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:2304.13692

Frustration of triplet interaction in spin-glass background

M. Bagherikalhor, B. Askari, Gholamreza Jafari

Published Apr 26, 2023• 0 citations
arXivHTMLPDF

Abstract

Parisi demonstrated in 1979 that pairwise interactions exhibit a glass spin phase when there is disorder. While he discovered an equilibrium solution of the Sherrington-Kirkpatrick (SK) spin-glass model and we know it as a continuous phase transition, the model dedicated to pairwise interactions and higher-order interactions has not been addressed. This research intends to determine whether this phase exists in triplet interactions. Due to the intractable nature of the three interacting spins alone, we employed a perturbation approach to provide an analytical solution for the triplet interactions in the background of the SK spin-glass model. Our results show the existence of this phase in the third-order interaction and a sudden transition that indicates a change in the nature of a glassy spin system transitioning from the continuous order to the first order. It causes a forward shift in the critical temperature by identifying the frustration of triplet interactions.

DOI
10.48550/arxiv.2304.13692

Related papers

Numerical simulations on spin glasses (invited)

A. P. Young

We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.

Journal of Applied Physics•Apr 15, 1985•14 citations

Stability of a crystal-field split spin glass

Peter Mottishaw, D Sherrington

A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.

Journal of Physics C Solid State Physics•Sep 20, 1985•46 citations

Inverse freezing in the fermionic spin glass model

Isabela Corrêa Berger

The present work studies the counterintuitive phenomenon of inverse transitions (IT). The IT are a class of phase transitions (reversible) in which the ordered phase appears at higher temperatures than the disordered one. Different models have been proposed to describe characteristics of IT. The goal of this work is just to propose a model of disordered interactions, which is able to present general characteristics of a inverse transition type, the inverse freezing (IF). The IF is a reversible transition from a paramagnetic phase (PM) to the spin glass order (SG) on heating. Within this context the van Hemmen SG model is analyzed in a fermionic formulation, in which the spin operators are written as a bilinear combination of fermionic creation and annihilation operators. In this model the random interactions Ji j introduce disorder and frustration, and the chemical potential m controls the average occupancy of fermions per site. The problem is analyzed for two different types of quenched disorders: bimodal (discrete) and gaussian (continuous). It is important to note that the disorder in the model analyzed here were treated without the use of the replica method and its complications. The results depend on the particular disorder. The behavior of the order parameters, entropy and occupation number are analyzed. Phase diagrams of temperature T versus chemical potential m can then be build. A reentrance transition is found for a certain range of m when the gaussian distribution is adopted. This reentrance transition is associated with the IF that occurs in a first-order transition from the paramagnetic to the spin glass phase when the temperature increases. On the other hand, models following disordered interactions of Sherrington-Kirkpatrick (SK) type suggest that the onset of IF can be a consequence of complex scenario for the free energy. In these models the replica method is used to treat the disorder and the solutions present this complex scenario. However, from the results obtained in present fermionic van Hemmen model, it can be concluded that the necessary conditions for a model exhibit IF are the combined effects of frustration and magnetic dilution (favoring states not interacting, controlled by m). Thus, the treatment of model presented in this work does not use the replica method or the complicated scenario of free energy and it is able to reproduces the characteristics of an inverse freezing. This is attributed to the presence of frustration and magnetic dilution. Keywords: Disorder. Quantum Spin Glass. Fermionic model. Inverse transitions

•Oct 3, 2012•0 citations

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

Alain H. Billoire, Ian A. Campbell

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature ${T}_{g}$. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at ${T}_{g}$. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.

arXiv:1105.1902•Physical Review B•Aug 12, 2011•15 citations

Ground-state energy and frustration of the Sherrington-Kirkpatrick model and related models

S. Kobe

Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin $e^{\infty}_0 = - 0.7637 \pm 0.0004$ is found from the $N$ dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter $x$ is introduced, which describes the fraction of antiferromagnetic interactions in these types of models. From the $x$ dependence of finite models it is concluded, that the the SK model $(x=0.5)$ assigns a transition between a ferromagnetic state $(x<0.5)$ to a spin-glass state $(x>0.5)$.

arXiv:cond-mat/0311657•Nov 28, 2003•6 citations

Studies of the phase diagram of randomly interacting fermionic systems

Oppermann, R., Bernd Rosenow

We present details of the phase diagrams of fermionic systems with random and frustrated interactions, emphasizing the important role of the chemical potential. The insulating fermionic Ising spin glass model is shown to reveal different entangled magnetic instabilities and phase transitions. We review tricritical phenomena related to the strong correspondence between charge and spin fluctuations, being controlled by quantum statistics. We compare the spin density diluted Sherrington-Kirkpatrick spin glass with classical spin 1 models such as the BEG model. We analyse in detail the infinite range model and show that spin glass order must decay discontinuously as the chemical potential exceeds a critical value, provided the temperature is below the tricritical one, and that the T=0 transition is of classical type. Parisi replica permutation symmetry breaking (RPSB) governs the thermal spin glass transitions and fermionic modifications of the SK-models AT-line emerge. RPSB takes place everywhere within the fermionic spin glass phase. Although the critical field theory of the quantum paramagnet to spin glass transition in metallic systems remains replica--symmetric at T=0, with only small corrections at low T from RPSB, the phase diagram is affected at O(T^0) by RPSB. Generalizing our results for the fermionic Ising spin glass we consider aspects of models with additional spin and charge quantum--dynamics such as metallic spin glasses.

arXiv:cond-mat/9803249•Mar 20, 1998•0 citations
Parisi demonstrated in 1979 that pairwise interactions exhibit a glass spin phase when there is disorder. While he discovered an equilibrium solution of the Sherrington-Kirkpatrick (SK) spin-glass model and we know it as a continuous phase transition, the model dedicated to pairwise interactions and higher-order interactions has not been addressed. This research intends to determine whether this phase exists in triplet interactions. Due to the intractable nature of the three interacting spins alone, we employed a perturbation approach to provide an analytical solution for the triplet interactions in the background of the SK spin-glass model. Our results show the existence of this phase in the third-order interaction and a sudden transition that indicates a change in the nature of a glassy spin system transitioning from the continuous order to the first order. It causes a forward shift in the critical temperature by identifying the frustration of triplet interactions.
We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.
The present work studies the counterintuitive phenomenon of inverse transitions (IT). The IT are a class of phase transitions (reversible) in which the ordered phase appears at higher temperatures than the disordered one. Different models have been proposed to describe characteristics of IT. The goal of this work is just to propose a model of disordered interactions, which is able to present general characteristics of a inverse transition type, the inverse freezing (IF). The IF is a reversible transition from a paramagnetic phase (PM) to the spin glass order (SG) on heating. Within this context the van Hemmen SG model is analyzed in a fermionic formulation, in which the spin operators are written as a bilinear combination of fermionic creation and annihilation operators. In this model the random interactions Ji j introduce disorder and frustration, and the chemical potential m controls the average occupancy of fermions per site. The problem is analyzed for two different types of quenched disorders: bimodal (discrete) and gaussian (continuous). It is important to note that the disorder in the model analyzed here were treated without the use of the replica method and its complications. The results depend on the particular disorder. The behavior of the order parameters, entropy and occupation number are analyzed. Phase diagrams of temperature T versus chemical potential m can then be build. A reentrance transition is found for a certain range of m when the gaussian distribution is adopted. This reentrance transition is associated with the IF that occurs in a first-order transition from the paramagnetic to the spin glass phase when the temperature increases. On the other hand, models following disordered interactions of Sherrington-Kirkpatrick (SK) type suggest that the onset of IF can be a consequence of complex scenario for the free energy. In these models the replica method is used to treat the disorder and the solutions present this complex scenario. However, from the results obtained in present fermionic van Hemmen model, it can be concluded that the necessary conditions for a model exhibit IF are the combined effects of frustration and magnetic dilution (favoring states not interacting, controlled by m). Thus, the treatment of model presented in this work does not use the replica method or the complicated scenario of free energy and it is able to reproduces the characteristics of an inverse freezing. This is attributed to the presence of frustration and magnetic dilution. Keywords: Disorder. Quantum Spin Glass. Fermionic model. Inverse transitions
A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature Tg{T}_{g}Tg​. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at Tg{T}_{g}Tg​. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.
Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin e0∞=−0.7637±0.0004e^{\infty}_0 = - 0.7637 \pm 0.0004e0∞​=−0.7637±0.0004 is found from the NNN dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter xxx is introduced, which describes the fraction of antiferromagnetic interactions in these types of models. From the xxx dependence of finite models it is concluded, that the the SK model (x=0.5)(x=0.5)(x=0.5) assigns a transition between a ferromagnetic state (x<0.5)(x<0.5)(x<0.5) to a spin-glass state (x>0.5)(x>0.5)(x>0.5).
We present details of the phase diagrams of fermionic systems with random and frustrated interactions, emphasizing the important role of the chemical potential. The insulating fermionic Ising spin glass model is shown to reveal different entangled magnetic instabilities and phase transitions. We review tricritical phenomena related to the strong correspondence between charge and spin fluctuations, being controlled by quantum statistics. We compare the spin density diluted Sherrington-Kirkpatrick spin glass with classical spin 1 models such as the BEG model. We analyse in detail the infinite range model and show that spin glass order must decay discontinuously as the chemical potential exceeds a critical value, provided the temperature is below the tricritical one, and that the T=0 transition is of classical type. Parisi replica permutation symmetry breaking (RPSB) governs the thermal spin glass transitions and fermionic modifications of the SK-models AT-line emerge. RPSB takes place everywhere within the fermionic spin glass phase. Although the critical field theory of the quantum paramagnet to spin glass transition in metallic systems remains replica--symmetric at T=0, with only small corrections at low T from RPSB, the phase diagram is affected at O(T^0) by RPSB. Generalizing our results for the fermionic Ising spin glass we consider aspects of models with additional spin and charge quantum--dynamics such as metallic spin glasses.