Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet

The anomalous thermodynamic properties of the paradigmatic frustrated spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLH) has remained an open topic of research over decades, both experimentally and theoretically. Here, we further the theoretical understanding based on the recently developed, powerful exponential tensor renormalization group method on cylinders and stripes in a quasi-one-dimensional (1D) setup, as well as a tensor product operator approach directly in 2D. The observed thermal properties of the TLH are in excellent agreement with two recent experimental measurements on the virtually ideal TLH material ${\mathrm{Ba}}_8{\mathrm{CoNb}}_6{\mathrm{O}}_{24}$. Remarkably, our numerical simulations reveal two crossover temperature scales, at $T_l/J\sim 0.20$ and $T_h/J\sim 0.55$, with $J$ the Heisenberg exchange coupling, which are also confirmed by a more careful inspection of the experimental data. We propose that in the intermediate regime between the low-temperature scale $T_l$ and the higher one $T_h$, the “rotonlike” excitations are activated with a strong chiral component and a large contribution to thermal entropies. Bearing remarkable resemblance to the renowned roton thermodynamics in liquid helium, these gapped excitations suppress the incipient ${120}^{\circ }$ order that emerges for temperatures below $T_l$.

Figure 1

Uniform TLH with nearest-neighbor (NN) coupling $J=1$ (which thus sets the unit of energy) and lattice spacing $a=1$, with three schematically depicted distinct regimes, separated by two crossover temperature scales, $T_l$ and $T_h$: an incipient ${120}^{\circ }$ ordered regime for $T<T_l$ (left), a paramagnetic regime for $T>T_h$ (right), an intermediate regime (center), which is explored in detail in this Rapid Communication. The thick black line indicates the 1D snake order adopted in the MPO-based XTRG. When the system is wrapped into a cylinder along the tilted left arrow, this is referred to as YC geometry. The clockwise oriented circles in the center of the system indicate chiral operators, $\chi \equiv 2^3·S_a·(S_b\times S_c)$, acting on the enclosing triangle of sites $(a,b,c)$ in the order of the arrows, as used for the calculation of chiral correlations between the triangle pair A-B.

Figure 2

Simulated thermodynamics in comparison to experimental measurements, Cui et al. (2018) [37] and Rawl et al. (2017) [36], as well as earlier numerical results. The YC and OS data are obtained via XTRG by retaining up to $D^{*}=1000$ multiplets $[D\sim 4000$ U(1) states], and by a TPO method [21] on infinite lattices, keeping up to 40 bond states. (a) Specific heat, $c_V$, results benchmarked against HTSE [5, 36] and experimental curves. (b) The thermal entropy $S$ vs $T$, together with the reconstructed Schwinger boson mean field (RSBMF) [38], and “roton” contributions [16]. (c) Uniform magnetic susceptibility $T{\chi }_0$ vs $T$, shown with BDMC data [7]. The left top inset compares ${\chi }_0$ to Curie-Weiss (CW) ${\chi }_0=C/(T+\theta )$ in a wide temperature range, where $C=1/4$ and $\theta =2.06$. In the right bottom inset we further compare various $T{\chi }_0$ values at $T=0.5$. The magnetic moment per Co is assumed $\simeq 2{\mu }_B$, with Landé factor $g\simeq 4.13$ [37].

Figure 3

(a)–(d) Structure factor on $\mathrm{YC}6\times 12$ lattice, i.e., with $q_y$ pointing along the direction of the cylinder, at temperatures $T=5$, 0.54, 0.2, and 0.1, respectively, [vertical gray lines in (e)]. (e) $S\left(q\right)$ vs $T$ at momenta $q=K$ and $M$ where the legend holds for both data sets. (f) $S_E$ vs $T$, where the tilted dashed lines indicate the logarithmic scaling $S_E=aln\left(\beta \right)+b$, where the slopes $a$ seen for the TLH are similar to that for the SLH (SC6 data). The vertical dashed line labels the low-temperature scale $T_l\sim 0.2$ for TLH and the only temperature scale $T_s\sim 0.6$ for SLH. $\mathrm{SC}6\times 12$ stands for a $W=6$, $L=12$ square cylinder, and $S_E$ scaling in the Heisenberg chain (length $L=200$) is also plotted as a comparison.

Figure 4

Chiral correlations on cylinders, YC5 and YC6 (for YC4, see Ref. [21]). The inset represents the eigenstates $\mathrm{\Psi }$ (and ${\mathrm{\Psi }}^{*}$) of the chiral operator $\chi$ (Fig. 1) with nonzero eigenvalues $\pm \sqrt{12}$. They have total spin $S=1/2$, and hence are superpositions of configurations with two-site singlet dimers (thick lines) whose signs are fixed in clockwise order (arrow). Having $\alpha =exp(2\pi i/3)$, this demonstrates the chiral nature.