1,721,022 research outputs found
Spherical 2+p spin-glass model: An analytically solvable model with a glass-to-glass transition
No full textWe present the detailed analysis of the spherical s+p spin-glass model with two competing interactions: among p spins and among s spins. The most interesting case is the 2+p model with p >= 4 for which a very rich phase diagram occurs, including, next to the paramagnetic and the glassy phase represented by the one step replica symmetry breaking ansatz typical of the spherical p-spin model, another two amorphous phases. Transitions between two contiguous phases can also be of a different kind. The model can thus serve as a mean-field representation of amorphous-amorphous transitions (or transitions between undercooled liquids of different structure). The model is analytically solvable everywhere in the phase space, even in the limit where the infinite replica symmetry breaking ansatz is required to yield a thermodynamically stable phase
Equilibrium dynamics of spin-glass systems
No full textWe 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, Horner, and Sommers [Z. Phys. B: Condens. Matter 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 ->infinity, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter Delta of the Sompolinsky theory
Complexity in mean-field spin-glass models: Ising p-spin
No full textThe 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, the latter accounting for the total number of solutions. The relevant TAP solutions counted by the non-SUSY complexity share the same features of the corresponding solutions in the Sherrington-Kirkpatrick model; in particular their Hessian has a vanishing isolated eigenvalue. The TAP solutions corresponding to the SUSY complexity, instead, are well separated minima
Thermodynamic first order transition and inverse freezing in a 3D spin glass
No full textWe present a numerical study of the random Blume-Capel model in three dimensions. The phase diagram is characterized by spin-glass-paramagnet phase transitions of both first and second order in the thermodynamic sense. Numerical simulations are performed using the exchange Monte Carlo algorithm, providing clear evidence for inverse freezing. The main features at criticality and in the phase coexistence region are investigated. We are not privy to other 3D short-range systems with quenched disorder undergoing inverse freezing. © 2010 The American Physical Society
Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume-Emery-Griffiths Models
No full textRenormalization group based on the Migdal-Kadanoff bond removal approach is often considered a simple and valuable tool to understand the critical behavior of complicated statistical mechanical models. In presence of quenched disorder, however, predictions obtained with the Migdal-Kadanoff bond removal approach quite often fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In an attempt to overcome this limitation we analyze the behavior of Ising and Blume-Emery-Griffiths models on more structured hierarchical lattices. We find that, apart from some exceptions, the failure is not limited to Midgal-Kadanoff cells but originates right from the hierarchization of Bravais lattices on small cells, and shows up also when in-cell loops are considered
Small-cluster renormalization group in Ising and Blume-Emery-Griffiths models with ferromagnetic, antiferromagnetic, and quenched disordered magnetic interactions
No full textThe Ising and Blume-Emery-Griffiths (BEG) models' critical behavior is analyzed in two dimensions and three dimensions by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under renormalization allow for the determination of the Néel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure does not yield a strong-disorder critical point corresponding to the transition to the spin-glass phase. This limitation is thoroughly analyzed and motivated. © 2014 American Physical Society
Marginal states in mean-field glasses
No full textWe study mean-field systems whose free-energy landscape is dominated by marginally stable states. We review and develop various techniques to describe such states, elucidating their physical meaning and the interrelation between them. In particular, we give a physical interpretation of the two-group replica symmetry-breaking scheme and confirm it by establishing the relation to the cavity method and to the counting of solutions of the Thouless-Anderson-Palmer equations. We show how these methods all incorporate the presence of a soft mode in the free-energy landscape and interpret the occurring order-parameter functions in terms of correlations between the soft mode and the local magnetizations. The general formalism is applied to the prototypical case of the Sherrington-Kirkpatrick-model, where we reexamine the physical properties of marginal states under a new perspective
Complexity of waves in nonlinear disordered media
No full textThe statistical properties of the phases of several modes nonlinearly coupled in a random system are investigated by means of a Hamiltonian model with disordered couplings. The regime in which the modes have a stationary distribution of their energies and in which the phases are coupled is studied for arbitrary degrees of randomness and energy. The complexity versus temperature and strength of nonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amount of disorder. Implications in random lasing, nonlinear wave propagation, and finite-temperature Bose-Einstein condensation are discussed
Statistical mechanical approach to secondary processes and structural relaxation in glasses and glass formers
No full textThe interrelation of dynamic processes active on separated time-scales in glasses and viscous liquids is investigated using a model displaying two time-scale bifurcations both between fast and secondary relaxation and between secondary and structural relaxation. The study of the dynamics allows for predictions on the system relaxation above the temperature of dynamic arrest in the mean-field approximation, that are compared with the outcomes of the equations of motion directly derived within the Mode Coupling Theory (MCT) for under-cooled viscous liquids. By varying the external thermodynamic parameters, a wide range of phenomenology can be represented, from a very clear separation of structural and secondary peak in the susceptibility loss to excess wing structures
The overlap parameter across an inverse first-order phase transition in a 3D spin-glass
No full textWe investigate the thermodynamic phase transition taking place in the Blume-Capel model in the presence of quenched disorder in three dimensions. In particular, performing exchange Monte Carlo simulations, we study the behaviour of the order parameters across the first-order phase transition and its related coexistence region. This transition is inverse freezing
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