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Higgs Amplitude Mode in a Two-Dimensional Quantum Antiferromagnet Near the Quantum Critical Point

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Higgs Amplitude Mode in a Two-Dimensional Quantum Antiferromagnet Near the Quantum Critical Point LETTERS PUBLISHED ONLINE: 4 JULY 2017 | DOI: 10.1038/NPHYS4182 Higgs amplitude mode in a two-dimensional quantum antiferromagnet near the quantum critical point Tao Hong1*, Masashige Matsumoto2, Yiming Qiu3, Wangchun Chen3,4, Thomas R. Gentile3, Shannon Watson3, Firas F. Awwadi5, Mark M. Turnbull6, Sachith E. Dissanayake1, Harish Agrawal7, Rasmus Toft-Petersen8, Bastian Klemke8, Kris Coester9, Kai P. Schmidt10 and David A. Tennant1 Spontaneous symmetry-breaking quantum phase transitions (2D) case, where the longitudinal susceptibility becomes infrared play an essential role in condensed-matter physics1–3. The divergent near the QCP, it has been debated whether the Higgs collective excitations in the broken-symmetry phase near the amplitude mode may not survive or it is still visible in terms of quantum critical point can be characterized by fluctuations a scalar susceptibility6,7,19–23. Indeed, the Higgs amplitude mode in of phase and amplitude of the order parameter. The phase 2D was evidenced by the scalar response for an ultracold atomic oscillations correspond to the massless Nambu–Goldstone gas near the superfluid to Mott-insulator transition13, although modes whereas the massive amplitude mode, analogous to the the observed spectral function is heavily damped. Note that Higgs boson in particle physics4,5, is prone to decay into a pair when the Nambu–Goldstone modes become gapped, there is no of low-energy Nambu–Goldstone modes in low dimensions2,6,7. such physical infrared singularity. In the following paper, we will Especially, observation of a Higgs amplitude mode in two demonstrate observation of a sharp Higgs amplitude mode through dimensions is an outstanding experimental challenge. Here, the longitudinal response being such a case in an S D 1=2 2D using inelastic neutron scattering and applying the bond- coupled-ladder compound C9H18N2CuBr4 (abbreviated as DLCB) operator theory, we directly and unambiguously identify the in the vicinity of a QCP in two dimensions. Higgs amplitude mode in a two-dimensional S D 1/2 quantum The quantum S D 1=2 Heisenberg antiferromagnetic two-leg antiferromagnet C9H18N2CuBr4 near a quantum critical point spin ladder is one of the cornerstone models in low-dimensional in two dimensions. Owing to an anisotropic energy gap, it quantum magnetism24,25. In the one-dimensional limit of isolated kinematically prevents such decay and the Higgs amplitude spin-1/2 ladders, the ground state consists of dressed valence-bond mode acquires an infinite lifetime. singlets on each rung of the ladder. Interestingly, the ground state The Higgs boson appears as the amplitude fluctuation of the con- as shown in Fig. 1a can be tuned by the inter-ladder coupling densed Higgs field in the Standard Model of particle physics. Since from the quantum disordered (QD) state, through the QCP, to the its discovery, there has been much interest in searching for similar renormalized classical regime of a long-range magnetically ordered Higgs boson-like particles in condensed-matter physics, such as (LRO) state26,27. In the QD phase, the magnetic excitations are triply in superconductors8–10, charge-density-wave systems11,12, ultracold degenerate magnons with a spin gap energy ∆ which vanishes bosonic systems13, and antiferromagnets14–16. Strictly speaking, only on approach to the QCP. In the LRO phase, the triplet modes superconductors are analogous to the particle physics from the evolve into two gapless Nambu–Goldstone modes reflecting spin point that the gauge field (photon) coupling to the condensate fluctuations perpendicular to the ordered moment, accompanied acquires its mass (Meissner effect) by means of symmetry breaking by a longitudinal mode (LM) reflecting spin fluctuations along (Anderson–Higgs mechanism). In a broad sense, nevertheless, the the ordered moment. The latter mode has a gap which grows excitation mode of the amplitude fluctuation of the order parameter continuously with the moment and is analogous to the Higgs is also termed as `Higgs amplitude mode' in condensed-matter amplitude mode. Such a LM is usually unstable and decays into physics17. These works provided new insights about the fundamental a pair of transverse modes, as observed in the S D 1=2 coupled 28,29 theories underlying these exotic materials. Heisenberg chain compound KCuF3, and has a finite lifetime . The Higgs amplitude mode is expected in the proximity of a In our previous work30,31, we have shown that the metal–organic quantum critical point (QCP) but can decay into a pair of low- compound DLCB is a unique spin ladder material where the inter- energy Nambu–Goldstone modes which makes it experimentally ladder coupling is sufficiently strong to drive the system into the difficult to detect. In three-dimensional (3D) systems, where the magnetically ordered phase. Figure 1b shows the molecular two- QCP is a Gaussian fixed point, the Higgs amplitude mode is leg ladder structure of DLCB. The collinear magnetic structure was well defined near the QCP18. In contrast, in the two-dimensional determined by the unpolarized neutron diffraction technique and 1Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA. 2Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan. 3National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA. 4Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA. 5Department of Chemistry, The University of Jordan, Amman 11942, Jordan. 6Carlson School of Chemistry and Biochemistry, Clark University, Worcester, Massachusetts 01610, USA. 7Instrument and Source Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA. 8Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin, Germany. 9Lehrstuhl für Theoretische Physik I, TU Dortmund, D-44221 Dortmund, Germany. 10Lehrstuhl für Theoretische Physik I, Staudtstrasse 7, Universität Erlangen-Nürnberg, D-91058, Germany. *e-mail: [email protected] 638 NATURE PHYSICS | VOL 13 | JULY 2017 | www.nature.com/naturephysics © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS4182 LETTERS a ↑↓〉 ↓↑〉 phase, and the two branches of transverse modes (TMs) would have = 1 (| − | ) √2 equal gap energies as predicted by the spin-wave theory. Therefore, J if the LM=Higgs amplitude mode is sufficiently close to TMs, it J αJ J J could become kinematically unable to decay in this region, and thus acquire an infinite lifetime. QD LRO Since TMs and the LM could be degenerate within the instru- mental resolution in DLCB, we carried out a follow-up unpolarized inelastic neutron-scattering (INS) study in an external magnetic field. In terms of the z component of total spin S, two species of TMs α D± D α have Sz 1 whereas the LM has Sz 0. To make Sz a good quan- 01c tum number, the field direction has to be aligned along the easy-axis b Jleg and a horizontal-field cryomagnet was employed for that purpose. Cu With an applied field µ0H, the Zeeman energy term is gµBµ0HSz , Br where g is the Landé g-factor and µB is the Bohr magneton. Thus, Jrung N ± C the energy shifts of TMs are expected to be gµBµ0H while the J H LM should remain unchanged. Consequently, if the splitting is large int enough, the LM could be identified by this Zeeman effect. Figure 2a shows the zero-field background-subtracted en- ergy scan at the magnetic zone centre q D .0.5, −0.5, 1.5/ and a b T D50 mK. The spectral lineshape was modelled by superposi- c tion of two double-Lorentzian damped harmonic-oscillator (DHO) models convolved with the instrumental resolution function32,33. | The two-dimensional spin-1/2 coupled two-leg spin ladder Figure 1 The best fit yields the gap energies of TMs (Sz D ±1) and the antiferromagnet. a, Schematic diagram of coupled two-leg square spin LM (Sz D0) as ∆TM D 0.34.3/ meV and ∆LM D 0.48.3/ meV, re- ladders, where the ground state can be tuned by the inter-ladder coupling spectively. At µ0H D 1 T in Fig. 2b, TMs (Sz D ±1) are split into αJ from the quantum disordered (QD) phase, through a quantum critical two branches. The observed quasielastic neutron scattering hinders point αc in two dimensions, to the long-range magnetically ordered (LRO) observation of TM (Sz D1) at µ0H D1.5 T in Fig. 2c. At µ0H D2 T phase. Blue circles stand for the spin-1/2 magnetic ions. The ellipses in Fig. 2d, TM (Sz D 1) is merged into the elastic line while represent a singlet valence bond of spins. b, The molecular two-leg ladder the LM becomes clearly visible and well distinguished from TM structure with the leg direction along the crystalline b axis and a (Sz D−1). Figure 2e summarizes the measured field dependences two-dimensional model for the magnetic interactions in C9H18N2CuBr4. of ∆TMs(Sz D±1) and ∆LM (Sz D0). ∆TMs (Sz D±1) as a function of Pink, red, and yellow bonds indicate the nearest-neighbour leg, rung, and field agree well with the Zeeman spectral splitting ±gµBµ0H using inter-ladder exchange constants, respectively. g D 2.15 and the LM is indeed field-independent within the ex- perimental uncertainties. The small discrepancy between data and the staggered moments point along the c∗ axis (≡z O) with a reduced calculations at 2 T is due to the occurrence of a spin-flop transition moment size of approximately 0.4µB (ref. 30). A minimal 2D spin- (Supplementary Fig. 3). The analysis also indicates that the peak interacting model was proposed based on the crystal structure, and profile of the Higgs amplitude mode in each field is limited by the the corresponding spin Hamiltonian can be written as: instrumental resolution within experimental uncertainty, as shown in Fig. 2f. In other words, decay of the LM=Higgs amplitude mode X H D J .λSx Sx C λSy Sy CSz Sz / (S D0) into a TM (S D1) and another TM (S D−1) is forbidden rung l1,i l2,i l1,i l2,i l1,i l2,i z z z l,i by the kinematic conditions.
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