O. Melchert

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

Abstract In the presented article we present an algorithm for the computation of ground state spin configurations for the 2 d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2 d systems, e.g. lattice graphs with up to N = 384 × 384 spins, can be analyzed very fast. Here, we investigate the critical behavior of the corresponding T = 0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.

Scaling behavior of domain walls at the T=0 ferromagnet to spin-glass transition

We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length L, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows to tune the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic groundstate. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the the average domain-wall energy (~L^\theta) and the average domain-wall length (~L^df). Performing a finite-size scaling analysis for systems with a side length up to L=512, we find that both exponents remain constant in the spin-glass phase, i.e. \theta ~- 0.28 and df~1.275. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of Stochastic Loewner evolutions (SLEs) via df-1=3/[4(3+\theta)]. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.

Negative-weight percolation

We describe a percolation problem on lattices (graphs, networks), with edge weights drawn from disorder distributions that allow for weights (or distances) of either sign, i.e. including negative weights. We are interested whether there are spanning paths or loops of total negative weight. This kind of percolation problem is fundamentally different from conventional percolation problems, e.g. it does not exhibit transitivity, hence no simple definition of clusters, and several spanning paths/loops might coexist in the percolation regime at the same time. Furthermore, to study this percolation problem numerically, one has to perform a non-trivial transformation of the original graph and apply sophisticated matching algorithms. Using this approach, we study the corresponding percolation transitions on large square, hexagonal and cubic lattices for two types of disorder distributions and determine the critical exponents. The results show that negative-weight percolation is in a different universality class compared to conventional bond/site percolation. On the other hand, negative-weight percolation seems to be related to the ferromagnet/spin-glass transition of random-bond Ising systems, at least in two dimensions.

Finite-temperature local protein sequence alignment: percolation and free-energy distribution.

Sequence alignment is a tool in bioinformatics that is used to find homological relationships in large molecular databases. It can be mapped on the physical model of directed polymers in random media. We consider the finite-temperature version of local sequence alignment for proteins and study the transition between the linear phase and the biologically relevant logarithmic phase, where the free energy grows linearly or logarithmically with the sequence length. By means of numerical simulations and finite-size-scaling analysis, we determine the phase diagram in the plane that is spanned by the gap costs and the temperature. We use the most frequently used parameter set for protein alignment. The critical exponents that describe the parameter-driven transition are found to be explicitly temperature dependent. Furthermore, we study the shape of the (free-) energy distribution close to the transition by rare-event simulations down to probabilities on the order 10(-64). It is well known that in the logarithmic region, the optimal score distribution (T=0) is described by a modified Gumbel distribution. We confirm that this also applies for the free-energy distribution (T>0). However, in the linear phase, the distribution crosses over to a modified Gaussian distribution.

Paths in the minimally weighted path model are incompatible with Schramm-Loewner evolution

We study numerically the geometrical properties of minimally weighted paths that appear in the negative-weight percolation (NWP) model on two-dimensional lattices assuming a combination of periodic and free boundary conditions (BCs). Each realization of the disorder consists of a random fraction (1−ρ) of bonds with unit strength and a fraction ρ of bond strengths drawn from a Gaussian distribution with zero mean and unit width. For each such sample, the path is forced to span the lattice along the direction with the free BCs. The path and a set of negatively weighted loops form a ground state (GS). A ground state on such a lattice can be determined performing a nontrivial transformation of the original graph and applying sophisticated matching algorithms. Here we examine whether the geometrical properties of the paths are in accordance with predictions of Schramm-Loewner evolution (SLE). Measuring the fractal dimension and reviewing Schramm’s left passage formula indicates that the paths cannot be described in terms of SLE.

Information-theoretic approach to ground-state phase transitions for two- and three-dimensional frustrated spin systems.

The information-theoretic observables entropy (a measure of disorder), excess entropy (a measure of complexity), and multi-information are used to analyze ground-state spin configurations for disordered and frustrated model systems in two and three dimensions. For both model systems, ground-state spin configurations can be obtained in polynomial time via exact combinatorial optimization algorithms, which allowed us to study large systems with high numerical accuracy. Both model systems exhibit a continuous transition from an ordered to a disordered ground state as a model parameter is varied. By using the above information-theoretic observables it is possible to detect changes in the spatial structure of the ground states as the critical point is approached. It is further possible to quantify the scaling behavior of the information-theoretic observables in the vicinity of the critical point. For both model systems considered, the estimates of critical properties for the ground-state phase transitions are in good agreement with existing results reported in the literature.

Upper critical dimension of the negative-weight percolation problem.

By means of numerical simulations, we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d=2 through 7, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of system-spanning loops of total negative weight. The resulting negative-weight percolation (NWP) problem is fundamentally different from conventional percolation, as we have seen in previous studies of this model for the two-dimensional case. Here, we characterize the transition for hypercubic systems, where the aim of the present study is to get a grip on the upper critical dimension d u of the NWP problem. For the numerical simulations, we employ a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. We characterize the loops via observables similar to those in percolation theory and perform finite-size scaling analyses, e.g., three-dimensional hypercubic systems with side length up to L=56 sites, in order to estimate the critical properties of the NWP phenomenon. We find our numerical results consistent with an upper critical dimension d u=6 for the NWP problem.

Inversion of real and complex phase shifts to potentials by the generalized Cox?Thompson inverse scattering method at fixed energy

The Cox?Thompson inverse scattering method at fixed energy has been generalized to treat complex phase shifts derived from experiments. New formulae for relating phase shifts to shifted angular momenta are derived. The method is applied to phase shifts of known potentials in order to test its quality and stability and, further, it is used to invert experimental n-? and n-12C phase shifts.

Mean-field behavior of the negative-weight percolation model on random regular graphs

We investigate both analytically and numerically the ensemble of minimum-weight loops in the negative-weight percolation model on random graphs with fixed connectivity and bimodal weight distribution. This allows us to study the mean-field behavior of this model. The analytical study is based on a conjectured equivalence with the problem of self-avoiding walks in a random medium. The numerical study is based on a mapping to a standard minimum-weight matching problem for which fast algorithms exist. Both approaches yield results that are in agreement on the location of the phase transition, on the value of critical exponents, and on the absence of any sizable indications of a glass phase. By these results, the previously conjectured upper critical dimension of d(u)=6 is confirmed.

Phase transitions in diluted negative-weight percolation models

We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of total negative weight. The resulting percolation problem is fundamentally different from conventional percolation, as we have seen in a previous study of this model for the undiluted case. Here, we investigate how the percolation transition is affected by additional dilution. We consider two types of dilution: either a certain fraction of edges exhibits zero weight, or a fraction of edges is even absent. We study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms. We perform a finite-size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary. We find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change in the universality class.