetry breaking. We also investigate the relevance of our result to the finite-dimensional systems by studying the renormalization group flow in the complex plane.   

Aging behavior in disordered and frustrated spin systems

Using Monte Carlo simulations we investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length $L(t)$ is determined numerically and the scaling behavior of various two-time quantities as a function of $L(t)/L(s)$ is discussed. For disordered Ising models deviations of $L(t)$ from the algebraic growth law show up. The generalized scaling forms as a function of $L(t)/L(s)$ reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets.   

Critical behavior of the 1D L{\'e}vy lattice spin-glass: from mean-field threshold to the effective lower critical dimension

By means of Monte Carlo numerical simulations we analyze the critical behavior of a one dimensional spin-glass model with diluted interactions decaying, in probability, as an inverse power of the distance: the L{\'e}vy lattice spin-glass. Varying the power $\rho$, corresponds to change the effective dimension from mean-field-like (small power $\rho<4/3$) to finite dimensional-like short-range models ($<4/3\rho<2$) and, eventually, to 1D short-range models ($\rho>2$), where no phase transition occurs. The bond diluteness drastically reduces the computational time and large sizes can be approached. The one dimensionality allows for studying long systems, e.g., long correlation lenghts in the critical region. The spin-glass critical behavior can, therefore, be studied in and out of the range of validity of the mean-field approximation. After reviewing the main results in the L{\'e}vy lattice model about the spin-glass transition and the nature of the spin-glass phase for different values of the effective dimension, we will present new results on the critical behavior at $\rho=2$, corresponding to the lower critical dimension, and compare them with old and recent renormalization group approaches in this limit.   

Spin glasses: Still frustrating after all these years?

Spin glasses are archetypal model systems to study the effects of frustration and disorder. Despite ongoing research spanning several decades, there remain many fundamental open questions, such as the existence of a spin-glass state in a field or the low-temperature structure of phase space for short-range systems. Novel applications across disciplines, as well as progress in algorithms and the advent of fast and cost-effective computers, have recently revived interest in the study of spin glasses. First, an overview of spin glasses will be given, followed by recent novel applications to fields as diverse as structural glasses and quantum computing.   

The de Almeida-Thouless line of the four-dimensional Ising spin glass

We present the results of a large scale numerical simulation of the four dimensional Edwards-Anderson model in an external field. Using the Janus computer, as well as standard CPU clusters, we simulate lattices of size up to L=16 at several values of the external field. Our analysis method departs from the standard one. In fact, it has been previously noticed that the spin-glass susceptibility (i.e. the spin-glass propagator at zero external momentum) behaves anomalously. Instead, one should focus on the propagator at small but non-vanishing wave-vector. Starting from this observation, we obtain a simple and powerful finite-size scaling method. Clear evidence for a de Almeida-Thouless line is found. We compute critical exponents, widely differing from the zero field case, with an accuracy of five percent. The shape of the de Almeida-Thouless line in the (T,h) plane follows the Fisher-Sompolinsky scaling. Discrepancies with previous work are explained in terms of very strong scaling corrections.   

Reentrance and ultrametricity in three-dimensional Ising spin glasses

We study the three-dimensional Edwards-Anderson Ising spin glass with bimodal disorder with a fraction of 22.8\% antiferromagnetic bonds. Parallel tempering Monte Carlo simulations down to very low temperatures show that for this fraction of antiferromagnetic bonds the phase diagram of the system is reentrant, in agreement with previous results. Furthemore, using a clustering analysis, we analyze the ultrametric properties of phase space for this model.   

Replica exchange simulations of the three-dimensional Ising spin glass: static and dynamic properties

We present the results of a large-scale numerical study of the equilibrium three-dimensional Ising spin glass with Gaussian disorder. Using replica exchange (parallel tempering) Monte Carlo, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the replica exchange Monte Carlo method. The correlation between static and dynamic observables for 5000 disorder realizations ($N \le 10^3$ spins) down to very low temperatures ($T \approx 0.2T_c$) is examined. Our results show that autocorrelation times are directly correlated with the roughness of the free energy landscape. We also discuss the size dependence of several static quantities.   

Monte Carlo Simulations of Random Frustrated Systems on Graphics Processing Units

We study the implementation of the classical Monte Carlo simulation for random frustrated models using the multithreaded computing environment provided by the the Compute Unified Device Architecture (CUDA) on modern Graphics Processing Units (GPU) with hundreds of cores and high memory bandwidth. The key for optimizing the performance of the GPU computing is in the proper handling of the data structure. Utilizing the multi-spin coding, we obtain an efficient GPU implementation of the parallel tempering Monte Carlo simulation for the Edwards-Anderson spin glass model. In the typical simulations, we find over two thousand times of speed-up over the single threaded CPU implementation.   

Monte Carlo simulations of the ${\rm LiHo_xY_{1-x}F_4}$ diluted dipolar magnet

Recent intriguing experimental results on ${\rm LiHo_xY_{1-x}F_4}$, a diluted dipolar magnet, along with new analytical insights, suggest that neither a mean-field treatment nor simulations using simplified versions of the underlying Hamiltonian adequately describe these materials. Not only does this imply that novel disordering mechanism might be present, it requires a detailed numerical analysis that incorporates all terms in the Hamiltonian. We present large-scale Monte Carlo simulations of the diluted dipolar magnet with competing interactions on a ${\rm LiHo}$ lattice with the inclusion of a random field term. For low concentrations of ${\rm Ho}$ atoms we reproduce the peculiar linear dependence of the transition temperature as a function of the random-field strength found in recent experimental results by Silevich {\em et al}.~[Nature {\bf 448}, 567 (2007)]. We then find a zero-temperature phase transition between the ferromagnetic and quasi-spin-glass phases, suggesting that it is the underlying spin-glass phase that dictates the above linear dependence of $T_c$ on the random field. For large concentrations we recover the quadratic dependence of the critical temperature as a function of the random field strength.   

Novel disordering mechanisms in dipolar spin glasses and ferromagnets

At and below the critical dimension the disordering of an ordered phase by a random field occurs via a collective effect of large domains at infinitesimal random field [Imry \& Ma, Phys.~Rev.~Lett.~{\bf 35}, 1399 (1975)]. At larger space dimensions the disordering requires a large random field, of the order of the interaction energy. In a random field, the lower critical dimension is 2 for Ising ferromagnets, whereas it is infinity for spin glasses. We have generalized the Imry-Ma argument for ferromagnets with competing interactions and an underlying spin-glass phase, and for dilute dipolar spin glasses. For dilute dipolar spin glasses we have found [EPL~{\bf 88}, 66002 (2009)] that the broad distribution of random fields dictates more efficient disordering of the glass phase, and domain sizes which depend explicitly on the concentration, i.e., do not obey simple scaling. Here we show that as a result of a competing spin-glass phase, the disordering of the ferromagnet occurs at a finite random field, which is yet much smaller than the interactions. Our results are verified numerically, explain the recently-observed peculiar linear dependence of $T_c$ on the random field strength [Nature~{\bf 448}, 567 (2007)], and predict a zero-temperature random-field driven transition between a ferromagnetic and a quasi spin glass phase.