Abstract
Spontaneous symmetrybreaking quantum phase transitions play an essential role in condensedmatter physics^{1,2,3}. The collective excitations in the brokensymmetry phase near the quantum critical point can be characterized by fluctuations of phase and amplitude of the order parameter. The phase oscillations correspond to the massless Nambu–Goldstone modes whereas the massive amplitude mode, analogous to the Higgs boson in particle physics^{4,5}, is prone to decay into a pair of lowenergy Nambu–Goldstone modes in low dimensions^{2,6,7}. Especially, observation of a Higgs amplitude mode in two dimensions is an outstanding experimental challenge. Here, using inelastic neutron scattering and applying the bondoperator theory, we directly and unambiguously identify the Higgs amplitude mode in a twodimensional S = 1/2 quantum antiferromagnet C_{9}H_{18}N_{2}CuBr_{4} near a quantum critical point in two dimensions. Owing to an anisotropic energy gap, it kinematically prevents such decay and the Higgs amplitude mode acquires an infinite lifetime.
Main
The Higgs boson appears as the amplitude fluctuation of the condensed Higgs field in the Standard Model of particle physics. Since its discovery, there has been much interest in searching for similar Higgs bosonlike particles in condensedmatter physics, such as in superconductors^{8,9,10}, chargedensitywave systems^{11,12}, ultracold bosonic systems^{13}, and antiferromagnets^{14,15,16}. Strictly speaking, only superconductors are analogous to the particle physics from the point that the gauge field (photon) coupling to the condensate acquires its mass (Meissner effect) by means of symmetry breaking (Anderson–Higgs mechanism). In a broad sense, nevertheless, the excitation mode of the amplitude fluctuation of the order parameter is also termed as ‘Higgs amplitude mode’ in condensedmatter physics^{17}. These works provided new insights about the fundamental theories underlying these exotic materials.
The Higgs amplitude mode is expected in the proximity of a quantum critical point (QCP) but can decay into a pair of lowenergy Nambu–Goldstone modes which makes it experimentally difficult to detect. In threedimensional (3D) systems, where the QCP is a Gaussian fixed point, the Higgs amplitude mode is well defined near the QCP^{18}. In contrast, in the twodimensional (2D) case, where the longitudinal susceptibility becomes infrared divergent near the QCP, it has been debated whether the Higgs amplitude mode may not survive or it is still visible in terms of a scalar susceptibility^{6,7,19,20,21,22,23}. Indeed, the Higgs amplitude mode in 2D was evidenced by the scalar response for an ultracold atomic gas near the superfluid to Mottinsulator transition^{13}, although the observed spectral function is heavily damped. Note that when the Nambu–Goldstone modes become gapped, there is no such physical infrared singularity. In the following paper, we will demonstrate observation of a sharp Higgs amplitude mode through the longitudinal response being such a case in an S = 1/2 2D coupledladder compound C_{9}H_{18}N_{2}CuBr_{4} (abbreviated as DLCB) in the vicinity of a QCP in two dimensions.
The quantum S = 1/2 Heisenberg antiferromagnetic twoleg spin ladder is one of the cornerstone models in lowdimensional quantum magnetism^{24,25}. In the onedimensional limit of isolated spin1/2 ladders, the ground state consists of dressed valencebond singlets on each rung of the ladder. Interestingly, the ground state as shown in Fig. 1a can be tuned by the interladder coupling from the quantum disordered (QD) state, through the QCP, to the renormalized classical regime of a longrange magnetically ordered (LRO) state^{26,27}. In the QD phase, the magnetic excitations are triply degenerate magnons with a spin gap energy Δ which vanishes on approach to the QCP. In the LRO phase, the triplet modes evolve into two gapless Nambu–Goldstone modes reflecting spin fluctuations perpendicular to the ordered moment, accompanied by a longitudinal mode (LM) reflecting spin fluctuations along the ordered moment. The latter mode has a gap which grows continuously with the moment and is analogous to the Higgs amplitude mode. Such a LM is usually unstable and decays into a pair of transverse modes, as observed in the S = 1/2 coupled Heisenberg chain compound KCuF_{3}, and has a finite lifetime^{28,29}.
In our previous work^{30,31}, we have shown that the metal–organic compound DLCB is a unique spin ladder material where the interladder coupling is sufficiently strong to drive the system into the magnetically ordered phase. Figure 1b shows the molecular twoleg ladder structure of DLCB. The collinear magnetic structure was determined by the unpolarized neutron diffraction technique and the staggered moments point along the c^{∗} axis () with a reduced moment size of approximately 0.4μ_{B} (ref. 30). A minimal 2D spininteracting model was proposed based on the crystal structure, and the corresponding spin Hamiltonian can be written as:
where l indexes the site of a ladder, i indexes rungs, and 1 and 2 stand for each leg. J_{rung}, J_{leg}, and J_{int} are the nearestneighbour rung, leg, and interladder exchange constants. λ (between 0 and 1) specifies an exchange anisotropy, with λ = 0 or 1 being the limiting case of Ising or Heisenberg interactions. Indeed, the observed magnon dispersions can be described by this model quantitatively^{30}. The observed spin gap energy Δ = 0.32(3) meV is due to a small Ising anisotropy. Further evidence for the twodimensionality is provided by a measurement of the interlayer dispersion (Supplementary Fig. 1b).
Importantly, the analysis of the spin Hamiltonian indicates that DLCB is close to a QCP^{30}, making DLCB an ideal S = 1/2 quantum antiferromagnet to investigate the Higgs amplitude mode. The more detailed examination of this QCP was analysed by tuning the interrung interactions (Supplementary Fig. 2). In the easyaxis case, the U(1) symmetry is not spontaneously broken in the ordered phase, and the two branches of transverse modes (TMs) would have equal gap energies as predicted by the spinwave theory. Therefore, if the LM/Higgs amplitude mode is sufficiently close to TMs, it could become kinematically unable to decay in this region, and thus acquire an infinite lifetime.
Since TMs and the LM could be degenerate within the instrumental resolution in DLCB, we carried out a followup unpolarized inelastic neutronscattering (INS) study in an external magnetic field. In terms of the z component of total spin S, two species of TMs have S_{z} = ±1 whereas the LM has S_{z} = 0. To make S_{z} a good quantum number, the field direction has to be aligned along the easyaxis and a horizontalfield cryomagnet was employed for that purpose. With an applied field μ_{0}H, the Zeeman energy term is gμ_{B}μ_{0}HS_{z}, where g is the Landé gfactor and μ_{B} is the Bohr magneton. Thus, the energy shifts of TMs are expected to be ±gμ_{B}μ_{0}H while the LM should remain unchanged. Consequently, if the splitting is large enough, the LM could be identified by this Zeeman effect.
Figure 2a shows the zerofield backgroundsubtracted energy scan at the magnetic zone centre q = (0.5, −0.5,1.5) and T = 50 mK. The spectral lineshape was modelled by superposition of two doubleLorentzian damped harmonicoscillator (DHO) models convolved with the instrumental resolution function^{32,33}. The best fit yields the gap energies of TMs (S_{z} = ±1) and the LM (S_{z} = 0) as Δ_{TM} = 0.34(3) meV and Δ_{LM} = 0.48(3) meV, respectively. At μ_{0}H = 1 T in Fig. 2b, TMs (S_{z} = ±1) are split into two branches. The observed quasielastic neutron scattering hinders observation of TM (S_{z} = 1) at μ_{0}H = 1.5 T in Fig. 2c. At μ_{0}H = 2 T in Fig. 2d, TM (S_{z} = 1) is merged into the elastic line while the LM becomes clearly visible and well distinguished from TM (S_{z} = −1).Figure 2e summarizes the measured field dependences of Δ_{TMs}(S_{z} = ±1) and Δ_{LM} (S_{z} = 0). Δ_{TMs} (S_{z} = ±1) as a function of field agree well with the Zeeman spectral splitting ±gμ_{B}μ_{0}H using g = 2.15 and the LM is indeed fieldindependent within the experimental uncertainties. The small discrepancy between data and calculations at 2 T is due to the occurrence of a spinflop transition (Supplementary Fig. 3). The analysis also indicates that the peak profile of the Higgs amplitude mode in each field is limited by the instrumental resolution within experimental uncertainty, as shown in Fig. 2f. In other words, decay of the LM/Higgs amplitude mode (S_{z} = 0) into a TM (S_{z} = 1) and another TM (S_{z} = −1) is forbidden by the kinematic conditions. However, because of the limited access of the reciprocal space using a horizontalfield cryomagnet, we could not map out the excitation spectra in the Brillouin zone (BZ).
Another straightforward way to unambiguously determine the nature of spin polarization of magnetic excitation spectra is by the polarized neutronscattering method. In general, however, it is fairly challenging to carry out the polarized INS because of significant loss of neutron intensity due to neutron polarization arrangements compared with unpolarized neutron arrangements. To compensate for that, the polarized neutron data were collected using a highflux cold neutron spectrometer. The polarization analysis was performed using the recently developed capability of wideangle ^{3}He spin filters^{34}. Figure 3a shows the θscans of the nuclear Bragg reflection (0,1, −1) at T = 1.4 K with the nonspinflip (NSF) and spinflip (SF) configurations, respectively. The flipping ratio F can be calculated as I_{NSF}/I_{SF} ≃ 43(1), which corresponds to an overall polarization efficiency of (F −1)/(F + 1) = 0.95. Furthermore, Fig. 3b shows the backgroundsubtracted θscans of the magnetic Bragg reflection (0.5,0.5, −0.5) at T = 1.4 K with the NSF and SF configurations, respectively. The scattering intensity in the NSF channel dominates over that in the SF channel, suggesting that the outofplane spin component is dominant, and thus confirming the determined orientation of the staggered moments.
With the high efficiency of neutron spin filters firmly established and the magnetic spin structure in DLCB well determined, we proceed to investigate the spin dynamics using polarized neutrons. The experiment was designed in such a way that TMs and the LM can be separated from each other in the SF and NSF configurations, respectively (for further details, see Methods). For the comparison purposes, Fig. 3c shows the backgroundsubtracted energy scan at q = (0.5,0.5, −0.5) and T = 1.4 K using unpolarized neutrons. Figure 3d, e shows the same energy scans with the SF and NSF configurations, respectively. And data were fitted to the same DHO model convolved with the instrumental resolution function. The spin gap energies of TMs and the LM/Higgs amplitude mode were obtained as Δ_{TMs} = 0.33(3) meV and Δ_{LM} = 0.48(3) meV, in excellent agreement with the results from the Zeeman effect. The spectral weight ratio between them is approximately 2.6:1.
After confirming the feasibility of such a challenging experiment, we managed to map out the excitation spectra in the BZ. Serving as a reference, Fig. 4a, b shows the falsecolour maps of the spinwave spectra along two highsymmetry directions in the reciprocallattice space using unpolarized neutrons. With the SF configuration, as expected, the magnetic excitation spectra in Fig. 4c, d are in excellent agreement with the calculations using SPINW^{35} in the linear spinwave theory (LSWT) approximation, and thus was confirmed as TMs. The Hamiltonian parameters of J_{leg} = 0.64 meV, J_{rung} = 0.70 meV, J_{int} = 0.19 meV, and λ = 0.95 provide the best agreement with experimental data. The modelled dispersion curves in Fig. 4c, d are close to the quantitative calculations using the highorder series expansions^{30}.
Since the onemagnon excitation of the LM is not predicted in LSWT, to analyse the experimental data with the NSF configuration, we employed the bondoperator theory (BOT) for the description of the lowenergy excitations in the vicinity of the QCP on DLCB. The detailed description of the harmonic BOT can be found in refs 36,37,38,39. The ordered moment is estimated as 38.5% of the saturation value and is consistent with the experimental value of 37(5)%. The exchange interaction parameters were extracted as J_{leg} = 0.57 meV, J_{rung} = 1.21 meV, J_{int} = 0.11 meV, and λ = 0.95 from the best fit. Note that the extracted parameters correspond to the renormalized parameters within the scope of harmonic BOT. Since DLCB is a weakly interacting ladder system, J_{rung} and J_{leg} are strongly renormalized (the former and latter are enhanced and reduced, respectively)^{40,41}. Therefore, J_{rung} is markedly larger than J_{leg}. The solid green lines in Fig. 4e, f shows the calculated LM (Higgs amplitude mode) with a gap energy at 0.48 meV. We notice that the BOT calculation in Fig. 4f increases monotonically and deviates from the experimental data at the zone boundary, which may originate from the fact that the BOT is a meanfield treatment and application of this technique to the lowdimensional system could be limited. For the lowenergy excitations, nevertheless, it works well in both QD and LRO phases. For instance, as shown in Fig. 2e, the agreement of the field dependence of the Zeeman energy term between the experimental data and the calculations by BOT is excellent.
Figure 4e, f shows the calculated excitation spectra by BOT, which reproduce the experimental data qualitatively. Thus, our conclusion that the nature of spin excitation observed in the NSF configuration is due to spin fluctuation along the staggered moment direction is fairly convincing. Note that the Higgs amplitude mode in DLCB is distinctly different from the longitudinal excitations in the S = 1/2 2D Heisenberg squarelattice (HSL) antiferromagnet Cu(DCOO)_{2} ⋅ 4D_{2}O (CFTD)^{42,43} because the S = 1/2 2D HSL antiferromagnet is far from the QCP and the observed longitudinal spectra in CFTD originate from the twomagnon continuum. Consequently, the spectral lineshapes are broadened. Recent theoretical work^{44,45} suggests that there is a prominent resonance, which was proposed as a Higgs resonance with finite lifetime, inside the continuum due to the attractive magnon–magnon interaction. Moreover, the grey lines in Fig. 4e, f are the calculated lower boundary of the twomagnon continuum in DLCB, which lies well above the Higgs amplitude mode. Hence, the spontaneous decay of the Higgs amplitude mode into a pair of TMs is forbidden due to violation of the laws of energy and momentum conservation, as evidenced by the wavevector dependence of the intrinsic linewidth Γ, which is limited by the instrumental resolution, as shown in Fig. 4g. It is worth pointing out that the Higgs amplitude mode is also evident from the excitation spectra with an external magnetic field applied perpendicularly to the easyaxis^{31}. In that case, the Higgs amplitude mode is stable at low fields and the decay occurs beyond the crossover with the lower boundary of the twomagnon continuum at ∼1.5 T.
In summary, the unique ability of neutron scattering to probe the spin polarization of dynamic spin paircorrelation functions allows one to distinguish the Higgs amplitude mode from the dominant transverse Nambu–Goldstone modes in the twodimensional S = 1/2 antiferromagnet DLCB. The transverse modes have a finite excitation gap energy due to a weak Ising anisotropy. The opening of the gap kinematically prevents the decay process from the Higgs amplitude mode to a pair of transverse modes. This leads to the long lifetime for the Higgs amplitude mode and makes it observable near the quantum critical point in two dimensions.
Note added in proof: Recently we became aware of an INS work^{46} that reports the Higgs amplitude mode in a 2D antiferromagnet Ca_{2}RuO_{4}.
Methods
Single crystal growth.
Deuterated single crystals were grown using a solution method^{47}. An aqueous solution containing a 1:1:1 ratio of deuterated (DMA)Br, (35DMP)Br, where DMA^{+} is the dimethylammonium cation and 35DMP^{+} is the 3,5dimethylpyridinium cation, and copper(II) bromide was allowed to evaporate over several days in a closed desiccator. A few drops of DBr were added to the solution to avoid hydrolysis of the Cu(II) ion.
Neutronscattering measurements.
Unpolarized neutronscattering measurements using a horizontalfield superconducting magnet with a dilution fridge insert were carried out on a cold neutron tripleaxis spectrometer (FLEXX)^{48} at HelmholtzZentrum Berlin. The sample consists of two coaligned deuterated single crystals with a total mass of 2.5 g and a 1.0° mosaic spread and was oriented in the (H − HL) reciprocallattice plane. Unpolarized inelastic neutronscattering measurements using a standard heliumflow cryostat were carried out on a cold neutron tripleaxis spectrometer (CTAX) at the High Flux Isotope Reactor, Oak Ridge National Laboratory. Polarized neutronscattering measurements using a standard heliumflow cryostat were performed on a highintensity multiaxis crystal spectrometer (MACS)^{49} at the NIST Center for Neutron Research. The peak flux at the sample position is approximately 5 × 10^{8} neutrons cm^{−2} s^{−1}. The sample assembly with three coaligned deuterated single crystals (a total mass of 3.5 g and a 1.0° mosaic spread) was oriented in the (HK − K) reciprocallattice plane. In all experiments, the final neutron energy was fixed at 3.0 meV and the energy resolutions of FLEXX and MACS at the elastic line are 0.10 meV and 0.15 meV, respectively. The background was determined at T = 15 K under the same instrument configurations, and has been subtracted.
Polarized neutron measurements.
In the experimental setup, both incident and outgoing neutron beams were polarized by nuclear spin polarized ^{3}He gas cells. NMRbased inversion of the ^{3}He polarization in the polarizer cell allows polarization of the incident beam parallel or antiparallel to the vertical axis at will. The overall transmission at the beginning of a polarized neutron setup (each run lasts about two days) is approximately 11% and reduced to approximately 5% before the ^{3}He gas cells change out. The initial flipping ratio F is about 43(1), indicating that the product of the polarizing efficiencies of the NSF cells PA = (F − 1)/(F + 1) is 95%. Typically, after the twoday operation, F is reduced to 20(1) and PA becomes 91%. Since F was always above 20, the polarization leakage effect is as small as 1/F (≤5%). To account for decay of the ^{3}He polarization and neutron transmission with time, the polarized neutron data were corrected by ^{3}He efficiency correction software as described in Supplementary Information.
The principles for polarized neutron scattering can be summarized as follows: phonons and structural scattering are seen in the NSF channel; components of spin fluctuations parallel to the direction of neutron polarization are seen in the NSF channel; components of spin fluctuations perpendicular to the direction of the neutron polarization are seen in the SF channel^{50}.
The sample was aligned in such a way that the [0,1,1] direction in the real space is vertical. Thus, the angle α between the vertical polarization and staggered moment direction is 17.6°. In this geometry with the NSF configuration, the large fraction (cos^{2}α ≃ 91%) is due to spin fluctuations along the direction of the staggered moment (LM/the Higgs amplitude mode), while the remaining 9% corresponds to the spin fluctuations perpendicular to the staggered moment (TMs) and is negligible. In contrast, in the SF configuration, TMs have accounted for 91% of the contribution and the Higgs amplitude mode is negligible. Therefore, by employing polarized INS, we are able to separate the Higgs amplitude mode from the TMs in the magnetic excitation spectra. Polarized neutron measurements cover half of Brillouin zone due to the fact that the polarizing efficiency becomes either significantly reduced or not available for small neutronscattering angles.
Data analysis.
The spectral lineshape in Figs 2a–d and 3d–e was fitted to the following doubleLorentzian damped harmonicoscillator model
where k_{B} is the Boltzmann constant, Δ is the peak position and Γ is the resolutioncorrected intrinsic excitation linewidth, that is, halfwidth at halfmaximum (HWHM) and convolved with the instrumental resolution function. The experimental resolution was calculated using the Reslib software^{51}.
For the falsecolour maps in Fig. 4a–f, data were obtained by combining a series of constantq scans along either the (H0.5 − 0.5) or (0.5 K − K) direction with a step size of 0.05 r.l.u. and simulations were convolved with the instrumental resolution function where the neutron polarization factor and the magnetic form factor for Cu^{2+} were included.
A detailed description about the determination of the lower boundary of twomagnon continuum can be found in ref. 31.
Measurement of the interlayer dispersion.
Additional unpolarized inelastic neutronscattering measurements were performed at MACS to investigate the possible interaction between the twodimensional layers in DLCB. For that purpose, a single crystal (∼2 g) with a 1.0° mosaic spread was aligned in the (HKH) reciprocallattice plane. Supplementary Fig. 1a shows the measured excitation spectrum along the leg direction and the observed dispersions are fully consistent with the calculations by LSWT as described in the main text. Along the interlayer direction, the dispersion is absent within the instrumental resolution (Supplementary Fig. 1b), indicating that DLCB is an excellent 2D spininteracting system.
Analysis of the energy gap by tuning the interrung interactions.
We assume that J_{rung} is fixed and the interrung interactions vary as J_{leg}^{∗} = RJ_{leg} and J_{int}^{∗} = RJ_{int}, where values of J_{rung}, J_{leg} and J_{int} are obtained as described in the main text. Supplementary Fig. 2 summarizes the calculations by BOT of the evolution of the spin gap energy as a function of the enhancement factor R. At small R, due to the Ising anisotropy, the triplet spin gap energy splits into a singlet (S_{z} = 0) and a doublet (S_{z} = ±1). When R initially increases, the spin gap energies of both the singlet and the doublet decrease. Spin gap of the singlet closes at the QCP while the doublet remains gapped. When R further increases, the softened singlet mode acquires a spin gap again and becomes the Higgs amplitude mode. The analysis indicates that the QCP is located at R_{c} = 0.923, which is close to the case in DLCB (R = 1) and thus confirms our conclusion in the main text. For R < R_{c}, the quantum disordered phase is stabilized, while the longrange ordered phase is stabilized for R > R_{c}. Calculation by BOT of emergence of the staggered moment size as a function of R is also shown in Supplementary Fig. 2.
The spinflop transition.
In DLCB, an application of a magnetic field along the easyaxis direction would lead to the spin reorientation, that is, spinflop transition. In the spinflop phase, S_{z} is no longer a good quantum number. To find out the critical field where the spin reorientation occurs, we measured the field dependence of several magnetic reflections, as shown in Supplementary Fig. 3, at FLEXX using a horizontalfield cryomagnet. The integratedpeak intensities are almost fieldindependent at low fields, then start to increase above 1.7 T, and finally become saturated above 2.5 T after the spin is flopped for the (0.5, −0.5, −1.5) and (1.5, −1.5, 2.5) magnetic reflections. From the fact that neutron scattering probes the components of spin fluctuation perpendicular to the transferred wavevector, the orientation of the ordered moment in the spinflop phase is 90 ° out of the horizontal plane, with the axis of rotation approximately along q = (1.5, −1.5,0.5). Overall, the above results confirm that in DLCB, when the field direction is aligned parallel to the easyaxis direction, S_{z} remains as a good quantum number at least up to 1.7 T.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.
Additional Information
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Acknowledgements
T.H. thanks C. D. Batista for the insightful discussion, Q. Ye for the initial neutron polarization setup and R. Erwin for the development of ^{3}He efficiency correction software. T.H. also thanks D. L. Q. Castro, Z. L. Lu and Z. Hüsges for the assistance during the experiment. One of the authors (M.M.) is supported by JSPS KAKENHI Grant Number 26400332. A portion of this research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. Access to MACS was provided by the Center for High Resolution Neutron Scattering, a partnership between NIST and NSF under Agreement No. DMR1508249.
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T.H. conceived the project. F.F.A. and M.M.T. prepared the samples. The polarization apparatus and corrections were provided by W.C., T.R.G. and S.W. T.H., Y.Q., H.A., R.T.P. and B.K. performed the neutronscattering measurements. T.H., M.M., D.A.T., S.E.D., K.C. and K.P.S. analysed the data. All authors contributed to writing of the manuscript.
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Correspondence to Tao Hong.
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Hong, T., Matsumoto, M., Qiu, Y. et al. Higgs amplitude mode in a twodimensional quantum antiferromagnet near the quantum critical point. Nature Phys 13, 638–642 (2017). https://doi.org/10.1038/nphys4182
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