Magnetic Excitations and Continuum of a Field-Induced Quantum Spin Liquid in $\alpha$-RuCl$_3$

TL;DR

This study uses terahertz spectroscopy to explore magnetic excitations in $ ext{α-RuCl}_3$, revealing a field-induced quantum spin liquid state with characteristic excitations and a broad continuum above a critical magnetic field.

Contribution

It provides detailed experimental evidence of magnetic excitations and continuum evolution in $ ext{α-RuCl}_3$ under magnetic field, elucidating the nature of the field-induced quantum spin liquid.

Findings

Development of a magnetic continuum below 62K

Spectral weight transfer at magnetic ordering temperature

Emergence of a broad continuum above 7T indicating a quantum spin liquid

Abstract

We report on terahertz spectroscopy of quantum spin dynamics in $\alpha$-RuCl$_3$, a system proximate to the Kitaev honeycomb model, as a function of temperature and magnetic field. An extended magnetic continuum develops below the structural phase transition at $T_{s2}=62$K. With the onset of a long-range magnetic order at $T_N=6.5$K, spectral weight is transferred to a well-defined magnetic excitation at $\hbar \omega_1 = 2.48$meV, which is accompanied by a higher-energy band at $\hbar \omega_2 = 6.48$meV. Both excitations soften in magnetic field, signaling a quantum phase transition at $B_c=7$T where we find a broad continuum dominating the dynamical response. Above $B_c$, the long-range order is suppressed, and on top of the continuum, various emergent magnetic excitations evolve. These excitations follow clear selection rules and exhibit distinct field dependencies, characterizing the dynamical properties of the field-induced quantum spin liquid.

Full text

Quantum spin liquids (QSLs) are exotic states of matter in which quantum fluctuations prevent conventional magnetic long-range order even at the lowest temperatures. The ground states of QSLs are highly-entangled and albeit disordered, can exhibit well-defined quasiparticles which are non-local and fractionalized Balents17 ; Balents10 . The Kitaev honeycomb model Kitaev06 is a representative example for the existence of a QSL, which hosts fractionalized Majorana fermions and flux excitations. Moreover, bound states of the fractional excitations and excitations of non-Abelian type can occur when the time-reversal symmetry is broken Trebst14 ; Knolle15 ; ThVojta17 , e.g, by applying an external magnetic field. Also, as this model is exactly solvable and thus provides quantitative understanding for the ground state as well as for the dynamical response of the QSL Kitaev06 ; Knolle14 , significant experimental efforts have been motivated to realize the QSL and to search for its unconventional excitations.

Strong spin-orbit couplings together with a proper crystal electric field are important ingredients for realizing the Kitaev honeycomb model in magnetic insulators Jackeli09 . Based on $5d$ or $4d$ ions with strong spin-orbit coupling, a Mott insulator with an effective spin-1/2 localized on a honeycomb lattice has been realized in the iridates Singh12 and recently in $\alpha$-RuCl3 Plumb14 ; Kubota15 ; Johnson15 ; Cao16 ; Park16 . In $\alpha$-RuCl3, besides the significant Kitaev term Sandilands15 ; Do17 that accounts for the bond-dependent Ising exchanges, other interactions, especially the Heisenberg and off-diagonal exchange interactions, have been suggested to be responsible for the magnetic zigzag order below $T_{N}=6.5$ K and for the observed sharp spin-wave-like excitations Banerjee16NM ; Banerjee16 ; Ran17 ; Winter16 ; Winter17 ; Janssen16 .

To realize a QSL in $\alpha$-RuCl3, obviously the long-range zigzag order has to be suppressed, e.g. by tuning the external parameters, such as pressure LilingSun17 ; WeiqiangYu17 or magnetic field Baek17 ; Hentrich17 ; Sears17 ; Wolter17 ; Zheng17 ; Kubota15 . When the external magnetic field is applied in the crystallographic $ab$ plane of $\alpha$-RuCl3, the magnetic zigzag order is suppressed at about $B_{c}=7$ T, with the emergence of a field-induced disordered phase, a putative QSL. However, whether the elementary spin excitations are gapped or gapless, which is important for identifying a QSL Balents17 , is still under debate for the field-induced disordered phase in $\alpha$-RuCl3 Baek17 ; Zheng17 , since a direct probe of the magnetic excitations in the disordered phase is missing. Moreover, the existence of exotic quasiparticles is yet to be proven for the time-reversal symmetry-broken phase in the applied magnetic field.

Here, by performing terahertz (THz) spectroscopy as a function of temperature and magnetic field, we are able to directly reveal the quantum spin dynamics in the different phases of $\alpha$-RuCl3. We resolve a magnetic continuum at zero field, which develops below the structural phase transition at $T_{s2}=62$ K. Across the magnetic ordering temperature $T_{N}=6.5$ K, the spectral weight of the continuum is partially shifted to two magnetic excitations. As a function of magnetic field applied in the crystallographic $ab$ plane, we provide spectroscopic evidence for a field-induced quantum phase transition at $B_{c}=7$ T. Approaching $B_{c}$ from below, the two magnetic excitations soften and we observe a remarkably broad continuum just at the critical field. Above $B_{c}$, various magnetic excitations show up signaling the opening of a spin gap. The hierarchy and distinct field dependencies of the emergent excitations characterize the dynamical properties, which could be viewed as a signature of many-body interactions and/or bound states of fractionalized excitations Trebst14 ; Knolle15 ; ThVojta17 in the field-induced QSL. Our experimental results also reveal a clear polarization dependence of these excitations, establishing a comprehensive characterization of the quantum spin dynamics of the QSL.

High-quality single crystals of $\alpha$-RuCl3 were grown using a vacuum sublimation method Do17 and have been characterized by various techniques Park16 ; Wolter17 ; Reschke17 ; Choi17 . The samples for the optical experiments have a typical $ab$ surface of $5\times 3$ mm2 and a thickness of 1 mm. Time-domain THz transmission measurements were performed with the THz wave vector $\mathbf{k}$ perpendicular to the crystallographic $ab$ plane, for 4–300 K in the spectral range 0.1–3.3 THz using a TPS Spectra 3000 spectrometer (TeraView Ltd.). Time domain signals were obtained for references (empty apertures) and samples, from which the power spectra could be evaluated via Fourier transformation. Field dependent THz absorption experiments were carried out in Voigt configuration, i.e. $\mathbf{k}\perp\mathbf{B}$, with a magneto-optic cryostat equipped with a 17 T superconducting magnet. The absorption spectra were recorded using a Sciencetech SPS200 Martin-Puplett type spectrometer with a 0.3 K bolometer and a rotatable polarizer in front of the sample.

THz response of $\alpha$-RuCl3 exhibits a very anomalous temperature dependence. Figure 1 shows the absorption coefficient as a function of temperature measured by heating and cooling the sample, for the photon energies $\hbar\omega_{1}=2.48$ meV and $\hbar\omega_{2}=6.38$ meV [see Fig. 2(a)]. For both frequencies we observe a markedly strong hysteresis between the heating and cooling curves for the temperature range from $T_{s2}=62$ K to $T_{s1}=166$ K. This confirms the structural transition between the high-temperature monoclinic and the low-temperature rhombohedral phases Park16 ; Baek17 ; Choi17 ; Reschke17 , and also indicates that above $T_{s2}$ the THz response mainly reflects strong structural fluctuations.

THz spectroscopy is sensitive to dynamical response in the vicinity of the $\Gamma$ point, where inelastic neutron scattering experiments have revealed a broad continuum of magnetic quasiparticles up to 120 K Banerjee16 ; Do17 . The evolution of this continuum with temperature was compared to an expected temperature dependence of the continuum of fractional Majorana fermions in the Kitaev model Do17 ; Yoshitake16 . To avoid the complication of the structural fluctuations above $T_{s2}$, we show in Fig. 2(b) the absorption spectra below $T_{s2}$ with the spectrum of 60 K taken as a reference. It has been indicated by other experimental techniques that below this temperature, an evident magnetic-field dependent behavior develops Baek17 ; Hentrich17 . Compared to the inelastic neutron scattering results, the THz spectroscopy reveals here a similar broad continuum extending over the spectral range up to 9 meV. With decreasing temperature the continuum develops rapidly and exhibits a broad maximum with the maximum position shifting to lower frequency. At 6 K (below $T_{N}$), the continuum is slightly reduced and the spectral weight is transferred to an excitation appearing at about 2.5 meV. The transfer of spectral weight to lower energies becomes more evident at lower temperatures. As seen from the 4.5 K spectrum, a very sharp peak shows up at 2.48 meV, while the continuum with maximum at 5–6 meV is strongly suppressed. This behavior is also clearly displayed in Fig. 1. While the absorption coefficient at 2.48 meV increases below $T_{N}$, it drops slightly around 6.38 meV.

To unambiguously identify the magnetic excitations, we calculate the ratio of transmission as measured at 4.5 and 7 K, below and right above the magnetic ordering temperature. As shown in Fig. 2(a), the ratio of transmission exhibits a sharp dip at 2.48 meV (mode M1) and a shallow minimum around 6.38 meV (band M2). Both modes, with frequencies comparable to the magnetic excitations determined in the inelastic neutron scattering experiments Banerjee16 ; Do17 , have been captured by recent exact-diagnolization calculations Winter17 of a model guided by ab-initio studies Winter16 ; Yadav16 ; Li16 ; Xiang16 , where in addition to the Kitaev interactions, the off-diagonal couplings are included together with the nearest- and third-neighbour Heisenberg exchange interactions. However, the nature of the two modes remains unsettled. The lower-lying mode M1 with a sharp absorption line, ascribed to the antiferromagnetic resonance in Ref. Little17 , corresponds to the single-magnon branch at the $\Gamma$ point in the calculations Winter17 . In contrast, the band M2 at higher energies is rather broad and weak. This mode may correspond to the higher-energy mode with small intensity in the linear spin-wave theory Winter17 . Since M1 and M2 arise below the antiferromagnetic phase transition, they should characterize the dynamical response of the zig-zag order below $T_{N}$. Above $T_{N}$, the modes M1 and M2 transfer their spectral weight to the continuum as observed in the THz spectra [Fig. 2(b)]. The temperature dependence of the continuum is consistent with the results of the neutron scattering study Do17 ; Banerjee16 , which has been discussed in the context of the Kitaev honeycomb model Yoshitake16 ; Banerjee16 . Alternatively, since the non-Kitaev terms are also important for $\alpha$-RuCl3 Ran17 ; Winter17 , one may speculate that this experimentally resolved continuum could be a two-magnon continuum, which can be excited due to the anharmonic terms of the magnon interactions Winter17 .

Figures 3(a) and 3(b) show the absorption spectra as a function of an external magnetic field applied in the $ab$ plane, for the THz magnetic field $\mathbf{h}^{\omega}\parallel\mathbf{B}$ and $\mathbf{h}^{\omega}\perp\mathbf{B}$, respectively. The spectra at 10 K in zero field are taken as the respective reference. For $\mathbf{h}^{\omega}\parallel\mathbf{B}$, the lower-lying mode M1,∥ softens and becomes sharper in finite fields, together with the softening of the broad band M2,∥ [see Fig. 3(c)]. At the critical field $B_{c}=7$ T, the dynamical response is dominated by a broad continuum, extending over the whole resolvable spectra range (with the oscillations resulting from multiple interference at the sample surfaces). On further increasing field, the broad continuum shifts to higher energies with three new excitations appearing. At 9 T, the three excitations are observed at 2.7, 4.4, and 8.1 meV, denoted by L1,∥, L2,∥, and L3,∥, respectively, and they can be well tracked and shift to higher energies with increasing magnetic field. While the two lower-lying modes L1,∥ and L2,∥ are well-defined and become narrower in higher fields, the high-energy excitation L3,∥ is a broad band and shifts out of the spectral range above 12 T. Due to the broad-band nature and small intensity, L3,∥ cannot be fully resolved, thus it could also be the higher-energy flank of the continuum, which shifts to higher energies in the magnetic fields.

The field dependence of these modes and the emergent continuum is represented in the contour plot of Fig. 3(c). The softening feature close to the critical field indicates that the spin gap is closed and another one is reopened, in agreement with the results obtained from other methods Sears17 ; Baek17 ; Hentrich17 . A very interesting feature of the field-induced disordered phase is the linear field dependence of the lower-lying mode L1,∥. As shown in Fig. 3(c) this mode follows $\hbar\omega=\hbar\omega_{0}+g^{\ast}\mu_{B}B\Delta S$ with an apparent $g$-value of $g^{\ast}=11.1(1)$ and the Bohr magneton $\mu_{B}$, assuming a magnetic-dipole transition, i.e. $\Delta S=1$. This suggests that a spin gap opens linearly with the magnetic field for the sector of the dynamical structure factor corresponding to $\mathbf{h}^{\omega}\parallel\mathbf{B}$. The linear opening of the spin gap is a feature expected for models beyond the Kitaev limit Song16 . The apparent $g^{\ast}$-factor revealed here is significantly larger than the value of $g_{ab}=2.5$ estimated from the magnetization measurements Kubota15 or $g_{ab}=2.27$ from the x-ray magnetic circular dichroism experiments Agrestini17 , which is a reminiscence of the existence of bound states and many-body interactions in the QSL, as for the Heisenberg spin-1/2 chains WangBethe ; Yang17 .

For $\mathbf{h}^{\omega}\perp\mathbf{B}$ [Fig. 3(b)], approaching the critical field from below, the lower-lying mode M1,⟂ first slightly hardens and then softens, while the band M2,⟂ softens monotonically. At the critical field, the spectrum is again dominated by a broad continuum extending the whole spectral range. Above $B_{c}$ one can identify three modes: the two low-lying sharp modes L1,⟂ and L2,⟂, and the higher-energy band L3,⟂, emerging on top of the continuum and shifting to higher energy together with the continuum. Both of the two lower-lying modes become sharper with increasing field, while the band L3,⟂ remains broad and thus could merely be a feature of the continuum shifting in the magnetic field, similar as for L3,∥. In addition, the continuum is suppressed in high fields and the spectral weight is transferred to the low-lying modes. In the contour plot of Fig. 3(d) one can readily see these field-dependent features.

The emergent magnetic excitations and continuum of the field-induced QSL exhibit clear contrast between $\mathbf{h}^{\omega}\perp\mathbf{B}$ and $\mathbf{h}^{\omega}\parallel\mathbf{B}$. Both the eigenenergies and their field dependencies are different for the two polarizations, revealing the contrast of the two different sectors of dynamical structure factors. For $\mathbf{h}^{\omega}\perp\mathbf{B}$ the lower-lying mode L1,⟂ has lower eigenenergies, and the absorption lines are much sharper and stronger than those of its counterpart L1,∥. Moreover, the eigenenergy of L1,⟂ does not follow a linear field dependence, but a slower increase with the increasing field. The slower increase is consistent with the results of low-temperature specific heat measurements (below 3 K) Wolter17 where the thermodynamics is presumably dominated by the lowest-lying excitations. In contrast, the mode L2,⟂ in the sector of $\mathbf{h}^{\omega}\perp\mathbf{B}$ can be described by the linear Zeeman term with $g^{\ast}=10.2(2)$, while the L2,∥ mode hardens evidently slower than linear with the field, although these two modes have similar linewidths. These modes could be the bound states of fractionalized excitations Knolle15 ; Trebst14 ; ThVojta17 , which are narrow and can evolve from the excitation continuum as the time-reversal symmetry is broken by the applied magnetic field KnolleThesis . For the higher-energy bands L3,∥ and L3,⟂, the field dependencies are also sensitive to the polarizations. Together with the underlying continuum or as a part of the continuum, their spectral weights are reduced in high magnetic fields.

To conclude, terahertz spectroscopy as a function of temperature and magnetic field reveals emergent magnetic excitations and continua in a field-induced QSL in $\alpha$-RuCl3. While two magnonic excitations evolving on top of a continuum are resolved in the magnetically ordered phase and well below the critical field, the QSL is dynamically characterized by an emergent continuum at the quantum criticality and various magnetic excitations with distinct dependencies on field and polarization above the quantum phase transition. Our results pave the way to understand the dynamical behavior of the QSL and provide constrains for a quantitative theoretical description, e.g. via the identification of a realistic model or by extending the Kitaev model with terms breaking the time-reversal symmetry.