Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of alpha-RuCl3

TL;DR

This study demonstrates that alpha-RuCl3 enters a gapless quantum spin liquid phase with gapless spin excitations when subjected to a magnetic field, revealing unconventional quantum magnetic behavior.

Contribution

It provides experimental evidence of a field-induced gapless quantum spin liquid phase in alpha-RuCl3, a key candidate for Kitaev physics, using combined susceptibility, specific heat, and NMR measurements.

Findings

Quantum phase transition to QSL at 7.5 T

Gapless spin excitations observed up to 16 T

Unconventional behavior not seen in Heisenberg or Kitaev magnets

Abstract

$\alpha$-RuCl$_3$ is a leading candidate material for theobservation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that $\alpha$-RuCl$_3$ undergoes a quantum phase transition to a QSL in a magnetic field of 7.5 T applied in the $ab$ plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of $\alpha$-RuCl$_3$.

Full text

The quantum spin liquid (QSL) is an exotic state of matter with long-range coherence but with no spontaneous breaking of translational or spin-rotational symmetry down to zero temperature Balents_nature464_199 . Such a state in two or higher dimensions has implications for phenomena ranging from high-temperature superconductivity Anderson1987 to quantum computation Kitaev_ap_2003 ; Nayak_RMP.80.1083 . QSLs have long been sought in systems with strong geometric frustration Anderson1973153 ; Kanoda_PRL_2003 ; Han_nature492_406 ; Greedan_RMP.82.53 , where magnetic order is destroyed by quantum fluctuations in a highly degenerate ground manifold. A more recent avenue to QSL formation is by competing interactions with combined spin and spatial anisotropies, as in the Kitaev model, where both gapped and gapless QSLs are realized exactly in a honeycomb-lattice spin-1/2 system Kitaev_ap_2006 .

A pure Kitaev Hamiltonian is hard to achieve in real materials. However, the compounds A2IrO3 (A = Na, Li) Khaliullin_prl102_017205 ; Khaliullin_prl105_027204 ; Singh_PhysRevLett.108.127203 ; Gretarsson_PhysRevLett.110.076402 ; Chun_np_2015 ; Kee_conmatphys_2016 and $\alpha$-RuCl3 Pollini_PhysRevB.53.12769 ; Plumb_PhysRevB.90.041112 ; Kee_PhysRevB.91.241110 ; Burch_Prl_2015 ; Kindo_PhysRevB.91.094422 ; Sandilands_PhysRevB.93.075144 ; Koirzsch_PRL.117.126403 are candidate systems for significant Kitaev-type interactions. In each case, the $4d$ (Ru3+) or $5d$ (Ir4+) ions form a Mott insulator on a honeycomb lattice, whose localized electrons have an effective spin $j_{\rm eff}=1/2$ due to strong spin-orbit coupling Khaliullin_prl102_017205 ; Kim_science_2009 ; Foyevtsova_PhysRevB.88.035107 ; Khaliullin_PhysRevB.94.064435 ; Haule_PhysRevLett.114.096403 . In $\alpha$-RuCl3 at zero field, a finite-energy continuum of magnetic excitations Banerjee_nm15_733 is suggestive of fractionalized (spinon or Majorana-fermion) excitations Coldea_PhysRevB.68.134424 ; Kitaev_ap_2006 ; Shankar_PhysRevLett.98.247201 ; Moessner_PhysRevLett.112.207203 ; Moessner_PhysRevB.92.115127 ; Nasu_np12_912 . However, the ground states in all cases have “zig-zag” magnetic order Ye_PhysRevB.85.180403 ; Choi_PhysRevLett.108.127204 ; Fletcher_JCSA_1967 ; Sears_PhysRevB.91.144420 ; Johnson_PhysRevB.92.235119 ; Cao_PRB.93.134423 , indicating the presence of significant non-Kitaev terms, whose exact nature continues to occupy many authors Kimchi_PhysRevB.84.180407 ; Mazin_PhysRevLett.109.197201 ; Khaliullin_prl110_097204 ; Rau_PRL.112.077204 ; Yamaji_PRL_2014 ; Perkins_PRB.90.155126 ; Winter_PRB_2016 ; Kim_PRB.93.155143 ; Perkins_PRB.94.085109 ; Wang_arxiv_2016 . While the large $T_{N}$ in Na2IrO3 Ye_PhysRevB.85.180403 ; Choi_PhysRevLett.108.127204 suggests subdominant Kitaev terms, the relatively low $T_{N}$ of $\alpha$-RuCl3 has sparked an intensive search for experimental Nagler_2016_arxiv ; LeeM_PRL.118.187203 ; Baek_arxiv_170201671 and theoretical Janssen_arxiv_2016 ; Yadav_arxiv_2016 evidence for “proximate Kitaev” behavior.

Here we report a nuclear-magnetic-resonance (NMR) investigation of high-quality single crystals of $\alpha$-RuCl3. With additional magnetic susceptibility and specific-heat measurements, we establish the phase diagram of Fig. 1. We demonstrate the presence of a field-induced QSL beyond the quantum phase transition at $\mu_{0}H_{c}\simeq$ 7.5 T. In the field range between 7.5 T and 16 T, this partially polarized QSL has a spin-lattice relaxation rate with power-law temperature dependence, indicating effectively gapless spin excitations with line-node dispersion.

Single crystals of $\alpha$-RuCl3 were grown by chemical vapor transport. The high quality of this batch of crystals is demonstrated in Ref. WenJS_prl_2016 and x-ray characterization of the NMR sample, shown in Sec. S1 of the Supplemental Material (SM) sm , demonstrated that it is a single domain and free of twinning. Magnetization measurements were performed in a 9T SQUID and the specific heat measured in a Quantum Design PPMS. 35Cl NMR spectra were collected by the spin-echo technique and the spin-lattice relaxation rate measured by the inversion-recovery method, as shown in Sec. S2 of the SM sm . The spin-recovery exponent, $\beta=1$ in the paramagnetic state ($T>20$ K in Fig. 1), also indicates a very high sample quality.

The magnetization, $M(T)$, is shown for a range of applied fields in Fig. 2(a), with its primary features, emphasized by the low-$T$ derivative, $dM/dT$, shown in the inset. At zero field, $\alpha$-RuCl3 has zig-zag antiferromagnetic (AFM) order (inset, Fig. 1) below $T_{N}\simeq$ 7.5 K (14 K) for crystals with ABC (AB) stacking along the $c$-axis Cao_PRB.93.134423 . The sharp phase transition at 7.5 K in both $dM/dT$ and the specific heat, $C_{v}$ [Fig. 2(b)], demonstrates the very high crystal quality. Our samples are almost exclusively ABC-stacked, with only a small admixture of AB stacking discernible through the weak anomaly in $C_{v}$. Our $C_{v}$ data include the phonon contribution; due to concerns over its subtraction and over the suitability of specific heat for this purpose, we do not attempt to use our data to analyze the magnetic response.

Fields applied in the $ab$-plane suppress $T_{N}$. This effect is especially strong from 7 T to 7.25 T, leading up to a field-induced quantum phase transition (QPT) at $\mu_{0}H_{c}\simeq$ 7.5 T. The QPT is observed clearly both in the disappearance of the peaks in $dM/dT$ and $C_{v}$ and in the dramatic changes in the NMR spectra [Fig. 2(c)], which we discuss next. We find no anisotropy in $M$ and $C_{v}$ as the field is rotated in the $ab$-plane. The NMR measurements shown in Fig. 2(c), and also Fig. 3, are fully representative of the generic in-plane response, as we demonstrate in Secs. S3 and S4 of the SM sm .

We collect the evidence that the phase at $H>H_{c}$ is a QSL. First, the vanishing peaks in both $dM/dT$ and $C_{v}$ (Fig. 2) demonstrate the absence of magnetic order. Second, the 35Cl line shapes for fields at and above 7.6 T contain none of the peaks corresponding to an ordered Cl environment. The 35Cl NMR spectra below 7.6 T [Fig. 2(c)] are very broad at low $T$, consistent in the absence of domain and twinning effects only with AFM order. By contrast, several sharp peaks are clearly resolvable for fields above 7.6 T in Figs. 2(c), 3(a), and 3(b). The center peak at $\Delta f\approx$ 0 ($\gamma H\approx$ 43.5 MHz) has a FWHM height of 0.5 MHz at 10.3 T at both $T=20$ K and 2 K [Fig. 3(a)], and shows no significant changes at 12 T [Fig. 3(b)]. Such narrow linewidths, and in particular their invariance upon cooling below 20 K, indicate a complete absence of magnetic order.

Further evidence is provided by the Knight shift, ${}^{35}K_{n}$, of the center peak, shown in Fig. 3(c). From its large values (6-7$\%$) below 10 K, the hyperfine field is strong at the 35Cl site and magnetic order, if present, is very unlikely to be missed by NMR. This statement remains true even for incommensurate or large-unit-cell ordered phases that are possible in a field Janssen_arxiv_2016 . Further, a Curie-Weiss (CW) fit to the high-temperature part of the data holds down to 2 K at fields of 8 T, with the CW temperature, $\theta\approx-45$ K [inset, Fig. 3(c)], supporting the absence of magnetic order. Because the NMR spectra show no evidence that the magnetic QPT is accompanied by a structural transition, which would cause much more dramatic peak shifts in Fig. 2(c), we conclude that the field-induced disordered phase is indeed a QSL.

Unlike $T_{N}$, the value of $H_{c}$ required to suppress magnetic order in a crystal with AB stacking is also around 8 T Johnson_PhysRevB.92.235119 . The similarity of $H_{c}$ values for ABC and AB stacking suggests that the field-induced suppression of magnetic order, and by extension the properties of the high-field QSL phase, are primarily two-dimensional (2D) in nature, rather than depending on interlayer coupling.

The spin excitations of the high-field QSL state are probed by the spin-lattice relaxation rate. We measure $1/^{35}T_{1}$ for each field at the sharp central peak marked by the red arrows in Figs. 2 and 3, and show the results in Fig. 4. Above 40 K, $1/^{35}T_{1}$ remains constant in both $T$ and $H$, which is typical of decorrelated spins at temperatures beyond their interaction energy scale. At 8 T, $1/^{35}T_{1}$ drops sharply on cooling below 10 K, and follows a power-law $T$-dependence, $1/^{35}T_{1}=bT^{\alpha}$, to the bottom of our measurement range ($T=1.5$ K). The same behavior holds at all fields up to our maximum of 16 T. The exponent $\alpha$, shown in the inset of Fig. 4, remains constant at $\alpha=3$ for fields between 8 and 12 T, and then increases to $\alpha=5$ at 16 T. While exponents above 5 suggest the opening of a full gap, the reliable extraction of power-law behavior with exponents as small as $\alpha=3$ demonstrates that the QSL in the field range $8<\mu_{0}H<16$ T has either gapless spin excitations or an anomalously small gap far below 2 K.

We extract also the characteristic temperature, $T^{*}$, marking the upper limit of the low-$T$, power-law regime, which is shown in Fig. 4 and also in Fig. 1. The increase of $T^{*}$ with field demonstrates that the phase of coherent QSL dynamics becomes increasingly robust, at least to 16 T. This occurs in tandem with an increasing field-driven polarization, observed in both $M$ and ${}^{35}K_{n}$, which leaves a decreasing component of the spin available to participate in the QSL. The related average moment per Ru ion has been measured directly Johnson_PhysRevB.92.235119 as 0.56$\mu_{B}$ at 8 T, 0.87$\mu_{B}$ at 16 T, and 1.22$\mu_{B}$ at 60 T. For comparison, the static moment in the zig-zag AFM state at $H=0$ is 0.4$\mu_{B}$. We note that the recovery exponent, $\beta$, decreases at $T<T^{*}$, suggesting that the QSL dynamics are very sensitive to any weak disorder. However, as discussed in Sec. S2 of the SM sm , our results contain no evidence that the QSL state itself could be a consequence of disorder, and verify rather its intrinsic nature.

Two recent experiments have also investigated $\alpha$-RuCl3 in fields exceeding $H_{c}$. Thermal conductivity measurements show a prominent low-$T$ peak whose magnitude grows linearly with $H-H_{c}$ LeeM_PRL.118.187203 , indicating a gapless and linearly dispersive excitation. By contrast, a different NMR study reports gapped behavior Baek_arxiv_170201671 , although the field is applied at an angle 30∘ out of the $ab$-plane. There is increasing evidence that different field orientations lead to qualitatively different properties in $\alpha$-RuCl3, as expected from the strong spin anisotropy (our own NMR investigation is presented in Sec. S4 of the SM sm ). However, the gaps reported by these authors are deduced only over a limited temperature range that, crucially, does not extend to $T=0$, and the finite $1/T_{1}$ observed at low $T$ is not consistent with a spin gap.

To interpret our results, we consider the definition $1/T_{1}T={\rm lim}_{\omega\to 0}\sum_{q}A_{\rm hf}(q)\,{\rm Im}\chi(q,\omega)/\omega$, where $\chi(q,\omega)$ is the dynamical susceptibility and $A_{\rm hf}(q)$ the hyperfine coupling. For a conventional magnetic system, this may be reexpressed as $1/T_{1}\sim\int_{0}^{\infty}\rho^{2}(E)n(E)[1+n(E)]dE$, where $\rho(E)$ is the magnon density of states (d.o.s.) and $n(E)$ the Bose distribution function, and this makes $1/T_{1}$ a sensitive probe of low-energy spin dynamics. Our result $1/T_{1}\sim T^{3}$ suggests that the density of magnetic states follows $\rho(E)\propto E$. In condensed matter, such a linear d.o.s. is more familiar from fermionic excitations, including line nodes in the gap at the Fermi surface of a 3D superconductor Monien and the 2D Dirac-cone dispersion Dora .

In conventional (Heisenberg) quantum magnets, the partial polarization is distributed uniformly and becomes complete at a finite saturation field, $H_{s}$. All spin excitations are bosonic and for $H>H_{s}$ they have a gap $\Delta\propto H-H_{s}$. Spatially anisotropic and frustrated systems show magnetization plateaus at low or intermediate fields, whose ground states are in general gapped Tamaka_PRL_102_2009 with bosonic excitations. Thus our discovery of gapless spin excitations at intermediate fields in $\alpha$-RuCl3 represents an extremely unconventional situation.

When a Hamiltonian anisotropic in spin space does not commute with the field, $H_{s}\rightarrow\infty$. Although the spatially isotropic Kitaev model has one gapless and linearly dispersive Majorana fermion at zero field, its spin excitations, consisting of Majorana fermions coupled to two massive, static flux quanta Kitaev_ap_2006 ; Shankar_PhysRevLett.98.247201 , are gapped. In Ref. Song_prl_2016 it was shown that the spin gap can vanish in a “generic Kitaev” system, meaning one in which the symmetry-allowed Heisenberg ($J$) and off-diagonal ($\Gamma$) terms are present at a perturbative level, and these authors obtained the highly suggestive result $1/T_{1}\propto T^{3}$. However, in an applied field, all the modes in both the pure ($K$) and generic (perturbative $K$-$J$-$\Gamma$) Kitaev models become gapped. Although a gapless Majorana quasiparticle could be responsible for the measured thermal conductivity LeeM_PRL.118.187203 , our NMR results show unequivocally that the spin excitations themselves are gapless, and thus such proximate Kitaev physics appears to be excluded.

We review scenarios allowing gapless spin excitations over a range of field strengths. As a result of its weak interplane interactions, $\alpha$-RuCl3 is effectively a 2D magnet Banerjee_nm15_733 and therefore its line nodes are interplane, connecting point nodes in the honeycomb layers. Bosonic excitations in (and beyond) 2D are either gapped or condense, leading to magnetic order, whereas gapless and disordered (i.e. critical) behavior over a finite field range is unknown. Turning to fermionic excitations, these imply a fractionalization and deconfinement of spinonic quasiparticles taking place at the QPT. In the Heisenberg chain and ladder (1D), gapless spinons appear at incommensurate wave vectors over a range of fields. In 2D, some spin-orbit-coupled magnetic systems may be represented by fermionic spinons, whose dispersion has Dirac cones. These cones provide the linear d.o.s. and remain stable, i.e. pinned at the Fermi level, over a finite range of fields applied in specific symmetry directions, whereas fields in generic directions open a gap (an example is shown in Sec. S6 of the SM sm ). For Majorana fermions, at least two flavors of Majorana cone are required for the system to host gapless spin excitations.

Concerning a microscopic model exhibiting such exotic physics, the effective magnetic Hamiltonian of the spin-orbit-coupled Mott insulator has focused attention on $K$-$J$-$\Gamma$-type models Rau_PRL.112.077204 ; Yamaji_PRL_2014 . Opinions on the terms and parameters describing $\alpha$-RuCl3 remain strongly divergent. Early efforts using a $J$-$K$ model with ferromagnetic $J$ and AFM $K$ Banerjee_nm15_733 have been supplanted by an exchange of signs and longer-ranged $J$ terms Winter_PRB_2016 ; Yadav_arxiv_2016 . Very recent studies Nagler_2016_arxiv ; Wang_arxiv_2016 ; WenJS_prl_2016 have turned to $K$-$\Gamma$ models with parameters far outside the perturbative regime of Ref. Song_prl_2016 . Although the effects of a magnetic field have yet to be investigated in detail, our own $K$-$\Gamma$ analysis (rln , Sec. S6 of the SM sm ) indicates robust QSL states whose gapless fermionic spinons have four dispersion cones for certain field directions and a very small gap for all in-plane fields.

In summary, we have observed a QPT to a field-induced QSL above $\mu_{0}H_{c}=7.5$ T in $\alpha$-RuCl3. We have shown by NMR measurements that this state has effectively gapless spin excitations over a broad field range. This result cannot be reconciled with the behavior of conventional quantum magnets or of the pure or generic Kitaev QSL. Thus our data suggest fractionalized spinon excitations in $\alpha$-RuCl3 and set a significant challenge to theory.

This work was supported by the National Science Foundation of China (Grant Nos. 11374364, 11374143, and 11674157), by the Ministry of Science and Technology of China (Grant No. 2016YFA0300504), and by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China (Grant No. 14XNLF08). JCZ and KJR contributed equally to this study.