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Physica Scripta

Image Restoration and Ising Spin Glasses in a Transverse Field

Jun‐ichi Inoue

Published Jan 1, 2003• 2 citations
Publisher

Abstract

Image restoration is formulated by the quantum Ising spin glass model. In terms of the Bayesian statistics, two types of restoration are possible, and each restoration is closely related to grand state searching via quantum or thermal annealing and calculation of local magnetizations at finite temperature , respectively. We evaluate these performances by analysis of the Sherrington–Kirkpatrick model in a transverse field and by employing the quantum Monte Carlo method. Effects of a transverse field on the quality of restoration are discussed.

DOI
10.1238/physica.topical.106a00070

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Image restoration is formulated by the quantum Ising spin glass model. In terms of the Bayesian statistics, two types of restoration are possible, and each restoration is closely related to grand state searching via quantum or thermal annealing and calculation of local magnetizations at finite temperature , respectively. We evaluate these performances by analysis of the Sherrington–Kirkpatrick model in a transverse field and by employing the quantum Monte Carlo method. Effects of a transverse field on the quality of restoration are discussed.
We introduce the transverse Ising model as a prototype for discussing quantum phase transitions. Mean field theory and its application to superconductivity by BCS are discussed, as well as real space renormalization group techniques in one dimension. Next, we introduce the Suzuki-Trotter formalism to show the correspondence between d-dimensional quantum systems and (d + 1)-dimensional classical systems. We then discuss transverse Ising spin glass models, namely the Sherrington-Kirkpatrick (SK) model, the Edwards-Anderson (EA) model and the ±J model. We briefly discuss the mean field, exact diagonalization and quantum Monte Carlo results for their phase diagrams. We discuss the question of replica symmetry restoration in quantum spin glasses due to the possibility of tunneling through the barriers. Finally, we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses.
We introduce the transverse Ising model as a prototype for discussing quantum phase transition. Next we introduce Suzuki-Trotter formalism to show the correspondence between ddd-dimensional quantum system with a (d+1)(d+1)(d+1)-dimensional classical system. We then discuss transverse Ising spin glass models, namely S-K model, E-A model, and the ±J\pm J±J model with Ising spin in transverse field. We briefly discuss the mean field, exact diagonalization, and quantum Monte Carlo results for their phase diagrams. Next we discuss the question of replica symmetry restoration in quantum spin glasses (due to tunnelling possibility through the barriers). Then we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses. We have also breifly discussed the possibility of Quantum Annealing in context of kinetically constrained systems. Mean-field calculation for BCS superconductivity, Real-space RG calculation for one dimensional transverse Ising system, scattering amplitude calculation for tunneling through asymmetric barrier (useful for quantum kinetically constrained system) are given in the appendices.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.
We study the three-dimensional quantum Ising spin glass in a transverse magnetic field following the evolution of the bond probability distribution under renormalization-group transformations. The phase diagram (critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ vs transverse field \ensuremath{\Gamma}) we obtain shows a finite slope near T=0, in contrast with the infinite slope for the pure case. Our results compare very well with the experimental data recently obtained for the dipolar Ising spin glass LiHo0.167{\mathrm{LiHo}}_{0.167}LiHo0.167​Y0.833{\mathrm{Y}}_{0.833}Y0.833​F4{\mathrm{F}}_{4}F4​, in a transverse field. This indicates that this system is more appropriately described by a model with short-range interactions than by an equivalent Sherrington-Kirkpatrick model in a transverse field.
The effect of a transverse field on the spin-glass freezing transition in the Sherrington-Kirkpatrick version of the quantum Ising model is studied using the thermofield dynamics (the real-time, finite-temperature quantum field theory) due to Takahashi and Umezawa (1975) as a substitute for the 'n-replica trick'. The field-dependent transition line to the spin-glass phase is found and the averaged local static metastable susceptibility in the transverse direction is evaluated in the simple mean-field approximation for the effective dynamic Lagrangian.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.