Spin-Orbit Coupling Enables Finite-Temperature Stabilisation of Triangular-Lattice Supersolid States

Researchers investigating geometrically frustrated triangular-lattice magnets have long sought to understand the emergence of spin supersolids. Seongjun Park, Sung-Min Park, and Yun-Tak Oh, from the Korea Advanced Institute of Science and Technology, alongside Hyun-Yong Lee of Korea University and Eun-Gook Moon, challenge the prevailing assumption that spin-orbit coupling (SOC) invariably destroys these exotic states. Their work, utilising spin-wave theory and density-matrix renormalization group simulations, demonstrates that while weak SOC can initially destabilise supersolidity, specific supersolid phases surprisingly persist at finite temperatures. This finding reveals a novel stabilisation mechanism where thermal fluctuations effectively oppose the gap-inducing effects of SOC, potentially unlocking pathways to realising robust supersolidity in real materials and associated phenomena like a significant magnetocaloric effect.

The team achieved a comprehensive understanding of the interplay between SOC, quantum fluctuations, and thermal effects in a triangular-lattice spin model, constructing a detailed phase diagram. Experiments show that at infinitesimally weak SOC, the spin-supersolid phase is destabilized, introducing a gap into its excitation mode. However, iDMRG simulations further demonstrate that the system undergoes a sequence of SOC-driven Quantum phase transitions into distinct magnetically ordered states, including the formation of a skyrmion lattice at sufficiently strong SOC.

Crucially, the research establishes that supersolidity can be thermodynamically stabilized at non-zero temperatures, even in the presence of SOC, a phenomenon not previously accounted for in theoretical analyses. This finite-temperature regime highlights a constructive role for thermal fluctuations, partially restoring the effective symmetry associated with supersolid coherence and offering a new perspective on its robustness in frustrated magnets. The resulting finite-temperature supersolids retain key responses, notably a giant magnetocaloric effect, suggesting potential relevance for real materials and broadening the possibilities for exploiting supersolid functionality. The work opens avenues for understanding how symmetry-allowed anisotropic interactions, thermal fluctuations, and magnetic fields cooperate to stabilize or reshape coherent quantum phases. The research establishes a unified SOC, field phase diagram for triangular-lattice magnets, providing a theoretical framework for interpreting recent experiments on Co-based triangular lattice antiferromagnets. This breakthrough demonstrates that even with moderate SOC, materials can still host supersolid phenomenology, and offers new possibilities for realizing spin-orbit-driven topological and supersolid states in frustrated quantum materials.

Spin-orbit coupling and supersolid phase stability are intricately

Researchers constructed effective Hamiltonians, HY and HV, to model the Y and V phases respectively, capturing symmetries of MY = Z6 and MV = Z3. These classical models allowed for extensive thermal transition structure analysis, leveraging established theoretical frameworks. The team engineered a renormalization-group (RG) analysis in two dimensions, revealing a Kosterlitz, Thouless (KT) phase for temperatures below πJ/2 when g6 equals zero. Inclusion of the sixfold anisotropy resulted in a specific RG flow equation, dg6/dl = 2 − 9T/πJ g6, demonstrating that g6 becomes irrelevant for temperatures exceeding 2πJ/9, thereby stabilizing the KT phase within the 2πJ/9 < T < πJ/2 window.

This analysis confirmed the finite-temperature continuation of the Y phase into a supersolid state exhibiting quasi-long-range order. Conversely, RG analysis of the V phase, governed by HV, revealed that the threefold anisotropy eliminates any stable KT phase, suppressing supersolidity at finite temperatures. LSWT calculations explicitly demonstrated the opening of a pseudo-Goldstone gap, confirming the lifting of continuous U(1) degeneracy upon the introduction of SOC. This work revealed a dichotomy in the thermodynamic fate of these modes; for the Y and Ψ phases, SOC-induced anisotropy became irrelevant at intermediate temperatures, allowing thermal fluctuations to restore quasi-long-range coherence. This led to the emergence of a finite-temperature spin-supersolid phase, bounded by a Berezinskii, Kosterlitz, Thouless transition. The study highlights a nuanced picture regarding the magnetocaloric effect, demonstrating that thermal stabilization of the Y and Ψ supersolids preserves entropic enhancement, even with SOC present, potentially explaining experimental observations in materials like Na2BaCo(PO4)2.

Spin-orbit coupling and thermal stability in supersolids

Experiments revealed that while infinitesimally weak SOC does induce a zero-temperature instability in the supersolid by opening a gap, specific supersolid states maintain thermodynamic stability at non-zero temperatures. The resulting finite-temperature supersolids retain crucial responses, notably a giant magnetocaloric effect, suggesting potential applications in real materials. Measurements confirm that at higher SOC levels, the system transitions into distinct magnetic orders, including a skyrmion lattice, thereby completing a unified phase diagram. The study demonstrates a constructive role of thermal fluctuations in partially restoring the effective symmetry associated with supersolid coherence, offering a new perspective on the robustness of supersolidity in frustrated magnets.

The team employed iDMRG on a Y-type cylinder with a circumference of W= 6 to carve out the ground-state phase diagram, examining two parameter-plane cuts of the model. Setting J/Jz= 0.6, relevant to Na2BaCo(PO4)2, the researchers generated a (h, JPD) plane at fixed JΓ = 0 and a (h, JΓ) plane at fixed JPD = 0. These panels display density maps of bipartite entanglement entropy, with overlaid dashed lines marking phase boundaries determined by energy density, magnetization, entanglement entropy, and their derivatives. Without SOC-induced interactions (JPD = JΓ = 0), iDMRG calculations identified five distinct phases: three spin-supersolid phases (Y, V, and Ψ), a up-up-down 1/3- magnetization plateau (UUD), and a polarized phase (P).

Spin configurations for each phase are illustrated, and the team applied linear spin wave theory (LSWT) to further map the ground-state phase diagram. The ground state manifold of each spin-supersolid phase is parameterized by a continuous variable φ, representing the angle of a global spin rotation about the z-axis, forming a 1-sphere, MY/V/Ψ S1. The team found that the three-sublattice orders (Y, V, and Ψ) remain robust against weak SOC perturbations, forming finite ground state phases.

Thermal Fluctuations Stabilise Spin Supersolidity at Temperature scales

This challenges the common expectation that SOC invariably gaps the low-energy modes and destroys these states. The study identified distinct magnetic orders emerging at larger SOC, including a skyrmion lattice, completing a comprehensive phase diagram. The researchers found that the ground state manifolds of three phases, Y, V, and Ψ, exhibit continuous U(1) degeneracy without SOC, which is distorted by discrete anisotropies when SOC is introduced. Furthermore, they observed that the system transitions into various ordered phases, such as uniformly-canted and canted stripe phases, and a quantum skyrmion-lattice phase, each characterized by unique spin textures and responses.

The skyrmion lattice is distinguished by a finite jump in scalar spin chirality. Acknowledging limitations, the authors note that their iDMRG calculations were initially performed on cylinders with limited circumference, potentially affecting the observed stability of certain phases. Future research could explore the behaviour of these systems in three dimensions or with different types of magnetic anisotropies. These findings are significant because they suggest a pathway for stabilizing supersolidity in real materials, potentially enabling applications leveraging the giant magnetocaloric effect observed in these finite-temperature supersolids, and broadening the scope of materials exhibiting such exotic magnetic behaviour.