Spin-orbit-induced Instability and Finite-Temperature Stabilization of a Triangular-lattice Supersolid

TL;DR

This work addresses whether spin-supersolid phases on a triangular lattice can survive spin-orbit coupling. By combining spin-wave theory and iDMRG with symmetry analysis, it shows that infinitesimal SOC gaps the would-be Goldstone mode at zero temperature, but thermal fluctuations at finite temperature can restore quasi-long-range order in Y- and Psi-like supersolids, while V-like phases are suppressed. The authors derive an effective finite-temperature description revealing a KT-like supersolid phase for the Y family but not for V, and they map SOC-driven transitions to a skyrmion-lattice state at larger SOC, yielding a unified SOC–field phase diagram. These results explain persistence of magnetocaloric effects in experiments and provide a framework for realizing SOC-driven topological and supersolid states in frustrated magnets.

Abstract

Geometrically frustrated triangular-lattice magnets provide fertile ground for realizing intriguing quantum phases such as spin supersolids. A common expectation is that spin-orbit coupling (SOC), which breaks continuous spin rotational symmetry, destabilizes these phases by gapping their low-energy modes. Revisiting this assumption, we map out the SOC-field phase diagram of a frustrated triangular-lattice magnet using spin-wave theory and infinite density-matrix renormalization group (iDMRG) simulations. We find that while infinitesimally weak SOC indeed drives a zero-temperature instability of the supersolid by opening a gap, certain supersolid states remain thermodynamically stable at non-zero temperatures. This reveals a previously unrecognized mechanism in which thermal fluctuations counteract SOC to stabilize supersolidity. The resulting finite-temperature supersolids retain key responses, including a giant magnetocaloric effect, highlighting their potential relevance to real materials. At larger SOC, the system transitions into distinct magnetic orders, including a skyrmion lattice, completing a unified phase diagram.

Figures (5)

• Figure 1: iDMRG phase diagrams of the triangular-lattice antiferromagnet with (a) the $\Gamma$-interaction and (b) the PD interaction at $J/J_z = 0.6$. Color encodes the maximal bipartite entanglement entropy of the ground state; dotted curves mark phase boundaries inferred from gradients of the energy density, magnetization, and entanglement entropy. The system is defined on an infinite cylinder with circumference $W=6$. The cross-hatched region corresponds to the spin-supersolid phase at zero temperature.
• Figure 2: (a)-(f) Illustrations of magnetic orderings. For states with three-sublattice order (a-c), we introduce sublattice labels (A, B, C) and use red, green, and blue to indicate the corresponding sublattices in the figures.
• Figure 3: Illustration of the ground state manifolds. (a) In the absence of SOC, the ground state manifolds of three phases form a $S^1$-manifold, reflecting the continuous $U(1)$ degeneracy. In the presence of SOC, this manifold is distorted by discrete anisotropies: a six-fold ($Z_6$) for $\overline{\rm Y}$ and $\overline{\Psi}$ as in (b) and a three-fold ($Z_3$) for $\overline{\rm V}$ as in (c).
• Figure 4: (a) Ground-state scalar spin chirality obtained from iDMRG as a function of $J_{\Gamma}$ at fixed field $h/J_{z}=2.0$ and $J_{\rm PD}=0$, showing the onset of chirality upon entering the SkX from the UUD state. (b) (gray and green) Pseudo-Goldstone gap $\Delta_{\rm PG}$ as a function of $J_{\rm PD}$ at $J_{\rm \Gamma} = 0$. (red and blue) $\Delta_{\rm PG}$ as a function of $J_{\rm \Gamma}$ at $J_{\rm PD} = 0$. (c and d) Change in the semiclassical energy density $\Delta\epsilon(\phi)=\epsilon(\phi)-\epsilon(\phi^{*})$ under a uniform spin rotation $U_{z}(\phi)$, obtained from LSWT. The angle $\phi^{*}$ denotes the position of local minima with respect to the $U(1)$ rotation. We fix $J_{\rm PD}=0$, with varying $J_{\Gamma}$ for the $\overline{\rm Y}$ phase ($B = 0.2\,\rm T$) and the $\overline{\rm V}$ phase ($B = 1.4\,\rm T$). We adopt the parameters $J = 0.075$ meV, $J_z = 0.125$ meV, and $h = g_z \mu_{\mathrm{B}} B$ with $g_z = 4.645$, appropriate for $\mathrm{Na}_2\mathrm{BaCo}{(\mathrm{PO}_4)}_2$sheng2022chi2024dynamicalSheng25
• Figure 5: Schematic finite-temperature phase diagrams and the evolution of the ground state manifold. Phase diagrams for (a) $\overline{\rm Y}$ and $\overline{\Psi}$ phases and (b) the $\overline{\rm V}$ phase as a function of SOC strength $g$. The supersolid phases are indicated by the regions with $\times$.