Monte Carlo study of the scaling of universal correlation lengths in three-dimensional O(n) spin models

Using an elaborate set of simulational tools and statistically optimized methods of data analysis we investigate the scaling behavior of the correlation lengths of three-dimensional classical O($n$) spin models. Considering three-dimensional slabs $S^1\times S^1\times\mathbb{R}$, the results over a wide range of $n$ indicate the validity of special scaling relations involving universal amplitude ratios that are analogous to results of conformal field theory for two-dimensional systems. A striking mismatch of the $n\to\infty$ extrapolation of these simulations against analytical calculations is traced back to a breakdown of the identification of this limit with the spherical model.Comment: 18 pages, 9 figures, REVTeX4, slightly shortened, updated critical exponent estimate

Universal amplitude ratios in finite-size scaling: three-dimensional Ising model

Motivated by the results of two-dimensional conformal field theory (CFT) we investigate the finite-size scaling of the mass spectrum of an Ising model on three-dimensional lattices with a spherical cross section. Using a cluster-update Monte Carlo technique we find a linear relation between the masses and the corresponding scaling dimensions, in complete analogy to the situation in two dimensions. Amplitude ratios as well as the amplitudes themselves appear to be universal in this case.Comment: 3 pages, 2 figures, proceedings of LATTICE99, Pis

Spin and chiral stiffness of the XY spin glass in two dimensions

We analyze the zero-temperature behavior of the XY Edwards-Anderson spin glass model on a square lattice. A newly developed algorithm combining exact ground-state computations for Ising variables embedded into the planar spins with a specially tailored evolutionary method, resulting in the genetic embedded matching (GEM) approach, allows for the computation of numerically exact ground states for relatively large systems. This enables a thorough re-investigation of the long-standing questions of (i) extensive degeneracy of the ground state and (ii) a possible decoupling of spin and chiral degrees of freedom in such systems. The new algorithm together with appropriate choices for the considered sets of boundary conditions and finite-size scaling techniques allows for a consistent determination of the spin and chiral stiffness scaling exponents.Comment: 6 pages, 2 figures, proceedings of the HFM2006 conference, to appear in a special issue of J. Phys.: Condens. Matte

Domain-wall excitations in the two-dimensional Ising spin glass

The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.Comment: 19 pages, 12 figures, 5 tables, accepted versio

Cluster Percolation in the Two-Dimensional Ising Spin Glass

Suitable cluster definitions have allowed researchers to describe many ordering transitions in spin systems as geometric phenomena related to percolation. For spin glasses and some other systems with quenched disorder, however, such a connection is missing to date. Using Monte Carlo simulations, we study the percolation properties of several classes of clusters occurring in the Edwards-Anderson Ising spin-glass model in two dimensions. The Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the ferromagnetic problem do percolate at a temperature that remains non-zero in the thermodynamic limit. On the Nishimori line, this location is accurately predicted by an argument due to Yamaguchi. More relevant for the spin-glass transition are clusters defined on the basis of the overlap of several replicas. We show that various such cluster types have percolation thresholds that shift to lower temperature by increasing the system size, in agreement with the zero-temperature spin-glass transition in two dimensions. The overlap is linked to the difference in density of the two largest clusters, thus supporting a picture where the spin-glass transition corresponds to an emergent density difference of the two largest clusters inside the percolating phase.Comment: 14 pages, 18 figures, 1 table, RevTeX 4.