Control of the Effective Free-Energy Landscape in a Frustrated Magnet by a Field Pulse

Thermal fluctuations can lift the degeneracy of a ground state manifold, producing a free-energy landscape without accidentally degenerate minima. In a process known as order by disorder, a subset of states incorporating symmetry breaking may be selected. Here, we show that such a free-energy landscape can be controlled in a nonequilibrium setting as the slow motion within the ground state manifold is governed by the fast modes out of it. For the paradigmatic case of the classical pyrochlore $XY$ antiferromagnet, we show that a uniform magnetic field pulse can excite these fast modes to generate a tunable effective free-energy landscape with minima at thermodynamically unstable portions of the ground state manifold.

Figure 1

Geometrical frustration for vector spins produces an accidentally degenerate GS manifold (blue ellipse) in many-body configuration space (gray box). $\mathcal{Q}$ parametrizes the GS manifold, and $q$ the fluctuations out of it.

Figure 2

(a) GS of Eq. (1) in a unit cell. 0–3 label the sublattices. Starting from a GS (blue solid arrows, corresponding to $\varphi =0$), one obtains another GS (blue broken arrows, $\varphi =\pi /2$) by rotating all spins with respect to their local threefold axes. Inset: Crystallographic axes (black arrows), pulse polarization (light blue arrow), and polarization angle ${\theta }_B$. (b) GS Néel order parameter argument $\varphi$ as a function of time $t$ after a pulse with polarization angle ${\theta }_B=0.15\pi$. Results from the simulation of Eq. (1) (blue) contrasted to a model with an inherently energetic (rather than emergent entropic) clock term (red). (c) Effective potential energy density $V(\varphi )/[6(1-j_{zz})]$ in the GS manifold from the toy model (black) and the model simulation in (b) (blue). Gray dashed lines mark the positions of ${\psi }_2$ states. (d) Potential minima ${\varphi }_{\min }$ as a function of pulse polarization angle ${\theta }_B$. Black solid lines show the two degenerate minima predicted by the toy model. Blue crosses mark the center position of the $\varphi$ oscillation extracted from simulation.

Figure 3

(a) Argument and (b) modulus of the ensemble-averaged complex Néel order parameter $⟨O⟩$ as a function of time $t$ at $k_{B}T={10}^{-3}{J}_{\pm }{S}^2$. Blue and orange lines are for pulse polarization angles ${\theta }_B=0.1\pi$ and $0.5\pi$, respectively. (b) also shows the ensemble average of modulus squared, $⟨|O{|}^2⟩$, for ${\theta }_B=0.5\pi$ (orange open circles). (c)(d) Histogram of the Néel order parameter argument $\text{arg}O$ as function of time $t$ at the same temperature. (e) Same as (a) but for ensemble average of action variables $I_a$ (dots) and $I_b$ (open circles) associated with the optical magnons $a$ and $b$. (f) Number of trajectories for which the total energy of the pseudo-Goldstone mode exceeds the potential maxima.

Figure 4

(a) Argument and modulus (inset) of the ensemble-averaged Néel order parameter $⟨O⟩$ versus time $t$ for various pulse polarizations at $k_{B}T={10}^{-3}{J}_{\pm }{S}^2$. $\text{arg}⟨O⟩=0$ at time $t=0$, but plots are shifted vertically for visibility. Dashed lines mark the predicted effective potential minima from the toy model. (b) Same as (a) but for fixed pulse polarization angle ${\theta }_B=0.15\pi$ at four different temperatures. (c) Histogram of Néel order parameter argument $\text{arg}O$ as a function of time $t$ for ${\theta }_B=0.15\pi$ at $k_{B}T=0.2J_{\pm }{S}^2$.