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Physical review. B, Condensed matter

Transverse Ising spin-glass model

Raimundo R. dos Santos, Rita Zorzenon dos Santos, Mauricio Kischinhevsky

Published Apr 1, 1985• 19 citations
Publisher

Abstract

We discuss the zero-temperature behavior of the transverse Ising spin-glass ($\ifmmode\pm\else\textpm\fi{}{J}_{0}$) model. The $d$-dimensional quantum model is shown to be equivalent to a classical ($d+1$)-dimensional Ising spin glass with correlated disorder. A renormalization-group treatment of the one-dimensional quantum model indicates the existence of a spin-glass phase. The Migdal-Kadanoff approximation is used to obtain the phase diagram of the quantum spin glass in two dimensions.

DOI
10.1103/physrevb.31.4694

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We discuss the zero-temperature behavior of the transverse Ising spin-glass (\ifmmode±\else\textpm\fiJ0\ifmmode\pm\else\textpm\fi{}{J}_{0}\ifmmode±\else\textpm\fiJ0​) model. The ddd-dimensional quantum model is shown to be equivalent to a classical (d+1d+1d+1)-dimensional Ising spin glass with correlated disorder. A renormalization-group treatment of the one-dimensional quantum model indicates the existence of a spin-glass phase. The Migdal-Kadanoff approximation is used to obtain the phase diagram of the quantum spin glass in two dimensions.
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.
A mean-field approximation is used to investigate the quantum transverse Ising spin-glass model in the Edwards-Anderson version. We introduce two order parameters, diagonal and off diagonal in replica indices, and discuss their physical significance. The effective free-energy function is obtained, and the critical value of the transverse field $\ensuremath{\Gamma}$ at zero temperature is calculated. We find the behavior of the order parameters near the zero-temperature critical point. The phase diagram in the $\ensuremath{\Gamma}$,TTT plane is also given.
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 classical transverse field Ising spin-glass model with short-range interactions is investigated beyond the mean-field approximation for a real d -dimensional lattice. We use an appropriate non-trivial modification of the Bethe-Peierls method recently formulated for the Ising spin-glass. The zero-temperature critical value of the transverse field and the linear susceptibility in the paramagnetic phase are obtained analytically as functions of dimensionality d . The phase diagram is also calculated numerically for different values of d . In the limit , known mean-field results are consistently reproduced.
We study the quantum transition at T=0 in the spin-1/2 Ising spin glass in a transverse field in two dimensions. The world line path integral representation of this model corresponds to an effective classical system in 2+1 dimensions, which we study by Monte Carlo simulations. Values of the critical exponents are estimated by a finite-size scaling analysis. We find that the dynamical exponent, z, and the correlation length exponent, \ensuremath{\nu}, are given by z=1.5\ifmmode\pm\else\textpm\fi{}0.05 and \ensuremath{\nu}=1.0\ifmmode\pm\else\textpm\fi{}0.1. Both the linear and nonlinear susceptibility are found to diverge at the critical point.
We present results of a Monte Carlo simulation study of the zero-temperature quantum phase transition of a three-dimensional short-range spin-1/2 Ising spin glass, driven by a transverse magnetic field. This quantum transition is equivalent to the finite-temperature transition in the (3+1)-dimensional anisotropic classical random Ising model that is the path integral of the quantum system. The critical exponents are estimated using a finite-size scaling analysis. The uniform linear susceptibility is finite at the transition, while the nonlinear susceptibility diverges. The results are more consistent with conventional, rather than activated, dynamic scaling in the quantum system.