Zero-temperature computer simulations are reported for the Sherrington-Kirkpatrick random Ising model of a spin glass, including an external field. Results are presented for the internal field distribution P(H), and for the ground state energy and magnetisation as functions of field. P(H) has a linear rise from H=0 for all external fields. The zero-temperature susceptibility chi (0) is close to unity when equilibrium states are examined, in agreement with Parisi's replica symmetry breaking theory and in conflict with linear response theory. The linear response result chi (0)=0 can be obtained by searching for metastable local energy minima close to the zero-field ground state in configuration space.Zero-temperature computer simulations are reported for the Sherrington-Kirkpatrick random Ising model of a spin glass, including an external field. Results are presented for the internal field distribution P(H), and for the ground state energy and magnetisation as functions of field. P(H) has a linear rise from H=0 for all external fields. The zero-temperature susceptibility chi (0) is close to unity when equilibrium states are examined, in agreement with Parisi's replica symmetry breaking theory and in conflict with linear response theory. The linear response result chi (0)=0 can be obtained by searching for metastable local energy minima close to the zero-field ground state in configuration space.

World Scientific lecture notes in physicsNov 1, 19860 citations

Stability and susceptibility in Parisi's solution of a spin glass model

D. J. Thouless, J. R. L. de Almeida, J. Michael Kosterlitz

The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.

Journal of Physics C Solid State PhysicsJun 20, 198085 citations

On the mean field theory of spin glasses

Jairo Rolim Lopes de Almeida

This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.

Jan 1, 19800 citations

Analytic and computational studies of spin glasses

Shreyans Jain

The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.

Jan 1, 19850 citations

Replica-symmetry breaking for the Ising spin glass in a transverse field

G. Büttner, Klaus D. Usadel

Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near ${\mathit{T}}_{\mathit{f}}$ , while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near ${\mathit{T}}_{\mathit{f}}$, while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.

Physical review. B, Condensed matterOct 1, 199037 citations

Spin-glass phase transition and behavior of nonlinear susceptibility in the Sherrington-Kirkpatrick model with random fields

C.V. Morais, Fábio M. Zimmer, Matheus J. Lazo +2 more

The behavior of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$ and its relation to the spin-glass transition temperature ${T}_{f}$ in the presence of random fields are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature ${T}_{f}$ can be traced by a divergence in the spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{SG}}$, which presents a term inversely proportional to the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$. As a result of a relation between ${\ensuremath{\chi}}_{\mathrm{SG}}$ and ${\ensuremath{\chi}}_{3}$, the latter also presents a divergence at ${T}_{f}$ , which comes as a direct consequence of ${\ensuremath{\lambda}}_{\mathrm{AT}}=0$ at ${T}_{f}$ . However, our results show that, in the presence of random fields, ${\ensuremath{\chi}}_{3}$ presents a rounded maximum at a temperature ${T}^{*}$ which does not coincide with the spin-glass transition temperature ${T}_{f}$ (i.e., ${T}^{*}>{T}_{f}$ for a given applied random field). Thus, the maximum value of ${\ensuremath{\chi}}_{3}$ at ${T}^{*}$ reflects the effects of the random fields in the paramagnetic phase instead of the nontrivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that ${\ensuremath{\chi}}_{3}$ still maintains a dependence on the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$, although in a more complicated way as compared with the case without random fields. These results are discussed in view of recent observations in the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound.The behavior of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$ and its relation to the spin-glass transition temperature ${T}_{f}$ in the presence of random fields are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature ${T}_{f}$ can be traced by a divergence in the spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{SG}}$, which presents a term inversely proportional to the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$. As a result of a relation between ${\ensuremath{\chi}}_{\mathrm{SG}}$ and ${\ensuremath{\chi}}_{3}$, the latter also presents a divergence at ${T}_{f}$, which comes as a direct consequence of ${\ensuremath{\lambda}}_{\mathrm{AT}}=0$ at ${T}_{f}$. However, our results show that, in the presence of random fields, ${\ensuremath{\chi}}_{3}$ presents a rounded maximum at a temperature ${T}^{*}$ which does not coincide with the spin-glass transition temperature ${T}_{f}$ (i.e., ${T}^{*}>{T}_{f}$ for a given applied random field). Thus, the maximum value of ${\ensuremath{\chi}}_{3}$ at ${T}^{*}$ reflects the effects of the random fields in the paramagnetic phase instead of the nontrivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that ${\ensuremath{\chi}}_{3}$ still maintains a dependence on the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$, although in a more complicated way as compared with the case without random fields. These results are discussed in view of recent observations in the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound.

arXiv:1606.01186Physical review. B./Physical review. BJun 17, 20167 citations

Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near

{\mathit{T}}_{\mathit{f}}

, while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.

The behavior of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$ and its relation to the spin-glass transition temperature

{T}_{f}

in the presence of random fields are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature

{T}_{f}

can be traced by a divergence in the spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{SG}}$, which presents a term inversely proportional to the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$. As a result of a relation between ${\ensuremath{\chi}}_{\mathrm{SG}}$ and ${\ensuremath{\chi}}_{3}$, the latter also presents a divergence at

{T}_{f}

, which comes as a direct consequence of ${\ensuremath{\lambda}}_{\mathrm{AT}}=0$ at

{T}_{f}

. However, our results show that, in the presence of random fields, ${\ensuremath{\chi}}_{3}$ presents a rounded maximum at a temperature

{T}^{*}

which does not coincide with the spin-glass transition temperature

{T}_{f}

(i.e.,

{T}^{*}>{T}_{f}

for a given applied random field). Thus, the maximum value of ${\ensuremath{\chi}}_{3}$ at

{T}^{*}

reflects the effects of the random fields in the paramagnetic phase instead of the nontrivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that ${\ensuremath{\chi}}_{3}$ still maintains a dependence on the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$, although in a more complicated way as compared with the case without random fields. These results are discussed in view of recent observations in the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound.