Bose-Einstein Condensation of Magnons and Spin Superfluidity in the Polar Phase of ${}^3\mathrm{He}$

The polar phase of ${}^3\mathrm{He}$, which is topological spin-triplet superfluid with the Dirac nodal line in the spectrum of Bogoliubov quasiparticles, has been recently stabilized in a nanoconfined geometry. We pump magnetic excitations (magnons) into the sample of polar phase and observe how they form a Bose-Einstein condensate, revealed by coherent precession of the magnetization of the sample. Spin superfluidity, which supports this coherence, is associated with the spontaneous breaking of U(1) symmetry by the phase of precession. We observe the corresponding Nambu-Goldstone boson and measure its mass emerging when applied rf field violates the U(1) symmetry explicitly. We suggest that the magnon BEC in the polar phase is a powerful probe for topological objects such as vortices and solitons and topological nodes in the fermionic spectrum.

Figure 1

Creation of coherently precessing state in the polar phase of ${}^3\mathrm{He}$ using cw NMR. (a) Absorption and dispersion signals recorded on a sweeping down magnetic field $H$ with $H_{\mathrm{rf}}=0.32\mu \mathrm{T}$ at $T=0.41T_c$, $P=6.9$ bar, and $\lambda =90°$. On the horizontal axis, the frequency shift ${\omega }_{\mathrm{rf}}-\gamma H$ is shown. (b) The tipping angle $\beta$ and the phase of precession $\alpha$ of magnetization $\mathbf{M}$ are determined from the records in the panel (a). Definition of the angles is given in the inset. Magnetization $\mathbf{M}$ is in a rotating frame of precession. Absorption and dispersion signals are proportional to $M_y=M\sin \beta \sin \alpha$ and $M_x=M\sin \beta \cos \alpha$, respectively. The relation between $\mathrm{\Delta }\omega$ and $\cos \beta$ is linear (dashed line) in accordance with Eq. (2). (c) Cw NMR absorption versus $H_{\mathrm{rf}}^{-1}$ at the fixed $\mathrm{\Delta }\omega /(2\pi )=246\mathrm{Hz}$ at $T=0.43T_c$, $P=7.1$ bar, and $\lambda =90°$. The solid line is a linear fit through zero.

Figure 2

Long-living coherent precession is demonstrated by a free induction decay (FID) signal recorded after turning off at $t=0$ the rf excitation at the frequency shift marked by the dashed line in Fig. 1. Frequency (filled circle) and amplitude (diamond) are obtained by fitting a sine wave to short sections of the raw signal. Oscillations on the amplitude are due to the excitation of the NG mode. Linear extrapolation of $\cos \beta$ to 1 in Fig. 1 is taken as zero frequency. The solid line shows the expected time dependence of $\mathrm{\Delta }\omega$ calculated from cw NMR data in Fig. 1, assuming relaxation only due to magnon loss. Much faster dephasing time for incoherent precession is demonstrated by a FID signal measured in normal ${}^3\mathrm{He}$ (inverted triangle).

Figure 3

The mass $\mathcal{M}$ of the pseudo-Nambu-Goldstone mode in magnon BEC supported by cw NMR in the polar phase of ${}^3\mathrm{He}$ as a function of $H_{\mathrm{rf}}$ (a), ${\mathrm{\Omega }}_P$ (b), and $\sin \beta$ (c) at $P=7.1$ bar and $\lambda =90°$. Symbols are experimental data, curves are theoretical predictions of Eq. (4) without fitting. Inset to panel (a) shows an example of excitation spectrum of magnon BEC measured as described in the text with $3\times {10}^{-6}$ relative peak-to-peak variation of the static field. A set of peaks (marked by dashed lines) is seen and $\mathcal{M}$ is given by the frequency of the largest peak, while the error bar is determined as the peak width. The Leggett frequency ${\mathrm{\Omega }}_P(T)$ is determined from cw NMR spectra at $\lambda =0$. Values of $\cos \alpha$, which depend on $H_{\mathrm{rf}}$ and $\mathrm{\Delta }\omega$, are calculated from absorption and dispersion signals, while $\sin \beta$ is determined from Eq. (2).

Figure 4

Dependence of coherent precession on the magnetic field orientation $\lambda$. (a) The mass $\mathcal{M}$ of the pseudo-NG mode. Symbols represent ranges of $\sin \beta$ over which the mass has been averaged. The curve is the theoretical dependence for $\sin \beta =0.45$ according to Eq. (4). For smaller $\beta$ theoretical line goes lower. The measurements have been done with $H_{\mathrm{rf}}$ increasing from $0.16\mu \mathrm{T}$ to $0.48\mu \mathrm{T}$ as $\lambda$ decreases from 90° to 65°, but are scaled in the plot with Eq. (4) to $H_{\mathrm{rf}}=0.71\mu \mathrm{T}$, which coincides with the data in Fig. 3. Temperature is between $0.44T_c$ and $0.49T_c$. (b) The relaxation rate of magnon BEC at two temperatures.