Nonlinear magnetization dynamics as a route to nonreciprocal phases, spin superfluidity, and analogue gravity

Abstract

The identification of platforms with independently tunable nonlinearity and non-Hermiticity promises a quantitative route to far-from-equilibrium universality across many-body systems. Here we show that a conventional ferromagnetic multilayer realizes this paradigm: balancing a dc drive against Gilbert damping stabilizes a chiral spin-superfluid limit cycle that spontaneously breaks spacetime-translation symmetry. The resulting superflow is intrinsically nonreciprocal: long-wavelength magnons of opposite chirality acquire asymmetric dispersions and propagate direction-selectively, realizing a spin-superfluid diode. This asymmetry is flow-borne - it reflects broken Galilean invariance and requires neither structural asymmetry nor finely tuned gain-loss balance. Linearized dynamics in the comoving superfluid frame are intrinsically pseudo-Hermitian and, in the long-wavelength sector, can be mapped to a (1+1)D wave equation on curved spacetime. Spatial modulation of the drive enables the generation of sonic horizons that parametrically squeeze magnons and produce Hawking-like particle-hole emission. Our results establish a tabletop route from nonlinear dissipative-driven magnetization dynamics to nonreciprocal transport, nonequilibrium phase transitions, and analogue-gravity kinematics.

Figures (4)

• Figure 1: A schematic depiction of the superfluid limit cycle (SFLC) state realized in our model and the resulting spin-wave dynamics. The state breaks both spatial and temporal translation symmetry and supports excitations of opposite angular momenta with asymmetric propagation speeds. Inset: a schematic of the heterostructure device wherein the SFLC can be realized. Each ferromagnetic layer is subjected to the applied field $B\hat{\mathbf{z}}$ and separated by a metallic spacer that mediates both interfacial symmetric and antisymmetric (DM) exchange $A$ and $D$, respectively. An antidamping torque of strength $J_s$ counteracts the intrinsic dissipation of the magnetization dynamics.
• Figure 2: The equilibrium phase diagram of the model in Eq. \ref{['eq: Fdiml']} superimposed with that of the SFLC. Given a dimensionless DMI $d$ and spiral pitch $k$, the color of the corresponding region indicates the existence (or non-existence) of either an energetically stable LESS (hatched green), an unstable LESS (unhatched green), or a HESS (red) - which are always unstable (see Appendix A for more details). The grayed out region corresponds to $|d|<1$ whereby the ground state is fully polarized $\mathbf{m}=\hat{\mathbf{z}}$. Outside of this region, the ground state is the stable LESS with pitch $k=-d$ indicated with the black dashed line. The yellow/green region corresponds to those values of $d$ (and corresponding values of $k$) where an SFLC of pitch $k$ can be stably supported. Crucially, the yellow region extends to the weak-DMI regime $|d|<1$ where equilibrium superfluid states are forbidden.
• Figure 3: The SFLC unwinding map defined by Eq. \ref{['eq: unwind']}. (a) The transverse components of $\mathbf{m}_{\text{LC}}(x,t)$ are shown in blue. The relative spatial variations are set by the SFLC pitch $k$. The gray arrow indicates the precessing dynamics at frequency $\Omega$ of the SFLC state. The first step of the unwinding map moves the system into a frame rotating at $\Omega$, rendering each spin stationary. The second step unwinds the spatial spiral of pitch $k$. (b) The magnetization is finally globally rotated so that the canted state (with polar angle $\cos\theta_\text{LC} = m_\text{LC}^z$) is aligned along the north pole. The Goldstone mode associated with phase fluctuations $\phi\mapsto \phi+\delta\phi$ maps onto transverse fluctuations $n^y\mapsto n^y+\delta\phi\,\sin\theta_\text{LC}$, indicated by a red arrow.
• Figure 4: (a) Wave-propagation asymmetry $c_{+}+c_{-}$ plotted throughout the SFLC stability phase diagram. The central region bounded by gray dashed lines indicates the set of drive strengths $J_{s}=\alpha\Omega$ and pitches $k$ where an SFLC is stably supported. Regions bounded by the purple/red lines and the dashed boundaries correspond to parameter sets where only the Goldstone mode is unstable, or equivalently, long-wavelength spin waves on top of the SFLC propagate unidirectionally. If $(k,J_{s}/\alpha)$ is in the red (purple) region, a spin-wave of momentum $k$ will relax toward the aligned (antialigned) state $\mathbf{m}=+\hat{\mathbf{z}}$ ($-\hat{\mathbf{z}}$); in the overlapping region, the final steady state depends on the initial canting angle. The black (A) and orange (B) markers indicate the parameter choices used in panels (i–l). (b) Sound velocities $c_{\pm}$ defined below Eq. \ref{['eq: asymdisp']} versus SFLC frequency $\Omega$; green (brown) curves correspond to $c_{+}$ ($c_{-}$), with shaded regions highlighting right- (left-)propagating unidirectional regimes. (c–g) Magnon dispersion atop the SFLC of pitch $k=-0.4$ for $\Omega=0.97,\,0.985,\,1,\,1.015,\,1.03$ (increasing left to right). (h) Anomalous correlator strength defined below Eq. \ref{['eq: GF']} for the $\Omega$ of (g). (i) Complex eigenvalue bands $\lambda^{(1)}_{q}$ (solid) and $\lambda^{(2)}_{q}$ (dashed) of the dynamical matrix $\mathbf{M}_{q}$ defined above Eq. \ref{['eq: rqsq']}, plotted for $q\in[-0.1,0.1]$. The color of each band corresponds to the SFLC parameters A and B in (a). (j) Spatial profiles of the local SFLC frequency and sound velocities in the scattering scenario considered, using the same color code as in (b). (k) Dispersion within regions A (top) and B (bottom) of the scattering geometry of (j). The color coding matches that in (c–g). The vertical (horizontal) dashed line indicates the momentum $q_{0}$ (frequency $\omega_0=\omega_p(q_{0})$) used for the initial condition Eq. \ref{['eq: IC']}. The yellow circle in A encloses $(q_{0},\omega_{0})$, while the three yellow circles in B identify the intersections of $\omega_0$ with the dispersion curves in region B, corresponding to the frequency-allowed scattering channels. (l) Wavepacket scattering visualized using the particle pseudocharge density defined in Eq. \ref{['eq: rhop']}. For (a–l), $d=0.25$ and, in (i,l), $\alpha=10^{-3}$. In (k–l), horizons are located at $\pm x_{0}=\pm 200$. The initial condition used in (l) is defined in and around Eq. \ref{['eq: IC']} with $q_{0}=0.01$ with $\chi$ being a Gaussian centered at $x_{0}=-1000$ and characteristic width $200$.