Ising magnets with mobile defects

Motivated by recent experiments on cuprates with low-dimensional magnetic interactions, a new class of two-dimensional Ising models with short-range interactions and mobile defects is introduced and studied. The non-magnetic defects form lines, which, as temperature increases, first meander and then become unstable. Using Monte Carlo simulations and analytical low- and high-temperature considerations, the instability of the defect stripes is monitored for various microscopic and thermodynamic quantities in detail for a minimal model, assuming some of the couplings to be indefinitely strong. The robustness of the findings against weakening the interactions is discussed as well. Received 22 August 2002 / Received in final form 4 October 2002 Published online 19 November 2002     

Interactions of several replicas in the random field Ising model

The replicated field theory of the random field Ising model involves the couplings of replicas of different indices. The resulting correlation functions involve a superposition of different types of long distance behaviours. However the n = 0 limit allows one to discuss the renormalization group properties in spite of this phenomenon. The attraction of pairs of replicas is enhanced under renormalization flow and no stable fixed point is found. Consequently, an instability occurs in the paramagnetic region, before one reaches the Curie line, signalling the onset of replica symmetry breaking. Received 28 July 2000     

Ground states versus low-temperature equilibria in random field Ising chains

Random walk arguments and exact numerical computations are used to study one-dimensional random field chains. The ground state structure is described with absorbing and non-absorbing random walk excursions. At low temperatures, the local magnetization follows the ground state except at regions where a local random field fluctuation makes thermal excitations easier. This is explained by the random walk picture, implying that the magnetization lengthscale is a product of the domain size and the thermal excitation scale. Received 16 October 2000 and Received in final form 7 June 2001     

Low temperature properties of the random field Potts chain

The random field q-states Potts model is investigated using exact groundstates and finite-temperature transfer matrix calculations. It is found that the domain structure and the Zeeman energy of the domains resembles for general q the random field Ising case (q = 2). This is also the expected outcome based on a random-walk picture of the groundstate. The domain size distribution is exponential, and the scaling of the average domain size with the disorder strength is similar for q arbitrary. The zero-temperature properties are compared to the equilibrium spin states at small temperatures, to investigate the effect of local random field fluctuations that imply locally degenerate regions. The response to field perturbations (`chaos') and the susceptibility are investigated. In particular for the chaos exponent it is found to be 1 for q = 2,..., 5. Finally for q = 2 (Ising case) the domain length distribution is studied for correlated random fields. Received 27 August 2002 Published online 19 December 2002 RID="a" ID="a"e-mail: rieger@lusi-sb.de     

Monte Carlo test of the Goldstone mode singularity in 3D XY model

Monte Carlo simulations of magnetization and susceptibility in the 3D XY model are performed for system sizes up to L=384 (significantly exceeding the largest size L=160 considered in work published previously), and fields h ≥ 0.0003125 at two different coupling constants β=0.5, and β=0.55 in the ordered phase. We examine the prediction of the standard theory that the longitudinal susceptibility χ_∥ has a Goldstone mode singularity such that χ_∥ ∝h^-1/2 holds when h↦0. Most of our results, however, support another theoretical prediction that the singularity is of a more general form χ_∥ ∝h^ρ-1, where 1/2<ρ<1 is a universal exponent related to the ∼h^ρ variation of the magnetization.     

Aperiodic extended surface perturbations in the Ising model

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power of the distance l from the free surface with an oscillating amplitude where follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent , the wandering exponent which governs the fluctuation of the sequence and the bulk correlation length exponent . Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the -plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed. Received 1 December 1998     

Efficient Hopfield pattern recognition on a scale-free neural network

Neural networks are supposed to recognise blurred images (or patterns) of N pixels (bits) each. Application of the network to an initial blurred version of one of P pre-assigned patterns should converge to the correct pattern. In the “standard" Hopfield model, the N “neurons” are connected to each other via N² bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with N². The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabási-Albert scale-free network, in which each newly added neuron is connected to only m other neurons, and at the end the number of neurons with q neighbours decays as 1/q ³. Although the quality of retrieval decreases for small m, we find good associative memory for 1 ≪ m ≪ N. Hence, these networks gain a factor N/m ≫ 1 in the computer memory and time. Received 12 January 2003 Published online 11 April 2003 RID="a" ID="a"e-mail: stauffer@thp.uni-koeln.de     

Multiscaling and multifractality in an one-dimensional Ising model

Scaling properties of the Gibbs distribution of a finite-size one-dimensional Ising model are investigated as the thermodynamic limit is approached. It is shown that, for each nonzero temperature, coarse-grained probabilities of the appearance of particular energy levels display multiscaling with the scaling length ℓ = 1/M _n, where n denotes the number of spins and M_n is the total number of energy levels. Using the multifractal formalism, the probabilities are argued to reveal also multifractal properties. Received 10 July 2000 and Received in final form 6 November 2000     

Persistence of opinion in the Sznajd consensus model: computer simulation

The density of never changed opinions during the Sznajd consensus-finding process decays with time t as 1/t ^θ. We find θ ≃ 3/8 for a chain, compatible with the exact Ising result of Derrida et al. In higher dimensions, however, the exponent differs from the Ising θ. With simultaneous updating of sublattices instead of the usual random sequential updating, the number of persistent opinions decays roughly exponentially. Some of the simulations used multi-spin coding. Received 22 August 2002 / Received in final form 12 November 2002 Published online 31 December 2002     

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w². Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the -expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w² and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed. Received 23 December 1998 and Received in final form 23 March 1999     

Ground state properties of a spin-3/2 model on a decorated square lattice

We present the construction of an optimum ground state for a quantum spin-3/2 antiferromagnet. The spins reside on a decorated square lattice, in which the basis consists of a plaquette of four sites. By using the vertex state model approach we generate the ground state from the same vertices as those used for the corresponding ground state on the hexagonal lattice. The properties of these two ground states are very similar. Particularly there is also a parameter-controlled phase transition from a disordered to a Néel ordered phase. In the regime of this transition, ground state properties can be obtained from an integrable classical vertex model. Received 28 June 1999     

Spin glasses on Bethe lattices for large coordination number

We study spin glasses on random lattices with finite connectivity. In the infinite connectivity limit they reduce to the Sherrington Kirkpatrick model. In this paper we investigate the expansion around the high connectivity limit. Within the replica symmetry breaking scheme at two steps, we compute the free energy at the first order in the expansion in inverse powers of the average connectivity (z), both for the fixed connectivity and for the fluctuating connectivity random lattices. It is well known that the coefficient of the 1/z correction for the free energy is divergent at low temperatures if computed in the one step approximation. We find that this annoying divergence becomes much smaller if computed in the framework of the more accurate two steps breaking. Comparing the temperature dependance of the coefficients of this divergence in the replica symmetric, one step and two steps replica symmetry breaking, we conclude that this divergence is an artefact due to the use of a finite number of steps of replica symmetry breaking. The 1/z expansion is well defined also in the zero temperature limit. Received 15 July 2002 Published online 31 December 2002     

Ground state phase diagram of a spin-2 antiferromagnet on the square lattice

We use the vertex state model approach to construct optimum ground states for a large class of quantum spin-2 antiferromagnets on the square lattice. Optimum ground states are exact ground states of the model which minimize all local interaction operators. The ground state contains two continuous parameters and exhibits a second order phase transition from a disordered phase with exponentially decaying correlation functions to a Néel ordered phase. The behaviour is very similar to that of the corresponding ground state of a quantum spin-3/2 model on the hexagonal lattice, which has been investigated in an earlier paper. Received 8 April 1999     

The travelling salesman problem on randomly diluted lattices: Results for small-size systems

If one places N cities randomly on a lattice of size L, we find that and vary with the city concentration p=N/L ², where is the average optimal travel distance per city in the Euclidean metric and is the same in the Manhattan metric. We have studied such optimum tours for visiting all the cities using a branch and bound algorithm, giving the exact optimized tours for small system sizes () and near-optimal tours for bigger system sizes (). Extrapolating the results for , we find that for p=1, and and with as . Although the problem is trivial for p=1, for it certainly reduces to the standard travelling salesman problem on continuum which is NP-hard. We did not observe any irregular behaviour at any intermediate point. The crossover from the triviality to the NP-hard problem presumably occurs at p=1. Received 15 April 2000     

Exact diagonalization of the S = 1/2 XY ferromagnet on finite bcc lattices to estimate T = 0 properties on the infinite lattice

In this paper finite bcc lattices are defined by a triple of vectors in two different ways - upper triangular lattice form and compact form. In Appendix A are lists of some 260 distinct and useful bcc lattices of 9 to 32 vertices. The energy and magnetization of the S = 1/2 XY ferromagnet have been computed on these bcc lattices in the lowest states for S _z = 0, 1/2, 1 and 3/2. These data are studied statistically to fit the first three terms of the appropriate finite lattice scaling equations. Our estimates of the T = 0 energy and magnetization agree very well with spin wave and series expansion estimates. Received 1st August 2000 and Received in final form 22 December 2000     

Phase transitions in generalized chiral or Stiefel's models

We study the phase transition in generalized chiral or Stiefel's models using Monte Carlo simulations. These models are characterized by a breakdown of symmetry O(N)/O(N-P). We show that the phase transition is clearly first order for when P=N and P=N-1, contrary to predictions based on the renormalization group in expansion but in agreement with a recent non perturbative renormalization group approach. Received 7 October 1999     

Phase transitions in three dimensional generalized xy models

Generalized xy lattice spin models consist of three-component unit vectors, associated with a D-dimensional lattice (say ), parameterized by usual spherical angles (θ_k,φ_k), and interacting via a ferromagnetic potential restricted to nearest neighbours, of the form here epsilon is a positive quantity setting energy and temperature scales. The models were recently introduced, and proven to support an ordering transition taking place at finite temperature when D=3; in turn, this transition had been investigated by different techniques for p=2,3,4, and found to belong to the same universality class as the xy model (i.e. p=1). More recently, it was rigorously proven that for sufficiently large p the transition becomes first order. Here we present a detailed analysis of the transitional properties of this class of models for selected values of p. For p=8 simulation results showed a second order phase transition belonging to the xy class of universality; they suggested tricritical behaviour for p=12, and gave evidence of first-order transitions for both p=16 and p=20.     

On the connection between off-equilibrium response and statics in non disordered coarsening systems

The connection between the out of equilibrium linear response function and static properties established by Franz, Mezard, Parisi and Peliti for slowly relaxing systems is analyzed in the context of phase ordering processes. Separating the response in the bulk of domains from interface response, we find that in order for the connection to hold the interface contribution must be asymptotically negligible. How fast this happens depends on the competition between interface curvature and the perturbing external field in driving domain growth. This competition depends on space dimensionality and there exists a critical value d _c = 3 below which the interface response becomes increasingly important eventually invalidating the connection between statics and dynamics as the limit d = 1 is reached. This mechanism is analyzed numerically for the Ising model with d ranging from 1 to 4 and analytically for a continuous spin model with arbitrary dimensionality. Received 10 July 2001     

The 2D XY model on a finite lattice with structural disorder: quasi-long-range ordering under realistic conditions

We present an analytic approach to study concurrent influence of quenched non-magnetic site-dilution and finiteness of the lattice on the 2D XY model. Two significant deeply connected features of this spin model are: a special type of ordering (quasi-long-range order) below a certain temperature and a size-dependent mean value of magnetisation in the low-temperature phase that goes to zero (according to the Mermin-Wagner-Hohenberg theorem) in the thermodynamic limit. We focus our attention on the asymptotic behaviour of the spin-spin correlation function and the probability distribution of magnetisation. The analytic approach is based on the spin-wave approximation valid for the low-temperature regime and an expansion in the parameters which characterise the deviation from completely homogeneous configuration of impurities. We further support the analytic considerations by Monte Carlo simulations performed for different concentrations of impurities and compare analytic and MC results. We present as the main quantitative result of the work the exponent of the spin-spin correlation function power law decay. It is non universal depending not only on temperature as in the pure model but also on concentration of magnetic sites. This exponent characterises also the vanishing of magnetisation with increasing lattice size.