arXiv subjects

D. Sherrington

Publications and source records attributed to D. Sherrington.

At least 19 recordsLinked to original sources

arXiv2010

Spin glass transition in geometrically frustrated antiferromagnets with weak disorder

We study the effect in geometrically frustrated antiferromagnets of weak, random variations in the strength of exchange interactions. Without disorder the simplest classical models for these systems have macroscopically degenerate ground states, and this degeneracy may prevent ordering at any temperature. Weak exchange randomness favours a small subset of these ground states and induces a spin-glass transition at an ordering temperature determined by the amplitude of modulations in interaction strength. We use the replica approach to formulate a theory for this transition, showing that it falls into the same universality class as conventional spin-glass transitions. In addition, we show that a model with a low concentration of defect bonds can be mapped onto a system of randomly located pseudospins that have dipolar effective interactions. We also present detailed results from Monte Carlo simulations of the classical Heisenberg antiferromagnet on the pyrochlore lattice with weak randomness in nearest neighbour exchange.

cond-mat.dis-nn↗

arXiv2006

On the strategy frequency problem in batch Minority Games

Ergodic stationary states of Minority Games with S strategies per agent can be characterised in terms of the asymptotic probabilities $ϕ_a$ with which an agent uses $a$ of his strategies. We propose here a simple and general method to calculate these quantities in batch canonical and grand-canonical models. Known analytic theories are easily recovered as limiting cases and, as a further application, the strategy frequency problem for the batch grand-canonical Minority Game with S=2 is solved. The generalization of these ideas to multi-asset models is also presented. Though similarly based on response function techniques, our approach is alternative to the one recently employed by Shayeghi and Coolen for canonical batch Minority Games with arbitrary number of strategies.

physics.soc-ph↗

arXiv2006

One-step replica symmetry breaking solution of the quadrupolar glass model

We consider the quadrupolar glass model with infinite-range random interaction. Introducing a simple one-step replica symmetry breaking ansatz we investigate the para-glass continuous (discontinuous) transition which occurs below (above) a critical value of the quadrupole dimension m*. By using a mean-field approximation we study the stability of the one-step replica symmetry breaking solution and show that for m>m* there are two transitions. The thermodynamic transition is discontinuous but there is no latent heat. At a higher temperature we find the dynamical or glass transition temperature and the corresponding discontinuous jump of the order parameter.

cond-mat.dis-nn↗

arXiv2005

Scaling and Renormalization Group in Replica Symmetry Breaking space: Evidence for a simple analytical solution of the SK model at zero temperature

Using numerical self-consistent solutions of a sequence of finite replica symmetry breakings (RSB) and Wilson's renormalization group but with the number of RSB-steps playing a role of decimation scales, we report evidence for a non-trivial T->0-limit of the Parisi order function q(x) for the SK spin glass. Supported by scaling in RSB-space, the fixed point order function is conjectured to be q*(a)=sqrt{π/2} a/ξerf(ξ/a) on 0\leq a\leq infty where x/T->a at T=0 and ξ\approx 1.13\pm 0.01. ξplays the role of a correlation length in a-space. q*(a) may be viewed as the solution of an effective 1D field theory.

cond-mat.dis-nn↗

arXiv2003

Modulated replica symmetry breaking schemes for antiferromagnetic spin glasses

We define modulated replica symmetry breaking (RSB)-schemes which combine tree- and wave-like structures. A modulated scheme and unmodulated RSB are applied at 1-step level to a semiconductor model with antiferromagnetic Korenblit-Shender interaction. By comparison of the free energies we find evidence that a T=0 phase transition in the ferrimagnetic phase leads to a transition between the different RSB-schemes. An embedding factor of Parisi block matrices with sublattice-asymmetrical size is employed as a new variational parameter in the modulated scheme.

cond-mat.dis-nn↗

arXiv2003

Dynamics of a spherical minority game

We present an exact dynamical solution of a spherical version of the batch minority game (MG) with random external information. The control parameters in this model are the ratio of the number of possible values for the public information over the number of agents, and the radius of the spherical constraint on the microscopic degrees of freedom. We find a phase diagram with three phases: two without anomalous response (an oscillating versus a frozen state), and a further frozen phase with divergent integrated response. In contrast to standard MG versions, we can also calculate the volatility exactly. Our study reveals similarities between the spherical and the conventional MG, but also intriguing differences. Numerical simulations confirm our analytical results.

cond-mat.dis-nn↗

arXiv2002

Fermionic SK-models with Hubbard interaction: Magnetism and electronic structure

Models with range-free frustrated Ising spin- and Hubbard interaction are treated exactly by means of the discrete time slicing method. Critical and tricritical points, correlations, and the fermion propagator, are derived as a function of temperature T, chemical potential μ, Hubbard coupling U, and spin glass energy J. The phase diagram is obtained. Replica symmetry breaking (RSB)-effects are evaluated up to four-step order (4RSB). The use of exact relations together with the 4RSB-solutions allow to model exact solutions by interpolation. For T=0, our numerical results provide strong evidence that the exact density of states in the spin glass pseudogap regime obeys ρ(E)=const |E-E_F| for energies close to the Fermi level. Rapid convergence of ρ'(E_F) under increasing order of RSB is observed. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of localized disordered fermionic systems in 2D. Beyond half filling we obtain a quadratic dependence of the fermion filling factor on the chemical potential. We find a half filling transition between a phase for U>μ, where the Fermi level lies inside the Hubbard gap, into a phase where μ(>U) is located at the center of the upper spin glass pseudogap (SG-gap). For μ>U the Hubbard gap combines with the lower one of two SG-gaps (phase I), while for μ<U it joins the sole SG-gap of the half-filling regime (phase II). We predict scaling behaviour at the continuous half filling transition. Implications of the half-filling transition between the deeper insulating phase II and phase I for delocalization due to hopping processes in itinerant model extensions are discussed and metal-insulator transition scenarios described.

cond-mat.dis-nn↗

arXiv2002

Coupled dynamics of sequence selection and compactification in mean-field hetero-polymers

We study a simple solvable model describing the genesis of monomer sequences for hetero-polymers (such as proteins), as the result of the equilibration of a slow stochastic genetic selection process which is assumed to be driven by the competing demands of functionality and reproducibility of the polymer's folded structure. Since reproducibility is defined in terms of properties of the folding process, one is led to the analysis of the coupled dynamics of (fast) polymer folding and (slow) genetic sequence selection. For the present mean-field model this analysis can be carried out using the finite-dimensional replica method, leading to exact results for (first- and second-order) transitions and to rich phase diagrams.

cond-mat.dis-nn↗

arXiv2001

Dynamics of the Batch Minority Game with Inhomogeneous Decision Noise

We study the dynamics of a version of the batch minority game, with random external information and with different types of inhomogeneous decision noise (additive and multiplicative), using generating functional techniques à la De Dominicis. The control parameters in this model are the ratio $α=p/N$ of the number $p$ of possible values for the external information over the number $N$ of trading agents, and the statistical properties of the agents' decision noise parameters. The presence of decision noise is found to have the general effect of damping macroscopic oscillations, which explains why in certain parameter regions it can effectively reduce the market volatility, as observed in earlier studies. In the limit $N\to\infty$ we (i) solve the first few time steps of the dynamics (for any $α$), (ii) calculate the location $α_c$ of the phase transition (signaling the onset of anomalous response), and (iii) solve the statics for $α>α_c$. We find that $α_c$ is not sensitive to additive decision noise, but we arrive at non-trivial phase diagrams in the case of multiplicative noise. Our theoretical results find excellent confirmation in numerical simulations.

cond-mat.dis-nn↗

arXiv2001

Correlated adaptation of agents in a simple market: a statistical physics perspective

We discuss recent work in the study of a simple model for the collective behaviour of diverse speculative agents in an idealized stockmarket, considered from the perspective of the statistical physics of many-body systems. The only information about other agents available to any one is the total trade at time steps. Evidence is presented for correlated adaptation and phase transitions/crossovers in the global volatility of the system as a function of appropriate information scaling dimension. Stochastically controlled irrationally of individual agents is shown to be globally advantageous. We describe the derivation of the underlying effective stochastic differential equations which govern the dynamics, and make an interpretation of the results from the point of view of the statistical physics of disordered systems.

cond-mat.dis-nn↗

arXiv1998

Enumeration of States in a Periodic Glass

We present an analytic enumeration of the metastable states, $N_s$, in a periodic long-range Josephson array frustrated by a transverse field. We find that the configurational entropy, $S_{conf} \equiv \ln N_s$, is extensive and scales with frustration, confirming that the non-random system is glassy. We also find that $S_{conf}$ is different from that of its disordered analogue, despite that fact that the two models share the same dynamical equations.

cond-mat.stat-mech↗

arXiv1998

Phase Diagram and Storage Capacity of Sequence Processing Neural Networks

We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation- and response functions, in the thermodynamic limit. We calculate the time translation invariant solutions of these equations, describing stationary limit-cycles, which leads to a phase diagram. The effective retarded self-interaction usually appearing in symmetric models is here found to vanish, which causes a significantly enlarged storage capacity of $α_c\sim 0.269$, compared to $α_\c\sim 0.139$ for Hopfield networks storing static patterns. Our results are tested against extensive computer simulations and excellent agreement is found.

cond-mat.dis-nn↗

arXiv1997

On the distribution of barriers in the spin glasses

We discuss a general formalism that allows study of transitions over barriers in spin glasses with long-range interactions that contain large but finite number, $N$, of spins. We apply this formalism to the Sherrington-Kirkpatrick model with finite $N$ and derive equations for the dynamical order parameters which allow ''instanton'' solutions describing transitions over the barriers separating metastable states. Specifically, we study these equations for a glass state that was obtained in a slow cooling process ending a little below $T_{c}$ and show that these equations allow ''instanton'' solutions which erase the response of the glass to the perturbations applied during the slow cooling process. The corresponding action of these solutions gives the energy of the barriers, we find that it scales as $τ^{6}$ where $τ$ is the reduced temperature.

cond-mat.dis-nn↗

arXiv1997

High-Temperature Dynamics of Spin Glasses

We develop a systematic expansion method of physical quantities for the SK model and the finite-dimensional $\pm J$ model of spin glasses in non-equilibrium states. The dynamical probability distribution function is derived from the master equation using a high temperature expansion. We calculate the expectation values of physical quantities from the dynamical probability distribution function. The theoretical curves show satisfactory agreement with Monte Carlo simulation results in the appropriate temperature and time regions. A comparison is made with the results of a dynamics theory by Coolen, Laughton and Sherrington.

cond-mat.dis-nn↗

arXiv1996

Macrodynamics of Disordered and Frustrated Systems

It is shown how the macroscopic non-equilibrium dynamics of a class of systems whose microscopic stochastic dynamics involves disordered and frustrated but range-free interactions can be well described by closed deterministic flow equations; this requires an appropriate choice of order parameters/function and ansatze.

cond-mat↗

arXiv1995

Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects

We develop further a recent dynamical replica theory to describe the dynamics of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution equations for macroscopic order parameters. We show how microscopic memory effects can be included in the formalism through the introduction of a dynamic order parameter function: the joint spin-field distribution. The resulting formalism describes very accurately the relaxation phenomena observed in numerical simulations, including the typical overall slowing down of the flow that was missed by the previous simple two-parameter theory. The advanced dynamical replica theory is either exact or a very good approximation.

cond-mat↗

arXiv1995

Dynamical Replica Theory for Disordered Spin Systems

We present a new method to solve the dynamics of disordered spin systems on finite time-scales. It involves a closed driven diffusion equation for the joint spin-field distribution, with time-dependent coefficients described by a dynamical replica theory which, in the case of detailed balance, incorporates equilibrium replica theory as a stationary state. The theory is exact in various limits. We apply our theory to both the symmetric- and the non-symmetric Sherrington-Kirkpatrick spin-glass, and show that it describes the (numerical) experiments very well.

cond-mat↗

arXiv1995

Possible Glassiness in a Periodic Long-Range Josephson Array

We present an analytic study of a periodic Josephson array with long-range interactions in a transverse magnetic field. We find that this system exhibits a first-order transition into a phase characterized by an extensive number of states separated by barriers that scale with the system size; the associated discontinuity is small in the limit of weak applied field, thus permitting an explicit analysis in this regime.

cond-mat↗