Spin Supersolid in Triangular Lattice
Overview
Studies the spin supersolid phase of the triangular-lattice antiferromagnet Na₂BaCo(PO₄)₂, combining spin-wave theory, tensor-network methods, and renormalization-group analysis to trace its stability against spin-orbit coupling and thermal fluctuations — the subject of “Spin-orbit-induced Instability and Finite-Temperature Stabilization of a Triangular-lattice Supersolid” (arXiv:2601.20963, 2026).
The code behind this splits into two parts: analysis scripts specific to this paper’s tensor-network and renormalization-group calculations, and the spin-wave-theory core, which was generalized beyond this one result and published separately as a reusable package (see Links).
Motivation
Spin-orbit coupling (SOC) is usually assumed negligible in Na₂BaCo(PO₄)₂, but symmetry says otherwise: any nonzero SOC breaks the continuous spin symmetry that supersolid order depends on, opening a gap in its low-energy excitations. This raises the central question — to what extent can supersolidity survive symmetry-allowed SOC, and what new phases emerge as it grows?
Method: Spin-Wave Component
- Start from a bilinear spin exchange Hamiltonian on a given lattice and magnetic structure, with the classical ground-state order fixed per exchange configuration.
- Expand around that order with a Holstein-Primakoff transformation to get a bosonic Hamiltonian, then diagonalize it in momentum space with a Bogoliubov (BdG) transformation to obtain the magnon band structure.
- Layer solvers for exchange-parameter optimization and ground-state energy on top of the diagonalization core.
Method: Tensor-Network Component
- Infinite density-matrix renormalization group (iDMRG) on an infinite Y-type cylinder maps the zero-temperature ground-state phase diagram across the exchange-anisotropy parameter planes.
- Identifies five phases without spin-orbit terms — three spin-supersolid phases (Y, V, Ψ) alongside an up-up-down plateau and a polarized phase — and tracks how growing spin-orbit coupling collapses the three-sublattice orders into modified magnetic unit cells, including stripe phases and, at larger coupling, a quantum skyrmion lattice identified by its scalar spin chirality.
Method: Renormalization-Group Component
- Maps the spin-orbit-induced pseudo-Goldstone gap onto a discrete ground-state anisotropy — sixfold for the Y and Ψ phases, threefold for the V phase — and builds an effective classical Hamiltonian for each.
- A standard two-dimensional Kosterlitz-Thouless RG analysis on the anisotropy coupling shows the sixfold term becomes irrelevant above a threshold temperature, while the threefold term stays relevant at every temperature.
Results
- Confirms the spin-orbit-induced pseudo-Goldstone gap directly via spin-wave theory, and reproduces the Y and V ground states found by iDMRG (linear spin-wave theory alone does not capture the Ψ phase, attributed to quantum fluctuations beyond linear order).
- Shows that the Y and Ψ phases regain a finite-temperature spin-supersolid window, bounded above by a Berezinskii-Kosterlitz-Thouless transition, once the sixfold anisotropy becomes RG-irrelevant — a stability window invisible to a zero-temperature analysis.
- Shows the V phase cannot support any finite-temperature supersolid once spin-orbit coupling is present, since its threefold anisotropy never becomes RG-irrelevant.
- Offers an explanation for the giant magnetocaloric effect observed in Na₂BaCo(PO₄)₂: the finite-temperature restoration of coherence keeps the entropy-enhancing soft modes active even though spin-orbit coupling is not negligible.
Links
- Spin-orbit-induced Instability and Finite-Temperature Stabilization of a Triangular-lattice Supersolid — arXiv:2601.20963 (2026)
- Linear-Spin-Wave-Theory — Python package for linear spin wave theory (LSWT) computation, generalized from this project’s spin-wave component