Excitation spectra of the spin-$\frac{1}{2}$ triangular-lattice Heisenberg antiferromagnet

We use series expansion methods to calculate the dispersion relation of the one-magnon excitations for the spin-$\frac{1}{2}$ triangular-lattice nearest-neighbor Heisenberg antiferromagnet above a three-sublattice ordered ground state. Several striking features are observed compared to the classical (large-$S$) spin-wave spectra. Whereas, at low energies the dispersion is only weakly renormalized by quantum fluctuations, significant anomalies are observed at high energies. In particular, we find rotonlike minima at special wave vectors and strong downward renormalization in large parts of the Brillouin zone, leading to very flat or dispersionless modes. We present detailed comparison of our calculated excitation energies in the Brillouin zone with the spin-wave dispersion to order $1∕S$ calculated recently by Starykh, Chubukov, and Abanov [Phys. Rev. B74, 180403(R) (2006)]. We find many common features but also some quantitative and qualitative differences. We show that at temperatures as low as $0.1J$ the thermally excited rotons make a significant contribution to the entropy. Consequently, unlike for the square lattice model, a nonlinear sigma model description of the finite-temperature properties is only applicable at temperatures $\ll 0.1J$. Finally, we review recent NMR measurements on the organic compound $\kappa \text{−}\left(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2{\mathrm{Cu}}_2\right(\mathrm{CN}{)}_3$. We argue that these are inconsistent with long-range order and a description of the low-energy excitations in terms of interacting magnons, and that therefore a Heisenberg model with only nearest-neighbor exchange does not offer an adequate description of this material.

Figure 1

Exchange interactions in the Heisenberg model (1) on the triangular lattice. In this paper we focus on results for the case $J_1=J_2\equiv J$.

Figure 2

The ground state energy per site $E_0∕N$, as a function of the angle $q$ between nearest neighbor spins along $J_2$ bonds, for the triangular-lattice model (i.e., $J_1=J_2\equiv J$). The minimum energy is obtained when $q=2\pi ∕3$, the same as for the classical model.

Figure 3

Reciprocal space of the triangular lattice including the hexagonal first Brillouin zone. Squares denote ordering wave vectors, circles denote wave vectors of the “roton” minima. The labeled points have coordinates $O=(0,0)$, $P=(2\pi ∕3,0)$, $A=(\pi ,0)$, $B=(\pi ,\pi ∕\sqrt{3})$, $C=(2\pi ∕3,2\pi ∕\sqrt{3})$, $Q=(4\pi ∕3,0)$, and $E=(0,\pi ∕\sqrt{3})$. Also shown is the path ABOCPQBE along which the magnon dispersion is plotted in Figs. 4, 5.

Figure 4

(Color online) Calculated spectra along ABOCPQBE of the Brillouin zone. Series extrapolation results (data points with error bars) are plotted together with naive sum of series with $t=2$ (solid green curve) and naive sum of ratio series with $t=1$ (short-dashed magenta) and $t=2$ (long-dashed blue).

Figure 5

(Color online) Magnon spectra along ABOCPQBE from series expansions compared with LSWT (dashed red line) and $\mathrm{SWT}+1∕S$ (solid green line).

Figure 6

(Color online) Projection plot showing the magnon energies obtained from series expansions in the triangular-lattice Brillouin zone. The path in Fig. 3 is also shown (dashed lines). The low-energy regions are located around the Brillouin zone center and corners.

Figure 7

(Color online) Projection plot showing the $\mathrm{SWT}+1∕S$ magnon energies in the triangular-lattice Brillouin zone. Note that the color scheme used here is different than in Fig. 6. The path in Fig. 3 is also shown (dashed lines). The low-energy regions are located around the Brillouin zone center and corners.

Figure 8

(Color online) Plots of magnon density of states for the series expansions, LSWT, and $\mathrm{SWT}+1∕S$ spectra.

Figure 9

(Color online) Highlight of regions in the Brillouin zone that contribute to the DOS peaks in the series calculations.

Figure 10

(Color online) Highlight of regions in the Brillouin zone that contribute to the DOS peaks in the $\mathrm{SWT}+1∕S$ calculations.

Figure 11

(Color online) Entropy of the triangular-lattice Heisenberg model due to the magnon excitations. Contributions to the entropy from different energy ranges are shown. The low energy magnons only dominate the entropy below $T=0.1J$.