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     

Hysteresis behavior of the random-field Ising model with 2-spin-flip dynamics: exact results on a Bethe lattice

We present an exact treatment of the hysteresis behavior of the zero-temperature random-field Ising model on a Bethe lattice when it is driven by an external field and evolved according to a 2-spin-flip dynamics. We focus on lattice connectivities z=2 (the one-dimensional chain) and z=3. For the latter case, we demonstrate the existence of an out-of-equilibrium phase transition, in contrast with the situation found with the standard 1-spin-flip dynamics. We discuss the influence of the degree of cooperativity of the (local) spin dynamics of the nonequilibrium response on the system.     

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     

Reparametrization invariance: a gauge-like symmetry of ultrametrically organised states

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit of infinite step replica symmetry breaking and is needed to derive this continuous gauge-like symmetry from the discrete permutation invariance of the n replicas. Goldstone's theorem and Ward identities can be deduced from the disappearance of the second (and higher order) variation of the longitudinal free energy. We recall also how these and other exact statements follow from permutation symmetry after introducing the concept of “infinitesimal" permutations. Received 21 July 2000     

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     

Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems

We show that the numerical method based on the off-equilibrium fluctuation-dissipation relation does work and is very useful and powerful in the study of disordered systems which show a very slow dynamics. We have verified that it gives the right information in the known cases (diluted ferromagnets and random field Ising model far from the critical point) and we used it to obtain more convincing results on the frozen phase of four-dimensional spin glasses. Moreover we used it to study the Griffiths phase of the diluted and the random field Ising models. Received 1 December 1998 and Received in final form 17 February 1999     

Thouless energy of a superconductor from non local conductance fluctuations

The three-dimensional bimodal random-field Ising model is investigated using the N-fold version of the Wang-Landau algorithm. The essential energy subspaces are determined by the recently developed critical minimum energy subspace technique, and two implementations of this scheme are utilized. The random fields are obtained from a bimodal discrete (±Δ) distribution, and we study the model for various values of the disorder strength Δ, Δ=0.5,1,1.5 and 2, on cubic lattices with linear sizes L=4–24. We extract information for the probability distributions of the specific heat peaks over samples of random fields. This permits us to obtain the phase diagram and present the finite-size behavior of the specific heat. The question of saturation of the specific heat is re-examined and it is shown that the open problem of universality for the random-field Ising model is strongly influenced by the lack of self-averaging of the model. This property appears to be substantially depended on the disorder strength.     

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     

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     

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 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     

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     

Frustrated Blume-Emery-Griffiths model

A generalised integer S Ising spin glass model is analysed using the replica formalism. The bilinear couplings are assumed to have a Gaussian distribution with ferromagnetic mean . Incorporation of a quadrupolar interaction term and a chemical potential leads to a richer phase diagram with transitions of first and second order. The first order transition may be interpreted as a phase separation, and contrary to what has been argued previously, it persists in the presence of disorder. Finally, the stability of the replica symmetric solution with respect to fluctuations in replica space is analysed, and the transition lines are obtained both analytically and numerically. Received 13 January 1997     

Dynamical linked cluster expansions with applications to disordered systems

Dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics. We discuss physical applications to systems with annealed and quenched disorder. Examples are the bond-diluted Ising model and the Sherrington-Kirkpatrick spin glass. We derive the rules and identify the full set of graphs that contribute to the series in the quenched case. This way it becomes possible to avoid the vague extrapolation from positive integer n to n = 0, that usually goes along with an application of the replica trick. Received 13 December 2001 Published online 25 June 2002     

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 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     

Critical Binder cumulant of two-dimensional Ising models

The fourth-order cumulant of the magnetization, the Binder cumulant, is determined at the phase transition of Ising models on square and triangular lattices, using Monte Carlo techniques. Its value at criticality depends sensitively on boundary conditions, details of the clusters used in calculating the cumulant, and symmetry of the interactions or, here, lattice structure. Possibilities to identify generic critical cumulants are discussed.     

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.     

Order-parameter fluctuations (OPF) in spin glasses: Monte Carlo simulations and exact results for small sizes

The use of parameters measuring order-parameter fluctuations (OPF) has been encouraged by the recent results reported in referenece [2,3] which show that two of these parameters, G and G _c, take universal values in the . In this paper we present a detailed study of parameters measuring OPF for two mean-field models with and without time-reversal symmetry which exhibit different patterns of replica symmetry breaking below the transition: the Sherrington-Kirkpatrick model with and without a field and the Ising p-spin glass (p = 3). We give numerical results and analyze the consequences which replica equivalence imposes on these models in the infinite volume limit. We give evidence for the transition in each system and discuss the character of finite-size effects. Furthermore, a comparative study between this new family of parameters and the usual Binder cumulant analysis shows what kind of new information can be extracted from the finite T behavior of these quantities. The two main outcomes of this work are: 1) Parameters measuring OPF give better estimates than the Binder cumulant for T _c and even for very small systems they give evidence for the transition. 2) For systems with no time-reversal symmetry, parameters defined in terms of connected quantities are the proper ones to look at. Received 20 September 2000 and Received in final form 10 January 2001