Finite-temperature properties of the extended Heisenberg model on a triangular lattice

We present numerical results for the $J_1\text{−}{J}_2$ Heisenberg model on a triangular lattice at finite temperatures $T>0$. In contrast to unfrustrated lattices we reach much lower $T\sim 0.15J_1$. In static quantities the novel feature is a quite sharp low-$T$ maximum in the specific heat. Dynamical spin structure factor $S(\mathbf{q},\omega )$ allows for the extraction of the effective spin-wave energies ${\omega }_{\mathbf{q}}\left(T\right)$ and their damping ${\gamma }_{\mathbf{q}}\left(T\right)$. While for $J_2=0$ our results are consistent with $T=0$ spin ordering, $J_2/J_1\sim 0.1$ induces additional frustration with a signature of spin-liquid ground state. In the latter case, the results for the spin-lattice relaxation rate indicate the low-$T$ accessible regime on $1/T_1\propto T^{\alpha }$ with $\alpha \ge 1$, as observed in recent spin-liquid materials on a triangular lattice.

Figure 1

Results for the specific heat $C_V\left(T\right)$ of the Heisenberg model on triangular lattice with $J_2=0$, as obtained by FTLM on systems with different number of sites $N=20–30$.

Figure 2

(a) The entropy density $s\left(T\right)$, and (b) specific heat $C_V\left(T\right)$ of the $J_1\text{−}{J}_2$ Heisenberg model on the triangular lattice for different $J_2$, and compared to the reference square-lattice result, as obtained on lattices with $N=30$ sites.

Figure 3

Uniform spin susceptibility ${\chi }_0\left(T\right)$ within the Heisenberg model on the triangular lattice with $J_2=0–0.2$ and on the square lattice.

Figure 4

Static spin structure factor $S_{\mathbf{q}}\left(T\right)$ for several characteristic $\mathbf{q}$ within the BZ, with $J_2=0–0.2$.

Figure 5

Frequency dependence of the damping function ${\gamma }_{\mathbf{q}}\left(\omega \right)$, for three characteristic wave vectors ${\mathbf{Q}}_K,{\mathbf{Q}}_M,{\mathbf{Q}}_X$, as evaluated on the $N=30$ lattice for $J_2=0.1$, and broadened with $\eta =0.05$.

Figure 6

(a) Static susceptibilities ${\chi }_{\mathbf{q}}^0\left(T\right)$, and (b) corresponding ${\omega }_{\mathbf{q}}\left(T\right)$ on TL for different $J_2=0,0.1,0.2$, presented for different $\mathbf{q}$: corner $K$, edge $M$, and inside $X$ BZ points, respectively. Inset shows the BZ and available $\mathbf{q}$ on the $N=30$ lattice.

Figure 7

dc damping ${\gamma }_{\mathbf{q}}^0\left(T\right)$ for different $J_2=0,0.1,0.2$ at two BZ-edge $\mathbf{q}$ for the $N=30$ lattice.

Figure 8

Spin-lattice relaxation rate $1/T_1$ vs $T$ for the HM on TL with different $J_2=0–0.2$. For comparison the result for the square lattice is also shown.