Abstract
In superfluids the order parameter, which describes spontaneous symmetry breaking, is an analogue of the Higgs field in the Standard Model of particle physics. Oscillations of the field amplitude are massive Higgs bosons, while oscillations of the orientation are massless NambuGoldstone bosons. The 125 GeV Higgs boson, discovered at Large Hadron Collider, is light compared with electroweak energy scale. Here, we show that such light Higgs exists in superfluid ^{3}HeB, where one of three NambuGoldstone spinwave modes acquires small mass due to the spin–orbit interaction. Other modes become optical and acoustic magnons. We observe parametric decay of BoseEinstein condensate of optical magnons to light Higgs modes and decay of optical to acoustic magnons. Formation of a light Higgs from a NambuGoldstone mode observed in ^{3}HeB opens a possibility that such scenario can be realized in other systems, where violation of some hidden symmetry is possible, including the Standard Model.
Introduction
The superfluid transition in ^{3}He, a fermionic isotope of helium, occurs due to formation of Cooper pairs with orbital momentum L=1 and spin S=1. The corresponding order parameter is a 3 × 3 matrix of complex numbers, which includes both spin and orbital degrees of freedom^{1}. Thus, besides fermionic quasiparticles, superfluid ^{3}He possesses 18 bosonic degrees of freedom, collective modes (oscillations) of the order parameter. Each mode has a relativistic spectrum , with a modespecific wave velocity c and a gap (or mass) ω_{0}. Modes with nonzero ω_{0} are Higgs modes and others are NambuGoldstone (NG) modes. Each NG mode corresponds to spontaneously broken continuous symmetry of the normal state.
In conventional superconductors with the order parameter of a single complex number, only the symmetry with respect to the change of the wavefunction phase is broken. This leads to one NG phase mode and oneamplitude Higgs mode, which was experimentally observed^{2,3,4}.
In unconventional Bphase of superfluid ^{3}He, the symmetry with respect to relative rotations of the spin and orbital spaces is additionally broken^{5}, and the order parameter in the zero magnetic field is
where Δ is the gap in the fermionic spectrum, Φ is the phase and R_{αi} is a rotation matrix, which connects spin and orbital degrees of freedom. The matrix R_{αi} is represented in terms of the rotation axis and angle θ. Parameters Φ, and θ determine a fourdimensional subspace of degenerate states. Thus, among 18 collective modes of ^{3}HeB (Fig. 1a), four are NG modes: oscillation of Φ is sound and oscillations of R_{αi} (or and θ) are spin waves. The other 14 modes are the Higgs modes with energy gaps of the order of Δ. These heavy Higgs modes have been investigated for a long time both theoretically^{6,7,8,9} and experimentally^{10,11,12,13,14}.
In superconductors and in the Standard Model, the NG bosons become massive due to the AndersonHiggs mechanism^{15,16,17}. In electrically neutral ^{3}He, these modes are gapless, when viewed from the scale of Δ∼100 MHz (set by the critical temperature T_{c}∼10^{−3} K). At lowenergy scale of ∼1 MHz, corresponding to the frequency of our NMR experiments, two weak effects become significant: spin–orbit interaction and applied magnetic field H (Fig. 1b).
Spin–orbit interaction lifts the degeneracy with respect to θ, and the minimum energy corresponds to the socalled Leggett angle θ_{L}=arccos(−1/4). This explicit violation of the symmetry of the Bphase leads to appearance of the gap for the θ mode. This mode becomes an additional, light, Higgs boson. The gap value Ω_{B} is called Leggett frequency, it is a measure of the spin–orbit interaction. At low temperatures .
Two other spinwave modes are oscillations of . In the magnetic field, the equilibrium state corresponds to and the field splits these two modes in the same way as in ferromagnets. One of the modes, the optical magnon, acquires the gap equal to the Larmor frequency ω_{L}=γH (where γ is the gyromagnetic ratio). Another one, the acoustic magnon, remains gapless, but its spectrum becomes quadratic. Such unusual form of the spectrum comes from a violation of the timereversal symmetry by the magnetic field (for general discussion of the NG modes with quadratic spectrum see ref. 18).
All three lowfrequency spinwave modes are described by the closed system of Leggett equations^{1}. In particle physics, such set of lowenergy modes, which includes NG modes and a light Higgs, is called the Little Higgs field^{19}. Formal definition of such field in ^{3}HeB is given in Supplementary Note 1.
In an NMR experiment, one follows the dynamics of magnetization M, or of the spin S of the sample. The motion of θ, corresponds to longitudinal spin waves, , while oscillations of correspond to transverse spin motion, δ SH. Optical magnons can be directly created with traditional transverse NMR. With a suitable coil system, one can also directly excite longitudinal spin oscillations, or light Higgs mode^{20}. Coupling to shortwavelength acoustic magnons is hard to achieve in a traditional NMR experiment with large excitation coils.
In this work, we use a technique, based on BoseEinstein condensate (BEC) of optical magnons^{21}, to probe interaction and conversion between all components of the little Higgs field in ^{3}HeB. As a result, we observe parametric decay of optical magnons to light Higgs bosons, and both parametric and direct conversion between optical and acoustic magnons. The measured mass of light Higgs and propagation velocity of acoustic magnons are close to the expected values. Thus, we experimentally confirm the little Higgs scenario in ^{3}HeB. The little Higgs field appears in quantum chromodynamics^{22}, where NG modes (pions) acquire light mass due to the explicit violation of the chiral symmetry, which is negligible at high energy, but becomes significant at low energy^{23}. The relatively small mass of the 125 GeV Higgs boson observed at the Large Hadron Collider suggests that it might be also the pseudoGoldstone (light Higgs) boson (see for example, ref. 24 and references therein).
Results
Suhl instability
As a tool to study dynamics of the little Higgs field in superfluid ^{3}HeB, we use trapped BECs of optical magnons (Fig. 2). The condensate is well separated from the container walls, where the strongest magnetic relaxation in ^{3}He usually occurs^{25}. Thus, tiny relaxation effects, connected to coupling of optical magnons to other components of the little Higgs field, can be observed.
When number of pumped magnons is low, slow exponential relaxation of the precession signal is determined by spin diffusion and energy losses in the NMR pickup circuit^{26} (Fig. 2b). We have found that above some threshold amplitude, the relaxation becomes much faster (Fig. 3a). The explanation is the Suhl instability^{27}, a wellknown nonlinear effect in magnets when a uniform precession of magnetization (here at the optical magnon frequency ω_{opt}) parametrically excites a pair of acoustic magnons with twice smaller frequency ω_{ac} and opposite kvectors: ω_{opt}=ω_{ac}(k)+ω_{ac}(−k). In the case of ^{3}HeB, both acoustic magnons and the light Higgs modes can be parametrically excited (Fig. 1b). The process occurs with conservation of energy and momentum. The threshold amplitude is inversely proportional to the coupling between decaying and excited waves, and proportional to the relaxation in the excited wave.
Mass of light Higgs
The measured threshold amplitude as a function of NMR frequency and pressure is plotted in Fig. 3b. The frequency dependence allows us to identify the decay channels. It is clear from Fig. 1b that the decay of the optical magnon to a pair of light Higgs bosons with the frequency ω_{Higgs}, ω_{opt}=ω_{Higgs}(k)+ω_{Higgs}(−k), is possible only when the precession frequency is larger then 2Ω_{B}. We see a pronounced drop of the threshold amplitude at this frequency: the threshold decreases by about an order of magnitude.
In Fig. 3d, the measured mass of light Higgs Ω_{B} is plotted as a function of pressure. Measurements are in a good agreement with values of the Leggett frequency from ref. 28.
Resonances of acoustic magnons
In addition to the sharp drop, connected with light Higgs mode, we find periodic modulation of the threshold amplitude as a function of the frequency of the precession (Fig. 3c). These periodic peaks originate from the parametric decay of the optical magnons in the BEC to acoustic magnons. The frequency dependence is explained by quantization of the magnon spectrum in the cylindrical container, which serves as a resonator for acoustic magnons. Consider a decay of the optical magnon with frequency ω_{opt} into acoustic magnons with frequency ω_{ac}=Nω_{opt}, where for the parametric excitation N=1/2. In the following discussion, we will use ω_{opt}=ω_{L}, since the difference is negligible for trapped optical magnons. By sweeping the magnetic field, we can change both magnon spectrum and magnon trap and observe resonances in the cell. The simple resonance condition for acoustic magnons in a cylinder with the radius R gives the distance between the resonances (Supplementary Note 2):
where c is the relevant spinwave velocity.
In Fig. 3e, the measured acoustic magnon resonance period is plotted as a function of pressure. The results are in a good agreement with equation (2), where values of the spinwave velocity are taken from our recent measurements^{29}.
Effect of quantized vortices
An additional relaxation mechanism for the magnon condensate is found when quantized vortices are formed in the sample. In the presence of these localized topological objects, the momentum k of the spinwave modes is not conserved, and one expects direct excitation of acoustic magnons by the optical mode. We can rotate the sample with angular velocities up to Ω=21 rad s^{−1} to create a cluster of rectilinear quantized vortices, which cross the whole experimental region including the magnon BEC (Fig. 4a). In this state, the relaxation rate, plotted as a function of the frequency in Fig. 4b, reveals several periodic sets of peaks. We attribute these peaks to resonances of acoustic magnons with frequencies ω_{L}, 2ω_{L} and so on.
In ^{3}HeB, the rotational symmetry of a vortex is spontaneously broken and the vortex core can be treated as a bound state of two halfquantum vortices, which can rotate around the vortex axis. Dynamics of the vortex is affected by the precessing magnetization^{30,31}. Precession of S and in the magnon BEC produces torsional oscillations of the vortex core. The fact that the equilibrium position of deviates from the vertical direction within the magnon BEC makes these oscillations unharmonic. As a result, acoustic magnons with frequencies Nω_{L} can be emitted.
The amplitudes of the various resonances depend on a distribution of vortex cores and the wave nodes of acoustic magnons. For example, an axially symmetric distribution of vortices can excite only symmetric waves, which means doubling of the observed resonance period. In our experiment, acoustic magnons (with wave length 5–10 μm) are emitted by vortices, which are within the magnon condensate. The distance between vortices 0.1–0.2 mm, is comparable with the size of the trapped condensate 0.2–0.4 mm. Thus, the amplitudes of resonances are sensitive to details, such as orderparameter texture, rotation and pressure, and we do not see all the harmonics at all pressures. Nevertheless, the resonance periods plotted in Fig. 4c follow the theoretical values (2) or their multiples (denoted as 2δf_{1} and so on).
Discussion
To summarize, we have observed the interplay of all three spinwave modes, which form a little Higgs field in superfluid ^{3}HeB. In particular, we have found two channels of parametric decay of optical magnons: to a pair of light Higgs bosons and to a pair of acoustic magnons. Although the search for similar resonant production of pairs of Standard Model Higgs bosons reported by the ATLAS collaboration^{32} has not succeeded yet, our results support the basic physical idea behind this effort. Another system where the light Higgs mode can be observed is the multicomponent condensate in cold gases^{33}, where interaction between components can be set up to produce the hidden symmetry.
We find that the lowenergy physics in superfluid ^{3}He has many common features of the Higgs scenario in Standard Model: both are described by the SU(2) and U(1) symmetry groups; the acoustic and optical magnons correspond to the doublet of W^{+} and W^{−} gauge bosons, which spectrum also splits in magnetic field^{34}; the light Higgs mode has parallel with the 125GeV Higgs boson. However, in addition, the ^{3}HeB has the highenergy sector with 14 heavy Higgs modes. This suggests that in the same manner, the 125GeV Higgs boson belongs to the lowenergy sector of particle physics, and if so, one may expect the existence of the heavy Higgs bosons at TeV scale.
We have demonstrated that the shortwavelength acoustic magnons can be emitted and detected with the BEC of optical magnons. Acoustic magnons can be lensed by nonuniform magnetic fields and the orderparameter texture, and thus might serve in future as a powerful local probe to study topological superfluidity of ^{3}He, including Majorana fermions on the boundaries of the superfluid and in the cores of quantized vortices.
Methods
The sample geometry and NMR setup
Superfluid ^{3}HeB is placed in a cylindrical container with the inner diameter of 2R=5.85 mm, made from fused quartz (Fig. 2). The container has closed top end and open bottom end, which provides the thermal contact to the nuclear demagnetization refrigerator. Static magnetic field is applied parallel to the container axis. A special coil creates a controlled minimum of the field magnitude along the axial direction. Transverse NMR coils, made from copper wire, are used to create and detect magnetization precession. Coils are part of a tuned tank circuit with the Q value of ∼130. Frequency tuning is provided by a switchable capacitance bank, installed at the mixing chamber of the dilution refrigerator. To improve signaltonoise ratio, we use a cold preamplifier, thermalized to liquid helium bath.
The measurements are performed at low temperatures T<0.2 T_{c}, where spinwave velocities and the Leggett frequency are temperatureindependent. Typically, we use T=130–350 μK, depending on pressure. The temperature is measured by a quartz tuning fork thermometer, installed at the bottom of the sample cylinder. The heat leak to the sample was measured in earlier work to be ∼12 pW (ref. 35). The measurements are performed at pressures 0–29 bar and in magnetic fields H=17–26 mT with corresponding NMR frequencies ω_{L}/2π=550–830 kHz.
Magnon trap
Minimum of the axial magnetic field forms a trapping potential for optical magnon quasiparticles in the axial direction. Trapping in the radial direction is provided by the spin–orbit interaction via the equilibrium distribution of the order parameter. In this geometry, it forms the socalled flareout texture: is parallel to H on the cell axis and tilted near walls because of boundary conditions^{36}. The combined magnetotextural trap is nearly harmonic with trapping length ∼0.3 mm in the radial direction and 1 mm in the axial direction (see Supplementary Note 3 for details).
Measurements of magnon BEC
Owing to the geometry, the coils couple only to optical magnons with k≈0. With a short radio frequncy (rf) pulse in the NMR coils, nonequilibrium optical magnons are created. At temperatures of our experiment the equilibration within the magnon subsystem proceeds much faster that the decay of magnon number, and the pumped magnons are condensed to the ground level of the trap within 0.1 s from the pulse. Manifestation of BoseEinstein condensation is the spontaneously coherent precession of the condensate magnetization^{37,38}, which induces current in the NMR coils. The amplified signal is recorded by a digital oscilloscope; an example record is in Fig. 2b. We then perform sliding Fourier transform of the signal with the window 0.3–1 s. In the resulting sharp peak in the spectrum, the frequency determines the BEC precession frequency ω_{opt}, while the amplitude (such as shown in Fig. 3a) is proportional to the square root of the number of magnons in the trap.
Rotation
The sample is installed in the rotating nuclear demagnetization refrigerator ROTA^{39}, and can be put in rotation together with the cryostat and the measuring equipment. The cryostat is properly balanced and suspended on active vibration isolation, and in rotation the heat leak to the sample remains <20 pW (ref. 35). Vortices are created by increasing angular velocity Ω from zero to a target value at temperature ∼0.7 T_{c}, where the mutual friction allows for fast relaxation of vortex configuration towards an equilibrium array^{40}. Further cooldown is performed in rotation.
Additional information
How to cite this article: Zavjalov, V. V. et al. Light Higgs channel of the resonant decay of magnon condensate in superfluid ^{3}HeB. Nat. Commun. 7:10294 doi: 10.1038/ncomms10294 (2016).
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Acknowledgements
We thank M. Krusius and V.S L’vov for useful discussions. This work has been supported in part by the EU 7th Framework Programme (FP7/20072013, Grant No. 228464 Microkelvin), the Academy of Finland (project no. 284594) and by the facilities of the Cryohall infrastructure of Aalto University. P.J.H. acknowledges financial support from the Väisälä Foundation of the Finnish Academy of Science and Letters, and S.A. acknowledges financial support from the Finnish Cultural Foundation.
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The experiments were conducted by S.A., P.J.H., V.V.Z. and V.B.E.; the theoretical analysis was carried out by G.E.V. and V.V.Z.; and the paper was written by V.V.Z., G.E.V. and V.B.E., with contribution from all the authors.
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Zavjalov, V., Autti, S., Eltsov, V. et al. Light Higgs channel of the resonant decay of magnon condensate in superfluid ^{3}HeB. Nat Commun 7, 10294 (2016). https://doi.org/10.1038/ncomms10294
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