Thermodynamic properties of an $S=\frac{1}{2}$ ring-exchange model on the triangular lattice

By using a numerically exact diagonalization technique and a block-extended version of the finite-temperature Lanczos method, we study thermodynamic properties of an $S=1/2$ Heisenberg model on the triangular lattice with an antiferromagnetic nearest-neighbor interaction $J$ and a four-spin ring-exchange interaction $J_{\mathrm{c}}$. Calculations are performed on small clusters under the periodic-boundary conditions. In contrast to the purely triangular case with $J_{\mathrm{c}}=0$, the specific heat exhibits a characteristic double-peak structure for $J_{\mathrm{c}}/J\gtrsim 0.04$. From the calculations of the entropy and the uniform magnetic susceptibility, it is shown that nonmagnetic excitations exist below the magnetic excitation for $J_{\mathrm{c}}/J\gtrsim 0.04$.

Figure 1

Schematic of the model described in Eq. (1) on the triangular lattice. Elementary parallelograms, where ${\widehat{P}}_{ijkl}$ and ${\widehat{P}}_{ijkl}^{†}$ act (indicated by circular arrows), are indicated.

Figure 2

Available momenta $\mathbf{k}=(k_x,k_y)$ for the $L=6\times 6$ cluster under the periodic boundary conditions. Solid lines denote the Brillouin-zone boundaries and light green circles indicate the 36 momenta inside the first Brillouin zone. High symmetric momenta, $\mathrm{\Gamma }$: (0,0), $K$: $(4\pi /3,0), M$: $(\pi ,\pi /\sqrt{3})$, and $K^{\prime }$: $(2\pi /3,2\pi /\sqrt{3})$, are also indicated.

Figure 3

(a) Specific heat $c\left(T\right)$, (b) entropy density $s\left(T\right)$, (c) uniform susceptibility $\chi \left(T\right)$, and (d) Wilson ratio $R_{\mathrm{W}}\left(T\right)$ at $J_{\mathrm{c}}/J=0.07$ for $L=4\times 4$, 18, $5\times 4, 6\times 4$, 30, and $6\times 6$ clusters (see Fig. 4). Solid lines are results obtained by the full exact diagonalization. Block-Lanczos parameters are $R_{\mathrm{B}}=24, M_{\mathrm{B}}=6$, and $N_{\mathrm{L}}=120$ for $L=4\times 5, R_{\mathrm{B}}=24, M_{\mathrm{B}}=8$, and $N_{\mathrm{L}}=160$ for $L=18, R_{\mathrm{B}}=8, M_{\mathrm{B}}=8$, and $N_{\mathrm{L}}=200$ for $L=6\times 4, R_{\mathrm{B}}=6, M_{\mathrm{B}}=8$, and $N_{\mathrm{L}}=320$ for $L=30$, and $R_{\mathrm{B}}=6, M_{\mathrm{B}}=4$ for $0⩽|S_z|⩽1, M_{\mathrm{B}}=6$ for $2⩽|S_z|⩽5, M_{\mathrm{B}}=6$ for $6⩽|S_z|$, and $N_{\mathrm{L}}=720$ for $L=6\times 6$.

Figure 4

Cluster structures used for the calculations. The periodic boundary conditions are imposed.

Figure 5

Semilog (left) and linear (right) plots of (a),(b) specific heat $c\left(T\right)$, (c),(d) entropy density $s\left(T\right)$, (e),(f) uniform susceptibility $\chi \left(T\right)$, and (g),(h) Wilson ratio $R_{\mathrm{W}}\left(T\right)$ for several values of $J_{\mathrm{c}}/J$, indicated in the figures, and $L=6\times 6$. Block-Lanczos parameters are $R_{\mathrm{B}}=6, M_{\mathrm{B}}=4$ for $0⩽|S_z|⩽1, M_{\mathrm{B}}=6$ for $2⩽|S_z|⩽5, M_{\mathrm{B}}=6$ for $6⩽|S_z|$, and $N_{\mathrm{L}}=720$.

Figure 6

Schematic figure of the algorithm to find a spin configuration $i$ for a given state label $j$. The figure should be read from bottom to top to compare with Algorithm 1. For a given set of $L, N_{\uparrow }$, and $j$, one of the shortest paths from the vertex $\left(\genfrac{}{}{0pt}{}{L}{L-N_{\uparrow }}\right)$ to the topmost vertex $\left(\genfrac{}{}{0pt}{}{0}{0}\right)$ of Pascal's triangle is assigned, and the path determines the spin configuration $i={\sum }_{\tilde{L}=1}^L{b}_{\tilde{L}}{2}^{\tilde{L}-1}={(b_L{b}_{L-1}...b_1)}_2$. The path goes rightward if $\tilde{j}>\left(\genfrac{}{}{0pt}{}{\tilde{L}-1}{{\tilde{N}}_{\uparrow }}\right)$ (indicated by magenta) or else leftward (indicated by green). The rightward (leftward) path from $\tilde{L}\mathrm{th}$ row to $\tilde{L}-1\mathrm{th}$ row implies that $b_{\tilde{L}}=1$ ($b_{\tilde{L}}=0$). The figure refers to the input $(L,N_{\uparrow },j)=(6,3,7)$, which results in the output $i={\left(010110\right)}_2=22$. The $(N_{\uparrow }+1)\times (N_{\downarrow }+1)=16$ vertices on the possible $\left(\genfrac{}{}{0pt}{}{6}{3}\right)=20$ shortest paths are highlighted with shaded blue color, and $b_{\tilde{L}}$ in $i$ is highlighted with boldface. Although a quick return is possible at $\tilde{L}=3$ in this example, according to the lines 9–11 of Algorithm 1, the remaining processes corresponding to the lines 12–15 of Algorithm 1 for $\tilde{L}⩽3$ are also shown here in this figure.

Figure 7

Schematic figure of the ${120}^{\circ }$ Néel ordered state on the triangular lattice. ${\left({\mathbf{r}}_i\right)}_{x\left(y\right)}$ denotes the $x\left(y\right)$ coordinate of ${\mathbf{r}}_i$.

Figure 8

Linear spin-wave dispersions for (a) the $J-J_{\mathrm{c}}$ model and (b) the $J-J^{\prime }$ model with several $J_{\mathrm{c}}/J$ and $J^{\prime }/J$ values indicated, respectively, in the figures. The horizontal axis is momentum $\mathbf{q}$ along the $\mathrm{\Gamma }$–$K$–$M$–$\mathrm{\Gamma }$ points in the (nonmagnetic) Brillouin zone, where $\mathrm{\Gamma }=(0,0), K=(4\pi /3,0)$, and $M=(\pi ,\pi /\sqrt{3})$. Thin vertical lines indicate the magnetic Brillouin-zone boundaries corresponding to the ${120}^{\circ }$ Néel order. The horizontal line at $\mathrm{\Omega }\left(\mathbf{q}\right)/J=1$ indicates the spin-wave excitation energy at $M$ point in the purely triangular case with $J_{\mathrm{c}}=J^{\prime }=0$.