Critical properties of short-range Ising spin glasses on a Wheatstone-bridge hierarchical lattice

Sebastião T. O. Almeida and Fernando D. Nobre

Phys. Rev. E 92, 022102 – Published 3 August 2015

Abstract

An Ising spin-glass model with nearest-neighbor interactions, following a symmetric probability distribution, is investigated on a hierarchical lattice of the Wheatstone-bridge family characterized by a fractal dimension $D \approx 3.58$ . The interaction distribution considered is a stretched exponential, which has been shown recently to be very close to the fixed-point coupling distribution, and such a model has been considered lately as a good approach for Ising spin glasses on a cubic lattice. An exact recursion procedure is implemented for calculating site magnetizations, $m_{i} = {〈 S_{i} 〉}_{T}$ , as well as correlations between pairs of nearest-neighbor spins, ${〈 S_{i} S_{j} 〉}_{T}$ ( ${〈 〉}_{T}$ denote thermal averages), for a given set of interaction couplings on this lattice. From these local magnetizations and correlations, one can compute important physical quantities, such as the Edwards-Anderson order parameter, the internal energy, and the specific heat. Considering extrapolations to the thermodynamic limit for the order parameter, such as a finite-size scaling approach, it is possible to obtain directly the critical temperature and critical exponents. The transition between the spin-glass and paramagnetic phases is analyzed, and the associated critical exponents $β$ and $ν$ are estimated as $β = 0.82 (5)$ and $ν = 2.50 (4)$ , which are in good agreement with the most recent results from extensive numerical simulations on a cubic lattice. Since these critical exponents were obtained from a fixed-point distribution, they are universal, i.e., valid for any coupling distribution considered.

Received 21 May 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022102

Authors & Affiliations

Sebastião T. O. Almeida^1,2,* and Fernando D. Nobre^1,3,†

¹Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro RJ, Brazil
²Colégio Técnico da Universidade Federal Rural do Rio de Janeiro, Seropédica, 23891-000 Rio de Janeiro RJ, Brazil
³National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro RJ, Brazil

^*Author to whom all correspondence should be addressed: stalme@cbpf.br
^†fdnobre@cbpf.br

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Two first steps in the generation of the Wheatstone-bridge hierarchical lattice investigated herein, which has been used in the literature to approach the cubic lattice [3]. In each step, one replaces a bond (a) by the unit cell shown in (b) (characterized by a fractal dimension $D \approx 3.58$ ), where the empty circles ( $μ$ and $ν$ ) represent external sites of the cell, whereas the black circles are internal sites to be decimated in the RG procedure. The same procedure carried in (b) leads to (c).
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Figure 2
Flux diagrams of two probability distributions $P (K_{i j})$ are represented in suitable variables [12, 18]. These distributions were constructed, respectively, from a Gaussian (with unit variance) and the stretched exponential of Eq. (2), for the coupling distributions of the Ising SG model of Eq. (1), on the HL defined according to Fig. 1. For each of these two distributions, three different temperatures were considered, namely $T > T_{c}$ (fluxes toward increasing abscissas), $T < T_{c}$ (fluxes toward decreasing abscissas), and $T = T_{c}$ . For the critical temperature associated with each distribution, we have used the estimates of Ref. [44], namely $(k T_{c} / J) = 0.9821$ (Gaussian distribution) and $(k T_{c} / J) = 0.948$ (stretched-exponential distribution). The blue arrow indicates the point associated with the fixed-point distribution $P^{*} (K_{i j})$ , from which one concludes that the distribution of Eq. (2) may be considered as a good approximation of the fixed-point distribution, whereas the Gaussian distribution presents a “larger distance” from $P^{*} (K_{i j})$ . In the variables used, ${[]}_{P}$ represent averages over the corresponding probability distributions.
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Figure 3
Basic unit cell showing the labeled internal sites to be used in the calculations, with $μ$ and $ν$ denoting the external sites; $Γ_{μ ν}$ , $h_{μ}$ , and $h_{ν}$ represent, respectively, effective coupling and fields acting upon the external spins of the basic cell.
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Figure 4
Data of the Edwards-Anderson order parameter [cf. Eq. (30)] for the Ising SG on a Wheatstone-bridge HL with $D \approx 3.58$ with different hierarchy levels $N$ . (a) The order parameter $q_{EA}$ is represented vs $(T - T_{c}) / T_{c}$ ; (b) a data collapse is exhibited, from which the best-collapse values for the critical exponents $β$ and $ν$ , as well as of the critical temperature $T_{c}$ , are obtained; the inset shows an enlargement of the critical region.
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Figure 5
The specific heat per spin, calculated from a numerical differentiation of the internal energy of Eq. (31) with respect to the temperature, at hierarchy level $N = 7$ , is exhibited in dimensionless variables. The dashed line is a guide to the eye, showing the expected tendency to zero temperature; the blue arrow indicates the critical temperature of Eq. (34). In the inset, we present the corresponding internal energy per spin in the temperature range in which the specific heat was computed.
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Figure 6
The dimensionless quantity $Γ_{SG}^{(1)}$ , defined in Eqs. (37) and (38), is exhibited vs $(T - T_{c}) / T_{c}$ for different hierarchy levels. In the inset, we present the dimensionless distance of the maximum of each curve from the critical temperature of Eq. (34), $a (L) = [\tilde{T} (L) - T_{c}] / T_{c}$ vs $L^{- 1} (L = 2^{N})$ ; the dashed line is a guide to the eye. Within our numerical accuracy, $\tilde{T} (L)$ should coincide with $T_{c}$ for a hierarchy level around $N = 10$ .
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