# Chiral excitations and the intermediate-field spin-liquid regime in the
Kitaev magnet $\alpha\text{-RuCl}_{3}\thickspace$

Anuja Sahasrabudhe<EMAIL_ADDRESS>Institute of Physics II,
University of Cologne, 50937 Cologne, Germany  Mikhail A. Prosnikov  High
Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen, The
Netherlands  Thomas C. Koethe  Philipp Stein  Institute of Physics II,
University of Cologne, 50937 Cologne, Germany  Vladimir Tsurkan  University of
Augsburg, 86159 Augsburg, Germany  Institute of Applied Physics, MD 2028,
Chisinau, Republic of Moldova  Alois Loidl  University of Augsburg, 86159
Augsburg, Germany  Markus Grüninger  Institute of Physics II, University of
Cologne, 50937 Cologne, Germany  Hamoon Hedayat<EMAIL_ADDRESS>Paul
H. M. van Loosdrecht<EMAIL_ADDRESS>Institute of Physics II, University
of Cologne, 50937 Cologne, Germany

(May 4 , 2023)

###### Abstract

In the Kitaev magnet $\alpha\text{-RuCl}_{3}$, the existence of a magnetic-
field-induced quantum spin liquid phase and of anyonic excitations are
discussed controversially. We address this elusive, exotic phase via helicity-
dependent Raman scattering and analyze the Raman optical activity of
excitations as a function of magnetic field and temperature. The hotly debated
field regime between $7.5\text{\,}\mathrm{T}$ and $10.5\text{\,}\mathrm{T}$ is
characterized by clear spectroscopic signatures such as a plateau of the Raman
optical activity of the dominant, chiral spin-flip excitation. This provides
direct evidence for the existence of a distinct intermediate field regime
compatible with a quantum spin liquid phase.

More than a decade ago, Kitaev proposed an exactly solvable model for a
spin-$\frac{1}{2}$ system on a honeycomb lattice with bond-directional
nearest-neighbour exchange couplings that hosts a quantum spin liquid (QSL)
ground state [1]. The collective excitations above this ground state are
gapless Majorana fermions and visons. In the presence of an external magnetic
field, the Majorana fermions are gapped. The gapped spectrum hosts a chiral
Majorana mode responsible for thermal edge transport and the emergent
excitations are non-abelian anyons [1].

Strong experimental efforts have been triggered by the possibility to achieve
dominant Kitaev exchange and the related exotic excitations in real materials
[2]. In this context, $\alpha\text{-RuCl}_{3}\thickspace$has been touted as a
potential candidate that could possess a QSL phase [3, 4, 5, 6, 7, 8], which
in turn has sparked a flurry of research in exploring its physical properties
[9, 10, 11, 12, 13]. Unlike an ideal QSL that remains disordered even at
vanishing temperature, the presence of additional exchange interactions in
$\alpha\text{-RuCl}_{3}\thickspace$yields antiferromagnetic zigzag (ZZ) order
below $7\text{\,}\mathrm{K}$ and for external magnetic fields below
$7\text{\,}\mathrm{T}$ [9, 10, 11, 14]. For strong enough magnetic fields a
field induced spin-polarized phase is observed [15, 16, 17]. It is still
possible though that between these high- and low-field phases, bond-
directional interactions take over in an intermediate field regime (IFR),
thereby preparing a conducive ground for the appearance of a QSL.

In this purported intermediate field regime, thermal conductivity measurements
[18, 19, 20] have displayed a direct signature of a chiral Majorana edge mode
(and hence of a QSL) – viz. a half-integer quantized plateau in the transverse
heat conductivity [21, 22, 23]. However, half-integer quantisation of the
thermal conductivity and consequently the role of Majorana fermions as heat
carriers has been challenged [24, 25, 26]. Meanwhile, thermodynamic
experiments like thermal expansion, magnetostriction, and the magnetocaloric
effect provide conflicting conclusions about the presence of the QSL phase in
the field regime of $7.5(5)\text{\,}\mathrm{T}11\text{\,}\mathrm{T}$ for field
applied parallel to the honeycomb plane [12, 27, 28, 29]. Recent evidence in
favor of the QSL phase comes from the observation of oscillations of the
thermal conductivity in this field regime [25].

When studied using spectroscopic experiments,
$\alpha\text{-RuCl}_{3}\thickspace$shows a low-energy multi-particle continuum
proposed to be of Majorana fermions [5, 6, 30]. Curiously, this continuum is
already present in the magnetically ordered phase in zero field. Upon
suppression of the antiferromagnetic ZZ phase in a large external magnetic
field, the continuum acquires a gap and the spectrum exhibits a sharp
excitation that is smoothly connected to a spin-flip excitation in the field-
polarized limit [15]. Altogether, the spectroscopic data reported thus far do
not provide any signature of a distinct IFR [8, 17, 31, 15, 16]. In Raman
scattering at $2\text{\,}\mathrm{K}$ in magnetic field, Wulferding et al. [32]
observed an excitation which they interpreted as a Majorana bound state.
However, the field was tilted by 18∘ out of the honeycomb plane and the
measurements were restricted to $10\text{\,}\mathrm{T}$. Hence the data could
not capture any possible difference between the high-field and the
intermediate field regime. Remarkably, this excitation has not been observed
in another Raman scattering study of $\alpha\text{-RuCl}_{3}\thickspace$in
high-magnetic fields [15], thus obstructing the clarity of opinion about the
presence of an IFR.

Raman scattering is expected to be an effective probe for emergent Majorana
fermions [6, 33, 34] and anyonic excitations. Experiments performed by Pinczuk
et al. have already demonstrated the strength of Raman scattering in probing
the emergent anyonic excitations of the fractional quantum Hall effect [35,
36, 37]. Recently, it has been suggested that these anyonic excitations are
chiral and may show fingerprints in helicity-dependent Raman scattering [38,
39]. Inspired by these ideas we performed helicity dependent Raman scattering
on $\alpha\text{-RuCl}_{3}$. The blueprint for this is to compare the Raman
intensities for circularly polarized light [40] in the four different
polarization geometries $I_{++},I_{--},I_{+-},I_{-+}$, where the first
(second) index denotes the circular polarization of the incident (scattered)
light. As the continuum and the magnetic excitations in
$\alpha\text{-RuCl}_{3}\thickspace$do not scatter in $I_{++}$ and $I_{--}$
(see Supplementary Information [41]), we focus on the Raman Optical Activity
(ROA) defined as

$\text{ROA}=\frac{I_{+-}-I_{-+}}{I_{+-}+I_{-+}}\ .$ (1)

ROA is a detectable quantity that is intimately connected with the chirality
of excitations [40, 42, 43]. Chiral excitations possess handedness and are
asymmetric under inversion. For example, let us consider magnons. Magnons in a
ferromagnet are chiral and are right circularly polarized with respect to the
direction of magnetization [44], while antiferromagnets have two degenerate
magnon branches with opposite chirality. In the particular case of a
ferromagnet, only one of the two incident polarizations $\sigma^{+}$ and
$\sigma^{-}$ yields a Raman signal and this shows $\text{ROA}=1$ [45, 46, 47].
In antiferromagnets, an external magnetic field can lift the degeneracy of
magnons which can again be detected using ROA [48].

In this letter, we focus on exploring the chirality of magnetic excitations in
$\alpha\text{-RuCl}_{3}\thickspace$through the associated ROA. For an external
magnetic field $B$ large enough to suppress the ZZ phase, we will show three
different trends in ROA. These will be pinpointed as characteristic
fingerprints of three distinct field-induced magnetic regimes. One stunning
example is the observation of a plateau of ROA = 1 of the dominant magnetic
excitation in the field range of
$7.5\text{\,}\mathrm{T}10.5\text{\,}\mathrm{T}$ which serves as spectroscopic
signature of a prominent intermediate field regime.

Figure 1: Schematic of experimental geometry for measuring the helicity-
resolved Raman intensities (a) $I_{+-}~{}$and (b) $I_{-+}~{}$as plotted in (c)
for different magnetic field strengths. Square, star, triangle, and circle
symbols denote the magnetic excitations M0, M1, M2, and M3, respectively.

For these experiments, high quality single crystals of
$\alpha\text{-RuCl}_{3}\thickspace$were prepared by the vacuum sublimation
method [7]. The sample is placed in a magneto-cryostat and cooled down to
$1.7\text{\,}\mathrm{K}$. The magnetic field is applied within the $ab$ plane,
perpendicular to a Ru-Ru bond. The light polarization is defined with respect
to the $z$ axis. Left (right) circular light is represented as $\sigma^{+}$
$(\sigma^{-})$ as it projects spin angular momentum $+\hbar$ $(-\hbar)$ onto
the quantisation axis. The experimental geometry is defined using the Porto
notation as $z(\sigma^{+}\sigma^{-})\bar{z}$ and
$z(\sigma^{-}\sigma^{+})\bar{z}$, see Fig. 1a, b. We refer to the
corresponding Raman intensities as $I_{+-}$ and $I_{-+}$, respectively. The
temperature-dependent measurements have been performed by varying the power of
the incident light so as to avoid a mechanical drift of the sample, see
Supplementary Information [41].

11footnotetext: The features M1, M2, and M3 correspond to $m_{1\alpha}$,
$m_{2\gamma}$, and $m_{2\alpha}$ in Ref. [15] and to M1, 2M, and 3M in Ref.
[32], while M0 corresponds to MB in Ref. [32]. Note that previous data were
reported in $I_{+-}~{}$[15] and $I_{-+}~{}$channels [32]. Due to the
pronounced ROA for intermediate fields, not all features are discussed in all
previous studies.

Figure 1c shows the helicity-dependent Raman response for $B>B_{\text{c}}$
(=$7.5\text{\,}\mathrm{T}$). Overall, the observed features agree with
previous Raman results [15, 32]. The data show the magnetic modes M0, M1, M2,
and M3, and a broad magnetic multi-particle continuum. The M0 peak has been
observed by Wulferding et al [32, 49]

in a tilted magnetic field up to $10\text{\,}\mathrm{T}$ with incident
$\sigma^{-}$ polarization. They attributed this feature to a singlet Majorana
bound state as its excitation energy remained insensitive to the magnetic
field. The M1 mode has been identified as a spin-flip excitation with $|\Delta
S|$ = $1$ in the fully spin-polarized limit, i.e., for infinite magnetic
field, while M2 has been attributed to a two-particle bound state [15, 32].
The feature M3 has been interpreted either as a two-particle excitation since
its energy at high fields roughly equals twice the energy of M1 [15] or as a
three-particle bound state [32]. We emphasize that this picture applies to the
high-field limit [15, 32, 16].

From the helicity dependence, it is immediately clear that the magnetic
features M0, M1, M2, and M3 exhibit chirality. The effect of magnetic field on
the chirality of these excitations allows us to distinguish different regimes.
To this end, Fig. 2a depicts the ROA on a color scale. We first focus on the
ROA of M1, which can be continuously traced down to $B_{c}$. Values for ROA
are obtained via Eq. (1), using the integrated intensities obtained by fitting
a Lorentzian to the M1 mode in $I_{+-}$ and $I_{-+}$ (see Supplementary
Information [41]). The result is depicted in Fig. 2b, which reveals three
distinct trends in the ROA of M1. We find a clear plateau with ROA $\approx$
$1$ from $7.5\text{\,}\mathrm{T}10.5\text{\,}\mathrm{T}$ in the IFR, a
continuous decrease in ROA in the transition regime from
$10.5\text{\,}\mathrm{T}15\text{\,}\mathrm{T}$, and ROA $\approx$ $0$ in the
high-field regime above $15\text{\,}\mathrm{T}$.

Figure 2: (a) ROA as a function of energy and field. Symbols denote the peak
energies of M0, M1, M2, and M3. (b) ROA of the M1 mode. The different trends
in the ROA of M1 can be used to determine the three distinct magnetic-field
regimes. Orange region: IFR with ROA = 1; color gradient: transition regime
(TR); white region: high-field regime (HFR) with ROA $\approx$ $0$. The grey
shaded region indicates the ordered ZZ phase. Dashed lines: guide to the eye.
(c) Integrated intensity of the M0 mode as a function of magnetic field.

Supporting evidence for the thus identified onset of the high-field regime at
$15\text{\,}\mathrm{T}$ emanates from the behavior of M0, M2, and M3, see Fig.
2a. The ROA of M2 crosses zero around $15\text{\,}\mathrm{T}$ and undergoes a
drastic change from ROA = $-1$ at $10.5\text{\,}\mathrm{T}$ to ROA = $+1$ at
$18\text{\,}\mathrm{T}$. Concerning M3, a clear peak is only identified above
$15\text{\,}\mathrm{T}$, while a consistent broad feature with ROA$>0$ is
visible from $10.5\text{\,}\mathrm{T}$ (see Fig. 1c, 2a). In contrast, M0 is
only detected below $15\text{\,}\mathrm{T}$ where it shows a negative ROA (cf.
Fig. 1c, 2c). As we do not observe any pronounced changes in ROA of these
features from $15\text{\,}\mathrm{T}$ to $30\text{\,}\mathrm{T}$, see
Supplementary Information [41], we identify $B\sim$15\text{\,}\mathrm{T}$$ as
the onset of the high-field regime.

Having identified the high-field regime, we now discuss the observations of
the IFR. Apart from a fully chiral M1, the only other chiral excitation that
is observed in this regime is M0. The energy (open squares in Fig. 2a) and the
integrated intensity (Fig. 2c) of the M0 mode From 7.5 to
$10.5\text{\,}\mathrm{T}$, the energy of M0 remains constant, while the
intensity shows a pronounced decrease. Above $10.5\text{\,}\mathrm{T}$, M2
starts to peak in the energy window of M0, requiring two Lorentzians to
account for the presence of both peaks. Simultaneously, the excitation energy
of M0 starts to increase linearly with $B$ with almost the same slope as M2.
In summary, all relevant excitations (M0, M1 and M2) show distinct changes at
$10.5\text{\,}\mathrm{T}$, providing clear spectroscopic evidence for the
existence of an IFR.

Next we present our results from the measurements of the power-dependent
spectra. Figure 3a exemplifies the power-dependent change of ROA at
$8\text{\,}\mathrm{T}$ and $11.5\text{\,}\mathrm{T}$, i.e., inside and just
above the IFR, respectively. For ease of visualization, we have superimposed
smoothened reference spectra taken at $10\text{\,}\mathrm{\SIUnitSymbolMicro
W}$. At $8\text{\,}\mathrm{T}$, the spectra show M1 and M0 around
$1.5\text{\,}\mathrm{meV}$ and $5\text{\,}\mathrm{meV}$, respectively, cf.
Fig. 2a. At $11.5\text{\,}\mathrm{T}$, the peaks are shifted to around
$3.2\text{\,}\mathrm{meV}$ and $6.0\text{\,}\mathrm{meV}$ while M2 rears its
head around $4.5\text{\,}\mathrm{meV}$. The color plot again pinpoints the
opposite chiralities of M1 and M0. The power dependence reveals a further key
difference between $8\text{\,}\mathrm{T}$ and $11.5\text{\,}\mathrm{T}$. In
the IFR at $8\text{\,}\mathrm{T}$, the chiralities of both M1 and M0 rapidly
decline with increasing laser power, i.e., sample temperature. In contrast,
the ROA of both modes is less sensitive to the laser power at
$11.5\text{\,}\mathrm{T}$. This is highlighted in Fig. 3c, showing the
normalised ROA of M1. At $40\text{\,}\mathrm{\SIUnitSymbolMicro W}$
($\approx$7\text{\,}\mathrm{K}$$), the ROA of M1 is suppressed by more than a
factor of two at $8\text{\,}\mathrm{T}$ but remains unaffected at
$11.5\text{\,}\mathrm{T}$. Figure 3b shows the smoothened spectrum of M1 as
observed in $I_{+-}$, while Fig. 3d and 3e show the energy and the linewidth
of M1 as a function of power for different fields, obtained by fitting M1 in
$I_{+-}$ with a Lorentzian. The energy of M1 hardens with increasing laser
power at $8\text{\,}\mathrm{T}$. Again, this behavior is found to be unique to
the IFR. Furthermore, the linewidth of M1 is much larger at
$8\text{\,}\mathrm{T}$ than at higher fields.

Figure 3: (a) Power dependence of ROA at $8\text{\,}\mathrm{T}$ and
$11.5\text{\,}\mathrm{T}$. Red (blue) areas represent larger scattering in
$I_{+-}$ ($I_{-+}$). As a reference, the red and blue solid lines show
$I_{+-}~{}$and $I_{-+}~{}$at $10\text{\,}\mathrm{\SIUnitSymbolMicro W}$. At
$8\text{\,}\mathrm{T}$, the M1 mode in the $I_{+-}~{}$channel shows maximum
power sensitivity around $40\text{\,}\mathrm{\SIUnitSymbolMicro W}$
($T\sim$7\text{\,}\mathrm{K}$$), while M0 in the $I_{-+}~{}$channel disappears
around $20\text{\,}\mathrm{\SIUnitSymbolMicro W}$
($T<$7\text{\,}\mathrm{K}$$). (b) Effect of laser power on M1 in $I_{+-}~{}$in
different field regimes. (c) Power-induced change in ROA of M1 at
$8\text{\,}\mathrm{T}$ and $11.5\text{\,}\mathrm{T}$. (d),(e) Power dependence
of energy and linewidth of the M1 mode in $I_{+-}~{}$channel.

Previous Raman scattering studies on $\alpha\text{-RuCl}_{3}\thickspace$in
high magnetic fields reported on $I_{+-}$ [15] and $I_{-+}$ [32]. Due to the
chiral character of the magnetic excitations, in particular with M0 and M1
showing opposite chiralities, the key to a comprehensive picture of the
magnetic Raman features and to a thorough understanding of the spectra lies in
the comparison of the $I_{+-}$ and $I_{-+}$ channels, i.e., in the ROA. In
particular our ROA data measured over a wide range of magnetic fields above
$B_{\text{c}}$ allow us to unravel the distinct spectroscopic characteristics
of the intermediate-field regime and to elucidate its properties.

The IFR is characterized by the presence of the two excitations M0 and M1 for
which we find opposite chiralities. Furthermore, ROA = $1$ for M1 within the
IFR. To illustrate the importance of M0 and M1 being chiral, we consider the
case of a simple paramagnet instead of a QSL above $7.5\text{\,}\mathrm{T}$,
where long-range zigzag order is suppressed by the magnetic field. The
external magnetic field competes with exchange interactions. In this
paramagnet scenario, it is feasible to neglect field-induced order at $B_{c}$
and assume that the time-average of the local magnetic moments tends to zero.
In this case, the Raman tensor $R$ of the excitations is defined by the
crystallographic space group $C2/m$ [30] or $R\bar{3}$ [50]. The Raman
intensity is given by $I_{is}=|e_{s}^{{\dagger}}\cdot R\cdot e_{i}|^{2}$,
where $e_{s}$ and $e_{i}$ are the electric field vectors of the scattered and
the incident light. Since neither of these crystallographic space groups is
chiral, we find $I_{+-}$ = $I_{-+}$ and ROA = 0, see Supplementary Information
[41]. The pronounced ROA of the magnetic excitations M0 and M1 in the IFR
indicates an unconventional ground state distinct from a conventional
paramagnet.

Previously, M0 has been attributed to a singlet Majorana bound state [32].
While our results do not provide any direct evidence for a Majorana character,
they support the interpretation in terms of a bound state to some extent.
Above $10.5\text{\,}\mathrm{T}$, i.e., above the IFR, the M0 mode hardens and
exhibits a strongly suppressed intensity, hinting at an unbinding process,
i.e., a decay into the multi-particle continuum. However, the large width of
M0 speaks against a strict bound-state scenario but rather suggests a
resonance within the continuum close to its lower edge.

The mode M1 has been identified as a spin-flip excitation in the high-field
regime [15]. Above $15\text{\,}\mathrm{T}$, the ROA of M1 is negligible. The
pronounced increase of the ROA with decreasing field and in particular the
plateau with ROA = $1$ in the IFR are clear signatures of a change of
character of this mode. Remarkably, the plateau is observed between
$7.5\text{\,}\mathrm{T}10.5\text{\,}\mathrm{T}$, consistent with the
intermediate field regime where a QSL phase has been proposed to exist [25,
19, 18, 20, 12]. The linewidth $\gamma$ of the M1 mode equals
$0.5\text{\,}\mathrm{meV}$ in the high-field regime where it corresponds to a
spin flip, see Fig. 3e. In the IFR, $\gamma$ is enhanced by more than a factor
of two. It is tempting to explain this increase of the linewidth in a scenario
based on Majorana fermions, where the local spins decay into Majoranas and
fluxes, giving rise to a substantial broadening of M1.

Lastly, we briefly address our observation of chiral multi-magnon excitations
M2 and M3. Topological magnons contributing to the thermal Hall effect are
expected to live in the field-polarized phase of a Kitaev-Heisenberg magnet
[51, 52, 53]. Even though a detailed analysis of the chiral origin of these
excitations is beyond the scope of this work, the chirality of these multi-
particle excitations hints at their topological nature and indicates an onset
of field polarization.

In conclusion, magnetic field and laser power dependent Raman optical activity
experiments show the presence of distinct field induced phases for
$\alpha\text{-RuCl}_{3}$. In particular, the upper limit of our identified
intermediate field regime is very close to the value of
$11\text{\,}\mathrm{T}$, above which the quantum oscillations intrinsic to a
QSL phase vanish [25]. The intermediate field regime shows intriguing
spectroscopic features. The most salient features are a fully chiral response
(ROA=1) of the strongest (M1) magnetic mode, a strong broadening of this mode
may suggest a fractional character of the fundamental magnetic excitations
specific for the IFR, and the presence of a resonance mode (M0) on the lower
edge of the magnetic Raman continuum which previously has been discussed in
terms of a Majorana bound state [32]. The laser power dependence (i.e.,
temperature dependence) of the spectra indicates that the ground state in the
IFR region is rather fragile as is demonstrated by the rapid decrease of the
fully chiral response upon increasing temperature. In contrast, the ROA of the
spectra in the high field regime is hardly affected by the increased
temperature. Though it is clear from the present and earlier experiments that
the IFR has an unconventional ground state which could very well be a quantum
spin-liquid state, its exact nature remains elusive at this point. The present
work shows the power of ROA experiments in materials showing unconventional
magnetism and sheds light on the intriguing chiral magnetic properties of the
Kitaev-Heisenberg magnet $\alpha\text{-RuCl}_{3}$, inspiring future research
to realize a field-induced spin-liquid phase in
$\alpha\text{-RuCl}_{3}\thickspace$and other Kitaev-like systems.

###### Acknowledgements.

We thank Aprem Joy, Ciarán Hickey, David Kaib, Roser Valentí, Daniel I.
Khomskii and Fulvio Parmigiani for insightful discussions. We thank Jonathan
Buhot and Clément Faugeras for supporting preliminary experimental
measurements. A S thanks Vivek Lohani for critical comments on the manuscript.
M P thanks Matija Čulo and Peter C. M. Christianen for support during the
experiment. The authors acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) via Project No.
277146847 – CRC 1238 (project B03), via Project No. 107745057 – TRR 180
(project F5), and via Project No. VA117/15-1. A L and V T acknowledge support
by the DFG through TRR 80. V T acknowledges the support via the project ANCD
20.80009.5007.19 (Moldova).

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