Flat spin-wave dispersion in a triangular antiferromagnet

The excitation spectrum of an $S=1∕2$ two-dimensional triangular quantum antiferromagnet is studied using $1∕S$ expansion. Due to the noncollinearity of the classical ground state significant and nontrivial corrections to the spin-wave spectrum appear already in the first order in $1∕S$ in contrast to the square lattice antiferromagnet. The resulting magnon dispersion is almost flat in a substantial portion of the Brillouin zone. Our results are in quantitative agreement with recent series expansion studies by Zheng et al. [Phys. Rev. Lett. 96, 057201 (2006); cond-mat/0608008 (unpublished)].

Figure 1

(Color online) BZ of a triangular antiferromagnet. Thin lines indicate the directions along which the cuts of dispersion are taken. Coordinates of the points are $P(2\pi ∕3,0)$, $A(\pi ,0)$, $Q(4\pi ∕3,0)$, $B(\pi ,\pi ∕\sqrt{3})$, $C(2\pi ∕3,2\pi ∕\sqrt{3})$, $F(0,2\pi ∕\sqrt{3})$, and $E(0,\pi ∕\sqrt{3})$. $\mathrm{Im}{\Sigma }_k^{\left(S\right)}\ne 0$ inside the shaded star-shaped area.

Figure 2

(Color online) Classical and renormalized spin-wave dispersions. Left panels: three-dimensional plots; right panels: color density plots. The dispersions are shown in the $OQCF$ quadrant of the BZ and are set to zero outside this region to ease the viewing. The classical dispersion (a), (b) is $1.5E_k$. The real part of the renormalized dispersion $E_{\mathrm{ren}}\left(k\right)$ [Eq. (12)] is shown in (c) and (d). The flat dispersion is in the blue (dark) region in (d). The imaginary part of the renormalized dispersion is shown in (e) and (f). Observe that it is numerically small throughout the BZ, and is zero outside the “star” region (see also Fig. 1).

Figure 3

(Color online) Classical and renormalized dispersions (ordinate) along (a) $FC$ and (b) $OF$ directions (abscissa). Red dashed line, classical dispersion. Black solid line, real part of the renormalized dispersion $E_{\mathrm{ren}}\left(k\right)$, Eq. (12). Blue dotted line, imaginary part of $E_{\mathrm{ren}}\left(k\right)$. The black dots are series expansion data from Ref. 14. No fitting parameter is involved in the comparison with the data.

Figure 4

(Color online) Same as in Fig. 3 but along (a) $OQ$ and (b) $PC$ directions.

Figure 5

(Color online) Same as in Fig. 3 but along (a) $AB$ and (b) $EB$ directions.