Effects of an external magnetic field on the gaps and quantum corrections in an ordered Heisenberg antiferromagnet with Dzyaloshinskii-Moriya anisotropy

We study the effects of external magnetic field on the properties of an ordered Heisenberg antiferromagnet with the Dzyaloshinskii-Moriya (DM) interaction. Using the spin-wave theory quantum correction to the energy, on-site magnetization, and uniform magnetization are calculated as a function of the field $H$ and the DM anisotropy constant $D$. It is shown that the spin-wave excitations exhibit an unusual field evolution of the gaps. This leads to various nonanalytic dependencies of the quantum corrections on $H$ and $D$. It is also demonstrated that, quite generally, the DM interaction suppresses quantum fluctuations, thus driving the system to a more classical ground state. Most of the discussion is devoted to the spin-$S$, two-dimensional square lattice antiferromagnet, whose $S=\frac{1}{2}$ case is closely realized in ${\mathrm{K}}_2{\mathrm{V}}_3{\mathrm{O}}_8$ where at $H=0$ the DM anisotropy is hidden by the easy-axis anisotropy but is revealed in a finite field. The theoretical results for the field dependence of the spin-excitation gaps in this material are presented and the implications for other systems are discussed.

Figure 1

(Color online) (a) $\mathbf{H}\parallel \mathbf{D}\parallel z$ configuration. The canting angle $\phi$ in the $x\text{−}y$ plane is induced by the DM interaction and does not depend on the field. The canting angle $\theta$ towards the field ($z$ axis) is given by Eq. (3). (b) $\mathbf{H}\perp \mathbf{D}$ configuration. The canting angle $\phi$ is given by Eq. (5).

Figure 2

(Color online) The difference of the in-plane canting angle $\phi$ and its $D=0$ limit ${\phi }_0$ vs field in the $\mathbf{H}\perp \mathbf{D}$ configuration for two representative values of $D=0.001$ (dashed lines) and $D=0.04$ (solid lines), $D$ is in units of $J$. Inset: the canting angle $\phi$ vs field. The asymptotic behavior of $\phi -{\phi }_0=-D{H}_s∕2(H-H_s)$ is shown by the dotted line. $D^{1∕3}$ values for both choices of $D$ are shown by the horizontal dashed lines.

Figure 3

(Color online) Linear spin-wave result, Eq. (13), for the magnon spectrum in a field $(D=0)$, for $H=0$, $H=H_s∕\sqrt{2}$, and $H=H_s$ along the main diagonal of the BZ. Inset: full BZ with the direction of the cut. Magnetic BZ is shown by the dashed diamond.

Figure 4

(Color online) Linear spin-wave result, Eq. (14), for the magnon spectrum in zero field for $D=0.04$, $D=0.4$, and $D=1.0$ along the main diagonal of the BZ.

Figure 5

(Color online) ${\Delta }_{00}∕4SJ$ vs $H∕H_s$ for $D=0.001$ and $D=0.04$. Inset: same plot in the region of small fields.

Figure 6

(Color online) Field dependence of ${\Delta }_{\pi \pi }∕4SJ$ vs $H∕H_s$ for $D=0.001$, Eq. (21). The gap values at $H=0$, $H=H_s∕\sqrt{2}$, and $H=H_s$ are marked by the horizontal dashed lines. Dotted lines are as follows: for $H<H_s$, the low- and intermediate-field approximations for the gap, Eq. (22); for $H>H_s$, the $D=0$ gap dependence ${\Delta }_{\pi \pi }=H-H_s$. Arrows mark $H=H_s∕\sqrt{2}$. Inset: same for $D=0.04$.

Figure 7

(Color online) $1∕S$ corrections to the ground-state energy (left axis) and the on-site magnetization (right axis) for $D=0$ (dashed) and $D=0.04$ (solid) vs $H∕H_s,S=1∕2$.

Figure 8

(Color online) Uniform magnetization $(\text{classical part}+1∕S\text{correction})$, right axis, and the $1∕S$ part to the uniform magnetization normalized to its classical part, left axis, vs $H∕H_s$. $D=0$ (dashed) and $D=0.04$ (solid). Arrows at the left axis show ${\lim }_{H\to 0}{\lim }_{D\to 0}$, Eq. (29), and ${\lim }_{D\to 0}{\lim }_{H\to 0}$, Eq. (30). $S=1∕2$.

Figure 9

(Color online) The DM-induced changes in the $1∕S$ corrections to (a) the on-site magnetization at $H=H_s∕\sqrt{2}$ and (b) the uniform magnetization at $H=H_s$. Exact results obtained by using the numerical integration in (a) Eq. (25) and (b) Eq. (27) (squares) and the asymptotic expressions (a) Eq. (32) and (b) Eq. (33) (solid lines) are shown. (a) $A=1∕2\pi S,B=0.48$, (b) $A=1∕3\pi S$. $S=1∕2$.

Figure 10

(Color online) The field-evolution of the gaps for the spin-flop field configuration $\mathbf{H}\parallel \mathbf{D}\parallel z$. The spin-flop transition occurs at $H_{SF}$, Eq. (41). Below the transition the gaps depend linearly on $H$, Eq. (43). Above the transition one of the gaps is zero (dashed) and the other corresponds to a uniform precession of the field-induced magnetization (solid), Eq. (45). The uniform precession mode for the $E=D=0$ case is shown by the dotted line $(\Delta =H)$. Inset: same results in the region of the small fields. Saturation field for ${\mathrm{K}}_2{\mathrm{V}}_3{\mathrm{O}}_8$ is $H_s=8SJ=37.6\mathrm{T}$.

Figure 11

(Color online) The projections of the spin arrangement on the $x\text{−}z$, $x\text{−}y$, and $y\text{−}z$ planes within the spin-rotation phase $0<H<H_{SR}$. While $\phi$ is the angle between the projections of spins on the $x\text{−}y$ plane and the $x$ axis, $\theta$ is the angle between the actual spin direction and the $z$ axis. $\mathbf{H}$ is chosen along the $y$ axis.

Figure 12

(Color online) ${\Delta }_{00}∕4SJ$ vs $H∕H_s$ for the in-plane field direction, $D=0.004$, and two values of $E$: $E=0.0017$ (solid), Eqs. (52, 55), and $E=0$ (dashed), Eq. (19). Inset: same plot in the region of small fields. The spin-rotation transition is marked by the dotted vertical line. Saturation field for ${\mathrm{K}}_2{\mathrm{V}}_3{\mathrm{O}}_8$ is $H_s=8SJ=37.6\mathrm{T}$.

Figure 13

(Color online) ${\Delta }_{\pi \pi }∕4SJ$ vs $H∕H_s$ for the in-plane field direction, $D=0.004$, and two values of $E$: $E=0.0017$ (solid), Eqs. (53, 56), and $E=0$ (dashed), Eq. (21). Inset: same plot in the region of small fields. Saturation field for ${\mathrm{K}}_2{\mathrm{V}}_3{\mathrm{O}}_8$ is $H_s=8SJ=37.6\mathrm{T}$.