Dynamic spin-glass behavior in a disorder-free,two-component model of quantum frustrated magnets

Motivated by the observation of a spin-glass transition in almost disorder-free Kagome antiferromagnets, and by the specific form of the effective low-energy model of the S = 1/2, trimerized Kagome antiferromagnet, we investigate the possibility to obtain a spin-glass behavior in two-component, disorder-free models. We concentrate on a toy-model, a modified Ashkin-Teller model in a magnetic field that couples only to one species of spins, for which we prove that a dynamic spin-glass behavior occurs. The dynamics of the magnetization is closely related to that of the underlying Ising model in zero field in which spins and pseudo-spins are intimately coupled. The spin-glass like history dependence of the magnetization is a consequence of the ageing of the underlying Ising model. Received 21 September 2001 and Received in final form 16 January 2002     

The Bethe lattice spin glass revisited

So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization. Received 27 September 2000     

Ultrametricity between states at different temperatures in spin-glasses

We prove the existence of correlations between the equilibrium states at different temperatures of the multi-p-spin spherical spin-glass models with continuous replica symmetry breaking: there is no chaos in temperature in these models. Furthermore, the overlaps satisfy ultrametric relations. As a consequence the Parisi tree is essentially the same at all temperatures with lower branches developing when lowering the temperature. We conjecture that the reference free energies of the clusters are also fixed at all temperatures as in the generalized random-energy model. Received 18 March 2002 / Received in final form 14 June 2002 Published online 1st October 2002 RID="a" ID="a"e-mail: tommaso.rizzo@inwind.it     

Aging dynamics of Edwards-Anderson spin glasses

We analyze by means of extensive computer simulations the out of equilibrium dynamics of Edwards-Anderson spin glasses in d = 4 and d = 6 dimensions with ± J interactions. In particular, we focus our analysis on the scaling properties of the two-time autocorrelation function in a wide range of temperatures from T = 0.07 T _c to T = 0.75 T _c in both systems. In both the 4 d and 6 d models at very low temperatures we study the effects of discretization of energy levels. Strong sub-aging behaviors are found. We argue that this is because in the times accessible to our simulations the systems are only able to probe activated dynamics through the lowest discrete energy levels and remain trapped around nearly flat regions of the energy landscape. For temperatures T ≥ 0.5 T _c in 4 d and 6 d we find logarithmic scalings that are compatible with simple dynamical ultrametricity. Nevertheless the behaviour of the systems, even in 6 d is very different from the mean field SK model results. Received 21 October 2002 / Received in final form 13 January 2003 Published online 11 April 2003 RID="a" ID="a"Associate researcher of the Abdus Salam International Centre for Theoretical Physics; e-mail: stariolo@if.ufrgs.br; http://www.if.ufrgs.br/stariolo RID="b" ID="b"Present address: The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy e-mail: mmontemu@ictp.trieste.it RID="c" ID="c"e-mail: tamarit@famaf.unc.edu.ar     

The <Emphasis Type="Bold">±</Emphasis><Emphasis Type="Bold">J</Emphasis> spin glass in Migdal-Kadanoff approximation

We study the low-temperature phase of the three-dimensional ± J Ising spin glass in Migdal-Kadanoff approximation. At zero temperature, T = 0, the properties of the spin glass result from the ground-state degeneracy and can be elucidated using scaling arguments based on entropy. The approach to the asymptotic scaling regime is very slow, and the correct exponents are only visible beyond system sizes around 64. At T > 0, a crossover from the zero-temperature behaviour to the behaviour expected from the droplet picture occurs at length scales proportional to T ^-2/ds where d_s is the fractal dimension of a domain wall. Canonical droplet behaviour is not visible at any temperature for systems whose linear dimension is smaller than 16 lattice spacings, because the data are either affected by the zero-temperature behaviour or the critical point behaviour. Received 18 February 2001     

A microscopic mechanism for rejuvenation and memory effects in spin glasses

Aging in spin glasses (and in some other systems) reveals astonishing effects of `rejuvenation and memory' upon temperature changes. In this paper, we propose microscopic mechanisms (at the scale of spin-spin interactions) which can be at the origin of such phenomena. Firstly, we recall that, in a frustrated system, the effective average interaction between two spins may take different values (possibly with opposite signs) at different temperatures. We give simple examples of such situations, which we compute exactly. Such mechanisms can explain why new ordering processes (rejuvenation) seem to take place in spin glasses when the temperature is lowered. Secondly, we emphasize the fact that inhomogeneous interactions do naturally lead to a wide distribution of relaxation times for thermally activated flips. `Memory spots' spontaneously appear, in the sense that the flipping time of some spin clusters becomes extremely long when the temperature is decreased. Such memory spots are capable of keeping the memory of previous ordering at a higher temperature while new ordering processes occur at a lower temperature. After a qualitative discussion of these mechanisms, we show in the numerical simulation of a simplified example that this may indeed work. Our conclusion is that certain chaos-like phenomena may show up spontaneously in any frustrated and inhomogeneous magnetic system, without impeding the occurrence of memory effects. Received 5 February 2001 and Received in final form 27 April 2001     

Generic replica symmetric field-theory for short range Ising spin glasses

Symmetry considerations and a direct, Hubbard-Stratonovich type, derivation are used to construct a replica field-theory relevant to the study of the spin glass transition of short range models in a magnetic field. A mean-field treatment reveals that two different types of transitions exist, whenever the replica number n is kept larger than zero. The Sherrington-Kirkpatrick critical point in zero magnetic field between the paramagnet and replica magnet (a replica symmetric phase with a nonzero spin glass order parameter) separates from the de Almeida-Thouless line, along which replica symmetry breaking occurs. We argue that for studying the de Almeida-Thouless transition around the upper critical dimension d = 6, it is necessary to use the generic cubic model with all the three bare masses and eight cubic couplings. The critical role n may play is also emphasized. To make perturbative calculations feasible, a new representation of the cubic interaction is introduced. To illustrate the method, we compute the masses in one-loop order. Some technical details and a list of vertex rules are presented to help future renormalisation-group calculations. Received 9 October 2001     

On the finite size corrections to some random matching problems

We get back to the computation of the leading finite size corrections to some random link matching problems, first adressed by Mézard and Parisi [J. Phys. France 48, 1451 (1987)]. In the so-called bipartite case, their result is in contradiction with subsequent works. We show that they made some mistakes, and correcting them, we get the expected result. In the non bipartite case, we agree with their result but push the analytical treatment further. Received 28 April 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: giorgio.parisi@roma1.infn.it RID="b" ID="b"e-mail: matthieu.ratieville@roma1.infn.it     

Neighborhood preferences in random matching problems

We consider a class of random matching problems where the distance between two points has a probability law which, for a small distance l, goes like l^r. In the framework of the cavity method, in the limit of an infinite number of points, we derive equations for p_k, the probability for some given point to be matched to its kth nearest neighbor in the optimal configuration. These equations are solved in two limiting cases: r = 0 -- where we recover p _k = 1/2^k, as numerically conjectured by Houdayer et al. and recently rigorously proved by Aldous -- and r→ + ∞. For 0 < r < + ∞, we are not able to solve the equations analytically, but we compute the leading behavior of p_k for large k. Received 14 February 2001     

Renormalization group approach to spin glass systems

A renormalization group transformation suitable for spin glass models and, more generally, for disordered models, is presented. The procedure is non-standard in both the nature of the additional interactions and the coarse graining transformation, that is performed on the overlap probability measure. Universality classes are thus naturally defined on a large set of models, going from and Gaussian spin glasses to Ising and fully frustrated models, and others. The proposed analysis is tested numerically on the Edwards-Anderson model in d = 4. Good estimates of the critical index ν and of T _c are obtained, and an RG flow diagram is sketched for the first time. Received 17 November 2000     

Perturbation theory by flow equations: dimerized and frustrated S = 1/2 chain

The flow equation method (Wegner, 1994) is used as continuous unitary transformation to construct perturbatively effective Hamiltonians. The method is illustrated in detail for dimerized and frustrated antiferromagnetic S =1/2 chains. The effective Hamiltonians conserve the number of elementary excitations which are S =1 magnons for the dimerized chains. The sectors of different number of excitations are clearly separated. Easy-to-use results for the gap, the dispersion and the ground state energies of the chains are provided.     

The glassy phase of Gallager codes

Gallager codes are the best error-correcting codes to date. In this paper we study them by using the tools of statistical mechanics. The corresponding statistical mechanics model is a spin model on a sparse random graph. The model can be solved by elementary methods (i.e. without replicas) in a large connectivity limit. For low enough temperatures it presents a completely frozen glassy phase (q _EA = 1). The same scenario is shown to hold for finite connectivities. In this case we adopt the replica approach and exhibit a one-step replica symmetry breaking order parameter. We argue that our ansatz yields the exact solution of the model. This allows us to determine the whole phase diagram and to understand the performances of Gallager codes. Received 9 April 2001     

Vitrification in a 2D Ising model with mobile bonds

A bond-disordered two-dimensional Ising model is used to simulate Kauzmann's mechanism of vitrification in liquids, by a Glauber Monte Carlo simulation. The rearrangement of configurations is achieved by allowing impurity bonds to hop to nearest neighbors at the same rate as the spins flip. For slow cooling, the theoretical minimum energy configuration is approached, characterized by an amorphous distribution of locally optimally arranged impurity bonds. Rapid cooling to low temperatures regularly finds bond configurations of higher energy, which are both a priori rare and severely restrictive to spin movement, providing a simple realization of kinetic vitrification. A supercooled liquid regime is also found, and characterized by a change in sign of the field derivative of the spin-glass susceptibility at a finite temperature. Received 3 August 2000 and Received in final form 9 March 2001     

Solvable lattice gas models of random hetero-polymers at finite density: I. Statics

We introduce -dimensional lattice gas versions of three common models of random hetero-polymers, in which both the polymer density and the density of the polymer-solvent mixture are finite. These solvable models give valuable insight into the problems related to the (quenched) average over the randomness in statistical mechanical models of proteins, without having to deal with the hard geometrical constraints occurring in finite-dimensional models. Our exact solution, which is specific to the -dimensional case, is compared to the results obtained by a saddle-point analysis and by the grand ensemble approach, both of which can also be applied to models of finite dimension. We find, somewhat surprisingly, that the saddle-point analysis can lead to qualitatively incorrect results. Received 15 June 1999 and Received in final form 14 October 1999     

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     

The kinetic spin-1 Blume-Capel model with competing dynamics

We have studied by means of Monte-Carlo simulation and exact finite-size analysis, the spin-1 Blume Capel model with Glauber and Kawasaki dynamics. The Kawasaki spin-exchange process transfers energy into the system from an external source. Some phase diagrams of the model are presented. For some parameter values, the system displays a kind of self-organization phenomenon within the disordered phase. Received 15 February 2000     

Comparing mean field and Euclidean matching problems

Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional systems. Our focus here is on minimum matching problems, because they are computationally tractable while both frustrated and disordered. We first study a mean field model taking the link lengths between points to be independent random variables. For this model we find perfect agreement with the results of a replica calculation, and give a conjecture. Then we study the case where the points to be matched are placed at random in a d-dimensional Euclidean space. Using the mean field model as an approximation to the Euclidean case, we show numerically that the mean field predictions are very accurate even at low dimension, and that the error due to the approximation is O(1/d ² ). Furthermore, it is possible to improve upon this approximation by including the effects of Euclidean correlations among k link lengths. Using k=3 (3-link correlations such as the triangle inequality), the resulting errors in the energy density are already less than at . However, we argue that the dimensional dependence of the Euclidean model's energy density is non-perturbative, i.e., it is beyond all orders in k of the expansion in k-link correlations. Received: 1st December 1997 / Revised: 6 May 1998 / Accepted: 30 June 1998     

Vertex cover problem studied by cavity method: Analytics and population dynamics

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c > e = 2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature β in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated. Received 11 November 2002 Published online 1st April 2003 RID="a" ID="a"e-mail: zhou@mpikg-golm.mpg.de     

Aging and non-linear glassy dynamics in a mean field model

The mean field approach of glassy dynamics successfully describes systems which are out-of-equilibrium in their low temperature phase. In some cases an aging behaviour is found, with no stationary regime ever reached. In the presence of dissipative forces however, the dynamics is indeed stationary, but still out-of-equilibrium, as inferred by a significant violation of the fluctuation dissipation theorem. The mean field dynamics of a particle in a random but short-range correlated environment, offers the opportunity of observing both the aging and driven stationary regimes. Using a geometrical approach previously introduced by the author, we study here the relation between these two situations, in the pure relaxational limit, i.e. the zero temperature case. In the stationary regime, the velocity (v)-force (F) characteristics is a power law v∼F ⁴, while the characteristic times scale like powers of v, in agreement with an early proposal by Horner. The cross-over between the aging, linear-response regime and the non-linear stationary regime is smooth, and we propose a parametrization of the correlation functions valid in both cases, by means of an “effective time”. We conclude that aging and non-linear response are dual manifestations of a single out-of-equilibrium state, which might be a generic situation. Received 7 May 2000 and Received in final form 22 August 2000