Nonreciprocal magnons and symmetry-breaking in the noncentrosymmetric antiferromagnet

TL;DR

This paper presents the first direct evidence of nonreciprocal magnons in a noncentrosymmetric antiferromagnet, caused by symmetry-breaking interactions, leading to unique spin-wave behaviors and potential applications in magnonic devices.

Contribution

It demonstrates the existence of nonreciprocal magnons in an antiferromagnet due to broken symmetry and competing interactions, revealing new spin-wave phenomena.

Findings

Observation of nonreciprocal magnon dispersion in $ ext{α}$-Cu$_2$V$_2$O$_7$

Detection of opposite magnonic Faraday rotation for counter-propagating modes

Magnetic-field-induced asymmetric energy shifts confirming symmetry breaking

Abstract

Magnons, the spin-wave quanta, are disturbances that embody a wave propagating through a background medium formed by ordered magnetic moments. In an isotropic Heisenberg system, these disturbances vary in a continuous manner around an ordered spin structure, thus requiring infinitesimal energy as a wavevector approaches a magnetic zone centre. However, competing anisotropic interactions arising from broken symmetry can favour a distinct static and dynamic spin state causing a shift of the minimum point of the magnon dispersion to a nonreciprocal wavevector. Here we report the first direct evidence of these nonreciprocal magnons in an antiferromagnet. In the antiferromagnet we investigated, namely, noncentrosymmetric $\alpha$-Cu$_2$V$_2$O$_7$, they are caused by the incompatibility between anisotropic exchange and antisymmetric Dzyaloshinskii-Moriya interactions resulting in competing collinear and helical spin structures, respectively. The nonreciprocity introduces the difference in the phase velocity of the counter-rotating 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.

Equations

$$\mathcal{H}$$

$$S(-\mathbf{q},-\hslash \omega )=e^{-\hslash \omega /k_{B}T}S(\mathbf{q},\hslash \omega ),$$

$$I(\mathbf{Q},\hslash \omega )={\left(\frac{\gamma r_0}{2}\right)}^{2}f(\mathbf{Q}{)}^2\sum _{\alpha ,\beta }({\delta }_{\alpha \beta }-{\hat{Q}}_{\alpha }{\hat{Q}}_{\beta })S^{\alpha \beta }(\mathbf{Q},\hslash \omega ),$$

$$\mathcal{H}=\sum _{i,j}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+\sum _{i,j}{G}_{i,j}(S_i^x{S}_j^x-S_i^y{S}_j^y-S_i^z{S}_j^z)\text{mbox}+{\mathbf{D}}_{i,j}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)-g_e{\mu }_B\sum _i{\mathbf{S}}_i\cdot \mathbf{B},$$

$${\tilde{R}}_1=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right)\text{ and }{\tilde{R}}_2=\left(\begin{array}{ccc}0& 0& -1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right).$$

$$S_j^{x^{\prime }}=\sqrt{\frac{S}{2}}(a_j+a_j^{†}),S_j^{y^{\prime }}=\mathrm{i}\sqrt{\frac{S}{2}}(a_j^{†}-a_j),S^{z^{\prime }}=S-a_j^{†}{a}_j,$$

$$a_{\mathbf{k}}^j=\sum _{\mathbf{k}}{e}^{-\mathrm{i}\mathbf{k}\cdot {\mathbf{R}}_j}{a}_j,\mathbf{k}=(0,k_y,0),$$

$$H=\left(\begin{array}{cccc}(G+J)-g_e{\mu }_{B}B& 0& 0& {\Delta }_{GJ}\cos \frac{k_y}{2}-D\sin \frac{k_y}{2}\\ 0& (G+J)+g_e{\mu }_{B}B& {\Delta }_{GJ}\cos \frac{k_y}{2}+D\sin \frac{k_y}{2}& 0\\ 0& -{\Delta }_{GJ}\cos \frac{k_y}{2}-D\sin \frac{k_y}{2}& (-G-J)+g_e{\mu }_{B}B& 0\\ -{\Delta }_{GJ}\cos \frac{k_y}{2}+D\sin \frac{k_y}{2}& 0& 0& (-G-J)-g_e{\mu }_{B}B\end{array}\right).$$

$$E_{val}^1$$

$$A^{\pm }$$

$${\Delta }_{GJ}$$

$$E_{vec}^1$$

$$E_{vec}^2$$

$$E_{vec}^3$$

$$E_{vec}^4$$

$$A_J^{\pm }=J\sqrt{\frac{1-\cos k_y}{2}}=J\left|\sin \frac{k_y}{2}\right|,$$

$$A_{JG}^{\pm }=\sqrt{\frac{G^2+6GJ+J^2-{\Delta }_{GJ}^2\cos k_y}{2}},$$

$$S_{x^{\prime },l}^j$$

$$S_{y^{\prime },l}^j$$

Full text

While symmetry plays a central role in imposing uniformity on the fundamental laws of nature feynman ; noether , symmetry breaking introduces “the texture of the world”Gross by adding layers of complexity to the physical laws. In condensed matter systems, symmetry and a lack of it determine the underlying interactions of the governing Hamiltonian. In particular, the absence of spatial inversion symmetry in magnetic systems causes the relativistic spin-orbit coupling, which gives rise to many intriguing phenomena such as the spin Hall effect Sinova , topological insulators Qi , multiferroics Fiebig , and noncentrosymmetric superconductors Bauer:2004da , to acquire antisymmetric Dzyaloshinskii-Moriya (DM) interactions Dzyaloshinsky ; Moriya . For noncentrosymmetric $\alpha$-Cu2V2O7, the crystal structure breaks spatial inversion symmetry Calvo1975 ; Robinson1987 , and the antiferromagnetic ordering below $T_{N}=33.4$ K ganatee ; Lee:2016 breaks time reversal symmetry. The simultaneous breaking of both symmetries sets the stage for the intertwining electric and magnetic properties Sannigrahi ; Lee:2016 and for the existence of toroidal moments Hayami2 ; Hayami3 ; Ederer .

In the magnetically ordered state, $S=1/2$ Cu2+ spins in $\alpha$-Cu2V2O7 align antiparallel along the crystallographic $a$-axis forming a collinear structure. In the presence of a magnetic field along the $c$-axis, weak ferromagnetism resulting from the DM-interaction-induced canted moments was observed. When the magnetic field is applied along the $a$-axis, two magnetic transitions appear: one transition at $\mu_{0}H_{c1}={}$6.5 T is characterized as the spin-flop transition whereas the other at $\mu_{0}H_{c2}{}={}$18.0 T is a result of the spin-flip ganatee2 . Combined density functional theory (DFT) calculations and Quantum Monte Carlo simulations suggest a complex spin-network shown in Fig. 1(d) Sannigrahi ; Banerjee ; ganatee2 . To the first approximation, the spin Hamiltonian for $\alpha$-Cu2V2O7 can be described by motome

[TABLE]

where the summation $\sum_{i,j}$ $(\sum_{k,l})$ is taken over the nearest, second-nearest, and third-nearest neighbours (nearest neighbours). The first term represents the isotropic exchange interactions, where $J_{1}$, $J_{2}$, and $J_{3}$ depicted in Fig. 1(d) are all antiferromagnetic with $J_{1}\sim J_{2}<J_{3}$ ganatee2 . The second term represents the anisotropic exchange interaction $G_{1}$, which arises from the multiorbital correlation effect caused by the relativistic spin-orbit coupling and multiorbital hybridization motome ; Hayami1 ; Hayami2 . The third term denotes the antisymmetric Dzyaloshinskii-Moriya interactions $\mathbf{D}_{1}$, which results from the absence of the inversion centre between the nearest-neighbour spins Dzyaloshinsky ; Moriya . The last term represents spins in an external magnetic field, where $g_{e}=-2$ is the electron spin $g$-factor, $\mu_{B}$ is the Bohr magneton, and $\mathbf{B}$ is the magnetic field applied parallel to the crystallographic $a$-axis. The anisotropic exchange interaction stabilizes the collinear antiferromagnetic spin structure and introduces an energy gap to the magnon excitations, whereas the $a$-component of the DM vector favours a helical spin structure in the $bc$-plane and determines the incommensurate wavevector of the helical modulation. Due to the competition between these two terms, spin fluctuations of the dynamic state may not be around the static spin structure, giving rise to distinct wavevectors for the ordered structure and the minimum of the dynamical spin-waves.

Spin dynamics of $\alpha$-Cu2V2O7 was investigated using single-crystal inelastic neutron scattering as described in Methods. An intensity contour map as a function of energy transfer $\hbar\omega$ and momentum transfer $\mathbf{Q}$ measured at the base temperature along $(0,k,0)$ shows two branches of spin-wave excitations symmetrically centred on both sides of the magnetic zone centre $(0,2,0)$, where a magnetic Bragg reflection is observed [Fig. 1(a)]. The two magnon modes correspond to the clockwise and anticlockwise rotation of spins bonded by $J_{1}$ with $G_{1}$ and $\mathbf{D}_{1}$ along the $[0\bar{1}\bar{1}]$ and $[0\bar{1}1]$ uniform zig-zag chains (Fig. S9). The degeneracy of these two modes is lifted by the DM interaction, which causes the observed symmetric shift of the magnon modes to the $-k$ (left) and $+k$ (right) side of the zone centre [Fig. 1(b)]. The $+k$ ($-k$) mode corresponds to the anticlockwise rotation along the $b$-axis but clockwise (anticlockwise) along the spin-chain directions (see Supplementary Material). Constant-energy contour maps of scattering intensity (Fig. S1) covering a wider range in the $(0kl)$ and $(hk0)$ scattering planes confirm that the nonreciprocity of the magnon dispersion, which was not observed in the recent inelastic neutrons scattering on a powder sample Banerjee , is only along $(0,k,0)$, which is consistent with linear spin-wave calculations [Figs. S1(g–j)], as will be discussed later. The energy of both modes increases steadily with roughly the same slope up to about 10 meV, and they cross at the zone centre at $\hbar\omega\sim 3$ meV. The absence of crossing avoidance suggests that the counter-rotating excitations are decoupled. The energy scan measured at the high-resolution cold neutron spectrometer SPINS yields the gap energy $\Delta=0.75(6)$ meV at $(0,1.75,0)$ [Fig. S2(d)]. The gap energy as a function of temperature correlates with the decrease of the order parameter and the peak width becomes broader, which is indicative of shorter life-time as temperature increases toward $T_{N}$ (Fig. S3). These results confirm that the excitations are due to the fluctuations of the ordered magnetic moments. The nonreciprocal magnons were recently observed in the field-induced ferromagnetic phase of noncentrosymmetric MnSi, where the shift of the single, non-degenerate magnon mode is asymmetric depending on the field direction sato , in contrast to the symmetric shift in the antiferromagnet at zero field observed in this study.

To quantitatively describe the observed magnon dispersion in $\alpha$-Cu2V2O7, we employed linear spin-wave calculations, which are described in Methods. The calculated magnons consist of 16 modes, denoted by the grey lines in Figs. 1(c), S4(a), and S4(b). However, only two modes, which were experimentally observed in Fig. 1(a) and theoretically confirmed in Figs. 1(b), S4(c), and S4(d), were selected to fit the measured dispersion. The data points in Fig. 1(c) shows the measured magnon dispersion along $(0,k,0)$ obtained from the constant-$\mathbf{Q}$ scans (Fig. S2). The dispersion along $(h,1.75,0)$ and $(0,2,l)$ (Fig. S4) as well as the field-dependence of the energy gap [Fig. 4(d)] were also measured and used in the global fit to obtain the relevant Hamiltonian parameters in Eq. 1. The ratio $J_{1}:J_{2}:J_{3}$ was fixed to the result obtained from the DFT calculations of $1.00:1.12:2.03$ ganatee2 . The minimal model, which includes three isotropic exchange interactions, the anisotropic exchange interaction $G_{1}$, and the uniform DM vector $\mathbf{D}_{1}=(D_{1a},0,0)$, is able to capture the magnon nonreciprocity along $(0,k,0)$ and the dispersion as shown by the solid lines in Figs. 1(c), S4(a), and S4(b). The obtained fitted parameters are $J_{1}=2.67(1)$ meV, $J_{2}=2.99$ meV, $J_{3}=5.42$ meV, $G_{1}=0.282(1)$ meV, and $D_{1a}=2.79(1)$ meV. The value of the DM parameter with $D_{1a}/J_{1}\sim 1$ is much higher than that measured in other $S=1/2$ Cu2+ spin systems Thio , suggesting the exceptionally strong spin-orbit coupling.

The symmetry between the anticlockwise and clockwise modes is broken in the presence of the applied magnetic field as the electron spins precess under the perpendicular field, and if the spin rotation is in the same (opposite) sense as (to) the spin precession, or anticlockwise (clockwise) rotation, the excitation energy becomes lower (higher) as illustrated for the spin-chain model in Supplementary Material. Experimentally, for $\alpha$-Cu2V2O7 when the applied field is positive along the $a$-axis ($\mu_{0}H=+6$ T), the $+k$ mode (clockwise rotation along the zig-zag chain) is shifted upward whereas the $-k$ mode (anticlockwise rotation) is shifted downward [Fig. 2(a)]; on the other hand, if the field is negative ($\mu_{0}H=-6$ T), the shift of the dispersion reverses [Fig. 2(b)]. Spin-wave calculations confirm the energy shift in the presence of the field as shown in Fig. S5. The energy scans at $+1$ T ($+2$ T) shown in Figs. 3(a) [3(b)] display the asymmetric shift of the gap energy at $(0,1.75,0)$ and $(0,2.25,0)$. Figure 3(c) depicts the same energy shift of the $+k$ and $-k$ modes when the field of the same magnitude is oppositely aligned. The gap energy as a function of field [Fig. 3(d)] shows a linear relation, consistent with the spin-chain model (Supplementary Material), with a negative slope for $(0,1.75,0)$ and positive slope for $(0,2.25,0)$. Extrapolating the linear relation to intersect the horizontal axis yields the critical field $\mu_{0}H_{c1}$ of $\pm$6.61(2) T, at which the energy gaps at $(0,2\mp 0.25,0)$ close and the spin-flop transition occurs ganatee2 .

When the applied field is increased from $+6$ T to $+10$ T, the magnetic Bragg reflection at $(0,2,0)$, indicated by the arrow in Fig. 2(a), moves to the incommensurate wavevectors $(0,2\pm\delta,0)$ where $\delta\sim 0.23$, as denoted by the pair of arrows in Figs. 2(c) and 2(d). The transfer of the Bragg intensity documented in Fig. 2(d) occurs at the spin-flop transition reported at $\mu_{0}H_{c1}=6.5$ T ganatee2 , and is consistent with the transition from the collinear spin structure to the helical spin structure with the majority of the spin component being in the $bc$-plane. The harmonic reflections at roughly $(0,2\pm 2\delta,0)$ indicated by the asterisks in Fig. 2(d) substantiate the incommensurate modulation of the helical structure. The spin-wave excitations in the spin-flop state show the minimum at $(0,2,0)$ [Fig. 2(c)] indicative of the reciprocal magnons. Hence, while the collinear spin structure below $\mu_{0}H_{c1}$ hosts the nonreciprocal magnons, the helical spin structure above $\mu_{0}H_{c1}$ gives rise to the reciprocal magnons with the polarisation most likely along the $a$-axis, highlighting the competitive nature of the anisotropic exchange and antisymmetric DM interactions.

The asymmetry between the $-k$ and $+k$ modes provides a great opportunity to verify the well-known detailed balance relation for the dynamical structure factor, which can be described by squires

[TABLE]

where $k_{B}$ is the Boltzmann constant, and the momentum transfer $\mathbf{q}$ is measured from the zone centre. The breaking of inversion symmetry in the $\alpha$-Cu2V2O7 crystal structure and of time-reversal symmetry due to the applied field introduces the asymmetry between the clockwise and anticlockwise magnon modes, with the result that $S(-\mathbf{q},\pm\hbar\omega)\neq S(+\mathbf{q},\pm\hbar\omega)$. The energy-momentum contour map was measured at 27 K and 10 T, where the system is in the collinear antiferromagnetic states as the critical field $\mu_{0}H_{c1}$ increases to $\sim 15$ T at $T\sim 25$ K (Fig. S10). The elevated temperature is necessary to populate magnons and facilitate the neutron-energy-gain (negative energy transfer) scattering process. Figure 4(a) illustrates the asymmetry upon the sign reversal of $\mathbf{q}$ and $\hbar\omega$ separately, reflecting broken inversion and time reversal symmetry, respectively. The detailed measurements of the constant-$\mathbf{Q}$ scans extended to negative energy shown in Figs. 4(b–d) display the asymmetry between the neutron-energy-gain and neutron-energy-loss scattering intensity satisfying the detailed balance relation in Eq. 2; the asymmetry is reversed when the momentum transfer changes from $+\mathbf{q}$ to $-\mathbf{q}$ and vice versa [Figs. 4(b) and 4(c)], and when the field direction is flipped [Figs. 4(b) and 4(d)].

The magnon nonreciprocity due to the DM interaction causes the clockwise and anticlockwise modes to acquire different phase velocities resulting in the rotation of the magnon polarisation, called the spontaneous magnonic Faraday effect. In contrast to a ferromagnet Nembach ; Kwon , the magnonic Faraday rotation in the antiferromagnet is opposite for the left- and right-propagating spin-waves, thus giving rise to the net nonreciprocal phase flow in thermal equilibrium. Taking the role of the DM interaction, an external electric field, which breaks inversion symmetry and lifts the degeneracy of the two counter-rotating modes, can cause magnon nonreciprocity and lead to the electric-field-induced magnonic Faraday effect, which may find applications in the spin-wave field-effect transistor Cheng:2016kv .

**Methods

**

Single crystals of $\alpha$-Cu2V2O7 were grown using the method described in Ref. ganatee, . Inelastic neutron scattering measurements were conducted to study magnetic excitations in $\alpha$-Cu2V2O7 using the thermal-neutron triple-axis spectrometer BT7 Lynn2012 , the cold-neutron triple-axis spectrometer SPINS, the Multi Axis Crystal Spectrometer (MACS) Rodriguez:2008be , and the Disk Chopper time-of-flight Spectrometer (DCS) Copley2003477 , all of which are located at the NIST Center for Neutron Research (NCNR), Gaithersburg, MD, USA. For the BT7 measurements, a single crystal of mass 1.39 g was aligned so that $(0,k,l)$ was in the scattering plane. The final energy was fixed at 14.7 meV with horizontal collimations of open – 80*′* – 80*′* – 120*′. A PG filter was placed after the sample to eliminate the higher order contamination. The crystal was cooled to the base temperature of 2 K using a 4He closed-cycle cryostat. A series of constant-$\mathbf{Q}$ scans was collected at the base temperature and at 50 K for background subtraction. The measurements at SPINS were performed using neutrons with a fixed final energy of 5 meV and horizontal collimations of open – 80′* – 80*′* – open, yielding the energy resolution of $\sim 0.5$ meV. A cold Be filter was placed in the scattered beam to cut off the neutron energy higher than 5 meV. A series of constant-$\mathbf{Q}$ scans was measured in the $(0kl)$ and $(hk0)$ planes. All scans were measured at the base temperature of 2.5 K. At MACS, contour maps at constant energies were measured in the $(0kl)$ and $(hk0)$ planes. The fixed final energy of 5 meV was selected using the vertically focusing, horizontally flat analyser with cold Be filters in the scattered beam. The base temperature of 1.7 K was achieved using a 4He cryostat. In the $(0kl)$ plane, the constant energy cuts were measured between 0 and 12 meV and the background was measured at 40 K and 80 K for $\hbar\omega{}={}$2.5 and 5.0 meV, whereas in the $(hk0)$ plane, the constant energy cuts were measured between 0 and 6 meV and the background was measured at 80 K for $\hbar\omega{}={}$2.0 and 5.0 meV. For the DCS experiment, the single crystal was measured in magnetic fields up to 10 T using neutrons with incident energy $E_{i}=5.98$ meV, yielding the instrumental energy resolution of $0.3$ meV at the elastic position. The crystal was mounted inside a vertical-field 10-T magnet and aligned so that the crystallographic $a$-axis was perpendicular to the scattering plane and parallel to the applied field. The magnet was equipped with a closed-cycle 4He cryostat, which can reach the base temperature of 1.5 K.

Linear spin-wave calculations were performed based on the collinear, antiferromagnetic spin structure and on the spin Hamiltonian in Eq. 1 using the standard Holstein-Primakoff transformation and Fourier transform implemented using a symbolic algebraic method. We choose $\hat{x}$, $\hat{y}$, and $\hat{z}$ to be along the crystallographic $a$-, $b$-, and $c$-axis, respectively. There are a total of 16 spins in the unit cell and hence 16 spin-wave modes as shown by the grey lines in Figs. 1(c), S4(a), and S4(b). The spin-wave energies are non-negative eigenvalues obtained from diagonalising a $32\times 32$ matrix. The spin-wave intensity for each mode was calculated using squires

[TABLE]

where $(\gamma r_{0}/2)^{2}=72.65\times 10^{-3}$ barn $/\mu_{B}^{2}$, and $f(\mathbf{Q})$ is the magnetic form factor for Cu2+ ions. $\sum_{\alpha,\beta}(\delta_{\alpha\beta}-\hat{Q}_{\alpha}\hat{Q}_{\beta})$ is the geometric factor, where $\alpha$ and $\beta$ are vector components $\hat{x}$, $\hat{y}$, or $\hat{z}$. $S^{\alpha\beta}(\mathbf{Q},\hbar\omega)$ is the dynamical structure factor, which is the spatial and temporal Fourier transform of the spin-pair correlation function $\braket{S^{\alpha}(0,0)S^{\beta}(\mathbf{r},t)}$ that can be calculated from the eigenvectors.

**Acknowledgements

** We thank Y. Motome for fruitful discussion. Work at Mahidol University was supported in part by the Thailand Research Fund Grant Number RSA5880037 and the Thailand Centre of Excellence in Physics. Work at NCNR was supported in part by the National Science Foundation under Agreement No. DMR-1508249. Work at IMRAM was partly supported by a Grant-In-Aid for Scientific Research (24224009) from the Japan Society for the Promotion of Science (JSPS), and by the Research Program “Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials”.

**Author Contributions

** G.G., Y.Z., Y.Q., L.W.H., N.P.B., and K.M. performed the neutron scattering experiments. P.P., T.J.S., and K.M. performed the linear spin-wave calculations. G.G., P.P., and K.M. wrote the manuscript. All authors commented on the final version of the manuscript.

Competing Interests The authors declare that they have no competing financial interests.

Correspondence Correspondence and requests for materials should be addressed to K. Matan (kittiwit.mat@mahidol.ac.th).

I Contour maps in $(0kl)$ and $(hk0)$ planes

Constant-energy contour maps of scattering intensity as a function of momentum transfer were measured using MACS covering a wide $\mathbf{Q}$ range in the $(0kl)$ and $(hk0)$ scattering planes to study the splitting of the excitations around the magnetic zone centres. Figs. S1(a–c) [S1(d–f)] show the folded contour maps at zero energy, 2.0 (2.5) meV, and 5.0 meV in the $(hk0)$ [$(0kl)$] plane. At $\hbar\omega{}={}$0 meV [Figs. S1(a) and S1(d)], the magnetic Bragg reflections, which are on top of the nuclear Bragg peaks, can be observed as a very intense single spot at all expected magnetic zone centres. At finite energy transfers, the contour maps show the splitting of the minimum point of the magnon dispersion around the zone centres along the $k$ direction. This splitting can be observed above the gap energy at $\hbar\omega{}={}$2.0 meV for the $(hk0)$ plane [Fig. S1(b)] and 2.5 meV for the $(0kl)$ plane [Fig. S1(e)] at all magnetic zone centres. At higher energies, the excitations become dispersive and form two ellipses which start to overlap at $\hbar\omega\sim 3$ meV. The ellipses become larger and more elongated along $k$ with increasing energy transfer as shown in Figs. S1(c) and S1(f) for $\hbar\omega=5.0$ meV. For comparison, the calculated scattering intensity maps plotted in Figs. S1(g–j) show good agreement with the measured contour maps at the corresponding energies. The calculated intensity is integrated using an energy width of $0.5$ meV with the mid-point at the labelled energy. The discrepancy between the measured and calculated results, i.e., the filled ovals for the measured data versus open circles for the calculated results, is due to the fact that the instrumental-resolution effect was ignored in the calculations.

II Constant-$\mathbf{Q}$ scans

All constant-$\mathbf{Q}$ scans, which were used to construct the contour maps in Fig. 1(a), were fitted using an empirical dispersion convoluted with the instrumental resolution function in order to extract the excitation energies as a function of $\mathbf{Q}$. Representative energy scans at several $\mathbf{Q}$’s around $(0,2,0)$ measured at SPINS are shown with the convoluted fit in Fig. S2. The fitting results including those obtained from fitting the BT7 data are plotted in Fig. 1(c).

III Temperature dependence of gap energy

The constant-$\mathbf{Q}$ scans at $(0,1.75,0)$ were performed at several temperatures below $T_{N}$ to study the evolution of the gap energy $\Delta$ as the temperature approaches $T_{N}$. The data were fitted and convoluted with the instrumental resolution function as shown by solid lines in Figs. S5(a–d). The obtained fitted gap energies were then plotted as a function of temperature in Fig. S3(e). The results exhibit a decrease of the gap energy, which correlates with the decline of the order parameter as $T_{N}$ is approached. The fit to the power law $\Delta\propto(T_{N}-T)^{\beta}$ yields $T_{N}=32.5(6)$ K, in agreement with $T_{N}$ obtained from the order parameter fit, and $\beta=0.23(5)$, consistent with the exponent describing the ordered moment $M$, which scales with the square root of the magnetic Bragg scattering intensity $(M\propto\sqrt{I})$. This result confirms that the magnetic excitations are due to the fluctuations of the ordered moments. Furthermore, as the temperature increases toward $T_{N}$, the peak width becomes broader, indicative of a shorter life-time of the excitations typical for the spin-waves.

IV Nonreciprocal magnons in an antiferromagnetic spin-chain

To illustrate the origin of the nonreciprocal magnons in $\alpha$-Cu2V2O7, we consider a simple model of an antiferromagnetic spin chain, where spin-waves can be analytically calculated. We hope that this calculation will provide the reader with some insight, which is probably lost in the complicated spin network of $\alpha$-Cu2V2O7 and the numerical calculations, into the underlying mechanism giving rise to the nonreciprocal magnons. We note that when only the nearest-neighbour spins connected by $J_{1}$ are considered, Cu2+ spins form a uniform zig-zag chain along the $[0\bar{1}\bar{1}]$ and $[0\bar{1}1]$ directions (Fig. S9), and hence the spin-chain model should qualitatively and sufficiently describe the magnon dispersion in $\alpha$-Cu2V2O7. The Dzyaloshinskii-Moriya (DM) interactions and anisotropic exchange interactions are introduced to the two-sublattice Heisenberg antiferromagnetic spin-chain model in order to study their effects on the elementary excitations of the system. The calculations show that the gap in the spin-wave spectrum is the result of the anisotropic exchange interactions and the nonreciprocity of the spin-wave spectrum is due to the DM interactions.

Spin-wave calculations

As with the Hamiltonian for $\alpha$-Cu2V2O7, the Hamiltonian for this spin-chain model is given by

[TABLE]

where the summation $\sum_{i,j}$ between spin operators is taken over the nearest-neighbor bonds (Fig. S6). The first and second terms represent the isotropic exchange interactions $J$ and anisotropic exchange interactions $G$, respectively. The third term is the antisymmetric DM interactions, and the last term represents the interactions between the spins and the external magnetic field $\mathbf{B}=(B,0,0)$. We note that the one-dimensional lattice spacing is equal to one.

We transform coordinates from the spin local coordinates (primed coordinates), in which the positive $z^{\prime}$-axis coincides with the spin direction, to the crystal coordinates (unprimed coordinates) using the following transformation matrices

[TABLE]

Using the Holstein-Primakoff transformation, the spin operators are transformed to the boson operators $a$ and $a^{{\dagger}}$ via

[TABLE]

where $j=1,2$ for the two sublattices. The spin Hamiltonian can then be written in terms of $a$ and $a^{{\dagger}}$ which now can be solved by introducing the Fourier transform

[TABLE]

where the summation runs from $\mathbf{-k}$ to $\mathbf{k}$. Using $(a_{\mathbf{k}}^{1},a_{\mathbf{k}}^{2},a_{\mathbf{-k}}^{1{\dagger}},a_{\mathbf{-k}}^{2{\dagger}})$ as a basis and noting that $a_{\mathbf{k}}^{1}\neq a_{\mathbf{-k}}^{1}$ for a noncentrosymmetric system, we can write the Hamiltonian as

[TABLE]

We then proceed to solve for the eigenvalues and eigenvectors of this matrix. The eigenvalues of this matrix are

[TABLE]

with the corresponding unnormalized eigenvectors

[TABLE]

Hence the following pairs of the operators ($a_{\mathbf{k}}^{1}$, $a_{\mathbf{-k}}^{2{\dagger}}$), ($a_{\mathbf{k}}^{2}$, $a_{\mathbf{-k}}^{1{\dagger}}$), ($a_{\mathbf{-k}}^{1{\dagger}}$, $a_{\mathbf{k}}^{2}$), and ($a_{\mathbf{-k}}^{2{\dagger}}$, $a_{\mathbf{k}}^{1}$) are mixed, and we can define the superpositions of these pairs as new boson operators $Y_{0}(k)$, $Y_{1}(k)$, $Y_{1}^{{\dagger}}(-k)$, and $Y_{0}^{{\dagger}}(-k)$; this procedure is equivalent to the Bogoliubov transformation. The eigenvalues are physically meaningful if they are real, i.e., the term inside the square root of $A^{\pm}$ is positive. The dispersion as a function of momentum transfer along $(0,k,0)$ is shown in Fig. S7(a). We note that the eigenvalues are linearly dependent on the magnetic field $B$, the effect of which is shown in Fig. S7(b).

When the parameters $G$ and $D$ are zero, we obtain

[TABLE]

which corresponds to the normal antiferromagnetic spin-wave spectrum, where the clockwise and anticlockwise modes are degenerate. Adding $G$ will open the gap (which can be seen from $G^{2}$ and $6GJ$ in $A^{\pm}$) but does not lift the degeneracy:

[TABLE]

and when $D$ is present, the degeneracy is lifted and the spin-wave spectrum will be shifted, in this case, along $k_{y}$ as shown in Fig. S7.

IV.1 Spin dynamics

We can calculate the spin component of the magnon excitations in the local spin coordinates for a particular wavevector $\mathbf{k}$. From Eqs. S2 and S3, we obtain

[TABLE]

where $l$ denotes magnon modes, and $l=1,2$ for the spin-chain model and $l=1,\ldots,16$ for $\alpha$-Cu2V2O7. Using the values of $a$ and $a^{{\dagger}}$ obtained from the eigenvectors, we can calculate and draw dynamical spin structures of the magnon excitations. We found that for the spin-chain model the $+k$ mode is the anticlockwise spin excitation, and the $-k$ mode is the clockwise excitations. Figure S8 shows the anticlockwise and clockwise spin rotations corresponding to the $+k$ mode and the $-k$ mode, respectively, for $k=(0,0.1,0)$ r.l.u., $J=1$, $G=0.005$, and $D=0.1$. When the magnetic field is present, the spins in the $+k(-k)$ mode rotate in the same (opposite) sense as the precession of an electron spin under a perpendicular magnetic field and thus require less (more) energy to excite one magnon [Fig. S7 (b)].

V Spin dynamics in $\alpha$-Cu2V2O7

To obtain the spin dynamics of the magnons in $\alpha$-Cu2V2O7, we follow the same procedure described in the spin-chain model. Out of the 16 magnon modes, we chose to draw the spin structure belonging to the two most intense magnon modes, i.e., the $+k$ and $-k$ modes as shown in Fig. 1. In $\alpha$-Cu2V2O7, spins along the zig-zag chain rotate clockwise for the $+k$ mode, and anticlockwise for the $-k$ mode. We note that the $\pm k$ modes refer to the shifts of the dispersion and do not imply the spin rotation. Because of the difference in the chain direction between $\alpha$-Cu2V2O7 and the spin-chain model, the shift of the anticlockwise and clockwise modes in $\alpha$-Cu2V2O7 is not the same as that in the spin-chain model. Figure S9 shows the spin dynamics of $\alpha$-Cu2V2O7 calculated using the fitted Hamiltonian parameters at $\mathbf{q}=(0,-0.25,0)$ (left column) and $(0,0.25,0)$ (right column). When the magnetic field is applied along the $+a$ direction, the spin rotation of the $+k$ mode is in the opposite sense to the spin precession in the $bc$-plane so it requires more energy to excite this mode, whereas for the $-k$ mode the spins rotate in the same sense as the spin precession. Hence the $+k$ mode shifts upward while the $-k$ mode shifts downward consistent with the measured (Fig. 2) and calculated (Fig. S5) magnon dispersion.