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Higgs mode and its decay in a two-dimensional antiferromagnet

Abstract

Condensed-matter analogues of the Higgs boson in particle physics allow insights into its behaviour in different symmetries and dimensionalities1. Evidence for the Higgs mode has been reported in a number of different settings, including ultracold atomic gases2, disordered superconductors3, and dimerized quantum magnets4. However, decay processes of the Higgs mode (which are eminently important in particle physics) have not yet been studied in condensed matter due to the lack of a suitable material system coupled to a direct experimental probe. A quantitative understanding of these processes is particularly important for low-dimensional systems, where the Higgs mode decays rapidly and has remained elusive to most experimental probes. Here, we discover and study the Higgs mode in a two-dimensional antiferromagnet using spin-polarized inelastic neutron scattering. Our spin-wave spectra of Ca2RuO4 directly reveal a well-defined, dispersive Higgs mode, which quickly decays into transverse Goldstone modes at the antiferromagnetic ordering wavevector. Through a complete mapping of the transverse modes in the reciprocal space, we uniquely specify the minimal model Hamiltonian and describe the decay process. We thus establish a novel condensed-matter platform for research on the dynamics of the Higgs mode.

Main

For a system of interacting spins, amplitude fluctuations of the local magnetization—the Higgs mode—can exist as well-defined collective excitations near a quantum critical point (QCP). We consider here a magnetic instability driven by the intra-ionic spin–orbit coupling, which tends towards a non-magnetic state through complete cancellation of orbital (L) and spin (S) moments when they are antiparallel and of equal magnitude5,6. This mechanism should be broadly relevant for d4 compounds of such ions as Ir(V), Ru(IV), Os(IV) and Re(III) with sizable spin–orbit coupling but remains little explored. We investigate the magnetic insulator Ca2RuO4, a quasi-two-dimensional antiferromagnet7 with nominally L = 1 and S = 1 (Fig. 1). Because the local symmetry around the Ru(IV) ion is very low8,9 (having only inversion symmetry), it is widely believed that the orbital moment is completely quenched by the crystalline electric field10,11,12,13, which is dominated by the compressive distortion of the RuO6 octahedra along the c-axis (Fig. 1). In the absence of an orbital moment, the nearest-neighbour magnetic exchange interaction is necessarily isotropic. Deviations from this behaviour are a sensitive indicator of an unquenched orbital moment. If this moment is sufficiently strong, it can drive Ca2RuO4 close to a QCP with novel Higgs physics.

Figure 1: Crystal, magnetic, and electronic structures of Ca2RuO4.
figure 1

Ca2RuO4 crystallizes in the orthorhombic Pbca space group, a distorted variant of the layered perovskite structure with a quasi-two-dimensional square lattice. For clarity, Ca ions are shown as small, light grey balls and oxygen ions, at all corners of the octahedra, are not shown. The distortion involves 2% compression of the RuO6 octahedra along the c-axis, and their rotation about the c-axis and tilting about an axis that lies in the ab plane8,9. The unit cell for the undistorted square lattice is shown as the blue square. (π,π) magnetic order develops below TN ≈ 110 K with the moment (orange arrow) aligned approximately along the b-axis. The compressive distortion of the RuO6 octahedra leads to the splitting Δ between the orbitals of xy and yz/zx symmetry. If Δ is much larger than the spin–orbit splitting (λ), the orbital degrees of freedom are completely quenched and a S = 1 Heisenberg magnet is obtained. In the other limit λ Δ, a non-magnetic singlet ground state is stabilized. These two distinct phases exhibit qualitatively different magnetic excitation spectra. See Supplementary Figs 1 and 2 for the evolution of the electronic structure and the spin-wave dispersions between these two limiting cases. Error bars denote one standard deviation.

Our comprehensive set of time-of-flight (TOF) inelastic neutron scattering (INS) data over the full Brillouin zone (Fig. 2a) indeed reveal qualitative deviations of the transverse spin-wave dispersion from those of a Heisenberg antiferromagnet. In particular, the global maximum of the dispersion is found at q = (0,0), in sharp contrast to a Heisenberg antiferromagnet, which has a minimum there (Fig. 1). This striking manifestation of orbital magnetism in Ca2RuO4 leads us to consider the limit of strong spin–orbit coupling described in terms of a singlet and a triplet separated in energy by λ (Fig. 1), which was estimated in earlier experiments14,15,16 to be in the range 75–100 meV. In this limit, the ground state is non-magnetic with zero total angular momentum, and therefore a QCP separating it from a magnetically ordered phase is expected as a matter of principle. Although this QCP can be pre-empted by an insulator–metal transition17,18 or rendered first-order by coupling to the lattice or other extraneous factors, it is sufficient that the system is reasonably close to the hypothetical QCP.

Figure 2: Spin-wave dispersions strongly deviating from the Heisenberg model.
figure 2

a, TOF INS spectra along high-symmetry directions measured at T = 5 K (see Supplementary Fig. 3 for more details). The momenta indicated on the top axis refer to the undistorted square lattice unit cell (see Fig. 1), which is doubled for the magnetic unit cell. The dotted line is from b for direct comparison between theory and experiment. b, The excitation spectra of the model in equation (1) calculated with the parameters E 25 meV, J 5.8 meV, α = 0.15, ε 4.0 meV and A 2.3 meV. The spectra were convolved with the instrumental resolution (4.2 meV full-width at half-maximum). Transverse and longitudinal modes are labelled as ‘T’ and ‘L’, respectively, and their motions are depicted. The T′ mode arises from back-folding of the T mode by the magnetic (π,π) scattering and thus its intensity vanishes when approaching the QCP. The L mode carries the Higgs amplitude oscillation. The latter is not subject to the back-folding because it is insensitive to the staggered vector field. The black arrows show the momentum- and energy-conserving decay process of the L mode into a pair of T modes. The total momentum of the pair is that of the L mode shifted by (π,π) momentum provided by the condensate (see Supplementary Information). c, Energy spectra for several q points along (0,0)–(π,0) obtained using an integration window of ±0.0475 in H and K after symmetrization. The integration ranges were doubled for q = (0,0) to obtain similar statistics. The red marks indicate the position of the longitudinal mode as calculated by theory. The black solid curves are a guide to the eye. Error bars denote one standard deviation.

To assess the proximity to the QCP and the possibility of finding the Higgs mode, we first reproduce the observed transverse spin-wave modes by applying the spin-wave theory19,20 to the following phenomenological Hamiltonian dictated by general symmetry considerations:

Here, denotes a pseudospin-1 operator describing the entangled spin and orbital degrees of freedom. This model includes single-ion terms (E and ε) of tetragonal (z c) and orthorhombic (x a) symmetries, correspondingly, as well as an XY-type exchange anisotropy (α > 0) and the bond-directional pseudodipolar interaction (A); note that its sign depends on the bond. Also symmetry allowed—but neglected here—are the Dzyaloshinskii–Moriya interaction (which can be gauged out by a suitable local coordinate transformation) and further-neighbour interactions. The coupling constants resulting from fits of the model to the measured spectra are provided in the caption of Fig. 2. We stress that this model gives the unique minimal description of the system, which we also derive explicitly starting from the microscopic electronic structure (see Supplementary Information).

We find that the single-ion term E overwhelms all other coupling constants, particularly the nearest-neighbour exchange coupling J, and thus confines the pseudospins to the ab basal plane. This accounts for the XY-like dispersion, which has a maximum at q = (0,0). This important aspect was missed in a recent INS study of Ca2RuO4, because the dispersion along the path (,)–(0,0) was not measured21. The large E implies unquenched spin–orbit coupling; in cubic symmetry E is λ itself (see Supplementary Fig. 1). This term also acts towards suppressing the magnetic order by favouring the singlet ground state—known in the literature as ‘spin nematic’22—and is responsible for the significant reduction of the moment size from the nominal 2μB for pure S = 1 moments to approximately 1.3μB (ref. 8). Other terms allow fine tuning of the spin-wave dispersions; the pseudodipolar term accounts for the weak dispersion along the magnetic zone boundary (,)-(π,0), and ε is responsible for gapping the transverse mode. The latter means that the Goldstone bosons acquire a finite mass through the explicit breaking of the continuous symmetry, the significance of which for the Higgs mode decay will be discussed later on. Our calculation (Fig. 2b) predicts an intense Higgs mode in this parameter regime, visible as a longitudinal spin wave, which heralds a proximate QCP. Although not evident in the image plot in Fig. 2a, a peak that matches well the predicted position of the Higgs mode is clearly seen in the energy spectra plotted in Fig. 2c.

We further pursue the Higgs mode using spin-polarized INS, using the scattering geometry that maximizes its neutron cross-section. We use the standard XYZ-difference method to filter out all non-magnetic and incoherent scattering signals and to resolve all three spin-wave polarizations: the longitudinal mode (L) oscillates along the crystallographic b-axis, and the transverse Goldstone modes (T and T′) along the a- and c-axes. Because our sample mosaic consisting of 100 crystals is ‘twinned’—that is, approximately half of them are rotated 90° about the c-axis with respect to the other half—we can distinguish only between in-plane (ab) and out-of-plane (c) polarized modes. However, this is sufficient to identify the Higgs mode (see Supplementary Information).

Figure 3a shows the measured (symbols with error bars) and calculated (solid lines) dynamical susceptibility at q = (0,0). We observe three peaks in total as expected, but not all of them were clearly seen in the TOF data (see also Supplementary Fig. 3) because their intensities are maximized in different scattering geometries. The highest-energy peak at approximately 52 meV is unambiguously identified as the Higgs mode by its magnetic and in-plane-polarized character, because the second in-plane-polarized mode at approximately 45 meV has already been identified as the T mode (Fig. 2). Further, the data are in excellent accord with the model calculation, which has no adjustable parameter after fitting the dispersion of the T modes. The intensity ratio between the L and T modes is 0.55 ± 0.11, which is a quantitative measure of the proximity to the QCP (Supplementary Fig. 7), at which the distinction between the L and T modes vanishes and their intensities become identical.

Figure 3: Identification of the magnetic modes with polarized INS and their comparison to model calculation.
figure 3

a,b, Imaginary part of the dynamic spin susceptibility obtained by normalizing the INS spectra measured at T = 2 K at q = (0,0) (a)(Supplementary Fig. 5) and q = (π, π) (b) (Supplementary Fig. 6) with respect to the orientation factor and the isotropic form factor of the Ru ion (Supplementary Fig. 4). Blue (red) symbols indicate in-plane (out-of-plane) polarized magnetic intensities. Solid symbols show data with the background removed by taking the difference between two spin-flip channels, and open symbols show data from a single spin-flip channel (see Supplementary Information). Solid lines show the calculated spectra, which were convolved using Gaussian functions with 0.19π and 2.5 meV full-width at half-maximum to account for the instrumental momentum and energy resolutions, respectively. The decay process of the L mode into T modes is described in the Supplementary Information. The shaded area indicates the spectral weight of the L mode. The intensities in a,b are in the same arbitrary units. The T′ mode arises from back-folding of the T mode by the magnetic (π,π) scattering. c, A feature consistent with the L mode in the unpolarized data; an H-cut obtained by integrating the data in the range 25 ≤ E ≤ 31 meV, −0.05 ≤ K ≤ 0.05, and 0 ≤ L ≤ 6. A quadratic background has been fitted and subtracted. The solid line is a Gaussian fit to single magnon peaks (T), and the shaded region indicates the intensity not accounted for by the T magnons but instead consistent with the longitudinal mode. Error bars denote one standard deviation.

Having established the existence of the Higgs amplitude mode, we now look at its long-wavelength behaviour. It is at the ordering wavevector where the stability of the Higgs mode critically depends on the dimensionality of the system. In three dimensions, earlier INS studies on a dimerized quantum magnet have established a well-defined Higgs mode4, which was then used to study its critical behaviour across a QCP23,24. In sharp contrast, our in-plane-polarized spectrum measured at q = (π,π) shows only one clear peak for the T mode at approximately 14 meV, followed by a broad magnetic intensity distribution in the energy range 20–50 meV, which is, however, well above the detection limit (Fig. 3b). The Higgs mode has decayed to the extent that a high-flux spin-polarized neutron spectrometer is required to detect its trace—although in retrospect a hint of this feature could be found in the unpolarized data shown in Fig. 3c.

However, it is also known that the response of the Higgs mode depends strongly on the symmetry of the probe being used. Therefore, its rapid decay in the longitudinal susceptibility measured by INS does not necessarily imply its instability in two dimensions. In fact, it has been shown in other two-dimensional systems, such as disordered superconductors3 and superfluids of cold atoms2, that the Higgs mode is clearly visible in the scalar susceptibility with its characteristic ω3 onset in the energy spectrum. Indeed, theory predicts a contrasting behaviour of the Higgs mode in the scalar and longitudinal susceptibilities; in the latter, the Higgs mode quickly loses its coherence by decaying into a pair of Goldstone modes25,26. This results in an infrared divergence in two dimensions and renders the Higgs mode elusive.

Conversely, the INS spectrum at q = (π, π) encodes detailed information on the decay process of the Higgs mode that is not available from other measurements. To model the decay process, we go beyond the harmonic approximation used in the spin-wave theory to include the coupling of the longitudinal mode to the two-magnon continuum (see Supplementary Information). The solid lines in Fig. 3 show the result of the final calculation, which give an excellent description of the data both at q = (0,0) and q = (π, π); the decay process (Fig. 2b) is kinematically restricted away from the ordering wavevector, and the Higgs mode is well identified at q = (0,0). Although continuum structures are generic to low-dimensional quantum magnets (for example, Haldane chains27,28), the present case is distinct because the continuum structure near q = (π, π) is derived from a Higgs mode that exists as a well-defined quasiparticle away from the ordering wavevector.

As mentioned earlier, we have a rather unusual situation where all the transverse modes are massive (gapped), as a result of the orthorhombic symmetry of the crystal structure parameterized by ε, which significantly modifies the Higgs mode decay process at low energies. The transverse gap cuts off the infrared singularity and the spectral weight piles up at non-zero energy. We illustrate this point in Fig. 4 by simulating the change in the longitudinal spectrum as the system approaches the QCP. At q = (π, π), the decay of the Higgs mode into a pair of minimum-energy transverse modes is still the dominant channel, which generates a ‘resonance’ at twice the energy of the gap. This resonance steals much of the spectral weight from the bare longitudinal mode, thus obscuring its spectral signature, especially near the QCP. As the system moves away from the QCP, the longitudinal mode progressively hardens and becomes weaker, and its spectral weight spans a larger energy range. The spectral evolution at q = (0,0) shows this trend with the decay process suppressed; the Higgs mode remains a well-defined excitation even away from the QCP, although its intensity quickly diminishes.

Figure 4: Evolution of the Higgs mode towards the QCP.
figure 4

Imaginary part of the longitudinal susceptibility calculated for several values of J/E, which can be expressed in terms of the condensate density ρ via the relation ρ = 1/2(1 − E/8J). The ρ values of 0.18, 0.23, 0.28, 0.33 approximately correspond to J/E = 0.20, 0.23, 0.28, 0.37, respectively. The dotted lines show the bare susceptibility before taking the decay process into account.

Now that we have established a two-dimensional material system, future studies can reveal further aspects of the Higgs mode. In particular, it is uncertain at this point whether the decay process considered above fully describes its dynamics. In addition to the multimagnon continuum27,28,29,30, other channels such as decays into vortex-like excitations are conceivable in two-dimensional planar magnets and require further investigation. It would be also interesting to compare the results presented herein with the spectra from resonant inelastic X-ray scattering, which can in principle access both the scalar and longitudinal susceptibilities. Finally, it is interesting to note that the Higgs boson in particle physics is detected through its decay products, such as pairs of photons, W and Z bosons, or leptons. The Higgs potential can be determined through the decay rates and branching ratios of these processes, which have been calculated to very high precision. Our study represents the first step towards a parallel development in condensed-matter physics.

Methods

Sample synthesis and characterization.

Single crystals of Ca2RuO4 were grown by the floating zone method with RuO2 self-flux31. The lattice parameters a = 5.409 Å, b = 5.505 Å, and c = 11.9312 Å were determined by X-ray powder diffraction, in good agreement with the parameters reported in the literature8 for the ‘S’ phase with short c-axis lattice parameter. The magnetic ordering temperature TN = 110 K was determined using magnetization measurements in a Quantum Design SQUID-VSM device. Polarized neutron diffraction measurements indicate that most of the array orders in the ‘A-centred’ magnetic structure with magnetic ordering vector Q = (1,0,0)8. The fraction of the sample with ordering vector Q = (0,1,0), that is, ‘B-centred’, is estimated to be less than 5%.

Time-of-flight inelastic neutron scattering.

For the TOF measurements, we co-aligned about 100 single crystals with a total mass of 1.5 g into a mosaic on Al plates. Approximately half of the crystals were rotated 90° about the c-axis from the other half (Supplementary Fig. 2). The in-plane and c-axis mosaicities of the aligned crystal assembly were 3.2° and 2.7°, respectively. The measurements were performed on the ARCS time-of-flight chopper spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratory, Tennessee, USA. The incident neutron energy was 100 meV. The Fermi chopper and T0 chopper frequencies were set to 600 and 90 Hz, respectively, to optimize the neutron flux and energy resolution. The measurements were carried out at T = 5 K. The sample was mounted with the (H,0,L) plane horizontal. The sample was rotated over 90° about the vertical c-axis with a step size of 1°. At each step data were recorded over a deposited proton charge of 3 Coulombs (45 minutes) and then converted into 4D S(Q, ω) using the HORACE software package32 and normalized using a vanadium calibration.

Polarized inelastic neutron scattering.

Preliminary triple-axis measurements, to reproduce the TOF results and determine the feasibility of the polarized experiment, were done on the thermal triple-axis spectrometer PUMA at the FRM-II, Garching, Germany. The measurements were done on crystals from the same batch as the ones used for the TOF experiment. To optimize the flux and energy resolution, double-focused PG (002) and Cu (220) monochromators, for measurements below and above 30 meV, respectively, and a double-focused PG (002) analyser were used, keeping kf = 2.662 Å−1 constant. For the polarized triple-axis measurement we remounted the crystals from the TOF experiment on Si plates and increased the number of crystals to obtain a total sample mass of 3 g. The mosaicity of this sample was 3.2° and 2.6° for in-plane and c-axis, respectively. The experiment was performed on the IN20 three-axis spectrometer at the Institute Laue-Langevin, Grenoble, France. For the XYZ polarization analysis, we used a Heusler (111) monochromator and analyser in combination with Helmholtz coils at the sample position. Throughout the experiment we used a fixed kf = 2.662 Å−1 and performed polarization analysis in energy and H scans at (π,π) and (0,0), keeping L as small as permitted by kinematic constraints. The measurements were carried out at T = 2 K.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. The polarized inelastic neutron scattering data are available at http://doi.ill.fr/10.5291/ILL-DATA.4-01-1431.

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Acknowledgements

We acknowledge financial support from the German Science Foundation (DFG) via the coordinated research programme SFB-TRR80, and from the European Research Council via Advanced Grant 669550 (Com4Com). The experiments at Oak Ridge National Laboratory’s Spallation Neutron Source were sponsored by the Division of Scientific User Facilities, US DOE Office of Basic Energy Sciences. J.C. was supported by GACR (project no. GJ15-14523Y) and by MSMT CR under NPU II project CEITEC 2020 (LQ1601).

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Contributions

M.K., G.H.R. and D.P.C. grew the single crystals. A.J., M.K. and J.P. characterized and co-aligned the crystals. A.J., M.K., J.P. and B.J.K. performed INS experiments and analysed the data. D.L.A., J.T.P. and A.I. supported the INS experiments. G.K. developed the theoretical model. J.C. and B.J.K. performed the numerical calculations. B.J.K. wrote the manuscript with contributions from G.K., J.C., B.K., J.P., A.J. and M.K. and discussions with all authors. B.J.K. and B.K. managed the project.

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Correspondence to B. Keimer or B. J. Kim.

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Jain, A., Krautloher, M., Porras, J. et al. Higgs mode and its decay in a two-dimensional antiferromagnet. Nature Phys 13, 633–637 (2017). https://doi.org/10.1038/nphys4077

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