1 Planar Thermal Hall Effect s in Kitaev Spin Liquid Candidate Na2Co2TeO 6

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Planar Thermal Hall Effects in Kitaev Spin Liquid Candidate

Na2Co2TeO6

Authors and Affiliations:

Hikaru Takeda1, Jiancong Mai1, Masatoshi Akazawa1, Kyo Tamura1, Jian Yan1, Kalimuthu

Moovendaran2, Kalaivanan Raju2, Raman Sankar2, Kwang-Yong Choi3, and Minoru Yamashita1

1 The Institute for Solid State Physics, University of Tokyo, Kashiwa, 277-8581, Japan

2 Institute of Physics, Academia Sinica, Taipei 11529, Taiwan

3 Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea

Abstract:

We investigate both the longitudinal thermal conductivity (  ) and the planar thermal Hall

conductivity ( ) in the Kitaev spin liquid candidate of Co-based honeycomb antiferromagnet

Na2Co2TeO6 in a magnetic field () applied along the  and ∗ axes. A finite  is resolved for

both field directions in the antiferromagnetic (AFM) phase below the Néel temperature of 27 K. The

temperature dependence of / shows the emergence of topological bosonic excitations. In

addition, the field dependence of  shows sign reversals at the critical fields in the AFM phase,

suggesting the changes in the Chern number distribution of the topological magnons. Remarkably, a

finite  is observed in  ∥ ∗ between the first-order transition field in the AFM phase and the

saturation field, which is prohibited in a disordered state by the two-fold rotation symmetry around

the ∗ axis of the honeycomb lattice, showing the presence of a magnetically ordered state that breaks

the two-fold rotation symmetry. Our results demonstrate the presence of topological magnons in this

compound in the whole field range below the saturation field.

Main texts:

Topology in condensed-matter physics is a powerful concept allowing one to characterize material

properties solely by the topological classification without details of materials. For conduction electrons

in a metal, the anomalous quantum Hall effect [1] is one of the most celebrated examples of such

topological phenomena, in which the topological invariant (called the Chern number) of conduction

electrons is responsible for the dissipationless quantized Hall current even in the zero field [2]. In an

insulator, topological effects on the heat carriers give rise to thermal Hall effects (THEs) as described

by [3,4]



=

ℏ∫ Ω()(), (1)

where Ω() is the energy distribution of the Berry curvature and () is given by the distribution

function of the elementary excitations at the energy  , demonstrating that one can reveal the

topological property of the charge-neutral excitations by the thermal Hall measurements.

For fermions, the Fermi distribution leads to quantized transport dictated by the sum of the Chern

numbers of the occupied bands below the Fermi energy. For example, in the quantum spin liquid (QSL)

given by the Kitaev Hamiltonian, the nontrivial topology of the Majorana fermions gives rise to the

half quantization of the thermal Hall conductivity () [5]. In fact, the half-quantized / in the

Kitaev QSL, which is suggested to be realized in Ru or Ir compounds [6], has been reported in α–

RuCl3 by several groups [7–10]. For bosons, on the other hand, their topological transport vanishes in

the zero-temperature limit, because the temperature dependence is governed by the Bose distribution

function [11]. Such topological THEs of bosons are proposed for topological magnons [12–15],

triplons [16], and skyrmions [17,18]. Indeed, the topological THE of magnons is reported in a lattice

of magnetic skyrmions lately [19].

Recently, new Kitaev QSL candidates have been put forward for 3d compounds [20–22]. The Co-

based honeycomb antiferromagnet Na2Co2TeO6 [23] constitutes a prominent candidate, in which the

high-spin d7 state of Co2+ ions forms two-dimensional honeycomb layers of eff = 1/2 Kramers

doublets with a sizable Kitaev interaction term estimated from the neutron scattering experiments [24–

27]. At zero field, zigzag [28–30] or triple-q [31,32] antiferromagnetic (AFM) order occurs below the

Néel temperature N=27 K. Magnetization measurements under in-plane magnetic

fields [25,30,33] show multiple phase transitions at three critical fields in the AFM phase, as well as a

possible emergence of a quantum disordered state between the AFM phase and the spin polarized

phase above the saturation field (sat ∼13 T). The longitudinal thermal conductivity measurements

performed under in-plane fields [33] and the thermal Hall conductivity measurements under  ∥

 [34,35] bear a close resemblance with those of α–RuCl3, implying a possible realization of the

Kitaev QSL in Na2Co2TeO6 under in-plane field above ∼10 T. However, the half-quantized /

under the in-plane field, the key signature of the Kitaev QSL, has been missing.

In this Letter, we address the planar THE of Na2Co2TeO6 under in-plane magnetic fields. Our salient

findings are (1) a finite planar  for both  ∥  and  ∥ ∗ , showing the emergence of

topological charge-neutral excitations in Na2Co2TeO6, (2) the vanishing temperature dependence of

/ in the zero-temperature limit, consistent with topological bosonic excitations rather than

Majorana fermions, and (3) multiple sign changes of  at critical fields in  ∥  , suggesting

changes of the Berry curvature of the topological magnons in different magnetic states.

Single crystals of Na2Co2TeO6 were synthesized by the self-flux method as described in Ref. [32].

Prior to the thermal-transport measurements, we confirmed the crystallographic directions by x-ray

diffraction measurements. Thermal-transport measurements were performed by the steady method as

described in Ref. [36] by using a variable temperature insert (VTI) for 2–60 K and a dilution

refrigerator (DR) for 0.1–3 K. The heat current Q and the magnetic field  were applied parallel to

either the  and ∗ axes of the sample (see Fig. S1 in Supplementary Materials (SM) for more

details). The magnetic susceptibility measurements were performed by using a SQUID magnetometer

to check the quality and the crystallographic directions of the sample. The reproducibility was

confirmed by repeating the  measurements in different single crystals (see Figs. S4 and S5 in SM).

Figure 1 shows the temperature dependence of the longitudinal thermal conductivity () measured

under  ∥ Q∥  (“zigzag” direction, perpendicular to the honeycomb bond, Fig. 1(a)) and  ∥ Q∥

∗ (“armchair” direction, parallel to the honeycomb bond, Fig. 1(b)). At zero field, the temperature

dependence of  shows a broad peak around 50 K, which is followed by a kink at N, as clearly

seen in the temperature dependence of / (see Fig. S2 in SM). The Néel transition of the sample

is also clearly seen as the peak in the temperature dependence of the magnetic susceptibility (Fig. S3

in SM). The good quality of the sample is also confirmed by the boundary-limited phonons observed

 at 15 T (Fig. S6 in SM) where the phonon thermal conduction is enhanced by suppressing a

magnon-phonon scattering by opening a magnon gap (see section IV in SM). This enhancement of

 at 15 T (Fig. 1) shows the presence of a strong spin-phonon coupling suppressing  at 0 T in

this compound. The residual of / in the zero-temperature limit is vanishingly small for the

whole field range we measured (see Fig. S7 in SM), showing the absence of itinerant gapless

excitations.

Figure 2 shows the field dependence of  at fixed temperatures. The field dependence of  is

negligible above N except for the small increase at the highest field. This increase of  could be

attributed to an increase of the phonon conduction caused by a field suppression effect on the spin

fluctuation. Below N,  starts to depend on the magnetic field with features at critical fields of

the AFM phases determined by the previous studies [25,30,33] (c1 and c2 for  ∥ , and c1

∗,

c2

∗, and c3

∗ for  ∥ ∗), as marked by arrows in Fig. 2. In  ∥ , the field dependence of d/d

shows a shoulder at c1 ∼7.6 T and a peak at c2 ∼9.8 T [33]. Corresponding to these critical

fields,  shows the peak at c1 and the dip at c2, as shown in Fig. 2(a). On the other hand, in

 ∥ ∗ , three critical fields (denoted as c1

∗ , c2

∗ , and c3

∗ ) are found in the field dependence of

d/d [33]. As shown in Fig. 2(b),  shows a sharp jump at c1

∗∼6 T with a large magnetic

hysteresis, demonstrating the first-order transition at c1

∗. This first-order transition is also observed

as the jump in the magnetization of our sample in  ∥ ∗ (Fig. S3(b) in SM) [25,30,33]. In contrast

to  in  ∥ , only small humps are observed at c2

∗ and c3

∗ in  ∥ ∗. As  of a magnetic

insulator is given by the sum of the phonon contribution 

ph and the magnon contribution 

mag, the

field dependence of  of this compound can be understood in terms of the field effect of the

magnon gap on 

ph and 

mag (see section VI in SM).

Our main finding is the finite planar  in the AFM phase for both  ∥  (

, Fig. 3(a)) and  ∥

∗ (

∗ , Fig. 3(b)). We averaged the data of 

/ obtained in the field-up and field-down

procedures except for that measured in the DR (< 2 K). On the other hand, we present only the

field-up data of 

∗/ to avoid mixing the hysteretic field dependence at c1

∗ (see Fig. S8 in SM

for all the data in each measurement). As shown in Fig. 3, whereas / is virtually absent above

N, a finite / is observed in the AFM phase below N in both field directions. In  ∥ , 

/

gradually increases up to c1 , which is followed by a sign change to negative 

 at c1 , and

another sign change to positive 

 at c2. In  ∥ ∗, 

∗ shows a sharp negative peak at the first-

order transition at c1

∗, which is followed by a constant negative 

∗ up to the highest fields. We

note that the sharp negative anomaly of 

∗ at c1

∗ at 2 K is more pronounced in the field-down

measurements (see Fig. S8 in SM). We also note that the planar THE observed in Na2Co2TeO6 is

completely different from “planar Hall effects” studied in ferromagnets [37] and Weyl semimetals [38]

which are symmetric with respect to the field direction. In sharp contrast, the planar THE observed in

this compound only comes from the asymmetric component in the transverse temperature difference

(see section I in SM).

To investigate the origin of the planar THE, we turn to the temperature dependence of / at the

selected fields (see arrows in Fig. 5). As shown in Fig. 4, the temperature dependence of / shows

a peak at 4–10 K, which is followed by a decrease of �/� to zero as  → 0 K for both field

directions. Remarkably, we find that, although / at 10 T features one phonon peak at

approximately 30 K, / at 7, 9, and 11 T shows the second lower-temperature peak at

approximately 5 K, which almost coincides with the peak of / at 7 and 11 T.

Let us first focus on the origin of the planar THE observed in Na2Co2TeO6, which could be caused by

Majorana fermions [5], phonons [39], or topological magnons [12–15]. The magnitude of / of

the Majorana fermions in the Kitaev QSL is expected to show the half-quantized value 

2D/=

(/12)(B

2/ℏ), where B is the Boltzmann constant, and ℏ the reduced Planck constant. This half-

quantized thermal Hall conductance per one layer corresponds to /= (

2D/)

⁄∼

8.47 ×10−4 W K-2 m-1 in this compound, where = 0.559 nm is the interlayer distance [26].

However, as shown in Fig. 3 and Fig. 4, / observed in Na2Co2TeO6 remains about one order of

magnitude smaller than the half-quantized value. In addition, although / of fermions should

stay constant in the zero-temperature limit, / of Na2Co2TeO6 approaches zero with lowering

temperature (Fig. 4), showing the bosonic nature of the elementary excitations. In α-RuCl3, it is

pointed out that the half-quantized / is suppressed in the low-quality samples with suppressed



ph [8,9]. This is clearly not the case for our sample in which the phonon with a long mean free path

(Fig. S6 in SM) is observed. Indeed, the magnitude of  of our sample at 15 T (Figs. 1(a) and 1(b))

and that of the sample showing the half-quantized  in α-RuCl3 below the Néel temperature [8],

in which 

ph without the strong magnon-phonon scattering specific to each materials can be

observed, are similar in these two materials (approximately 6 W K-1 m-1 at 5 K, for example). It is also

pointed out that a good thermalization between the Majorana fermions and phonons is necessary to

observe the half-quantized / in the Kitaev QSL [40,41]. Although an accurate estimation for

the Majorana-phonon coupling in Na2Co2TeO6 is not possible at this moment, this coupling should be

related to the spin-phonon coupling that gives rise to the positive magneto-thermal conductivity. As

shown in Fig. 2, the thermal conductivity at 5 K increases by a factor of 8 under the in-plane field of

15 T, which is much larger than that in α-RuCl3 at a similar temperature and field [42]. Therefore, it

is implausible that the suppressed / is caused by a much smaller coupling between the Majorana

fermions and phonons in Na2Co2TeO6 than that in α-RuCl3. Furthermore, a finite  is prohibited

in  ∥ ∗ by the two-fold rotation symmetry around the ∗ axis in a Kitaev-Heisenberg

paramagnet [13–15], demonstrating the presence of a long-range ordered state that breaks the two-

fold rotation symmetry in  ∥ ∗. Therefore, we conclude that the emergence of Majorana fermions

is unlikely in Na2Co2TeO6.

As shown in Fig. 4, the planar THE develops below N, suggesting a dominant contribution from

topological magnons in Na2Co2TeO6. The abrupt sign changes of  observed at the magnetic phase

boundaries in the AFM phase (Fig. 5(b)) also suggest that the planar THE arises from the intrinsic

Berry phase effect of the topological magnons, because the sign changes can be most naturally

attributed to a redistribution of the Berry curvature of the magnons by the magnetic transitions. In fact,

the numerical calculations demonstrate sign changes of  of the magnons at the magnetic phase

transitions in the Kitaev-Heisenberg model [13,15,43]. On the other hand, it would be unlikely that

the abrupt sign changes are caused by a reversal of scattering direction upon collisions of magnons

with extrinsic impurities at the magnetic transitions.

Remarkably, a recent report of  done in  ∥  [35] shows a very different temperature

dependence from our planar THE. In  ∥  , a large  emerges far above N with a similar

temperature dependence with that of . This good scaling between  and  is regarded as

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1PlanarThermalHallEffectsinKitaevSpinLiquidCandidateNa2Co2TeO6AuthorsandAffiliations:HikaruTakeda1,JiancongMai1,MasatoshiAkazawa1,KyoTamura1,JianYan1,KalimuthuMoovendaran2,KalaivananRaju2,RamanSankar2,Kwang-YongChoi3,andMinoruYamashita11TheInstituteforSolidStatePhysics,UniversityofTokyo,Kashiwa,277-...

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1 Planar Thermal Hall Effect s in Kitaev Spin Liquid Candidate Na2Co2TeO 6

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