Tunable rotons in square-lattice antiferromagnets under strong magnetic fields

Excitation spectra of square lattice Heisenberg antiferromagnets in magnetic fields are investigated by the spin-wave theory. It is pointed out that a rotonlike structure appears in a narrow range of magnetic fields as a result of strong nonlinear effects. It is shown that the energy gap and the mass of the “roton” are quite sensitive to the magnetic field: the roton gap softens rapidly and eventually closes as a precursor of a quantum phase transition. The possibility of the experimental observation of the roton and a new ground state after its softening are discussed.

Figure 1

(a) Nonlinear spin-wave spectra ${\overline{\epsilon }}_{\mathbf{k}}$ for the $S=1/2$ system along the highly symmetric line calculated for several magnetic fields. The magnetic field $h$ corresponding to each line is shown on the right side. The spectrum along the $\Gamma \text{−}M$ line near the $M$ point varies drastically for small changes of $h$. There are apparent minima at $P$ point along the $X\text{−}{X}^{\prime }$ line at finite fields. (b) Enlarged ${\overline{\epsilon }}_{\mathbf{k}}$ for $S=1/2$ SLHAF along the $\Gamma \text{−}M$ line near the $M$ point [$\pi (1-\eta ,1-\eta ),0\le \eta \le 0.20$]. (c) Spectral weight $A(\mathbf{k},\omega )$ of (b) $(S=1/2,h=0.7568)$. We notice a sizable broadening by magnon decay, but we still see the roton feature in the part colored orange [$A(\mathbf{k},\omega )\gtrsim 2$].

Figure 2

(a) Contour plot of ${\overline{\epsilon }}_{\mathbf{k}}$ $(S=1/2,h=0.7568)$ of the whole Brillouin zone. The energy corresponding to each contour is shown on the right side. (b) Enlarged contour plot of ${\overline{\epsilon }}_{\mathbf{k}}$ near the $M$ point. It is now clear that an energy minimum appears near the $M$ point. (c) Enlarged ${\overline{\epsilon }}_{\mathbf{k}}$ along the line $\pi ({\eta }_{\mathrm{rot}}-\eta ,{\eta }_{\mathrm{rot}}+\eta )$, where ${\eta }_{\mathrm{rot}}$ denotes a roton wave vector $\pi {\eta }_{\mathrm{rot}}(1,1)$, perpendicular to the $\Gamma \text{−}M$ line. Note the sharpness of the minimum along this direction; the half width is on the order of 0.01.

Figure 3

(a) Nonlinear spin wave spectra ${\overline{\epsilon }}_{\mathbf{k}}$ for the various $S$ including $S=\infty$ (LSW) at $h=0.7568$. The spin magnitude $S$ for each line is shown on the right side. Spectra become more classical for larger $S$, but no qualitative changes are observed. It is also suggested that the quantum effects are stronger along the $\Gamma \text{−}M$ line than the others.

Figure 4

(a) Roton gap ${\Delta }_{\mathrm{rot}}$ along the $\Gamma \text{−}M$ line as a function of the field $h$. Solid diamonds, circles, and squares represent results for $S=1/2,S=1$ rotons, and shallow rotons, which appears only for $S=1/2$. Each roton gap decreases monotonically as the field increases. (b) Roton wave vector along the $\Gamma \text{−}M$ line [$\pi (1-\eta ,1-\eta ),0\le \eta \le 1$] as a function of $h$. The triangles and gray line correspond to $k_{\mathrm{rot}}$ and $k_0$. We notice that $k_{\mathrm{rot}}$ appears in the neighborhood of $k_{\mathrm{th}}$. Little change in $k_{\mathrm{rot}}$ is observed for the shallow roton. (c) Roton masses as functions of $h$, where solid (open) points correspond to $m_{\perp }^{*}$ $\left(m_{\parallel }^{*}\right)$. The masses are quite small, and nonmonotonic behavior is observed for the shallow roton. (d) Zoomed region of (c) is shown for clarity. The order of the ordinates is changed by two digits.

Figure 5

Self-energy causing strong renormalization near the roton wave vector.

Figure 6

(a) Roton gap ${\Delta }_{\mathrm{rot}}$ as a function of $k_0^4$. Solid diamonds and circles represent $S=1/2$ and $S=1$, and lines show fittings using Eq. (25). The $S=1/2$ roton gap decreases about two times faster than the $S=1$ roton gap. (b) Perpendicular mass $m_{\perp }^{*}$ as a function of $k_0^{-2}$; it scales linearly with $k_0^{-2}$. (c) Parallel mass $m_{\parallel }^{*}$ plotted as $m_{\parallel }^{*}{k}_0^2$ vs $k_0$. We see that Eq. (27) gives a good approximation to $m_{\parallel }^{*}$ for sufficiently small $k_0$.