Nonequilibrium selection of magnetic order in a driven triangular XY antiferromagnet

Yuan Wan and Roderich Moessner

Phys. Rev. B 98, 184432 – Published 29 November 2018

Abstract

We show that a weak periodic drive removes the accidental degeneracy in the ground state of the XY antiferromagnet on the triangular lattice in a uniform static magnetic field. The underlying mechanism involves adding a small periodically modulated component to the magnetic field, which influences finite-frequency modes to generate an effective potential for the accidental pseudo-Goldstone mode. This selection can be arranged to compete with the degeneracy lifting via the thermal order by disorder mechanism. This yields a nonequilibrium phase transition as the relative strength of the two mechanisms is tuned by varying temperature. This proposal may be amenable to experimental realization, in particular, as applying the field is noninvasive, and no complex bath engineering is required.

Received 9 August 2018
Revised 8 November 2018

DOI:https://doi.org/10.1103/PhysRevB.98.184432

Physics Subject Headings (PhySH)

Frustrated magnetism Magnetic order

Antiferromagnets Nonequilibrium systems

Langevin equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuan Wan^1,2,* and Roderich Moessner³

¹Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
²Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
³Max-Planck Institute for the Physics of Complex Systems, D-00187 Dresden, Germany

^*yuan.wan@iphy.ac.cn

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Vol. 98, Iss. 18 — 1 November 2018

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Images

Figure 1
Top row: Equilibrium magnetic orders in triangular XY antiferromagnet subject to a static, in-plane magnetic field $(B_{0})$ along the spin $x$ axis. XY spins in sublattice A, B, and C are colored in red, green, and blue. Bottom row: Driving the system with a weak, time-periodic, in-plane magnetic field $(B_{1})$ orthogonal to $B_{0}$ dynamically stabilizes the fan states. From left to right, the three columns show the specific spin configurations at $B_{0} / J = 1, 3, 7$ . Inset: Equilibrium phase diagram at fixed, low temperature $T$ . As $B_{0} / J$ increases, the system is successively in the Y phase (magenta), up-up-down (UUD, cyan), 2:1 (yellow), and, finally, fully polarized phase (black), where all spins are aligned with the field. Open circles mark the positions of the states given in $(a_{1}), (b_{1}), (c_{1})$ in the phase diagram. In a Y state $(a_{1})$ , the spins in one sublattice are antialigned with the field, whereas the spins in the other two sublattices form symmetric angles with the field. In a UUD state $(b_{1})$ , the spins in one sublattice are anti-aligned with the field, whereas the spins in the other two are aligned with the field. In a 2:1 state $(c_{1})$ , the spins in one sublattice form one angle with the field, whereas the spins in the other two sublattices form a different angle with the field. In a fan state $(a_{2}, b_{2}, c_{2})$ , spins in one of the three sublattices are aligned with the static field while spins in the other two sublattices form symmetric angles with the static field.
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Figure 2
Dynamical stabilization of the fan states due to periodic driving. The top, middle, and bottom rows, respectively, correspond to $B_{0} / J = 1$ , 3, and 7. Column (a): The one-dimensional degenerate ground state space (solid lines) in configuration space spanned by the sublattice spin angles $ϕ_{A, B, C}$ . Columns (b) and (c): The intrinsic frequency $ω$ of the two optical magnons and their magnetic dipole moments in the $y$ direction $m^{y}$ as functions of the ground-state coordinate $s$ . High- and low-frequency modes are colored in gold and purple, respectively. Dashed lines mark the driving frequency. Column (d): Driving-induced effective potential $V_{eff}$ . Driving amplitude $B_{1} = 0.05 J$ for all three cases. Column (e): Time-evolution of the sublattice spin orientation $ϕ_{A, B, C}$ after the driving is switched on at time $t = 0$ . Colored arrows show the sketch of the magnetic orders. Top row: The degenerate ground-state space at $B_{0} / J = 1$ contains two connected components (dark and light blue). $1 \sim 6$ mark the Y states. Circles with the same label mark identical states thanks to the periodicity $ϕ \to ϕ + 2 π$ . Filled (open) circles are on the front (back) surfaces of the box. Fan states are at midpoints of two neighboring Y states. Results in $(b \sim d)$ are for the component colored in darker blue. Driving frequency $Ω = 2.6 \sqrt{J / I}$ . Middle row: The ground state space at $B_{0} / J = 3$ contains topological singular points at the UUD states, labeled as $1 \sim 3$ , where the ground state space self-intersects $(a_{2})$ . Fan states are midpoints of two neighboring UUD states. The driving frequency $Ω = 3.2 \sqrt{I J}$ . Bottom row: The ground state space at $B_{0} / J = 7$ has a single connected component. $1 \sim 6$ label the 2:1 states. The fan states are the midpoints of two neighboring 2:1 states. The driving frequency $Ω = 2.6 \sqrt{J / I}$ . The time-profile of the periodic driving field is shown in the inset of $(e_{3})$ .
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Figure 3
Nonequilibrium phase transitions resulted from the competition between periodic drive and thermal fluctuations. Top, middle, and bottom rows show the results with static fields $B_{0} / J = 1, 3, 7$ , respectively. We set the driving amplitude $B_{1} = 0.05 J$ unless stated otherwise. Column (a): Value of order parameter $Ψ_{x}$ on the complex plane for the fan state (open circles) and the competing equilibrium orders (closed circles). Column (b): Average kinetic energy density $〈 K 〉 / N$ as a function of bath temperature $T$ . The kinetic energy density due to the wave vector $| q | = 0$ modes (closed circles) and $| q | > 0$ modes (open circles) are shown separately. For $| q | > 0$ modes, the thermal contribution $T / 2$ is subtracted to highlight the heating effect. Column (c): Order parameter $ζ_{3}$ as a function of $T$ for different system sizes $L$ . Columns (d) and (e): Histograms of the order parameter angle $3 \arg Ψ_{x}$ and the sublattice magnetization orientation $ϕ_{A, B, C}$ near the transition. The same color code corresponds to the same temperature. Note we do not distinguish the three sublattices in the latter histogram. Top row: Driving frequency $Ω = 2.6 \sqrt{I J}$ . The arrow in $(c_{1})$ marks the crossing point of the order parameter curves for different system sizes. The inset of $(c_{1})$ shows the scaling of $T_{c}$ with $B_{1}^{2}$ (open circles). The dashed line is the linear scaling relationship based on the crude estimate given in the main text. In $(d_{1})$ and $(e_{1})$ , the solid cyan line shows the histogram of $3 \arg Ψ_{x}$ and $ϕ_{A, B, C}$ obtained by randomly drawing ground states with uniform probability density. Middle row: $Ω = 3.2 \sqrt{I J}$ . The inset of $(b_{2})$ shows the enlarged view of the low-temperature part of the data. Arrow in $(c_{2})$ marks the crossing point of data curves for different system sizes. Bottom row: Driving frequency $Ω = 2.6 \sqrt{I J}$ .
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Figure 4
Sketch of the evolution of the nonequilibrium effective energy landscape, $V_{neq}$ , as a function of the ground-state coordinate $s$ . The curves are shifted vertically or better visibility. Filled circles mark the minima of $V_{neq}$ . (a) Evolution of $V_{neq}$ for $B_{0} / J = 1$ . Both connected components of the ground-state space are shown. The minima jump from fan states to Y states as $T$ increases across $T_{c}$ . At $T_{c}$ , the fan states and Y states are degenerate, and the overall magnitude of $V_{neq}$ is also relatively small. (b) Same as (a) but for $B_{0} / J = 3$ . As $T$ increases, the minima of $V_{neq}$ , originally at fan states at low $T$ , split into pairs. Meanwhile, $V_{eff}$ develop metastable minima at the UUD states, which eventually become the global minima. (c) Same as (a) but for $B_{0} / J = 7$ . The splitting of minima is similar to (b), but $V_{neq}$ shows no metastable states.
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