Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0312611

Tranverse Ising Model, Glass and Quantum Annealing

Bikas K. Chakrabarti, Arnab Das

Published Dec 23, 2003• 2 citations
arXivHTMLPDF

Abstract

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 $d$-dimensional quantum system with a $(d+1)$-dimensional classical system. We then discuss transverse Ising spin glass models, namely S-K model, E-A model, and the $\pm J$ 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.

DOI
10.48550/arxiv.cond-mat/0312611

Related papers

Phase Transitions in Disordered Quantum Systems: Transverse Ising Models

Bikas K. Chakrabarti, Arnab Das

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.

Texts and readings in physical sciences•Jan 1, 2012•1 citation

Image Restoration and Ising Spin Glasses in a Transverse Field

Jun‐ichi Inoue

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.

Physica Scripta•Jan 1, 2003•2 citations

Phase diagram of the quantum Ising spin glass in a transverse field

Klaus D. Usadel, G. Büttner, T. K. Kopeć

A recent calculation of Yokota on the phase boundary for the Sherrington-Kirkpatrick version of the quantum Ising spin glass in a transverse field seems to lead to unphysical results. Various other approaches on this subject are quoted and compared with the exact Trotter-Suzuki method to resolve the controversies about the critical line in this model.

Physical review. B, Condensed matter•Dec 1, 1991•22 citations

Cavity-field approach to quantum spin glasses: The Ising spin glass in a transverse field

L. De Cesare, Katarzyna Walasek, K. Walasek

The cavity-field method, recently used for classical spin glasses, is extended to study the infinite-range Ising spin-glass model in a transverse field, by means of the Trotter-Suzuki transformation. Within this approach, which relies on well-known mathematics and deals with quantities that have transparent physical significance by avoiding the replica method, the quantum version of the classical Thouless-Anderson-Palmer equation is obtained. The phase diagram of the model is also obtained and full agreement with previous replica-method results is found.

Physical review. B, Condensed matter•Apr 1, 1992•16 citations

Quantum transverse Ising spin-glass model in the mean-field approximation

Katarzyna Walasek, K. Lukierska‐Walasek

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 $\ensuremath{\Gamma}$,$T$ plane is also given.

Physical review. B, Condensed matter•Oct 1, 1986•32 citations

Landau theory of quantum spin glasses of rotors and Ising spins

N. Read, Subir Sachdev A, J. Ye

We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.

arXiv:cond-mat/9412032•Physical review. B, Condensed matter•Jul 1, 1995•165 citations
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.
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.
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.
The cavity-field method, recently used for classical spin glasses, is extended to study the infinite-range Ising spin-glass model in a transverse field, by means of the Trotter-Suzuki transformation. Within this approach, which relies on well-known mathematics and deals with quantities that have transparent physical significance by avoiding the replica method, the quantum version of the classical Thouless-Anderson-Palmer equation is obtained. The phase diagram of the model is also obtained and full agreement with previous replica-method results is found.
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 consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.