Vanishing Wilson ratio as the hallmark of quantum spin-liquid models

We present numerical results for finite-temperature $T>0$ thermodynamic quantities, entropy $s\left(T\right)$, uniform susceptibility ${\chi }_0\left(T\right)$, and the Wilson ratio $R\left(T\right)$, for several isotropic $S=1/2$ extended Heisenberg models, which are prototype models for planar quantum spin liquids. We consider in this context the frustrated $J_1\text{−}{J}_2$ model on kagome, triangular, and square lattice, as well as the Heisenberg model on a triangular lattice with the ring exchange. Our analysis reveals that typically in the spin-liquid parameter regimes the low-temperature $s\left(T\right)$ remains considerable, while ${\chi }_0\left(T\right)$ is reduced consistent mostly with a triplet gap. This leads to vanishing $R(T\to 0)$, being the indication of a macroscopic number of singlets lying below triplet excitations. This is in contrast to the $J_1\text{−}{J}_2$ Heisenberg chain, where $R(T\to 0)$ either remains finite in the gapless regime, or the singlet and triplet gap are equal in the dimerized regime.

Figure 1

Results in the $J_1\text{−}{J}_2$ Heisenberg chain for (a) entropy $s\left(T\right)$, (b) susceptibility ${\chi }_0\left(T\right)$, and (c) Wilson ratio $R\left(T\right)$, as obtained via FTLM on $N=30$ sites for different $J_2=0.0-1.0$. The dashed lines at $J_2=0$ represent the extension to $N\to \infty$, while for $J_2=0.2,0.3$ they denote modified $R\left(T\right)$ evaluated with reduced $\tilde{s}\left(T\right)$. The inset in (c) represents a sketch of $J_1$ (solid line) and $J_2$ (dashed line) in a 1D Heisenberg chain.

Figure 2

(a) Lowest triplet ${\epsilon }_t$ and singlet ${\epsilon }_s$ excitations vs $J_2$, as obtained via DMRG in the chain of $N=60$ sites, with the inset showing the scaling of ${\epsilon }_{t/s}$ vs $1/N$ for $J_2=0.2$. (b) Corresponding g.s. spin correlations $\langle S_i^z{S}_j^z\rangle$ on particular bonds.

Figure 3

$s\left(T\right)$, ${\chi }_0\left(T\right)$, and $R\left(T\right)$ within the $J_1\text{−}{J}_2$ HM on KL, obtained via FTLM on $N=36$ sites, for different $|J_2|\le 0.2$. The inset in (c) represents a sketch of the $J_1$ (solid line) and $J_2$ (broken line) connections in KL.

Figure 4

Finite-size comparison of $R\left(T\right)$ within the SL regimes for (a) basic KL model with $J_2=0$ and (b) $J_1\text{−}{J}_2$ model on TL. Results are obtained via FTLM on lattices with $N=18–36$ sites.

Figure 5

Lowest triplet excitation ${\epsilon }_t$ and (nondegenerate) singlet excitations ${\epsilon }_{s,i}$ ($i=1,2,\cdots ,6$) vs $J_2$ for different planar $J_1\text{−}{J}_2$ HM models on: (a) KL, (b) TL, and (d) SQL, as obtained with ED on $N=36$ sites, and (c) ${\epsilon }_t$ and ${\epsilon }_{s,i=1,2,3}$ vs $J_r$ on TL with ring exchange, obtained on $N=28$ sites.

Figure 6

$s\left(T\right)$, ${\chi }_0\left(T\right)$, and $R\left(T\right)$ within $J_1\text{−}{J}_2$ HM on TL, obtained via FTLM on $N=36$ sites for different $J_2\le 0.3$. The dashed line for $J_2=0.3$ represents results using reduced $\tilde{s}\left(T\right)$. The inset in (c) represents a sketch of the $J_1$ (solid line) and $J_2$ (broken line) connections in TL.

Figure 7

$s\left(T\right)$, ${\chi }_0\left(T\right)$, and $R\left(T\right)$ within HM on TL, including ring exchange, obtained via FTLM on $N=28$ sites for different $0\le J_r\le 0.2$. The inset in (c) represents a sketch of the $J$ (solid line) connections and the ring exchange $J_r$ (circle) in TL.

Figure 8

(a) $s\left(T\right)$, (b) ${\chi }_0\left(T\right)$, and (c) $R\left(T\right)$ within $J_1\text{−}{J}_2$ HM on SQL, obtained via FTLM on $N=36$ sites for different $0\le J_2\le 1.0$. In (d) $R\left(T\right)$ results are presented within the expanded intermediate regime $0.4\le J_2\le 0.6$.