658 research outputs found
Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations
We present an adaptive algorithm which optimizes the statistical-mechanical
ensemble in a generalized broad-histogram Monte Carlo simulation to maximize
the system's rate of round trips in total energy. The scaling of the mean
round-trip time from the ground state to the maximum entropy state for this
local-update method is found to be O([N log N]^2) for both the ferromagnetic
and the fully frustrated 2D Ising model with N spins. Our new algorithm thereby
substantially outperforms flat-histogram methods such as the Wang-Landau
algorithm.Comment: 6 pages, 5 figure
Dynamics of the Wang-Landau algorithm and complexity of rare events for the three-dimensional bimodal Ising spin glass
We investigate the performance of flat-histogram methods based on a
multicanonical ensemble and the Wang-Landau algorithm for the three-dimensional
+/- J spin glass by measuring round-trip times in the energy range between the
zero-temperature ground state and the state of highest energy. Strong
sample-to-sample variations are found for fixed system size and the
distribution of round-trip times follows a fat-tailed Frechet extremal value
distribution. Rare events in the fat tails of these distributions corresponding
to extremely slowly equilibrating spin glass realizations dominate the
calculations of statistical averages. While the typical round-trip time scales
exponential as expected for this NP-hard problem, we find that the average
round-trip time is no longer well-defined for systems with N >= 8^3 spins. We
relate the round-trip times for multicanonical sampling to intrinsic properties
of the energy landscape and compare with the numerical effort needed by the
genetic Cluster-Exact Approximation to calculate the exact ground state
energies. For systems with N >= 8^3 spins the simulation of these rare events
becomes increasingly hard. For N >= 14^3 there are samples where the
Wang-Landau algorithm fails to find the true ground state within reasonable
simulation times. We expect similar behavior for other algorithms based on
multicanonical sampling.Comment: 9 pages, 12 figure
Wang-Landau sampling for quantum systems: algorithms to overcome tunneling problems and calculate the free energy
We present a generalization of the classical Wang-Landau algorithm [Phys.
Rev. Lett. 86, 2050 (2001)] to quantum systems. The algorithm proceeds by
stochastically evaluating the coefficients of a high temperature series
expansion or a finite temperature perturbation expansion to arbitrary order.
Similar to their classical counterpart, the algorithms are efficient at thermal
and quantum phase transitions, greatly reducing the tunneling problem at first
order phase transitions, and allow the direct calculation of the free energy
and entropy.Comment: Added a plot showing the efficiency at first order phase transition
Performance Limitations of Flat Histogram Methods and Optimality of Wang-Landau Sampling
We determine the optimal scaling of local-update flat-histogram methods with
system size by using a perfect flat-histogram scheme based on the exact density
of states of 2D Ising models.The typical tunneling time needed to sample the
entire bandwidth does not scale with the number of spins N as the minimal N^2
of an unbiased random walk in energy space. While the scaling is power law for
the ferromagnetic and fully frustrated Ising model, for the +/- J
nearest-neighbor spin glass the distribution of tunneling times is governed by
a fat-tailed Frechet extremal value distribution that obeys exponential
scaling. We find that the Wang-Landau algorithm shows the same scaling as the
perfect scheme and is thus optimal.Comment: 5 pages, 6 figure
Optimized broad-histogram simulations for strong first-order phase transitions: Droplet transitions in the large-Q Potts model
The numerical simulation of strongly first-order phase transitions has
remained a notoriously difficult problem even for classical systems due to the
exponentially suppressed (thermal) equilibration in the vicinity of such a
transition. In the absence of efficient update techniques, a common approach to
improve equilibration in Monte Carlo simulations is to broaden the sampled
statistical ensemble beyond the bimodal distribution of the canonical ensemble.
Here we show how a recently developed feedback algorithm can systematically
optimize such broad-histogram ensembles and significantly speed up
equilibration in comparison with other extended ensemble techniques such as
flat-histogram, multicanonical or Wang-Landau sampling. As a prototypical
example of a strong first-order transition we simulate the two-dimensional
Potts model with up to Q=250 different states on large systems. The optimized
histogram develops a distinct multipeak structure, thereby resolving entropic
barriers and their associated phase transitions in the phase coexistence region
such as droplet nucleation and annihilation or droplet-strip transitions for
systems with periodic boundary conditions. We characterize the efficiency of
the optimized histogram sampling by measuring round-trip times tau(N,Q) across
the phase transition for samples of size N spins. While we find power-law
scaling of tau vs. N for small Q \lesssim 50 and N \lesssim 40^2, we observe a
crossover to exponential scaling for larger Q. These results demonstrate that
despite the ensemble optimization broad-histogram simulations cannot fully
eliminate the supercritical slowing down at strongly first-order transitions.Comment: 11 pages, 12 figure
Quantitative Determination of Temperature in the Approach to Magnetic Order of Ultracold Fermions in an Optical Lattice
We perform a quantitative simulation of the repulsive Fermi-Hubbard model using an ultracold gas trapped in an optical lattice. The entropy of the system is determined by comparing accurate measurements of the equilibrium double occupancy with theoretical calculations over a wide range of parameters. We demonstrate the applicability of both high-temperature series and dynamical mean-field theory to obtain quantitative agreement with the experimental data. The reliability of the entropy determination is confirmed by a comprehensive analysis of all systematic errors. In the center of the Mott insulating cloud we obtain an entropy per atom as low as 0.77k(B) which is about twice as large as the entropy at the Neel transition. The corresponding temperature depends on the atom number and for small fillings reaches values on the order of the tunneling energy
Properties and Detection of Spin Nematic Order in Strongly Correlated Electron Systems
A spin nematic is a state which breaks spin SU(2) symmetry while preserving
translational and time reversal symmetries. Spin nematic order can arise
naturally from charge fluctuations of a spin stripe state. Focusing on the
possible existence of such a state in strongly correlated electron systems, we
build a nematic wave function starting from a t-J type model. The nematic is a
spin-two operator, and therefore does not couple directly to neutrons. However,
we show that neutron scattering and Knight shift experiments can detect the
spin anisotropy of electrons moving in a nematic background. We find the mean
field phase diagram for the nematic taking into account spin-orbit effects.Comment: 13 pages, 11 figures. (v2) References adde
Transition matrix Monte Carlo method for quantum systems
We propose an efficient method for Monte Carlo simulation of quantum lattice
models. Unlike most other quantum Monte Carlo methods, a single run of the
proposed method yields the free energy and the entropy with high precision for
the whole range of temperature. The method is based on several recent findings
in Monte Carlo techniques, such as the loop algorithm and the transition matrix
Monte Carlo method. In particular, we derive an exact relation between the DOS
and the expectation value of the transition probability for quantum systems,
which turns out to be useful in reducing the statistical errors in various
estimates.Comment: 6 pages, 4 figure
Universal Statistical Behavior of Neural Spike Trains
We construct a model that predicts the statistical properties of spike trains
generated by a sensory neuron. The model describes the combined effects of the
neuron's intrinsic properties, the noise in the surrounding, and the external
driving stimulus. We show that the spike trains exhibit universal statistical
behavior over short times, modulated by a strongly stimulus-dependent behavior
over long times. These predictions are confirmed in experiments on H1, a
motion-sensitive neuron in the fly visual system.Comment: 7 pages, 4 figure
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