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

C. V. Morais, F. M. Zimmer, M. J. Lazo, S. G. Magalhães, and F. D. Nobre

Phys. Rev. B 93, 224206 – Published 17 June 2016

Abstract

The behavior of the nonlinear susceptibility $χ_{3}$ 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 $λ_{AT}$ (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 $χ_{SG}$ , which presents a term inversely proportional to the replicon $λ_{AT}$ . As a result of a relation between $χ_{SG}$ and $χ_{3}$ , the latter also presents a divergence at $T_{f}$ , which comes as a direct consequence of $λ_{AT} = 0$ at $T_{f}$ . However, our results show that, in the presence of random fields, $χ_{3}$ 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 $χ_{3}$ 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 $χ_{3}$ still maintains a dependence on the replicon $λ_{AT}$ , 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 ${LiHo}_{x} Y_{1 - x} F_{4}$ compound.

Received 28 April 2016
Revised 2 June 2016

DOI:https://doi.org/10.1103/PhysRevB.93.224206

Physics Subject Headings (PhySH)

Spin glasses

Replica methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. V. Morais^1,*, F. M. Zimmer², M. J. Lazo³, S. G. Magalhães⁴, and F. D. Nobre⁵

¹Instituto de Física e Matemática, Universidade Federal de Pelotas, 96010-900 Pelotas, Rio Grande do Sul, Brazil
²Departamento de Fisica, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Rio Grande do Sul, Brazil
³Programa de Pós-Graduação em Física, Instituto de Matemática, Estatística e Física, Universidade Federal do Rio Grande, 96.201-900, Rio Grande, Rio Grande do Sul, Brazil
⁴Instituto de Fisica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, Rio Grande do Sul, Brazil
⁵Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Rio de Janeiro, Brazil

^*carlosavjr@gmail.com

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Vol. 93, Iss. 22 — 1 June 2016

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Images

Figure 1
The 1S-RSB parameter $δ \equiv q_{1} - q_{0}$ and the eigenvalue $λ_{AT}$ are presented versus the dimensionless temperature $T / J$ , for typical values of $Δ / J$ . The inset shows the SG parameters $q_{1}$ and $q_{0}$ separately versus the dimensionless temperature. The freezing temperature $T_{f}$ is identified with the onset of RSB, where $λ_{AT} = 0$ , or equivalently, where the parameter $δ$ becomes nonzero. The arrows indicate the temperature $T^{*}$ , where $χ_{3}$ presents a rounded maximum, showing that $T^{*} > T_{f}$ . Due to the usual numerical difficulties, the low-temperature results [typically $(T / J) < 0.05$ )] correspond to smooth extrapolations from higher-temperature data.
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Figure 2
Magnetic susceptibility $χ_{1}$ versus $T / J$ for different values of $Δ / J$ . The arrows indicate the onset of the RSB solution ( $λ_{AT} = 0$ ), defining the temperature $T_{f}$ . Below $T_{f}$ , solid and dotted lines indicate linear susceptibilities computed using 1S-RSB ( $χ_{RSB}$ ) and RS ( $χ_{RS}$ ) solutions, respectively. Due to the usual numerical difficulties, the low-temperature results [typically $(T / J) < 0.05]$ correspond to smooth extrapolations from higher-temperature data.
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Figure 3
The susceptibility $χ_{3}$ as a function of $T / J$ for different values of $Δ / J$ . The arrows indicate the onset of the RSB solution ( $λ_{AT} = 0$ ), defining the temperature $T_{f}$ . Below $T_{f}$ , solid and dotted lines indicate 1S-RSB and RS solutions, respectively. The temperature $T^{*}$ , where $χ_{3}$ presents a rounded maximum, is estimated in each case shown. In the inset we exhibit the $χ_{3}$ behavior without the RF.
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Figure 4
The quantities appearing in the denominator of Eq. (16), $γ = λ_{AT} + {(β J)}^{2} I_{1}$ , are presented versus $T / J$ for $(Δ / J) = 0.1$ . The arrows locate the freezing temperature $T_{f}$ . The inset on the right shows in detail the behaviors of $γ$ and $1 / γ$ , for $(Δ / J) = 0.1$ , in the region where the RS solution is stable (to the left of this region one should use RSB); the quantity $1 / γ$ presents a rounded maximum, which is directly related with that found in $χ_{3}$ . The inset on the left shows in detail the behaviors of $γ$ and $λ_{AT}$ , for $Δ / J = 0.0$ , which are responsible for the divergence of $χ_{3}$ in the absence of RFs.
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Figure 5
Phase diagram $T / J$ versus $Δ / J$ showing the paramagnetic and SG phases. The freezing temperature $T_{f}$ , signaling the onset of RSB, defines the SG phase for $T < T_{f}$ . For completeness, we also present the line associated with the maximum of $χ_{3}$ , defining the temperature $T^{*}$ (dashed line). The possibility of two paramagnetic phases ( ${PM}_{1}$ and ${PM}_{2}$ ) is discussed in the text.
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