One-dimensional Spin Glass Research Articles

Analysis of temporal correlation in heart rate variability through maximum entropy principle in a minimal pairwise glassy model

In this work we apply statistical mechanics tools to infer cardiac pathologies over a sample of M patients whose heart rate variability has been recorded via 24 h Holter device and that are divided in different classes according to their clinical status (providing a repository of labelled data). Considering the set of inter-beat interval sequences {mathbf {r}(i) } = { r_1(i), r_2(i), ldots , }, with i=1,ldots ,M, we estimate their probability distribution P(mathbf {r}) exploiting the maximum entropy principle. By setting constraints on the first and on the second moment we obtain an effective pairwise (r_n,r_m) model, whose parameters are shown to depend on the clinical status of the patient. In order to check this framework, we generate synthetic data from our model and we show that their distribution is in excellent agreement with the one obtained from experimental data. Further, our model can be related to a one-dimensional spin-glass with quenched long-range couplings decaying with the spin–spin distance as a power-law. This allows us to speculate that the 1/f noise typical of heart-rate variability may stem from the interplay between the parasympathetic and orthosympathetic systems.

Ultrametric probe of the spin-glass state in a field

We study the ultrametric structure of phase space of one-dimensional Ising spin glasses with random power-law interaction in an external random field. Although in zero field the model in both the mean-field and non-mean-field universality classes shows an ultrametric signature [Phys. Rev. Lett. 102, 037207 (2009)], when a field is applied ultrametricity seems only present in the mean-field regime. The results for the non-mean-field case in an external field agree with data for spin glasses studied within the Migdal-Kadanoff approximation. Our results therefore suggest that the spin-glass state might be fragile to external fields below the upper critical dimension.

One-dimensional infinite-component vector spin glass with long-range interactions

We investigate zero and finite temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number m of spin components where the interactions decay with a power, \sigma, of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent \theta on \sigma. A good fit to this dependence is \theta =3/4-\sigma. This implies that the upper critical value of \sigma is 3/4, corresponding to the lower critical dimension in the d-dimensional short-range version of the model. For finite temperatures the large m saddle-point equations are solved self-consistently which gives access to the correlation function, the order parameter and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, \sigma =5/8, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.

Spin glasses in the nonextensive regime

Spin systems with long-range interactions are "nonextensive" if the strength of the interactions falls off sufficiently slowly with distance. It has been conjectured for ferromagnets and, more recently, for spin glasses that, everywhere in the nonextensive regime, the free energy is exactly equal to that for the infinite range model in which the characteristic strength of the interaction is independent of distance. In this paper we present the results of Monte Carlo simulations of the one-dimensional long-range spin glasses in the nonextensive regime. Using finite-size scaling, our results for the transition temperatures are consistent with this prediction. We also propose and provide numerical evidence for an analogous result for dilutedlong-range spin glasses in which the coordination number is finite, namely, that the transition temperature throughout the nonextensive regime is equal to that of the infinite-range model known as the Viana-Bray model.

New insights from one-dimensional spin glasses

The concept of replica symmetry breaking found in the solution of the mean-field Sherrington-Kirkpatrick spinglass model has been applied to a variety of problems in science ranging from biological to computational and even financial analysis. Thus it is of paramount importance to understand which predictions of the mean-field solution apply to non-mean-field systems, such as realistic short-range spin-glass models. The one-dimensional spin glass with random power-law interactions promises to be an ideal test-bed to answer this question: Not only can large system sizes—which are usually a shortcoming in simulations of high-dimensional short-range system—be studied, by tuning the power-law exponent of the interactions the universality class of the model can be continuously tuned from the mean-field to the short-range universality class. We present details of the model, as well as recent applications to some questions of the physics of spin glasses. First, we study the existence of a spin-glass state in an external field. In addition, we discuss the existence of ultrametricity in short-range spin glasses. Finally, because the range of interactions can be changed, the model is a formidable test-bed for optimization algorithms.

Study of the de Almeida–Thouless Line Using Power-Law Diluted One-Dimensional Ising Spin Glasses

We test for the existence of a spin-glass phase transition, the de Almeida-Thouless line, in an externally applied (random) magnetic field by performing Monte Carlo simulations on a power-law diluted one-dimensional Ising spin glass for very large system sizes. We find that a de Almeida-Thouless line occurs only in the mean-field regime, which corresponds, for a short-range spin glass, to dimension d larger than 6.

The Power of Quantum Systems on a Line

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.

Dilute One-Dimensional Spin Glasses with Power Law Decaying Interactions

We introduce a diluted version of the one-dimensional spin-glass model with interactions decaying in probability as an inverse power of the distance. In this model, varying the power corresponds to changing the dimension in short-range models. The spin-glass phase is studied in and out of the range of validity of the mean-field approximation in order to discriminate between different theories. Since each variable interacts only with a finite number of others the cost for simulating, the model is drastically reduced with respect to the fully connected version, and larger sizes can be studied. We find both static and dynamic indications in favor of the so-called replica symmetry breaking theory.

Exact Lyapunov exponent of the harmonic magnon modes of one-dimensional Heisenberg-Mattis spin glasses

A mapping is developed between the linearized equation of motion for the dynamics of the transverse modes at $T=0$ of the Heisenberg-Mattis model of one-dimensional (1D) spin glasses and the (discretized) random wave equation. The mapping is used to derive an exact expression for the Lyapunov exponent (LE) of the magnon modes of spin glasses and to show that it follows anomalous scaling at low magnon frequencies. In addition, through numerical simulations, the differences between the LE and the density of states of the wave equation in a discrete 1D model of randomly disordered media (those with a finite correlation length) and that of continuous media (with a zero correlation length) are demonstrated and emphasized.

Distribution of local Lyapunov exponents in spin-glass dynamics

We investigate the statistical properties of local Lyapunov exponents which characterize magnon localization in the one-dimensional Heisenberg-Mattis spin glass HMSG at zero temperature by means of a connection to a suitable version of the Fokker-Planck FP equation. We consider the local Lyapunov exponents LLEs ,i n particular, the case of instantaneous LLE. We establish a connection between the transfer-matrix recursion relation for the problem and an FP equation governing the evolution of the probability distribution of the instantaneous LLE. The closed-form stationary solutions to the FP equation are in excellent accord with numerical simulations for both the unmagnetized and magnetized versions of the HMSG. Scaling properties for nonstationary conditions are derived from the FP equation in a special limit in which diffusive effects tend to vanish, and also shown to provide a close description to the corresponding numerical-simulation data. I. INTRODUCTION The analytic treatment of quenched disordered systems in condensed-matter physics invokes many concepts from statistical theory. Among these, we shall be concerned in this paper with the connection between the probability distribution of Lyapunov characteristic exponents for the equilibrium problem of low-lying magnetic excitations in spin glasses, and the Fokker-Planck FP equation, 1 which is a key element in the description of nonequilibrium stochastic

Spin and chirality orderings of the one-dimensional Heisenberg spin glass with long-rangepower-law interaction

The ordering of the one-dimensional Heisenberg spin glass interacting via the long-rangepower-law interaction is studied by Monte Carlo simulations. Particular attention is paid to thepossible occurrence of ‘spin–chirality decoupling’ for appropriate values of the power-law exponentσ. Our result suggests that, for intermediate values ofσ, the chiral-glass order occurs at finite temperatures while the standard spin-glass orderoccurs only at zero temperature.

Scaling behavior of a one-dimensional correlated disordered electronic system

A one-dimensional diagonal tight binding electronic system with correlated disorder is investigated. The correlation of the random potential is exponentially decaying with distance and its correlation length diverges as the concentration of "wrong sign" approaches to 1 or 0. The correlated random number sequence can be generated easily with a binary sequence similar to that of a one-dimensional spin glass system. The localization length (LL) and the integrated density of states (IDOS) for long chains are computed. A comparison with numerical results is made with the recently developed scaling technique results. The Coherent Potential Approximation (CPA) is also adopted to obtain scaling functions for both the LL and the IDOS. We confirmed that the scaling functions show a crossover near the band edge and establish their relation to the concentration. For concentrations near to 0 or 1 (longer correlation length case), the scaling behavior is followed only for a very limited range of the potential strengths.

A random walk to local minima and saddle points on a potential energy surface. A strategy based on simulated annealing

We demonstrate how the Metropolis simulated annealing method, a global stochastic minimizer, can also be used as an all-purpose extremizer capable of locating all the local minima and saddle points on a given surface, in addition to the global minimum. The demonstration is done with reference to the well studied Müller-Brown potential and a one-dimensional spin glass with random nearest-neighbour interaction. The relevance of the present strategy in the construction of reaction paths is discussed.

Monte Carlo simulation of a quasi one-dimensional spin glass

A recently introduced model of a quasi one-dimensional spin glass by means of Monte Carlo simulations was studied. The model consists of linear chains of Ising spins with ferromagnetic nearest-neighbour interchain coupling of strength, K, in addition to infinite-range random interactions. The Edwards-Anderson order parameter and the field-cooled and zero-field-cooled susceptibilities are evaluated for various values of K.

Dynamics of one-dimensional Heisenberg spin glasses in the high-field limit.

This paper reports the results of a study of the distribution and localization of the magnon modes in one-dimensional Heisenberg spin glasses with nearest-neighbor interactions. The analysis is limited to high fields and frequencies near the precession frequency. Both symmetric and asymmetric distributions of exchange interactions of the form P(J)\ensuremath{\propto}\ensuremath{\Vert}J${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ (\ensuremath{\alpha}1) are treated in detail. The results of approximate calculations based on the coherent-exchange approximation are shown to be in good agreement with numerical data obtained by applying mode-counting techniques to arrays of ${10}^{7}$ spins. Particular emphasis is placed on the qualitative differences in the behavior that arise depending on whether the average values of ${\mathit{J}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{J}}^{\mathrm{\ensuremath{-}}2}$ are zero, nonzero, or infinite.

Magnons in a one-dimensional spin glass: The high-field and zero-field limits.

The high-field and zero-field behavior of the linearized magnetic excitations (harmonic magons) in a one-dimensional \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass are studied. In the high-field limit---a field strong enough to ensure complete alignment of the ground state---the density of states (DOS), the inverse localization length (ILL), and the dynamic structure factor are calculated over the interval -4JE-H4J (H is applied field) by employing the coherent-exchange approximation (CEA), negative-eigenvalue counting, and matrix diagonalization. In the low-energy regime (0.001E-H0.1J), the CEA closely approximates the exact results reproducing, in particular, the anomalous power-law behavior of the DOS \ensuremath{\rho}(E)\ensuremath{\sim}(E-H${)}^{1/3}$ and the ILL 1/L(E)\ensuremath{\sim}(E-H${)}^{2/3}$ for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)${\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}$ and 1/L(E)=g(c)${\mathit{E}}^{2/3}$.

Decay of the remanent magnetization in the asymmetric spin chain

The dynamics of the one-dimensional spin glass with asymmetric interactions between neighboring spins is considered. We confine ourselves to discrete couplings with values ±J. We show that the algebraic decay of the remanent magnetization of the infinite ±J-spin chain at zero temperature is only valid for symmetric couplings. Our analytical investigations as well as computer simulations show stretched exponential decay for any finite concentration of antisymmetric bonds. Thus, the asymmetric ±J-spin chain shows an asymmetry-induced phase transition at zero temperature.

Upper and Lower Bounds to the Free Energy Density of a One-dimensional Spin Glass Model

In this paper, a one-dimensional short-ranged spin-glass model in a random magnetic field is studied. Noticing convexity of the partition function, we use Jensen's inequality to obtain a lower bound to the averaged free energy density and the slope inequality for an upper bound. These bounds are derived under very general conditions and they can be used as tests for various approximate methods such as the replica method.

Distribution and localization of the harmonic magnon modes in a one-dimensional Heisenberg spin glass.

The harmonic magnon modes in a one-dimensional Heisenberg spin glass having nearest-neighbor exchange interactions of fixed magnitude and random sign are investigated. The Lyapounov exponent is calculated for chains of ${10}^{7}$--${10}^{8}$ spins over the interval 0\ensuremath{\le}\ensuremath{\omega}\ensuremath{\le}4J. In the low-frequency regime, \ensuremath{\omega}\ensuremath{\lesssim}0.1J, an anomalous behavior for the density of states \ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}1/3}$ is established, consistent with earlier results obtained by Stinchcombe and Pimentel using transfer-matrix techniques; at higher frequencies, gaps appear in the spectrum. At low frequencies, the localization length diverges as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}2/3}$. A formal connection is established between the spin glass and the one-dimensional discretized Schr\"odinger equation. By making use of the connection, it is shown that the theory of Derrida and Gardner, which was developed for weak potential disorder, can account quantitatively for the distribution and localization of the low-frequency magnon modes in the spin-glass model.

Transfer-matrix scaling for anomalous dynamics of a vector spin-glass chain.

The transverse dynamics of a one-dimensional vector spin glass at zero temperature is treated by a new transfer-matrix scaling technique, and also by evaluating static critical properties related to dynamics by a crossover argument. A nontrivial dynamic exponent is found which implies that the system at zero temperature has dynamic critical behavior w\ensuremath{\propto}${k}^{3/2}$, in contradiction to the hydrodynamic behavior normally assumed and to the results of previous work.