Mean-field phase diagram and spin-glass phase of the dipolar kagome Ising antiferromagnet

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Mean-field phase diagram and spin-glass phase of the dipolar kagome Ising antiferromagnet

Leticia F. Cugliandolo, Laura Foini, and Marco Tarzia

Phys. Rev. B 101, 144413 – Published 8 April 2020

Abstract

We derive the equilibrium phase diagram of the classical dipolar Ising antiferromagnet at the mean-field level on a geometry that mimics the two-dimensional kagome lattice. Our mean-field treatment is based on the combination of the cluster variational Bethe-Peierls formalism and the cavity method, developed in the context of the glass transition, and is complementary to the Monte Carlo simulations realized in a recent paper [Hamp et al., Phys. Rev. B 98, 144439 (2018)]. Our results confirm the nature of the low-temperature crystalline phase which is reached through a weakly first-order phase transition. Moreover, they allow us to interpret the dynamical slowing down observed in the work of Hamp et al. (referenced above) as a remnant of a spin-glass transition taking place at the mean-field level (and expected to be avoided in two dimensions).

Received 8 October 2019
Accepted 17 March 2020

DOI:https://doi.org/10.1103/PhysRevB.101.144413

Physics Subject Headings (PhySH)

Frustrated magnetism Glass transition Phase transitions

Kagome lattice Spin glasses

Cavity methods

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Leticia F. Cugliandolo

Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4, Place Jussieu, Tour 13, 5ème étage, 75252 Paris Cedex 05, France and Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France

Laura Foini

Institut de Physique Théeorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France

Marco Tarzia

Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4, Place Jussieu, Tour 13, 5ème étage, 75252 Paris Cedex 05, France and Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France

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Issue

Vol. 101, Iss. 14 — 1 April 2020

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Images

Figure 1
Sketch of a small portion of a (rooted) random regular graph (RRG) of triangular plaquettes in presence of a cavity (the dashed black plaquette $β$ ). Each up-type triangular plaquette of the RRG is connected to three down-type triangular plaquettes and each down-type triangular plaquette is connected to up down-type triangular plaquettes. The graph looks locally like a tree since typical loops are very large (the typical size of the loops diverges as $log N_{Δ}$ ). The first-nearest-neighboring plaquettes ( $β, γ$ , and $δ$ ) of the central (red) triangle ( $α$ ) are drawn in black, the second-nearest-neigboring plaquettes in blue, and the third-nearest-neighboring plaquettes in green. The mean-field Bethe-Peierls approximation consists in discarding the fact that the two spins inside the green circle are in fact the same spin on the original kagome lattice. Moreover the dipolar coupling is cut off beyond the second-nearest-neighboring plaquettes (i.e., fifth-nearest-neighbor spins).
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Figure 2
Sketch of the sublattice structure introduced to describe the ordered crystalline phase proposed in [48] and detected in [47]. The four up-type plaquette sublattices are denoted by $(A, B, C, D)$ , and the four down-type plaquette sublattices are denoted by $(a, b, c, d)$ . The green circles represent the $+ 1$ spins in one of the sixfold-degenerate ground-state configurations. Such sublattice structure is associated to 24 different kinds of cavities, denoted by $(A_{n}, B_{n}, C_{n}, D_{n})$ , and $(a_{n}, b_{n}, c_{n}, d_{n})$ , with $n = 1, 2, 3$ , obtained by removing one of the three neighboring plaquettes to each of the eight kinds of sublattice plaquettes. For example, the plaquette $(a_{1})$ is obtained by removing the bottom neighboring plaquette $(C)$ from the plaquette $(a)$ , and it is connected to the plaquettes $(A_{3})$ on the right and $(B_{2})$ on the left, obtained, respectively, by removing the right neighboring plaquette $(a)$ from the plaquette of type $(A)$ and the left neighboring plaquette $(a)$ from the plaquette of type $(B)$ .
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Figure 3
Average energy density $〈 e 〉$ (top left), magnetization of the polarized spins $| m_{pol} |$ (top right), specific heat (per plaquette) $c = \partial 〈 e 〉 / \partial T$ (bottom left), and average entropy density $〈 s 〉$ (bottom right) as a function of the temperature $T$ for $J = 0.5$ and $D = 1$ . Data in the paramagnetic phase are shown in blue, in the crystal phase in black, in the supercooled paramagnetic phase in magenta. The red circles are obtained by solving the 1RSB equations in the spin-glass phase (with $m = 1$ ). The vertical dashed black and red lines correspond to the first-order transition to the crystal state at $T_{m}$ and the continuous transition to the spin-glass phase at $T_{sg}$ , respectively. The gray curves correspond to the crystal solution in the metastable region and end at the spinodal point ( $T_{sp}$ , vertical black dotted line). The green dotted vertical line gives the position of the modulation instability of the homogeneous solution $T_{\mod}$ . The orange dashed curves correspond to the (unstable) RS solution of the equation below the spin-glass transition point, and show the entropy crisis in the manner of Kauzmann of the RS solution (at $T_{s = 0}$ ).
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Figure 4
Mean-field phase diagram of the model in the $D - T$ plane for $J = 0.5$ , showing the position of the different phases and the transition lines. The black continuous line corresponds to the first-order melting transition $T_{m} (D)$ , where the free energies of the paramagnetic and the crystalline solutions cross. The black dotted line gives the spinodal point at which the crystalline solution appears discontinuously. The green dotted line corresponds to the modulation instability of the paramagnetic solution $T_{\mod} (D)$ . The red continuous line is the continuous spin-glass transition $T_{sg} (D)$ , where the spin-glass susceptibility diverges.
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