Spin vestigial orders in extended Heisenberg-Kitaev models near hidden SU(2) points: Application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Na</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Co</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>TeO</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>
Niccolò Francini, Lukas Janssen
Abstract
The honeycomb magnet ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ has recently been argued to realize an approximate hidden SU(2) symmetry that can be understood by means of a duality transformation. Using large-scale classical Monte Carlo simulations, we study the finite-temperature phase diagram of the pertinent Heisenberg-Kitaev-$\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model near the hidden-SU(2)-symmetric point, in the presence of a six-spin ring exchange perturbation. At low temperatures, the model features collinear single-$\mathbf{q}$ zigzag and noncollinear triple-$\mathbf{q}$ ground states, depending on the sign of the ring exchange coupling. We show that in the vicinity of the hidden-SU(2)-symmetric point, the magnetic long-range orders melt in two stages. The corresponding finite-temperature transitions are continuous and fall into two-dimensional (2D) Ising and 2D Potts universality classes, respectively. The two fluctuation-induced phases at intermediate temperatures spontaneously break spin rotational and lattice translational symmetries, respectively, but both leave time-reversal symmetry intact. They are characterized by finite expectation values of a real, symmetric, traceless, second-rank tensor and are naturally understood as vestigial orders of the underlying magnetic states. We identify these vestigial orders as ${\mathbb{Z}}_{3}$ spin nematic and ${\mathbb{Z}}_{4}$ spin current density wave phases, respectively. For increasing ring exchange perturbations, the width of the vestigial phases decreases, eventually giving rise to a direct first-order transition from the magnetically ordered phase to the disordered paramagnet. We propose the ${\mathbb{Z}}_{4}$ spin current density wave phase, which is the vestigial phase of the primary triple-$\mathbf{q}$ magnetic order, as a natural candidate for the paramagnetic 2D long-range-ordered state observed in ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ in a small window above the antiferromagnetic ordering temperature.