Chiral phase transition and thermal Hall effect in an anisotropic spin model on the kagome lattice

We present a study of the thermal Hall effect in the extended Heisenberg model with $XXZ$ anisotropy in the kagome lattice. This model has the particularity that, in the classical case, and for a broad region in parameter space, an external magnetic field induces a chiral symmetry breaking: the ground state is a doubly degenerate $q=0$ order with either positive or negative net chirality. Here, we focus on the effect of this chiral phase transition in the thermal Hall conductivity using linear-spin-waves theory. We explore the topology and calculate the Chern numbers of the magnonic bands, obtaining a variety of topological phase transitions. We also compute the magnonic effect to the critical temperature associated with the chiral phase transition ($T_c^{SW}$). Our main result is that, the thermal Hall conductivity, which is null for $T>T_c^{SW}$, becomes nonzero as a consequence of the spontaneous chiral symmetry breaking at low temperatures. Therefore, we present a simple model where it is possible to “switch” on/off the thermal transport properties introducing a magnetic field and heating or cooling the system.

Figure 1

(a) The kagome lattice with Bravais vectors $\vec{a}=(1,0)$ and $\vec{b}=(\frac{1}{2},\frac{\sqrt{3}}{2})$. First, second, and third nearest-neighbor exchange couplings, $J_1, J_2$, and $J_3$, respectively, are indicated. We only consider $J_3$ as indicated in the figure. (b),(c) Two possible plaquette arrangements of the canted ${120}^{\circ }$ ground state.

Figure 2

Fixing $J_2=0.8, J_3=0.2$ (a) magnon spectrum for $h/S=0.6, \mathrm{\Delta }=0.7$. The bands do not touch at any point in the BZ. (b) $C_n$ for each magnon band as a function of $\mathrm{\Delta }$, for $\theta =1.5$ (in radians). (c) Distance between magnon bands at different points in the BZ as a function of $\mathrm{\Delta }$. When the distance is zero, we identify a closing of the local gap between bands.

Figure 3

Fixing $J_2=0.8, J_3=0.2, \mathrm{\Delta }=0.7$ (a) $C_n$ as a function of the external magnetic field $h/S$. (b) Details of the highlighted area from panel (a).

Figure 4

Absolute value of the chirality parameter as a function of temperature from LSW for $J_2=0.8, J_3=0.2$ and (a) $\mathrm{\Delta }=0.7, h/S=0.6$ and different values of $S$; (b) $S=1, \theta =1.5$ and five values of $\mathrm{\Delta }$; the inset zooms in to show that for higher $\mathrm{\Delta }, T_c^{SW}$ is lower; (c) $S=1, \mathrm{\Delta }=0.7$ and three values of $h$. (d) $|{\chi }_{△}|$ vs temperature from MC simulations, $J_2=0.8, \mathrm{\Delta }=0.7, h/S=0.6$, and $J_3=0.1,0.4,0.7$. Vertical black arrows indicate the critical temperature ${\mathcal{T}}_c^{SW,\infty }$ obtained in the classical limit $S\to \infty$. Calculations where done for a discretization of $10000\times 10000$ points in the BZ.

Figure 5

For $S=1, J_2=0.8, J_3=0.2$ (a) ${\kappa }_{xy}$ as a function of $h$ taking $\mathrm{\Delta }=0.7$ for two different temperatures $T=0.6$ and $T=1$; (b) thermal conductivity ${\kappa }_{xy}$ as a function of temperature for different values of $\mathrm{\Delta }$, for $\theta =1.5$; (c) ${\kappa }_{xy}$ as a function of temperature obtained from the magnon spectrum of the two possible classical ground states with opposite scalar chirality for $\mathrm{\Delta }=0.7,h/S=0.6$.

Figure 6

Classical configuration for one plaquette: with (a) positive and (b) negative scalar chirality. The numbers 1, 2, and 3 indicate spins from the three different triangular sublattices from the kagome lattice.