Spin Glass Ising Models Research Articles

Many-body localization: construction of the emergent local conserved operators via block real-space renormalization

A fully many-body localized (FMBL) quantum disordered system is characterized by the emergence of an extensive number of local conserved operators that prevents the relaxation towards thermal equilibrium. These local conserved operators can be seen as the building blocks of the whole set of eigenstates. In this paper, we propose to construct them explicitly via block real-space renormalization. The principle is that each renormalization group step diagonalizes the smallest remaining blocks and produces a conserved operator for each block. The final output for a chain of N spins is a hierarchical organization of the N conserved operators with layers. The system size nature of the conserved operators of the top layers is necessary to describe the possible long-range order of the excited eigenstates and the possible critical points between different FMBL phases. We discuss the similarities and the differences with the strong disorder RSRG-X method that generates the whole set of the 2N eigenstates via a binary tree of N layers. The approach is applied to the long-range quantum spin-glass Ising model, where the constructed excited eigenstates are found to be exactly like ground states in another disorder realization, so that they can be either in the paramagnetic phase, in the spin-glass phase or critical.

Untangling complex networks: Risk minimization in financial markets through accessible spin glass ground states

Recurrent international financial crises inflict significant damage to societies and stress the need for mechanisms or strategies to control risk and tamper market uncertainties. Unfortunately, the complex network of market interactions often confounds rational approaches to optimize financial risks. Here we show that investors can overcome this complexity and globally minimize risk in portfolio models for any given expected return, provided the margin requirement remains below a critical, empirically measurable value. In practice, for markets with centrally regulated margin requirements, a rational stabilization strategy would be keeping margins small enough. This result follows from ground states of the random field spin glass Ising model that can be calculated exactly through convex optimization when relative spin coupling is limited by the norm of the network’s Laplacian matrix. In that regime, this novel approach is robust to noise in empirical data and may be also broadly relevant to complex networks with frustrated interactions that are studied throughout scientific fields.

The tricritical point in the quantum Ising S = 1 spin glass withbiaxial crystal-field effects

We study a quantum generalization of the infinite-rangeSherrington-Kirkpatrick spin-glass model with biaxial crystal-fieldeffects described by two uniaxial anisotropy parameters Dx andDy. For spin dimensionality S = 1 we report an analytical andnumerical analysis in the (T, Dx, Dy) parameter space (with Tbeing the temperature). For D ≡ Dx = Dy the model effectivelybecomes classical and identical with the crystal-field-split spin-glassIsing model (introduced by Ghatak and Sherrington) showinga discontinuous phase transition to the spin-glass phase on a portionof the T-D line.

Simulating Ising spin glasses on a quantum computer

A linear-time algorithm is presented for the construction of the Gibbs distribution of configurations in the Ising model, on a quantum computer. The algorithm is designed so that each run provides one configuration with a quantum probability equal to the corresponding thermodynamic weight. The partition function is thus approximated efficiently. The algorithm neither suffers from critical slowing down nor gets stuck in local minima. The algorithm can be applied in any dimension, to a class of spin-glass Ising models with a finite portion of frustrated plaquettes, diluted Ising models, and models with a magnetic field.

Kinetic lattice models of disorder

We study a class of stochastic Ising (or interacting particle) systems that exhibit a spatial distribution of impurities that change with time. It may model, for instance, steady nonequilibrium conditions of the kind that may be induced by diffusion in some disordered materials. Different assumptions for the degree of coupling between the spin and the impurity configurations are considered. Two interesting well-defined limits for impurities that behave autonomously are (i) the standard (i.e., quenched) bond-diluted, random-field, random-exchange, and spin-glass Ising models, and (ii) kinetic variations of these standard cases in which conflicting kinetics simulate fast and random diffusion of impurities. A generalization of the Mattis model with disorder that describes a crossover from the equilibrium case (i) to the nonequilibrium case (ii) and the microscopic structure of a generalized heat bath are explicitly worked out as specific realizations of our class of models. We sketch a simple classification of transition rates for the time evolution of the spin configuration based on the critical behavior that is exhibited by the models in case (ii). The latter are shown to have an exact solution for any lattice dimension for some special choice of rates.

Critical behavior of three-dimensional Ising model of spin glass (invited)

First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.

Series expansions for a spin-glass model

Series expansions are investigated for a spin-glass Ising model with nearest-neighbour interactions J which can be randomly positive or negative. For the high-temperature phase the following conclusions result: (i) the free energy has a singularity at about w = µ-1/2 (w = tanh β J) (where µ is the self-avoiding walk limit), (ii) the magnetic susceptibility corresponds to uncoupled spins, (iii) the second derivative of susceptibility is simply related to the susceptibility of the standard Ising model and has a singularity at w = wc1/2 (where wc refers to the standard model).