Nonreciprocal Magnons and Symmetry-Breaking in the Noncentrosymmetric Antiferromagnet

Inelastic neutron scattering measurements were performed to study spin dynamics in the noncentrosymmetric antiferromagnet $\alpha \text{−}{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$. For the first time, nonreciprocal magnons were experimentally measured in an antiferromagnet. These nonreciprocal magnons are caused by the incompatibility between anisotropic exchange and antisymmetric Dzyaloshinskii-Moriya interactions, which arise from broken symmetry, resulting in a collinear ordered state but helical spin dynamics. The nonreciprocity introduces the difference in the phase velocity of the counterrotating modes, causing the opposite spontaneous magnonic Faraday rotation of the left- and right-propagating spin waves. The breaking of spatial inversion and time reversal symmetry is revealed as a magnetic-field-induced asymmetric energy shift, which provides a test for the detailed balance relation.

Figure 1

[(a)–(c)] The zero-field magnetic excitations in $\alpha \text{−}{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$. (a) The contour map, which is constructed from a series of constant-$\mathbf{Q}$ scans taken at SPINS for $k<2$ and at BT7 for $k>2$, shows two dispersive branches of the excitations centered around (0, 1.75, 0) and (0, 2.25, 0). The BT7 contour map represents the intensity difference between the data measured at 2 and 50 K. An arrow denotes the magnetic Bragg reflection at (0, 2, 0). (b) The contour map of the calculated dynamical structure factor $S(\mathbf{Q},\hslash \omega )$ representing the spin-wave intensity along $(0,k,0)$ was calculated using the obtained fit parameters. (c) The spin-wave dispersions of all 16 branches are shown in grey, while red and blue lines represent the most intense branches. The data points were obtained from resolution-convolution fitting of the constant-$\mathbf{Q}$ scans. Error bars denote standard derivations throughout the article. (d) The spin network of $\alpha \text{−}{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ is formed by three dominant exchange interactions $J_1$, $J_2$, and $J_3$. Red and blue spheres represent two spin sublattices, which are parallel and antiparallel to the $a$ axis, respectively. Arrows and spin labels denote the order of the cross product. (e) The magnetic excitations of the spin-chain model in an applied magnetic field along the $x$ axis [Eq. (S1)] were calculated with $S=1/2$, $J=1\mathrm{meV}$, $G=0.26\mathrm{meV}$, $D=1.0\mathrm{meV}$, and $H=1\mathrm{T}$ (Supplemental Material [25]). Dynamical spin structures, which show the counterclockwise and clockwise rotation of the fluctuating spins around the field direction ($+x$ direction), correspond to the solid red and dashed blue modes, respectively.

Figure 2

Spin-wave excitations in a magnetic field at 1.5 K. (a)–(c) show the energy-momentum contour maps measured at 6, $-6$, and 10 T, respectively. Dashed lines denote the most intense branches resulting from the spin-wave calculations. (d) shows the magnetic Bragg peak measured along $(0,k,0)$ below and above the spin-flop transition (similar data measured at different fields are reported in Ref. [22]). Arrows denote the magnetic Bragg peaks, and asterisks (*) indicate the second harmonic reflections.

Figure 3

Constant-$\mathbf{Q}$ scans were measured at (0, 1.75, 0) and (0, 2.25, 0) at (a) 1 and (b) 2 T. The dashed and dotted lines denote positions of the energy gap. In (c), the scan at (0, 1.75, 0) and $-5\mathrm{T}$ is compared with that at (0, 2.25, 0) and $+5\mathrm{T}$, which shows the same shift in energy. All (0, 1.75, 0) data are shifted for clarity. The gap energy as a function of field at (0, 1.75, 0) and (0, 2.25, 0) is shown in (d). The critical field is labeled by ${\mu }_0{H}_c$, at which the energy gap vanishes.

Figure 4

Detailed balance relation of the scattering intensity. (a) The energy-momentum contour map of the scattering intensity was measured at $T=27\mathrm{K}$ with (a) ${\mu }_{0}H=+10\mathrm{T}$ and [(b) and (c)] ${\mu }_{0}H=\pm 5\mathrm{T}$, which put the system in the collinear antiferromagnetic state in the phase diagram (Fig. S10). The solid lines denote the constant-$\mathbf{Q}$ scans, which were performed to investigate the negative-energy-transfer and positive-energy-transfer excitations for (b) $\mathbf{Q}=(0,1.75,0)$ and ${\mu }_{0}H=+5\mathrm{T}$, (c) $\mathbf{Q}=(0,2.25,0)$ and ${\mu }_{0}H=+5\mathrm{T}$, and (d) $\mathbf{Q}=(0,1.75,0)$ and ${\mu }_{0}H=-5\mathrm{T}$.