arXiv subjects

L. Leuzzi

Publications and source records attributed to L. Leuzzi.

48 records · Page 3Linked to original sources

arXiv2006

On Equilibrium Dynamics of Spin-Glass Systems

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical $2+p$ spin glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, Hörner and Sommers [Z. für Physik {\bf 92}, 257 (1993)] dynamical solution of the spherical $p$-spin spin glass model, reproducing a stable static limit that coincides, in the case of a one step Replica Symmetry Breaking Ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off-equilibrium. We formally extend our analysis to any number of Replica Symmetry Breakings $R$. In the limit $R\to\infty$ both formulations lead to the Parisi anti-parabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The new formulation does not contain the additional order parameter $Δ$ of the Sompolinsky theory.

cond-mat.dis-nn↗

arXiv2004

The spherical $2+p$ spin glass model: an exactly solvable model for glass to spin-glass transition

We present the full phase diagram of the spherical $2+p$ spin glass model with $p\geq 4$. The main outcome is the presence of a new phase with both properties of Full Replica Symmetry Breaking (FRSB) phases of discrete models, e.g, the Sherrington-Kirkpatrick model, and those of One Replica Symmetry Breaking (1RSB). The phase, which separates a 1RSB phase from FRSB phase, is described by an order parameter function $q(x)$ with a continuous part (FRSB) for $x<m$ and a discontinuous jump (1RSB) at $x=m$. This phase has a finite complexity which leads to different dynamic and static properties.

cond-mat.dis-nn↗

arXiv2004

Complexity in Mean-Field Spin-Glass Models: Ising $p$-spin

The Complexity of the Thouless-Anderson-Palmer (TAP) solutions of the Ising $p$-spin is investigated in the temperature regime where the equilibrium phase is one step Replica Symmetry Breaking. Two solutions of the resulting saddle point equations are found. One is supersymmetric (SUSY) and includes the equilibrium value of the free energy while the other is non-SUSY. The two solutions cross exactly at a value of the free energy where the replicon eigenvalue is zero; at low free energy the complexity is described by the SUSY solution while at high free energy it is described by the non-SUSY solution. In particular the non-SUSY solution describes the total number of solutions, like in the Sherrington-Kirkpatrick (SK) model. The relevant TAP solutions corresponding to the non-SUSY solution share the same feature of the corresponding solutions in the SK model, in particular their Hessian has a vanishing isolated eigenvalue. The TAP solutions corresponding to the SUSY solution, instead, are well separated minima.

cond-mat.stat-mech↗

arXiv2004

Quenched Computation of the Complexity of the Sherrington-Kirkpatrick Model

The quenched computation of the complexity in the Sherrington-Kirkpatrick model is presented. A modified Full Replica Symmetry Breaking Ansatz is introduced in order to study the complexity dependence on the free energy. Such an Ansatz corresponds to require Becchi-Rouet-Stora-Tyutin supersymmetry. The complexity computed this way is the Legendre transform of the free energy averaged over the quenched disorder. The stability analysis shows that this complexity is inconsistent at any free energy level but the equilibirum one. The further problem of building a physically well defined solution not invariant under supersymmetry and predicting an extensive number of metastable states is also discussed.

cond-mat.dis-nn↗

arXiv2003

On Spin-Glass Complexity

We study the quenched complexity in spin-glass mean-field models satisfying the Becchi-Rouet-Stora-Tyutin supersymmetry. The outcome of such study, consistent with recent numerical results, allows, in principle, to conjecture the absence of any supersymmetric contribution to the complexity in the Sherrington-Kirkpatrick model. The same analysis can be applied to any model with a Full Replica Symmetry Breaking phase, e.g. the Ising $p$-spin model below the Gardner temperature. The existence of different solutions, breaking the supersymmetry, is also discussed.

cond-mat.dis-nn↗

arXiv2003

The Complexity of the Spherical $p$-spin spin glass model, revisited

Some questions concerning the calculation of the number of ``physical'' (metastable) states or complexity of the spherical $p$-spin spin glass model are reviewed and examined further. Particular attention is focused on the general calculation procedure which is discussed step-by-step.

cond-mat.dis-nn↗

arXiv2003

Complexity of the Sherrington-Kirkpatrick Model in the Annealed Approximation

A careful critical analysis of the complexity, at the annealed level, of the Sherrington-Kirkpatrick model has been performed. The complexity functional is proved to be always invariant under the Becchi-Rouet-Stora-Tyutin supersymmetry, disregarding the formulation used to define it. We consider two different saddle points of such functional, one satisfying the supersymmetry [A. Cavagna {\it et al.}, J. Phys. A {\bf 36} (2003) 1175] and the other one breaking it [A.J. Bray and M.A. Moore, J. Phys. C {\bf 13} (1980) L469]. We review the previews studies on the subject, linking different perspectives and pointing out some inadequacies and even inconsistencies in both solutions.

cond-mat.dis-nn↗

arXiv2001

The disordered Backgammon model

In this paper we consider an exactly solvable model which displays glassy behavior at zero temperature due to entropic barriers. The new ingredient of the model is the existence of different energy scales or modes associated to different relaxational time-scales. Low-temperature relaxation takes place by partial equilibration of successive lower energy modes. An adiabatic scaling solution, defined in terms of a threshold energy scale $\eps^*$, is proposed. For such a solution, modes with energy $\eps\gg\eps^*$ are equilibrated at the bath temperature, modes with $\eps\ll\eps^*$ remain out of equilibrium and relaxation occurs in the neighborhood of the threshold $\eps\sim \eps^*$. The model is presented as a toy example to investigate conditions related to the existence of an effective temperature in glassy systems and its possible dependence on the energy sector probed by the corresponding observable.

cond-mat.stat-mech↗

arXiv2001

The 3-SAT problem with large number of clauses in $\infty$-replica symmetry breaking scheme

In this paper we analyze the structure of the UNSAT-phase of the overconstrained 3-SAT model by studying the low temperature phase of the associated disordered spin model. We derive the $\infty$ Replica Symmetry Broken equations for a general class of disordered spin models which includes the Sherrington - Kirkpatrick model, the Ising $p$-spin model as well as the overconstrained 3-SAT model as particular cases. We have numerically solved the $\infty$ Replica Symmetry Broken equations using a pseudo-spectral code down to and including zero temperature. We find that the UNSAT-phase of the overconstrained 3-SAT model is of the $\infty$-RSB kind: in order to get a stable solution the replica symmetry has to be broken in a continuous way, similarly to the SK model in external magnetic field.

cond-mat.stat-mech↗

arXiv2001

Exactly Solvable Model Glass with a Facilitated Dynamics

A model glass with fast and slow processes is studied. The statics is simple and the facilitated slow dynamics is exactly solvable. The main features of a fragile glass take place: Kauzmann transition, Vogel-Fulcher law, Adam-Gibbs relation and aging. The time evolution can be so slow that a quasi-equilibrium occur at a time dependent effective temperature. The same effective temperature is derived from the Fluctuation-Dissipation ratio, which supports the applicability of out of equilibrium thermodynamics.

cond-mat.stat-mech↗

arXiv2001

The K-sat problem in a simple limit

We compute the thermodynamic properties of the 3-satisfiability problem in the infinite connectivity limit. In this limit the computations can be strongly simplified and the thermodynamical properties can be obtained with an high accuracy. We find evidence for a continuous replica symmetry breaking in the region of high number of clauses, $α> α_c$.

cond-mat.dis-nn↗

arXiv2000

Thermodynamics of a Tiling Model

A particular, two-dimensional, tiling model, composed by the so called Wang tiles has been studied at finite temperature by Monte Carlo numerical simulations. In absence of any thermal bath the Wang tiles give the opportunity of building a very large number of non-periodic tilings. We can construct a local Hamiltonian such that only perfectly matched tilings are ground states with zero energy. This Hamiltonian has a very large degeneracy. The thermodynamic behaviour of such a system seems to show a continuous phase transition at non zero temperature. An order parameter with non-trivial features is proposed. Under the critical temperature the model exhibits aging properties. The fluctuation-dissipation theorem is violated.

cond-mat.dis-nn↗