Bulletin of the American Physical Society
APS March Meeting 2013
Volume 58, Number 1
Monday–Friday, March 18–22, 2013; Baltimore, Maryland
Session F29: Focus Session: Spin Glasses: Advances, Algorithms, and Applications |
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Sponsoring Units: GSNP Chair: Jonathan Machta, University of Massachusetts Amherst Room: 337 |
Tuesday, March 19, 2013 8:00AM - 8:36AM |
F29.00001: Is there a de Almeida-Thouless line in finite-dimensional spin glasses? Invited Speaker: Peter Young The question of whether there is a line of transitions in a magnetic field in an Ising spin glass (the de Almeida-Thouless, or AT, line) is important for two reasons: (i) its existence or otherwise is a major difference between the ``droplet'' and ``replica symmetry breaking (RSB)'' pictures of the spin glass state, and (ii) the spin glass in a field is argued to be quite similar to structural glasses, and, in this analogy, the spin glass AT line corresponds to the ``ideal glass'' transition of structural glasses. ``Standard'' finite-size scaling (FSS) methods do not find evidence for an AT line in three- or four-dimensional spin glasses. However, these results have been called into question by Leuzzi et al., Phys. Rev. Lett. 103, 267201 (2009) who perform a ``non-standard'' FSS analysis, in which they state that one should not include fluctuations at $k=0$ since these are argued to have larger corrections to FSS than $k > 0$ fluctuations. Using the ``non-standard'' analysis Leuzzi et al. find an AT line in four dimensions and also in a one-dimensional long-range model which is a proxy for four dimensions. In this talk I will describe results of large-scale Monte Carlo simulations for one-dimensional models which are proxies for three and for four dimensions, analyzed using both the ``standard'' and ``non-standard'' FSS approaches. I will also briefly discuss the merits of these two approaches to FSS. [Preview Abstract] |
Tuesday, March 19, 2013 8:36AM - 8:48AM |
F29.00002: Low-temperature behavior of the spin overlap distribution in one-dimensional long-range diluted spin glasses Matthew Wittmann, Helmut G. Katzgraber, J. Machta, A. P. Young Computer simulations of the spin glass state find that the \emph{average} order parameter distribution $P(q)$ has a weight in the region of small overlap $q$ which does not appear to decrease with size for the range of sizes that can be studied. This is in agreement with the ``Replica Symmetry Breaking'' (RSB) picture as opposed to the droplet picture which predicts $P(0)=0$ in the thermodynamic limit. Recently, a detailed study [1] has been made of peaks in $P(q)$ for \emph{individual samples} of a three-dimensional spin glass to gain more understanding of the situation. Here we pursue a similar approach but for long-range models in one dimension for which the interactions fall off with a power of the distance. Varying the power is analogous to varying the space dimension of a short-range model, so we can conveniently study models which are proxies for a \emph{range} of space dimensions. We will present results on the nature of the peaks in $P(q)$ for individual samples for several such models, and interpret them in terms of the RSB and droplet pictures. References: [1] Phys. Rev. Lett. 109, 177204 (2012) [Preview Abstract] |
Tuesday, March 19, 2013 8:48AM - 9:00AM |
F29.00003: Numerical evidence against both mean field and droplet scenarios of the Edwards-Anderson model Julio F. Fernandez, Juan J. Alonso From tempered Monte Carlo simulations, we have obtained accurate probability distributions $p(q)$ of the spin-overlap parameter $q$ for finite Edwards-Anderson (EA) and Sherrington-Kirkpatrick (SK) spin-glass systems at low temperatures. Our results for $p(q)$ follow from averages over $10^5$ disordered samples of linear sizes $L=4-8$ and over $15 \; 000$ samples for $L=10$. In both the SK and EA models, at temperatures as low as $0.2T_{sg}$, where $T_{sg}$ is the transition temperature, $p(q)$ varies insignificantly with $L$. This does not fit the trend that the droplet model predicts for large $L$. We have also calculated correlation functions, $F(q_1,q_2)$, from which rms deviations, $\delta p$, over different realizations of quenched disorder, as well as thermal fluctuations, $w$, of $q$ values, follow. Our numerical results for $\delta p$ and $w$ scale as $\sqrt{L}$ and $1/L$, respectively, in the SK model. This fits in well with mean field predictions. On the other hand, our data for $w$ and $\delta p$ vary little, if at all, for the EA model. [Preview Abstract] |
Tuesday, March 19, 2013 9:00AM - 9:12AM |
F29.00004: Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model Burcu Yucesoy, Helmut G. Katzgraber, Jonathan Machta The three and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. [Preview Abstract] |
Tuesday, March 19, 2013 9:12AM - 9:24AM |
F29.00005: Overlap distributions in two-dimensional spin glasses A. Alan Middleton Numerical results are presented for overlaps of configurations of two-dimensional Ising spin glasses. At low temperatures, the correlation length greatly exceeds the system size, so that spin-spin correlations are relatively long range and domain wall energies exhibit sensitive dependence to temperature,as seen in the low temperature phase of three-dimensional spin glasses. Exact sampling algorithms are used so that there is no doubt of equilibration. High statistics runs are carried out, with tens of thousands of samples of size $L^2=256^2$ simulated. The results of the size-dependent spin overlap distribution $P(q)$ are evaluated using statistics recently developed by Yucesoy, Katzgraber and Machta. The statistics for two-dimensional models at low temperature are found to be quite similar to those of three-dimensional spin glasses at finite temperatures below the spin-glass transition. [Preview Abstract] |
Tuesday, March 19, 2013 9:24AM - 9:36AM |
F29.00006: Monte Carlo Simulation of three dimensional Edwards Anderson model with multi-spin coding and parallel tempering using MPI and CUDA Sheng Feng, Ye Fang, Ka-Ming Tam, Bhupender Thakur, Zhifeng Yun, Karen Tomko, Juana Moreno, Jagannathan Ramanujam, Mark Jarrell The Edwards Anderson model is a typical example of random frustrated system. It has been a long standing problem in computational physics due to its long relaxation time. Some important properties of the low temperature spin glass phase are still poorly understood after decades of study. The recent advances of GPU computing provide a new opportunity to substantially improve the simulations. We developed an MPI-CUDA hybrid code with multi-spin coding for parallel tempering Monte Carlo simulation of Edwards Anderson model. Since the system size is relatively small, and a large number of parallel replicas and Monte Carlo moves are required, the problem suits well for modern GPUs with CUDA architecture. We use the code to perform an extensive simulation on the three-dimensional Edwards Anderson model with an external field. [Preview Abstract] |
Tuesday, March 19, 2013 9:36AM - 9:48AM |
F29.00007: Extremal Optimization for Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds Stefan Boettcher Using the Extremal Optimization heuristic (EO),\footnote{S. Boettcher \& A.G. Percus, {\it PRL} {\bf 86}, 5211 (2001)} ground states of the SK-spin glass are studied with bonds $J$ distributed according to a Levy distribution $P(J)\propto1/|J|^{1+\alpha}$ with $|J|>1$ and $1<\alpha<4.$ The variation of the energy densities with $\alpha$, their finite-size corrections, their fluctuations, and their local field distribution are analyzed and compared with those for the SK model with Gaussian bonds.\footnote{S. Boettcher, {\it Philosophical Magazine} {\bf 92}, 34 (2012)} We find that the energies attain universally the Parisi-energy of the SK when the second moment of $P\left(J\right)$ exists ($\alpha>2$). They compare favorably with recent one-step replica symmetry breaking predictions well below $\alpha=2$. Near $\alpha=2$, the simulations deviate significantly from theoretical expectations. The finite-size corrections exponent $\omega$ decays from the putative SK value $\omega_{SK}=\frac{2}{3}$ already well above $\alpha=2$. The exponent $\rho$ for the scaling of ground state energy fluctuations with system size decays linearly from its SK value for decreasing $\alpha$ and vanishes at $\alpha=1$. [Preview Abstract] |
Tuesday, March 19, 2013 9:48AM - 10:00AM |
F29.00008: Equilibrium and nonequilibrium properties of Boolean decision problems on scale-free graphs with competing interactions with external biases Zheng Zhu, Juan Carlos Andresen, Katharina Janzen, Helmut G. Katzgraber We study the equilibrium and nonequilibrium properties of Boolean decision problems with competing interactions on scale-free graphs in a magnetic field. Previous studies at zero field have shown a remarkable equilibrium stability of Boolean variables (Ising spins) with competing interactions (spin glasses) on scale-free networks. When the exponent that describes the power-law decay of the connectivity of the network is strictly larger than 3, the system undergoes a spin-glass transition. However, when the exponent is equal to or less than 3, the glass phase is stable for all temperatures. First we perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and show, in agreement with analytical calculations, that the system exhibits a de Almeida-Thouless line. Furthermore, we study avalanches in the system at zero temperature to see if the system displays self-organized criticality. This would suggest that damage (avalanches) can spread across the whole system with nonzero probability, i.e., that Boolean decision problems on scale-free networks with competing interactions are fragile when not in thermal equilibrium. [Preview Abstract] |
Tuesday, March 19, 2013 10:00AM - 10:12AM |
F29.00009: Self-organized criticality in glassy spin systems requires long-range interactions Juan Carlos Andresen, Ruben S. Andrist, Helmut G. Katzgraber, Vladimir Dobrosavljevic, Gergerly T. Zimanyi We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established in spin models only for the infinite-range Sherringtion-Kirkpatrick Ising spin-glass model [PRL 83, 1034 (1999)]. We study the avalanche and the magnetization jump distribution triggered by an external magnetic field in the short-range Edward-Anderson Ising spin glass for various space dimensions, between 2 and 8. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of infinite space dimensions (or equivalently of long-ranged interaction), and is not a generic property of spin-glass models in finite space dimensions. [Preview Abstract] |
Tuesday, March 19, 2013 10:12AM - 10:24AM |
F29.00010: Minimal spanning trees at the percolation threshold: a numerical calculation Sean Sweeney, A. Alan Middleton Through computer simulations on a hypercubic lattice, we grow minimal spanning trees (MSTs) in up to five dimensions and examine their fractal dimensions. Understanding MSTs is imporant for studying systems with quenched disorder such as spin glasses. We implement a combination of Prim's and Kruskal's algorithms for finding MSTs in order to reduce memory usage and allow for simulation of larger systems than would otherwise be possible. These fractal objects are analyzed in an attempt to numerically verify predictions of the perturbation expansion developed by T.~S.~Jackson and N.~Read for the pathlength fractal dimension $d_{s}$ of MSTs on percolation clusters at criticality [T.~S.~Jackson and N.~Read, Phys.\ Rev.\ E \textbf{81}, 021131 (2010)]. Examining these trees also sparked the development of an analysis technique for dealing with correlated data that could be easily generalized to other systems and should be a robust method for analyzing a wide array of randomly generated fractal structures. [Preview Abstract] |
Tuesday, March 19, 2013 10:24AM - 10:36AM |
F29.00011: Are the diluted antiferromagnet in a field and the random-field Ising model in the same universality class? Helmut G. Katzgraber, Bjoern Ahrens, Alexander K. Hartmann We perform large-scale Monte Carlo simulations using the Chayes-Machta and parallel-tempering algorithms to study the critical behavior of both the diluted antiferromagnet in a field (30\% dilution) and the random-field Ising model with Gaussian random fields for different field strengths. For small fields, analytical calculations by Cardy [Phys.~Rev.~B 29, 505 (1984)] predict that both models should share the same universality class. However, a detailed finite-size scaling analysis of both the Binder cumulant and the two-point finite-size correlation length suggests that even in the limit of small fields both models are not in the same universality class. Therefore, care should be taken when interpreting (experimental) data for diluted antiferromagnets in a field using the random-field Ising model. Finally, we present approximate analytical expressions based on our numerical data for the phase boundaries of both models. [Preview Abstract] |
Tuesday, March 19, 2013 10:36AM - 10:48AM |
F29.00012: Universality in the three-dimensional random-field Ising model Victor Martin-Mayor, Nikolaos Fytas We present the results of a large scale numerical simulation of the three-dimensional random-field Ising model at zero temperature. A combination of graph theoretical algorithms with a proper re-weighting scheme allows us to obtain data for systems with linear sizes $L \leq 192$ and extreme ensembles of disorder realizations, up to $5\times 10^{7}$. Three types of field distributions are considered, namely the Gaussian, the Poissonian, and the double Gaussian for two values of its width. In particular, for the double Gaussian case we choose parameters such that the distribution of random fields is bimodal. Our finite-size scaling analysis, based on the quotients method and universal quantities, indicates the existence of a unique random fixed-point. Therefore, the random-field Ising model is ruled by a single universality class, in disagreement with early mean-field theory predictions and the current opinion in the literature. The complete set of critical exponents characterizing this universality class is given, including the correction-to-scaling exponent $\omega$ and the violation of hyper-scaling exponent $\theta$. Finally, discrepancies with previous works are explained in terms of strong scaling corrections. [Preview Abstract] |
Tuesday, March 19, 2013 10:48AM - 11:00AM |
F29.00013: Competing Antiferromagnetic and Spin-Glass phases in a hollandite structure Yanier Crespo Hernandez, Alexei Andreanov, Nicola Seriani We introduce a simple model to explain recent experimental results on spin freezing in a hollandite-type structure. We argue that geometrical frustration of the lattice with antiferromagnetic (AFM) interactions is responsible for the appearance of a spin-glass phase in presence of disorder. We check our predictions numerically using parallel tempering on a model that considers Ising spins and nearest-neighbor AFM interactions. The proposed model presents a rich phenomenology: in absence of disorder two ground states are possible, depending on the strength of the interactions, namely an AFM or a geometrically frustrated phase. Remarkably for any set of AFM couplings having an AFM ground state in the clean system, there exist a critical value of the disorder for which the ground state is replaced by a spin-glass one while maintaining all couplings AFM. To the best of our knowledge in the literature there is not a model that presents this kind of transition considering just short-range AFM interactions. Therefore we argue that this model would be useful to understand the relation between AFM coupling, disorder and the appearance of spin glasses phase. [Preview Abstract] |
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