Temporal correlations and persistence in the kinetic Ising model: the role of temperature

We study the statistical properties of the sum S _t = dt'σ _t', that is the difference of time spent positive or negative by the spin σ _t, located at a given site of a D-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of S_t and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ( T > T _c), criticality ( T = T _c), and low temperature ( T < T _c). We discuss in particular the question of the temperature dependence of the persistence exponent , as well as that of the spectrum of exponents (x), in the low temperature phase. The probability that the temporal mean S _t/t was always larger than the equilibrium magnetization is found to decay as t ^{- - ?}. This yields a numerical determination of the persistence exponent in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model. Received 4 December 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     

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     

Acceleratingly growing scale-free networks with tunable degree exponents

In this paper, we analytically study the probabilistic accelerating network [M.J. Gagen, J.S. Mattick, Phys. Rev. E 72 (2005) 016123] in its accelerating regimes by using mean field theory. In the growing network, the number of links added with each new node is a nonlinearly increasing function aN^β(t) where N(t) is the number of nodes present at time t. It is found that the network appears to have a power-law degree distribution for large degree with tunable degree exponents (ranging from 3.0 to theoretically infinity) and the degree exponent γ depends only on the parameter β as . The analytical results are found to be in good agreement with those obtained by extensive numerical simulations.     

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     

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     

Election results and the Sznajd model on Barabasi network

The network of Barabasi and Albert, a preferential growth model where a new node is linked to the old ones with a probability proportional to their connectivity, is applied to Brazilian election results. The application of the Sznajd rule, that only agreeing pairs of people can convince their neighbours, gives a vote distribution in good agreement with reality Received 19 September 2001 and Received in final form 2 November 2001     

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     

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     

Universal properties of shortest paths in isotropically correlated random potentials

We consider the optimal paths in a d-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential. We study how the universal scaling exponents, the roughness and the energy fluctuation exponent, depend on the strength of the disorder correlations. Our numerical results using Dijkstra's algorithm to determine the optimal path in directed as well as undirected lattices indicate that the correlations become relevant if they decay with distance slower than 1/r in d = 2 and 3. We show that the exponent relation 2ν - ω = 1 holds at least in d = 2 even in case of correlations. Both in two and three dimensions, overhangs turn out to be irrelevant even in the presence of strong disorder correlations. Received 20 December 2002 / Received in final form 10 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: schorr@lusi.uni-sb.de     

Persistence distributions in a conserved lattice gas with absorbing states

Extensive simulations are performed to study the persistence behavior of a conserved lattice gas model exhibiting an absorbing phase transition from an active phase into an inactive phase. Both the global and the local persistence exponents are determined in two and higher dimensions. The local persistence exponent obeys a scaling relation involving the order parameter exponent of the absorbing phase transition. Furthermore we observe that the global persistence exponent exceeds its local counterpart in all dimensions in contrast to the known persistence behavior in reversible phase transitions. Received 27 August 2001 and Received in final form 15 November 2001     

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     

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     

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     

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     

Dynamical invariants in the deterministic fixed-energy sandpile

The non-ergodic behavior of the deterministic Fixed Energy Sandpile (DFES), with Bak-Tang-Wiesenfeld (BTW) rule, is explained by the complete characterization of a class of dynamical invariants (or toppling invariants). The link between such constants of motion and the discrete Laplacians properties on graphs is algebraically and numerically clarified. In particular, it is possible to build up an explicit algorithm determining the complete set of independent toppling invariants. The partition of the configuration space into dynamically invariant sets, and the further refinement of such a partition into basins of attraction for orbits, are also studied. The total number of invariant sets equals the graphs complexity. In the case of two dimensional lattices, it is possible to estimate a very regular exponential growth of this number vs. the size. Looking at other features, the toppling invariants exhibit a highly irregular behavior. The usual constraint on the energy positiveness introduces a transition in the frozen phase. In correspondence to this transition, a dynamical crossover related to the halting times is observed. The analysis of the configuration space shows that the DFES has a different structure with respect to dissipative BTW and stochastic sandpiles models, supporting the conjecture that it lies in a distinct class of universality.     

Excitation energy as a basic variable to control nuclear disassembly

Thermodynamical features of Xe system is investigated as functions of temperature and freeze-out density in the frame of lattice gas model. The calculation shows different temperature dependence of physical observables at different freeze-out density. In this case, the critical temperature when the phase transition takes place depends on the freeze-out density. However, a unique critical excitation energy %and the same excitation reveals regardless of freeze-out density when the excitation energy is used as a variable instead of temperature. Moreover, the different behavior of other physical observables with temperature due to different ρ_f vanishes when excitation energy replaces temperature. It indicates that the excitation energy can be seen as a more basic quantity to control nuclear disassembly. Received: 25 November 1998 /Revised version: 20 January 1999     

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