Phase diagram of dipolar-coupled XY moments on disordered square lattices

Dominik Schildknecht, Laura J. Heyderman, and Peter M. Derlet

Phys. Rev. B 98, 064420 – Published 22 August 2018

Abstract

The effects of dilution disorder and random-displacement disorder are analyzed for dipolar-coupled magnetic moments confined in a plane, which were originally placed on the square lattice. In order to distinguish the different phases, new order parameters are derived and parallel tempering Monte Carlo simulations are performed for a truncated dipolar Hamiltonian to obtain the phase diagrams for both types of disorder. We find that both dilution disorder and random-displacement disorder give similar phase diagrams, namely, disorder at small enough temperatures favors a so-called microvortex phase. This can be understood in terms of the flux closure present in dipolar-coupled systems.

Received 20 April 2018
Revised 6 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.064420

Physics Subject Headings (PhySH)

Frustrated magnetism Phase diagrams

Classical spin models Disordered systems

Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Dominik Schildknecht^1,2,3,*, Laura J. Heyderman^2,3, and Peter M. Derlet¹

¹Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
²Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
³Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland

^*dominik.schildknecht@psi.ch

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Vol. 98, Iss. 6 — 1 August 2018

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Images

Figure 1
(a) The degenerate ground state of the square-lattice dXY system is defined within a $2 \times 2$ magnetic unit cell via an angle-degeneracy parameter $ϕ$ . (b) Possible vectors $\vec{M}$ are shown as given in Eq. (2). The black solid circle indicates $| \vec{M} | = 1$ , which is fulfilled for the ground-state manifold depicted in (a). The arrows correspond to the four striped phases (light blue) and the four microvortex phases (dark blue). A pictograph is given to associate the vectors with their respective phases. The light blue dot in the middle corresponds to the paramagnetic phase.
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Figure 2
(a) The temperature dependence of the three order parameters discussed in Sec. 2 are shown. These are obtained with parallel tempering Monte Carlo simulations for the nondisordered tdXY system with a system size of $L = 16$ averaged over four independent runs. The same data plotted on a logarithmic temperature scale is shown in the inset. (b) The microvortex order parameter is shown for the three system sizes studied. The same data are plotted on a logarithmic temperature scale in the inset to highlight the transition at $T = 0$ . (c) A configuration is shown, which was obtained by our simulations at temperature $1.8 \times 10^{- 6}$ . Since the microvortex order parameter is very homogeneous across the system, it is likely to originate from the Goldstone mode.
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Figure 3
Each of the order parameters for the dilution-disordered tdXY system on the square lattice as a function of dilution rate $p$ and temperature $T$ for three different system sizes ( $L = 16, 32, 48$ ) obtained via parallel tempering Monte Carlo simulations averaged over 32 disorder realizations.
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Figure 4
The phase diagram for the diluted square lattice tdXY system as a function of dilution rate and temperature derived via Binder cumulant crossings, superimposed on the corresponding $M_{m v}$ data of Fig. 3 for $L = 48$ . The filled (open) markers give $T_{c}$ ( $p_{c}$ ) with red dots for $| \vec{M} |$ , orange triangles for $M_{s}$ , and violet diamonds for $M_{m v}$ , respectively. Region “ $m v$ ” corresponds to the microvortex phase, region “ $s$ ” to the striped phase, and region “ $para$ ” to the paramagnetic phase.
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Figure 5
(a) Each of the order parameters for random-displacement disordered tdXY systems on the square lattice as a function of the width of the random displacement $σ$ and temperature $T$ for three different system sizes ( $L = 16, 32, 48$ ) obtained via parallel tempering Monte Carlo simulations averaged over 32 realizations. (b) The phase diagram of the same system as (a) as a function of temperature and width of the random displacement. This is derived via Binder cumulant crossing and superimposed on the corresponding $M_{m v}$ data for $L = 48$ of (a). The assignment of colors, markers, and regions is the same as in Fig. 4. The region “fs” not present in the dilution case denotes region dominated by finite-size effects.
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