Multicanonical Study of the 3D Ising Spin Glass

Berg, Bernd A.; Çelik, Tarık; Hansmann, Ulrich H. E.

doi:10.1209/0295-5075/22/1/012

Cited by 40 publications

(32 citation statements)

References 19 publications

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“…A more sophisticated method is the multicanonical method [52], which is based on Monte Carlo simulations as well, but incorporates a reweighting scheme to speed up the simulation. Very low lying states of some systems up to size 123 have been obtained [53,54,55,56,571. Heuristics which are able to find low energy states but not true ground states unless the systems are very small can be found in Refs.…”

Section: Genetic Cluster-exact Approximationmentioning

confidence: 99%

Optimization Algorithms in Physics

Hartmann¹,

Rieger²

2001

249

297

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PrefaceThis book is an interdisciplinary book: it tries to teach physicists the basic knowledge of combinatorial and stochastic optimization and describes to the computer scientists physical problems and theoretical models in which their optimization algorithms are needed. It is a unique book since it describes theoretical models and practical situation in physics in which optimization problems occur, and it explains from a physicists point of view the sophisticated and highly efficient algorithmic techniques that otherwise can only be found specialized computer science textbooks or even just in research journals. Traditionally, there has always been a strong scientific interaction between physicists and mathematicians in developing physics theories. However, even though numerical computations are now commonplace in physics, no comparable interaction between physicists and computer scientists has been developed. Over the last three decades the design and the analysis of algorithms for decision and optimization problems have evolved rapidly. Most of the active transfer of the results was to economics and engineering and many algorithmic developments were motivated by applications in these areas. The few interactions between physicists and computer scientists were often successful and provided new insights in both fields. For example, in one direction, the algorithmic community has profited from the introduction of general purpose optimization tools like the simulated annealing technique that originated in the physics community. In the opposite direction, algorithms in linear, nonlinear, and discrete optimization sometimes have the potential to be useful tools in physics, in particular in the study of strongly disordered, amorphous and glassy materials. These systems have in common a highly non-trivial minimal energy configuration, whose characteristic features dominate the physics a t low temperatures. For a theoretical understanding the knowledge of the so called "ground states" of model Hamiltonians, or optimal solutions of appropriate cost functions, is mandatory. To this end an efficient algorithm, applicable to reasonably sized instances, is a necessary condition. The list of interesting physical problems in this context is long, it ranges from disordered magnets, structural glasses and superconductors through polymers, membranes, and proteins to neural networks. The predominant method used by physicists to study these questions numerically are Monte Carlo simulations and/or simulated annealing. These methods are doomed to fail in the most interesting situations. But, as pointed out above, many useful results in optimization algorithms research never reach the physics community, and interesting computational problems in physics do not come to the attention of algorithm designers. We therefore think that there is a definite need VI Preface to intensify the interaction between the computer science and physics communities. We hope that this book will help to extend the bridge between these two groups. Since one...

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Section: Genetic Cluster-exact Approximationmentioning

confidence: 99%

Optimization Algorithms in Physics

Hartmann¹,

Rieger²

2001

249

297

View full text Add to dashboard Cite

show abstract

“…Finally, by re-weighting with exp[−E/T + β(E)E − α(E)], the canonical distribution can be reconstructed. With regards to spin glasses, this ensemble has proven to be useful in particular for the investigation of ground state properties [19,20,21] (for this problem see also the above mentioned works [111,10,281,269,276,125,288]). Cluster algorithms as suggested by Swendsen and Wang [284] have become very useful in the investigation of pure systems (see [285] for an overview).…”

Section: Numerical Recipesmentioning

confidence: 99%

Monte Carlo Studies of Ising Spin Glasses and Random Field Systems

Rieger

1995

Annual Reviews of Computational Physics II

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We review recent numerical progress in the study of finite dimensional strongly disordered magnetic systems like spin glasses and random field systems. In particular we report in some details results for the critical properties and the non-equilibrium dynamics of Ising spin glasses. Furthermore we present an overview over recent investigations on the random field Ising model and finally of quantum spin glasses.

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“…The method also has been successfully applied to some complex systems, such as spin glass models [7,[14][15][16][17][18] and protein folding problems [8], for which the energy landscape is very rough and the conventional canonical Monte Carlo simulation gets easily trapped in local minima at low temperature. In the multicanonical method, we have to estimate the density of states g(E) first, then perform a random walk with a probability of 1 g(E) to make the histogram flat in the desired region in the phase space, such as between two peaks of the canonical distribution at the first-order transition temperature.…”

Section: Introductionmentioning

confidence: 99%

Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram

2001

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We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at essentially any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but can also estimate the Gibbs free energy and entropy, quantities which are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both 1st and 2nd order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 × 200 and Ising models on lattices up to 256 × 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a 3D ±J spin glass model we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space. 05.50.+q, 64.60.Cn, 02.70.Lq

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Multicanonical Study of the 3D Ising Spin Glass

Cited by 40 publications

References 19 publications

Optimization Algorithms in Physics

Optimization Algorithms in Physics

Monte Carlo Studies of Ising Spin Glasses and Random Field Systems

Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram

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