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:1606.01186 · Physical review. B./Physical review. B

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, S. G. Magalhães, Fernando Dantas Nobre

Published Jun 17, 2016• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.93.224206

Related papers

Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

S. G. Magalhães, C.V. Morais, Fábio M. Zimmer +2 more

The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glass model, in the presence of a uniform transverse field $\mathrm{\ensuremath{\Gamma}}$, as well as of a longitudinal random field ${h}_{i}$, which follows a Gaussian distribution characterized by a width proportional to $\mathrm{\ensuremath{\Delta}}$. The interactions are infinite-ranged, and the 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 applied fields is analyzed. This study is motivated by experimental investigations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility ${\ensuremath{\chi}}_{3}$, which have led to a considerable experimental and theoretical debate. We have analyzed two physically distinct situations, namely, $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ considered as independent, as well as these two quantities related, as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature ${T}_{f}$, with such phase being characterized by a nontrivial ergodicity breaking; moreover, ${T}_{f}$ decreases by increasing $\mathrm{\ensuremath{\Gamma}}$ towards a quantum critical point at zero temperature. The situation where $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ are related $[\mathrm{\ensuremath{\Delta}}\ensuremath{\equiv}\mathrm{\ensuremath{\Delta}}(\mathrm{\ensuremath{\Gamma}})]$ appears to reproduce better the experimental observations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$. In this later case, by increasing $\mathrm{\ensuremath{\Gamma}}$ gradually, ${\ensuremath{\chi}}_{3}$ becomes progressively rounded, presenting a maximum at a temperature ${T}^{*}$ $({T}^{*}>{T}_{f})$, with both the amplitude of the maximum and the value of ${T}^{*}$ decreasing gradually. Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature ${T}^{*}$, one for ${T}_{f}<T<{T}^{*}$, and another one for $T>{T}^{*}$. It is argued that the conventional paramagnetic state corresponds to $T>{T}^{*}$, whereas the temperature region ${T}_{f}<T<{T}^{*}$ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.

arXiv:1701.04373•Physical review. B./Physical review. B•Feb 1, 2017•6 citations

Complete replica solution for the transverse field Sherrington-Kirkpatrick spin glass model with continuous-time quantum Monte Carlo method

Annamária Kiss, Gergely Zaránd, Izabella Lovas

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.

arXiv:2306.07337•Physical review. B./Physical review. B•Jan 25, 2024•9 citations

Effects of a Gaussian random field in the Sherrington-Kirkpatrick spin glass

R. F. Soares, Fernando Dantas Nobre, J. R. L. de Almeida

The infinite-range Ising spin glass in the presence of a Gaussian random field is investigated by means of the replica method. The replica-symmetric solution and its instability are discussed; it is shown that the Almeida-Thouless instability regions are always reduced by the random field. The crossover exponent \ensuremath{\varphi} associated with the irreversibility line is shown to be nonuniversal in the presence of a random field, varying in a range from 1 to 3. Some of our results are compared with recent experimental measurements performed in the diluted antiferromagnet ${\mathrm{Fe}}_{\mathit{x}}$${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$.

Physical review. B, Condensed matter•Sep 1, 1994•32 citations

Stability of the replica-symmetric saddle point in general mean-field spin-glass models

Katharina Janzen, Andreas Engel

Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a $2^n\times 2^n$ matrix with the number $n$ of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit $n\to 0$ within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.

arXiv:1008.1733•Journal of Statistical Mechanics Theory and Experiment•Dec 1, 2010•4 citations

Stability of the quantum Sherrington-Kirkpatrick spin glass model

A. P. Young

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

arXiv:1707.07107•Physical review. E•Sep 11, 2017•29 citations

Spin-glass transition in a magnetic field: A renormalization group study

Iveta R. Pimentel, T. Temesvári, C. De Dominicis

We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.

arXiv:cond-mat/0204615•Physical review. B, Condensed matter•Jun 6, 2002•50 citations
The behavior of the nonlinear susceptibility \ensuremathχ3{\ensuremath{\chi}}_{3}\ensuremathχ3​ and its relation to the spin-glass transition temperature Tf{T}_{f}Tf​ 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 \ensuremathλAT{\ensuremath{\lambda}}_{\mathrm{AT}}\ensuremathλAT​ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature Tf{T}_{f}Tf​ can be traced by a divergence in the spin-glass susceptibility \ensuremathχSG{\ensuremath{\chi}}_{\mathrm{SG}}\ensuremathχSG​, which presents a term inversely proportional to the replicon \ensuremathλAT{\ensuremath{\lambda}}_{\mathrm{AT}}\ensuremathλAT​. As a result of a relation between \ensuremathχSG{\ensuremath{\chi}}_{\mathrm{SG}}\ensuremathχSG​ and \ensuremathχ3{\ensuremath{\chi}}_{3}\ensuremathχ3​, the latter also presents a divergence at Tf{T}_{f}Tf​, which comes as a direct consequence of \ensuremathλAT=0{\ensuremath{\lambda}}_{\mathrm{AT}}=0\ensuremathλAT​=0 at Tf{T}_{f}Tf​. However, our results show that, in the presence of random fields, \ensuremathχ3{\ensuremath{\chi}}_{3}\ensuremathχ3​ presents a rounded maximum at a temperature T∗{T}^{*}T∗ which does not coincide with the spin-glass transition temperature Tf{T}_{f}Tf​ (i.e., T∗>Tf{T}^{*}>{T}_{f}T∗>Tf​ for a given applied random field). Thus, the maximum value of \ensuremathχ3{\ensuremath{\chi}}_{3}\ensuremathχ3​ at T∗{T}^{*}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 \ensuremathχ3{\ensuremath{\chi}}_{3}\ensuremathχ3​ still maintains a dependence on the replicon \ensuremathλAT{\ensuremath{\lambda}}_{\mathrm{AT}}\ensuremathλAT​, 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 LiHoxY1\ensuremath−xF4{\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}LiHox​Y1\ensuremath−x​F4​ compound.
The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glass model, in the presence of a uniform transverse field $\mathrm{\ensuremath{\Gamma}}$, as well as of a longitudinal random field hi{h}_{i}hi​, which follows a Gaussian distribution characterized by a width proportional to $\mathrm{\ensuremath{\Delta}}$. The interactions are infinite-ranged, and the 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 applied fields is analyzed. This study is motivated by experimental investigations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility ${\ensuremath{\chi}}_{3}$, which have led to a considerable experimental and theoretical debate. We have analyzed two physically distinct situations, namely, $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ considered as independent, as well as these two quantities related, as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature Tf{T}_{f}Tf​, with such phase being characterized by a nontrivial ergodicity breaking; moreover, Tf{T}_{f}Tf​ decreases by increasing $\mathrm{\ensuremath{\Gamma}}$ towards a quantum critical point at zero temperature. The situation where $\mathrm{\ensuremath{\Delta}}$ and $\mathrm{\ensuremath{\Gamma}}$ are related $[\mathrm{\ensuremath{\Delta}}\ensuremath{\equiv}\mathrm{\ensuremath{\Delta}}(\mathrm{\ensuremath{\Gamma}})]$ appears to reproduce better the experimental observations on the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$. In this later case, by increasing $\mathrm{\ensuremath{\Gamma}}$ gradually, ${\ensuremath{\chi}}_{3}$ becomes progressively rounded, presenting a maximum at a temperature T∗{T}^{*}T∗ (T∗>Tf)({T}^{*}>{T}_{f})(T∗>Tf​), with both the amplitude of the maximum and the value of T∗{T}^{*}T∗ decreasing gradually. Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature T∗{T}^{*}T∗, one for Tf<T<T∗{T}_{f}<T<{T}^{*}Tf​<T<T∗, and another one for T>T∗T>{T}^{*}T>T∗. It is argued that the conventional paramagnetic state corresponds to T>T∗T>{T}^{*}T>T∗, whereas the temperature region Tf<T<T∗{T}_{f}<T<{T}^{*}Tf​<T<T∗ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities.
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.
The infinite-range Ising spin glass in the presence of a Gaussian random field is investigated by means of the replica method. The replica-symmetric solution and its instability are discussed; it is shown that the Almeida-Thouless instability regions are always reduced by the random field. The crossover exponent \ensuremath{\varphi} associated with the irreversibility line is shown to be nonuniversal in the presence of a random field, varying in a range from 1 to 3. Some of our results are compared with recent experimental measurements performed in the diluted antiferromagnet Fex{\mathrm{Fe}}_{\mathit{x}}Fex​${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$F2{\mathrm{F}}_{2}F2​.
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a 2n×2n2^n\times 2^n2n×2n matrix with the number nnn of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit n→0n\to 0n→0 within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the Lévy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.