Physical review. B, Condensed matter

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

Katarzyna Walasek, K. Lukierska‐Walasek

Published Oct 1, 1986• 32 citations

Abstract

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.

DOI: 10.1103/physrevb.34.4962

Related papers

Replica symmetry breaking in the transverse-field Ising spin-glass model: Two fermionic representations

Eduardo Muccillo Monteiro Santos, Alba Theumann

arXiv:cond-mat/0505522Physical Review BJul 19, 20054 citations

Classical transverse Ising spin glass with short-range interactions beyond the mean-field approximation

Katarzyna Walasek, K. Lukierska‐Walasek, L. De Cesare +1 more

arXiv:cond-mat/9804033Journal of Physics A Mathematical and GeneralJul 10, 19983 citations

Quantum Critical Behavior of the Infinite-range Transverse Ising Spin Glass : An Exact Numerical Diagonalization Study

Parongama Sen, Purusattam Ray, Bikas K. Chakrabarti

arXiv:cond-mat/9705297May 29, 19970 citations

Scaling theory for the quantum spin-glass transition

M. A. Contínentino, B. Boechat, Raimundo R. dos Santos

Physical review. B, Condensed matterNov 1, 199410 citations

Landau theory of quantum spin glasses of rotors and Ising spins

N. Read, Subir Sachdev A, J. Ye

arXiv:cond-mat/9412032Physical review. B, Condensed matterJul 1, 1995165 citations

Soluble model to treat the quantum spin glass

J. Roberto Viana, Yamilles Nogueira, J. Ricardo de Sousa

Physical review. B, Condensed matterSep 19, 20028 citations

Physical review. B, Condensed matter

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

Katarzyna Walasek, K. Lukierska‐Walasek

Published Oct 1, 1986• 32 citations

Publisher

Abstract

\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.

DOI: 10.1103/physrevb.34.4962

Related papers

Replica symmetry breaking in the transverse-field Ising spin-glass model: Two fermionic representations

Eduardo Muccillo Monteiro Santos, Alba Theumann

We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of $n$ replicas is divided in $r$ blocks of $m$ replicas each. We present results for different values of the block-size parameter $m$ . The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal ${S}_{i}^{z}$ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of $n$ replicas is divided in $r$ blocks of $m$ replicas each. We present results for different values of the block-size parameter $m$. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal ${S}_{i}^{z}$ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.

arXiv:cond-mat/0505522Physical Review BJul 19, 20054 citations

Classical transverse Ising spin glass with short-range interactions beyond the mean-field approximation

Katarzyna Walasek, K. Lukierska‐Walasek, L. De Cesare +1 more

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.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.

arXiv:cond-mat/9804033Journal of Physics A Mathematical and GeneralJul 10, 19983 citations

Quantum Critical Behavior of the Infinite-range Transverse Ising Spin Glass : An Exact Numerical Diagonalization Study

Parongama Sen, Purusattam Ray, Bikas K. Chakrabarti

We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field $Γ$ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of $Γ$ and for system sizes $N \le 16$ . We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of $Γ$ . The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at $Γ_c \simeq 1.5$ . Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get $z=2.1 \pm 0.1$ which is in contradiction with a recent estimate ( $z=4$ ). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for $Γ< Γ_c$ .We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field $Γ$ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of $Γ$ and for system sizes $N \le 16$. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of $Γ$. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at $Γ_c \simeq 1.5$. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get $z=2.1 \pm 0.1$ which is in contradiction with a recent estimate ($z=4$). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for $Γ< Γ_c$.

arXiv:cond-mat/9705297May 29, 19970 citations

Scaling theory for the quantum spin-glass transition

M. A. Contínentino, B. Boechat, Raimundo R. dos Santos

We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.

Physical review. B, Condensed matterNov 1, 199410 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.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/9412032Physical review. B, Condensed matterJul 1, 1995165 citations

Soluble model to treat the quantum spin glass

J. Roberto Viana, Yamilles Nogueira, J. Ricardo de Sousa

The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams $(T$ versus $\ensuremath{\Omega},$ and T versus ${J}_{o}/J,$ where $\ensuremath{\Omega}$ is the transverse field, ${J}_{o}$ and J are the ferromagnetic and random exchange interactions, respectively) are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field $\ensuremath{\Omega}$ is to destroy the ordered phases. In the $(T,\ensuremath{\Omega})$ plane our results are compared with those obtained by the replica-symmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams $(T$ versus $\ensuremath{\Omega},$ and T versus ${J}_{o}/J,$ where $\ensuremath{\Omega}$ is the transverse field, ${J}_{o}$ and J are the ferromagnetic and random exchange interactions, respectively) are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field $\ensuremath{\Omega}$ is to destroy the ordered phases. In the $(T,\ensuremath{\Omega})$ plane our results are compared with those obtained by the replica-symmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.

Physical review. B, Condensed matterSep 19, 20028 citations

\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.

We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of

n

replicas is divided in

r

blocks of

m

replicas each. We present results for different values of the block-size parameter

m

. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal

{S}_{i}^{z}

operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.

We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field

Γ

at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of

Γ

and for system sizes

N \le 16

. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of

Γ

. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at

Γ_c \simeq 1.5

. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get

z=2.1 \pm 0.1

which is in contradiction with a recent estimate (

z=4

). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for

Γ< Γ_c

The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams

(T

versus $\ensuremath{\Omega},$ and T versus

{J}_{o}/J,

where $\ensuremath{\Omega}$ is the transverse field,

{J}_{o}

and J are the ferromagnetic and random exchange interactions, respectively) are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field $\ensuremath{\Omega}$ is to destroy the ordered phases. In the $(T,\ensuremath{\Omega})$ plane our results are compared with those obtained by the replica-symmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.