Spin vestigial orders in extended Heisenberg-Kitaev models near hidden SU(2) points: Application to ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$

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-$\mathrm{\Gamma }\text{−}{\mathrm{\Gamma }}^{\prime }$ 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.

Figure 1

(a) Finite-temperature phase diagram of Heisenberg-Kitaev-$\mathrm{\Gamma }\text{−}{\mathrm{\Gamma }}^{\prime }$ model as function of ring-exchange perturbation $J_{⎔}$ from classical Monte Carlo simulations. Here the bilinear couplings have been fixed as $(J,K,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })=(-1/9,-2/3,8/9,-4/9)$, such that $J_{⎔}=0$ corresponds to the hidden-SU(2)-symmetric point. At high temperatures, the model is in the disordered paramagnetic phase, characterized by an isotropic distribution of the dual magnetization $\tilde{\mathbf{M}}$, see upper inset [measured at $(J_{⎔},T)=(0.2,0.422)$]. At low temperatures, the model features magnetically ordered collinear single-$\mathbf{q}$ zigzag and noncollinear triple-$\mathbf{q}$ phases for $J_{⎔}>0$ and $J_{⎔}<0$, respectively. These are characterized by dual magnetizations $\tilde{\mathbf{M}}$ along the three cubic basis vectors and four cubic diagonals, respectively, as illustrated by the distributions shown in the two lower insets [measured at $(J_{⎔},T)=(0.4,0.405)$ and $(-0.05,0.233)$]. For $0<|J_{⎔}|\lesssim 0.1$ near the hidden-SU(2)-symmetric point, the magnetic orders melt in two stages, with vestigial intermediate ${\mathbb{Z}}_3$ spin nematic and ${\mathbb{Z}}_4$ spin current density wave phases emerging at finite temperatures. Black line for $J_{⎔}>0$ represents fit according to Eq. (25). Other lines are guides to the eye. (b) Close-up view of vicinity of hidden-SU(2)-symmetric point, showing the spin vestigial phases.

Figure 2

(a) Four-sublattice structure used to define the ${\mathcal{T}}_1{\mathcal{T}}_4$ duality transformation. Dashed parallelogram indicates a corresponding unit cell, consisting of two sites per sublattice. The different colors indicate the different sublattices. The lower left inset indicates the projection of the cubic axes ${\mathbf{e}}_x, {\mathbf{e}}_y$, and ${\mathbf{e}}_z$ onto the honeycomb plane. The latter is spanned by the $\left[11\overline{2}\right]$ and $\left[\overline{1}10\right]$ axes, as indicated in the lower right inset. (b) Representative spin configuration of collinear single-$\mathbf{q}$ zigzag state, arising from the duality transformation of a Néel state with staggered magnetization along the [001] axis. Arrows correspond to spin directions projected onto the honeycomb plane. (c) Same as (b), but for the noncollinear triple-$\mathbf{q}$ state, arising from the duality transformation of a Néel state with staggered magnetization along the [111] axis. (d) Illustration of the ${\mathbb{Z}}_3$ spin nematic order, which is the vestigial order of the single-$\mathbf{q}$ zigzag state shown in (b). Different colors indicate inequivalent bonds. The ${\mathbb{Z}}_3$ spin nematic order respects time-reversal symmetry, but breaks $C_3^{*}$ rotational symmetry. The corresponding unit cell (dashed parallelogram) coincides with the crystallographic unit cell. (e) Same as (d), but for the ${\mathbb{Z}}_4$ spin current density wave order, which is the vestigial order of the triple-$\mathbf{q}$ state shown in (c). Different colors indicate inequivalent plaquettes. The ${\mathbb{Z}}_4$ spin current density wave order respects time-reversal symmetry but breaks translational symmetry by doubling the length of both lattice vectors, corresponding to an eight-site unit cell (dashed parallelogram).

Figure 3

(a) Dual magnetization $\tilde{M}$ as function of temperature $T$ for $J_{⎔}=0.01>0$ and fixed lattice size $L=32$. (b) Same as (a) but for $e_{\mathrm{g}}$ composite order parameter $Q_{e_{\mathrm{g}}}$ indicating single-$\mathbf{q}$ zigzag order at low temperature, corresponding to a dual magnetization along the cubic axes directions. (c) Same as (a) but for $t_{2\mathrm{g}}$ composite order parameter $Q_{t_{2\mathrm{g}}}$, indicating the absence of triple-$\mathbf{q}$ order at low temperature. [(d)–(f)] Same as (a)–(c) but for $J_{⎔}=-0.01<0$, indicating triple-$\mathbf{q}$ order at low temperature, corresponding to a dual magnetization along the space diagonals in the cubic basis, and the absence of single-$\mathbf{q}$ zigzag order.

Figure 4

(a) Binder cumulant $U_{\tilde{M}}$ of dual magnetization as function of temperature $T$ for fixed $J_{⎔}=0.01$ and different lattice sizes. Inset shows vicinity of crossings. Vertical dashed line indicates the critical temperature in the thermodynamic limit. (b) Same as (a) but for Binder cumulant $U_Q^{e_{\mathrm{g}}}$ of $e_{\mathrm{g}}$ composite order parameter. (c) Critical temperatures $T_{\mathrm{c}}$ as a function of inverse system size $1/L$, obtained from the crossings of the Binder cumulant, indicating two different transitions in thermodynamic limit and the presence of an intermediate phase.

Figure 5

(a) Histogram of dual magnetization ${\tilde{M}}^2$ near the critical temperature $T_{\mathrm{c}}=0.5576\left(4\right)$ for $J_{⎔}=1.4$, indicating first-order behavior. (b) Finite-size scaling of correlation-length ratio $R_{\xi }=\xi /L$ as function of $(T-T_{\mathrm{c}1})L^{\theta }$ [Eq. (23)], using $\theta =1.99$. (c) Exponent $\theta$ from finite-size scaling as function of system size $1/L$, indicating $\theta =1.99\left(3\right)$, consistent with a first-order transition. [(d)–(f)] Same as (a)–(c) but near the critical temperature $T_{\mathrm{c}}=0.490\left(1\right)$ for $J_{⎔}=0.8$, showing that the first-order behavior weakens for decreasing $J_{⎔}$. Nevertheless, the extrapolation to the thermodynamic limit shown in (f) yields $\theta =1.89\left(4\right)$ [this is also the value used in (e)], still consistent with first-order behavior.

Figure 6

(a) Critical finite-size scaling of correlation-length ratio $R_{\xi }=\xi /L$ as function of $(T-T_{\mathrm{c}1})L^{1/\nu }$ for fixed $J_{⎔}=0.05$ and different lattice sizes $L$, using $1/\nu =1.02$ and $T_{\mathrm{c}1}=0.302$, indicating a continuous finite-temperature transition. (b) Same as (a), but for the Binder cumulant $U_{\tilde{M}}$ of dual magnetization. (c) Finite-size scaling of correlation-length exponent $1/\nu$ as function of system size $1/L$, indicating $1/\nu =1.02\left(5\right)$ in the thermodynamic limit, consistent with classical 2D Ising criticality.

Figure 7

Critical finite-size scaling of Binder cumulant $U_Q^{e_{\mathrm{g}}}$ as function of $(T-T_{\mathrm{c}2})L^{1/\nu }$ for fixed $J_{⎔}=0.05$ and different lattice sizes $L$, using $1/\nu =6/5$ and $T_{\mathrm{c}2}=0.310$, consistent with a continuous finite-temperature transition in the 2D three-state Potts universality class.

Figure 8

Schematic renormalization group flow diagram emerging from our numerical results. H and L denote the fully stable high-temperature and low-temperature fixed points, respectively. The third fully stable fixed point V describes the corresponding spin vestigial phase, namely the ${\mathbb{Z}}_3$ spin nematic $({\mathbb{Z}}_4$ spin current density wave) for positive (negative) ring exchange perturbation. The Ising and three-state (four-state) Potts fixed points govern the continuous transitions in and out of the spin vestigial phase, respectively. The direct first-order transition for sizable positive (negative) ring exchange perturbation between the magnetically ordered single-$\mathbf{q}$ zigzag (triple-$\mathbf{q})$ phase and the disordered paramagnet is governed by the discontinuity fixed point D, which features an infrared relevant direction with associated eigenvalue of the stability matrix $\theta =d=2$.

Figure 9

(a) Finite-size crossover scaling of maxima of magnetic susceptibility ${\chi }_{\tilde{M}}{L}^{-{\gamma }_0/{\nu }_0}$ of dual magnetization as function of $|J_{⎔}|L^{\varphi /{\nu }_0}$, using ${\gamma }_0/{\nu }_0=1.82, \varphi /{\nu }_0=2/3$, and $J_{⎔}>0$. (b) Same as (a), but for $J_{⎔}<0$. (c) Same as (a), but for the susceptibility of the $e_{\mathrm{g}}$ composite order parameter, using ${\gamma }_0/{\nu }_0=1.64$. (d) Same as (b), but for the susceptibility of the $t_{2\mathrm{g}}$ composite order parameter, using ${\gamma }_0/{\nu }_0=1.55$.