arXiv subjects

A. K. Hartmann

Publications and source records attributed to A. K. Hartmann.

53 records · Page 3Linked to original sources

arXiv2005

Reduction of Two-Dimensional Dilute Ising Spin Glasses

The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. This allows, in combination with a graph-theoretical matching algorithm, to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent y_2. A value of y_2=-0.281(3) is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions d=3,...,7.

cond-mat.dis-nn↗

arXiv2005

Determining Energy Barriers by Iterated Optimization: The Two-Dimensional Ising Spin Glass

Energy barriers determine the dynamics in many physical systems like structural glasses, disordered spin systems or proteins. Here we present an approach, which is based on subdividing the configuration space in a hierarchical manner, leading to upper and lower bounds for the energy barrier separating two configurations. The fundamental operation is to perform a constrained energy optimization, where the degree of constraintness increases with the level in the hierarchy. As application, we consider Ising spin glasses, where the energy barrier which needs to be surmounted in order to flip a compact region of spins of linear dimension $L$ are expected to scale as $L^ψ$. The exponent $ψ$ is very hard to estimate from experimental and simulation studies. By using the new approach, applying efficient combinatorial matching algorithms, we are able to give the first non-trivial numerical bounds $0.25 < ψ< 0.54$ for the two-dimensional Ising spin glass.

cond-mat.dis-nn↗

arXiv2004

Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0

A new algorithm is presented, which allows to calculate numerically the partition function Z_q of the d-dimensional q-state Potts models for arbitrary real values q>0 at any given temperature T with high precision. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the distribution of the case q=1 (graph percolation), where the exact result Z_1=1 is known. As application, d=2 and d=3-dimensional ferromagnetic Potts models are studied, and the critical values q_c, where the transition changes from second to first order, are determined. Large systems of sizes N=1000^2 respectively N=100^3 are treated. The critical value q_c(d=2)=4 is confirmed and q_c(d=3)=2.35(5) is found.

cond-mat.stat-mech↗

arXiv2004

Energy size effects of two-dimensional Ising spin glasses

We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for non-free boundary conditions is governed by the stiffness exponent ~ -0.28 describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.

cond-mat.dis-nn↗

arXiv2004

Influence of Collision Cascade Statistics on Pattern Formation of Ion-Sputtered Surfaces

Theoretical continuum models that describe the formation of patterns on surfaces of targets undergoing ion-beam sputtering, are based on Sigmund's formula, which describes the spatial distribution of the energy deposited by the ion. For small angles of incidence and amorphous or polycrystalline materials, this description seems to be suitable, and leads to the classic BH morphological theory [R.M. Bradley and J.M.E. Harper, J. Vac. Sci. Technol. A 6, 2390 (1988)]. Here we study the sputtering of Cu crystals by means of numerical simulations under the binary-collision approximation. We observe significant deviations from Sigmund's energy distribution. In particular, the distribution that best fits our simulations has a minimum near the position where the ion penetrates the surface, and the decay of energy deposition with distance to ion trajectory is exponential rather than Gaussian. We provide a modified continuum theory which takes these effects into account and explores the implications of the modified energy distribution for the surface morphology. In marked contrast with BH's theory, the dependence of the sputtering yield with the angle of incidence is non-monotonous, with a maximum for non-grazing incidence angles.

cond-mat.mtrl-sci↗

arXiv2004

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.

cond-mat.stat-mech↗

arXiv2004

Domain-Wall Energies and Magnetization of the Two-Dimensional Random-Bond Ising Model

We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration $p\in[0,1]$ of antiferromagnetic and $(1-p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension $L$ up to 700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at $p_c \sim 0.103$ compared to the concentration $p_n\sim 0.109$ at the Nishimory point, which means that the phase diagram of the model exhibits a reentrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.

cond-mat.dis-nn↗

arXiv2003

Generating droplets in two-dimensional Ising spin glasses by using matching algorithms

We study the behavior of droplets for two dimensional Ising spin glasses with Gaussian interactions. We use an exact matching algorithm which enables study of systems with linear dimension L up to 240, which is larger than is possible with other approaches. But the method only allows certain classes of droplets to be generated. We study single-bond, cross and a category of fixed volume droplets as well as first excitations. By comparison with similar or equivalent droplets generated in previous works, the advantages but also the limitations of this approach are revealed. In particular we have studied the scaling behavior of the droplet energies and droplet sizes. In most cases, a crossover of the data can be observed such that for large sizes the behavior is compatible with the one-exponent scenario of the droplet theory. Only for the case of first excitations, no clear conclusion can be reached, probably because even with the matching approach the accessible system sizes are still too small.

cond-mat.dis-nn↗

arXiv2003

Glassy behavior induced by geometrical frustration in a hard-core lattice gas model

We introduce a hard-core lattice-gas model on generalized Bethe lattices and investigate analytically and numerically its compaction behavior. If compactified slowly, the system undergoes a first-order crystallization transition. If compactified much faster, the system stays in a meta-stable liquid state and undergoes a glass transition under further compaction. We show that this behavior is induced by geometrical frustration which appears due to the existence of short loops in the generalized Bethe lattices. We also compare our results to numerical simulations of a three-dimensional analog of the model.

cond-mat.stat-mech↗

arXiv2002

Large-Scale, Low-Energy Excitations in the Two-Dimensional Ising Spin Glass

We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two-dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, d_s, and stiffness exponent, theta', for "droplet" excitations, are in reasonable agreement with estimates from "domain wall" calculations, and so support the predictions of the "droplet theory". Restricting our analysis to small lattices, we do not find an effective value of theta' close to -0.47 as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give theta' approx -0.47 for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the "TNT" scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.

cond-mat.dis-nn↗

arXiv2002

Corrections to Scaling are Large for Droplets in Two-Dimensional Spin Glasses

The energy of a droplet of linear extent l in the droplet theory of spin glasses goes as l^θ for large l. It is shown by numerical studies of large droplets in two-dimensional systems that this formula needs to be modified by the addition of a scaling correction l^{-ω} in order to accurately describe droplet energies at the length scales currently probed in numerical simulations. Using this simple modification it is now possible to explain many results with the droplet model which have been found in simulations of three-dimensional Ising spin glasses.

cond-mat.dis-nn↗

arXiv2002

Hiding solutions in random satisfiability problems: A statistical mechanics approach

A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem (3SAT). The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.

cond-mat.dis-nn↗

arXiv2001

Specific-Heat Exponent of Random-Field Systems via Ground-State Calculations

Exact ground states of three-dimensional random field Ising magnets (RFIM) with Gaussian distribution of the disorder are calculated using graph-theoretical algorithms. Systems for different strengths h of the random fields and sizes up to N=96^3 are considered. By numerically differentiating the bond-energy with respect to h a specific-heat like quantity is obtained, which does not appear to diverge at the critical point but rather exhibits a cusp. We also consider the effect of a small uniform magnetic field, which allows us to calculate the T=0 susceptibility. From a finite-size scaling analysis, we obtain the critical exponents ν=1.32(7), α=-0.63(7), η=0.50(3) and find that the critical strength of the random field is h_c=2.28(1). We discuss the significance of the result that αappears to be strongly negative.

cond-mat.dis-nn↗

arXiv2001

Lower Critical Dimension of Ising Spin Glasses

Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal (+- J) distributions of the disorder are calculated using a ``matching'' algorithm, which allows large system sizes of up to N=480^2 spins to be investigated. We study domain walls induced by two rather different types of boundary-condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy'', which we denote by DE. For Gaussian disorder, we find a power-law behavior DE ~ L^θ, with θ=-0.266(2) and θ=-0.282(2) for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension d_c. For the +-J model, we obtain a different result, namely the domain-wall energy saturates at a nonzero value for L\to \infty, so θ= 0, indicating that the lower critical dimension for the +-J model exactly d_c=2.

cond-mat.dis-nn↗

arXiv2000

Ordered phase in the two-dimensional randomly coupled ferromagnet

True ground states are evaluated for a 2d Ising model with random near neighbor interactions and ferromagnetic second neighbor interactions (the Randomly Coupled Ferromagnet). The spin glass stiffness exponent is positive when the absolute value of the random interaction is weaker than the ferromagnetic interaction. This result demonstrates that in this parameter domain the spin glass like ordering temperature is non-zero for these systems, in strong contrast to the 2d Edwards-Anderson spin glass.

cond-mat.dis-nn↗

arXiv1998

Universality in Three Dimensional Random-Field Ground States

We investigate the critical behavior of three-dimensional random-field Ising systems with both Gauss and bimodal distribution of random fields and additional the three-dimensional diluted Ising antiferromagnet in an external field. These models are expected to be in the same universality class. We use exact ground-state calculations with an integer optimization algorithm and by a finite-size scaling analysis we calculate the critical exponents nu, beta, and gamma-bar. While the random-field model with Gauss distribution of random fields and the diluted antiferromagnet appear to be in same universality class, the critical exponents of the random-field model with bimodal distribution of random fields seem to be significantly different.

cond-mat.stat-mech↗

arXiv1995

Cluster-Exact Approximation of Spin Glass Groundstates

We present an algorithm which calculates groundstates of Ising spin glasses approximately. It works by randomly selecting clusters of spins which exhibit no frustrations. The spins which were not selected, contribute to the local fields of the selected spins. For the spin--cluster a groundstate is exactly calaculated by using graphtheoretical methods. The other spins remain unchanged. This procedure is repeated many times resulting in a state with low energy. The total time complexity of this scheme is approximately cubic. We estimate that the groundstate energy density of the infinite system for the +/- J model is -1.400 +/- 0.005 (2d) and -1.766 +/- 0.002 (3d). The distribution of overlaps for selected systems is calculated in order to characterize the algorithm.

cond-mat↗