Thermal-transport studies of kagomé antiferromagnets

A kagomé structure consists of the corner-sharing network of triangles (figure 1(a)). Although this structure has been used as a woven pattern of Japanese bamboo baskets since ancient times, it is quite recently that the magnetic ground state realized in a kagomé structure started to attract tremendous attention in physics.

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The focus of the scientific interest is whether a quantum disordered state of spins, termed as a quantum spin liquid (QSL [1]), is realized for antiferromagnetic Heisenberg spins of $S=1/2$ on a kagomé lattice, and, if it is the case, what kind of a QSL emerges.

The reason why a kagomé structure is promising to realize a QSL is the strong effect of the geometrical frustration caused by the corner-sharing network of the triangles. This geometrical frustration enhances the quantum fluctuations of the constituting spins, giving rise to a melting of magnetic long-ranged ordered states by the enhanced quantum fluctuations even at the absolute zero temperature. In fact, no magnetic ordered state has been found in kagomé even in the large-scale numerical simulations [2]. This is in sharp contrast to a less-frustrated triangular lattice (figure 1(b)) where the 120° ordered state is known to be stabilized [3, 4].

A variety of QSLs have been suggested as the ground state in the kagomé Heisenberg antiferromagnet (KHA), including the resonating-valence-bond state [5], Z₂ spin liquids [2, 6, 7], chiral spin liquids [8], fermionic spin liquids [9], and Dirac spin liquids [10, 11]. The key to clarify the ground state is to reveal the elementary excitation by experiments in candidate materials, because these different QSLs are characterized by different quasiparticles.

How can we find these elementary exicatations by experiments? The magnetic excitations in QSLs have been studied by thermodynamic measurements and by spectoscopies such as NMR, µSR, and neutrons. These are direct and powerful methods, but are often obscured by impurity effects that can become dominant at low temperatures.

Recently, thermal-transport measurements have attracted attention to clarify the low-lying spin excitations. In magnetic insulators, not only phonons but also magnetic excitations can transport heat, which have been observed in various low-dimensional quantum magnets [12–14]. There are several advantages in thermal-transport measurements to study the low-lying spin excitations. First, the thermal conductivity (${{\kappa}_{xx}}$) is exclusively sensitive to itinerant excitations. One can thus avoid the effects of localized excitations by impurities even at very low temperatures. Second, ${{\kappa}_{xx}}$ directly reflects the mean free path of the elementary excitations. Therefore, one can estimate the quality of the sample by the magnitude of ${{\kappa}_{xx}}$. Third, because thermal conductivity measurement is a directional probe, one can study the anisotropy of the low-lying excitation by measuring ${{\kappa}_{xx}}$ along different directions. Finally, by applying a magnetic field, one can measure the thermal Hall conductivity (${{\kappa}_{xy}}$) which reflects further details of the low-lying excitation such as the Berry phase.

In particular, thermal Hall measurements have attracted increasing attention as a very powerful probe to reveal the elementary excitation in spin liquids [15, 16]. A thermal Hall effect (THE) is usually caused by the Lorentz force acting on the mobile electrons in a metal, which is known as the Righi-Leduc effect. Therefore, no THE is expected in an insulating QSL candidate material. However, even for charge neutral excitations, it has been shown that a Hall effect occurs in analog to the anomalous Hall effect [17, 18]. In an insulator, this Hall effect is observed by the transverse thermal gradient in response to an applied heat current, i.e. THE. As discussed below, ${{\kappa}_{xy}}$ is given by the relation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{\kappa}_{xy}}}{T}\propto \frac{k_{{\rm B}}^{2}}{\hbar}\int{f\left(E \right)\Omega \left(E \right)}{\rm d}E,\nonumber \end{align} \tag{ 1 }$$

where $f(E)$ is the distribution function of the elementary excitations and $\Omega (E)$ is the Berry curvature. Therefore, ${{\kappa}_{xy}}$ reflects the detail of the elementary excitation, for example, bosons or fermions, and the band structure.

In fact, the THEs have been observed in insulating ferromagnets [19–21] as a magnon Hall effect. In these ferromagnets, the Dzyaloshinskii–Moriya (DM) interaction provides a Berry curvature on the magnon bands, giving rise to a THE given by equation (1). THEs in insulators have also been observed in various materials, for example ${{\kappa}_{xy}}$ of quantum spins in the QSL candidate Tb₂Ti₂O₇ [22], ${{\kappa}_{xy}}$ of itinerant Majorana fermions in the Kitaev material α-RuCl₃ [23–25], ${{\kappa}_{xy}}$ of phonons in Tb₃Ga₅O₁₂ [26] and Ba₃CuSb₂O₉ [27], and ${{\kappa}_{xy}}$ in the cuprates [28]. THEs of charge neutral excitations have also been actively studied theoretically from magnon THE in antiferromagnets [29, 30] to triplon THE [31], quantized THE in the Kitaev QSL [32, 33], non-quantized THE of Schwinger bosons [34], fermionic U(1) spin liquids [35, 36] and gapless Dirac QSL [37].

In this article, we introduce our thermal-transport measurements of kagomé materials, volborthite [38] and Ca kapellasite [39]. In both compounds, the magnetic ground state is not a disordered state. However, the Néel temperature ${{T}_{{\rm N}}}$ is suppressed well below the temperature corresponding to the spin interaction energy $J$, providing a wide temperature range of a spin liquid state (${{T}_{{\rm N}}}<T<J/{{k}_{{\rm B}}}$) to study the elementary excitations.

First, we shall show the spin thermal conduction from the measurements of the longitudinal thermal conductivity, ${{\kappa}_{xx}}$ [15, 16]. We find that the spin contribution becomes bigger in crystals with higher ${{\kappa}_{xx}}$, showing that spin thermal conduction is easily suppressed by defects in crystals. In high-quality crystals of volborthite, clear changes in the magnetic thermal conduction are observed at the multiple magnetic transitions in volborthite.

We then report our successful measurements of ${{\kappa}_{xy}}$ both in volborthite [15] and Ca kapellasite [16]. In a sharp contrast to the case of ${{\kappa}_{xx}}$, crystals with smaller ${{\kappa}_{xx}}$ are better to measure ${{\kappa}_{xy}}$ because of larger transverse temperature difference (see equation (3)). However, ${{\kappa}_{xy}}$ is similar in volborthite and in Ca kapellasite despite the very different magnitude of ${{\kappa}_{xx}}$, showing that ${{\kappa}_{xy}}$ does not depend on the degree of the defects. Remarkably, we find that our numerical calculations by the Schwinger-boson mean field theory well reproduce the observed ${{\kappa}_{xy}}$ both qualitatively and quantitatively, suggesting that the elementary excitations in these kagomé materials are well described by bosonic spin excitations [6, 16, 40].

Figure 2 shows pictures of our thermal-transport measurement probe and a schematic of setup. A heat current ($Q||x$) and a magnetic field ($B||z$) are applied within the $ab$ plane and along the $c$ axis of the sample, respectively. Three thermometers (Cernox^™ (CX-1050) thermometers for measurements above 2 K and RuO₂ thermometers below 2 K) and one heater are attached to the sample so that both longitudinal ($\newcommand{\e}{{\rm e}} \Delta {{T}_{x}}\equiv {{T}_{{\rm High}}}-{{T}_{L1}}$) and transverse ($\newcommand{\e}{{\rm e}} \Delta {{T}_{y}}\equiv {{T}_{L1}}-{{T}_{L2}}$) thermal differences can be measured simultaneously (figure 2(a)). All thermometers were carefully calibrated in magnetic fields by a calibrated thermometer placed outside the magnetic field. We confirmed that the magnetoresistance of all the Cernox thermometers are identical to the previous work [41] within our resolution.

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The longitudinal thermal conductivity, ${{\kappa}_{xx}}$, and the thermal Hall conductivity, ${{\kappa}_{xy}}$, are obtained by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{wt}\left(\begin{matrix} Q \nonumber \\ 0 \nonumber \\ \end{matrix} \right)=\left(\begin{matrix} {{\kappa}_{xx}} & {{\kappa}_{xy}} \nonumber \\ -{{\kappa}_{xy}} & {{\kappa}_{xx}} \nonumber \\ \end{matrix} \right)\left(\begin{matrix} \Delta {{T}_{x}}/L \nonumber \\ \Delta {{T}_{y}}/w \nonumber \\ \end{matrix} \right),\nonumber \end{align} \tag{ 2 }$$

where $t$ is the thickness of the sample and $L$ ($w$) is the distance between the thermal contacts for ${{T}_{High}}$ and ${{T}_{L1}}$ (${{T}_{L1}}$ and ${{T}_{L2}}$), respectively. To cancel the longitudinal response in $\Delta {{T}_{y}}$ owing to misalignment of the contacts, the asymmetric component with respect to the field direction is estimated as $\newcommand{\e}{{\rm e}} \Delta T_{y}^{{\rm Asym}}\left(B \right)\equiv \left(\Delta {{T}_{y}}\left(+B \right)-\Delta {{T}_{y}}(-B) \right)/2$. To measure a small thermal Hall signal from an insulator, we carefully checked the background Hall signal from the measurement cell as described in the supplementary material⁶ (stacks.iop.org/JPhysCM/32/074001/mmedia).

An experimental requirement to measure ${{\kappa}_{xy}}$ is if $\Delta T_{y}^{{\rm Asym}}$ is larger than the resolution limit of the thermometer (typically $\delta T/T\sim {{10}^{-5}}$ for CX-1050). From equation (2), $\Delta {{T}_{y}}$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{T}_{y}}=-\frac{{{\kappa}_{xy}}}{{{\left({{\kappa}_{xx}} \right)}^{2}}}\frac{Q}{t}.\nonumber \end{align} \tag{ 3 }$$

Therefore, for a given ${{\kappa}_{xy}}$, requirements to resolve $\Delta {{T}_{y}}$ are

• 1.  
  A thinner sample with smaller ${{\kappa}_{xx}}$ is better.
• 2.  
  Applying $Q$ as large as possible is important. Because the maximum of $Q$ is often limited by the increase of the sample temperature above the base temperature, reducing the thermal resistance between the sample and the heat bath is necessary.

Volborthite, Cu₃V₂O₇(OH)₂·2H₂O, is a green transparent insulator (figure 2(b)) in which Cu²⁺ ions form a two dimensional distorted kagomé structure with inequivalent exchange interactions (figure 2(d)). The magnetic susceptibility ($\chi$) shows a typical temperature dependence of a frustrated magnet with a broad peak around 20 K (figure 2(f)). The fitting of the temperature dependence by a high-T expansion gives an estimate of the spin interaction energy $J\sim 84$ K [38]. Whereas the heat capacity ($C$) measurements of the powder samples show no sign of a magnetic order down to 1 K [42], that of single crystals show two peaks at 0.89 and 1.10 K [43], indicating a long-range magnetic order below ${{T}_{{\rm N}}}=1.1$ K. The heat capacity [44] and NMR [45] measurements have shown the multiple ordered phases—phase I' ($B<4.5$ T, 0.89 K $<T<{{T}_{{\rm N}}}$), phase I ($B<4.5$ T, $T<0.89$ K), phase II ($4.5<B<26$ T, $T<{{T}_{N}}$), and phase III (26 T $<B$, not shown)—in the $H$–$T$ phase diagram (figure 2(e)). The magnetic structure in the phase II has been suggested as a spin-density-wave (SDW) state [46, 47]. On the other hand, the magnetic structure in phase I and I' is unidentified. This is because a large single crystal appropriate for neutron scattering experiments is not available. Neutron scattering experiments have been performed only for powder sample [48] in which the spin Hamiltonian is shown to be different because of the different lattice structure [43]. As a result, the detail of the magnetic interactions of volborthite is yet to be clarified. The magnetization measurement up to high fields have found a wide one-third magnetization plateau [46], showing that the effective Hamiltonian of volborthite is not a simple kagomé Heisenberg antiferromagnet. First-principles calculations suggest that the effective Hamiltonian can be captured by a coupled frustrated chain model [50] or a coupled-trimers model [51]. The coupled-trimers model is supported by the recent magnetostriction measurement up to 45 T [49].

Figure 2(f) shows the temperature dependence of $C/T$ and $\chi$ of the samples synthesized in ISSP. We confirmed that the zero-field heat capacity data is almost the same with the previous report [43]. Below ~15 K, $C/T$ becomes much larger than that of the lattice heat capacity estimated by the Zn analogue compound in [42], showing the dominant magnetic contribution at low temperatures. Both the temperature dependence of $\chi$ and that of $C/T$ above ${{T}_{{\rm N}}}$ show a finite residual in the zero-temperature extrapolation (figure 2(f)), suggesting gapless excitations in the spin liquid state.

3.1. ${{{\kappa}_{xx}}}$ measurements in volborthite

Single crystals of volborthite have an arrowhead-like shape with one twin boundary in the middle (figure 2(b)). All single crystals used in the thermal conductivity measurements were detwinned by cutting in half at the middle as shown in figure 2(c).

Figure 3(a) shows the temperature dependence of ${{\kappa}_{xx}}/T$ of volborthite measured in several single crystals. Whereas a large sample dependence is observed in the magnitude of ${{\kappa}_{xx}}/T$, all data shows a similar temperature and field dependence. ${{\kappa}_{xx}}/T$ shows a broad peak at 3–4 K. This peak is suppressed by a magnetic field. This field suppression is observed to be larger for crystals with higher ${{\kappa}_{xx}}$. An increase of ${{\kappa}_{xx}}/T$ is observed below ${{T}_{{\rm N}}}\sim 1$ K (2 K) at zero field (15 T).

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The longitudinal thermal conductivity of an insulator consists of the phonon contribution $\kappa _{xx}^{{\rm ph}}$ and the spin contribution $\kappa _{xx}^{{\rm sp}}$. Both are given by the kinetic equation $\newcommand{\e}{{\rm e}} \kappa _{xx}^{{\rm ph(sp})}={{C}_{{\rm ph}({\rm sp})}}{{v}_{{\rm ph}({\rm sp})}}{{\ell}_{{\rm ph}({\rm sp})}}/3$, where ${{C}_{{\rm ph(sp)}}}$ is the phonon (spin) heat capacity, ${{v}_{{\rm ph(sp)}}}$ the sound (spin-wave) velocity, and $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}({\rm sp})}}$ the phonon (spin) mean free path. Given that ${{\kappa}_{xx}}$ at high temperature (~10 K) is dominated by $\kappa _{xx}^{{\rm ph}}$ as discussed below and that both ${{C}_{{\rm ph}}}$ and ${{v}_{{\rm ph}}}$ are independent of the impurity density, the different magnitude of ${{\kappa}_{xx}}$ of the volborthite samples directly reflects the magnitude of $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}$ in different crystals. Therefore, the sample with larger ${{\kappa}_{xx}}$ contains smaller defects.

The field suppression effect observed above ${{T}_{{\rm N}}}$ is attributed to the resonance scattering effect on $\kappa _{xx}^{{\rm ph}}$ and a field suppression effect on $\kappa _{xx}^{{\rm sp}}$ [15]. The resonance scattering is caused by absorption of the phonon energy into the Zeeman gap $g{{\mu}_{{\rm B}}}B$, where $g$ is the g factor (${{g}_{{\rm c}}}=2.28$ for volborthie [46]) and ${{\mu}_{{\rm B}}}$ is the Bohr magneton [52]. This resonance scattering is the most effective when the Zeeman energy is equal to the peak of the Debye distribution function ($\sim 4{{k}_{{\rm B}}}T$). Therefore, the decrease of ${{\kappa}_{xx}}/T$ from 0 to 15 T is mainly by the decrease of $\kappa _{xx}^{{\rm ph}}$ by the resonance scattering at higher temperatures ($T>6$ K). On the other hand, this resonance scattering is less effective at low temperatures (${{T}_{{\rm N}}}<T\lesssim 3$ K). Therefore, the decrease of ${{\kappa}_{xx}}/T$ by applying 15 T at this temperature range is attributed to the field suppression effect on $\kappa _{xx}^{{\rm sp}}$, providing an estimation of the fraction of the spin contribution in ${{\kappa}_{xx}}$. This spin contribution estimated by the decrease of ${{\kappa}_{xx}}/T$ at the low temperatures is larger for samples with larger ${{\kappa}_{xx}}/T$. This smaller fraction of $\kappa _{xx}^{{\rm sp}}$ in the smaller-${{\kappa}_{xx}}$ crystals shows that the magnetic thermal conduction is more easily scattered by defects in crystals compared to the phonon thermal conduction.

To reveal the unknown ordered phases I and I' and the magnetic interactions of volborthite by detecting $\kappa _{xx}^{{\rm sp}}$, ${{\kappa}_{xx}}$ of the sample #3 was measured down to 60 mK (figure 3(b)). As shown in figure 3(b), ${{\kappa}_{xx}}/T$ shows two kinks at the temperatures corresponding to the two phase transitions shown by the two peaks in $C/T$; ${{\kappa}_{xx}}/T$ is slightly decreased upon entering phase I' which is followed by the increase of ${{\kappa}_{xx}}/T$ in phase I.

Here, we first discuss the increase of ${{\kappa}_{xx}}/T$ in phase I. An increase in ${{\kappa}_{xx}}/T$ in a magnetic ordered phase can be caused by an increase of $\kappa _{xx}^{{\rm ph}}$ owing to a decrease of the spin-phonon scattering and/or the appearance of $\kappa _{xx}^{{\rm sp}}$ by the magnons. Because the phonon wavelength becomes longer at lower temperatures, $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}$ increases at lower temperatures until being limited by the sample size. This boundary-limited phonons give an upper limit of $\kappa _{xx}^{\,ph}$. We estimate this upper limit (the dashed line in figure 3(b)) by ${{C}_{{\rm ph}}}/{{T}^{3}}=0.34$ mJ K⁻⁴ Cu-mol⁻¹, ${{v}_{{\rm ph}}}=2.47\times {{10}^{3}}$ m s⁻¹, and $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}={{10}^{-4}}$ m, where the value of ${{C}_{{\rm ph}}}/{{T}^{3}}$ is estimated from the lattice data in [42], ${{v}_{{\rm ph}}}$ is evaluated by ${{C}_{{\rm ph}}}/{{T}^{3}}$ by the Debye relation, and $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}$ by the effective sample size ($\sim \sqrt{w\cdot t}$) of sample #3. As shown in figure 3(b), ${{\kappa}_{xx}}/T$ of the sample #3 in the phase I is much larger than the boundary-limited $\kappa _{xx}^{{\rm ph}}$, showing the dominant contribution of $\kappa _{xx}^{{\rm sp}}$ in phase I. Therefore, the increase of ${{\kappa}_{xx}}/T$ in phase I is attributed to the appearance of $\kappa _{xx}^{{\rm sp}}$ by the magnons. At the lowest temperature, ${{\kappa}_{xx}}/T$ approaches asymptotically to a linear temperature dependence of $\kappa _{xx}^{{\rm sp}}/T\propto T$ (the solid line in figure 3(b)). As is the case of $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}$, $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm sp}}}$ also becomes saturated at lower temperatures. Given that spin wave velocity is temperature independent, the asymptotic approach of $\kappa _{xx}^{{\rm sp}}/T\propto T$ shows the linear temperature dependence of ${{C}_{{\rm sp}}}/T$. This linear temperature dependence of ${{C}_{{\rm sp}}}/T$ (thus ${{C}_{{\rm sp}}}\propto {{T}^{2}}$) is consistent with the 2D magnon contribution with a linear dispersion.

The reason for the decrease in ${{\kappa}_{xx}}/T$ in phase I' is not clear at present. Because $C/T$ clearly increases in phase I', the decrease in ${{\kappa}_{xx}}/T$ should be attributed to the decrease of the mean free path of phonons or spins. Given that $\newcommand{\e}{{\rm e}} {{\ell}_{{\rm ph}}}$ usually increases in a magnetic ordered phase by reduced spin-phonon scatterings, the decrease in phase I' may be attributed to a suppression of $\kappa _{xx}^{{\rm sp}}$. NMR measuremensts have reported that the spin state becomes inhomogeneous in phase I' from the deviation of the NMR relaxation curve from a single exponential decay [47]. Therefore, one possible scenario for this suppression is that short-ranged magnetic domains are formed in phase I', which then give rise to additional scatterings of the magnons by the domain boundaries.

To get further insight into the unknown magnetic structure in phase I, the field dependence of ${{\kappa}_{xx}}$ in phases I and II was measured in sample #2 (figure 4). In this measurement, both ${{\kappa}_{xx}}$ along the $a$ axis and the $b$ axis were measured in sample #2 to investigate if there is an anisotropy in ${{\kappa}_{xx}}$ expected in the anisotropic coupled chain model suggested by the first-principles calculation [50].

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As shown in figures 4(a) and (b), the increase of ${{\kappa}_{xx}}/T$ observed in phase I at zero field is shifted to higher temperatures at higher fields, which is consistent with the increase of ${{T}_{{\rm N}}}$ in phase II. At lower temperatures, ${{\kappa}_{xx}}/T$ in phase I shows a similar linear temperature dependence with that in phase II, implying that the magnons near the zero energy have a similar linear energy dispersion in both phases although the location in the momentum space might be different.

The field dependence of ${{\kappa}_{xx}}(H)$ normalized by the zero-field value ${{\kappa}_{xx}}(0)$ at fixed temperatures is shown in figures 4(c) and (d). In phase I', ${{\kappa}_{xx}}(H)$ increases by fields, which is followed by a rapid increase upon entering phase II (see the data at 1.0 K in figures 4(c) and (d)). Whereas the field increase effect remains at lower temperatures in phase I' and phase I, the field dependence in phase II disappears at lower tempetures (see the data at 0.2 K in figures 4(c) and (d)). At 0.2 K, ${{\kappa}_{xx}}(H)$ shows a sharp drop from phase I to phase II, indicating a first-order phase transition between the two phases.

A change of the magnon spectrum by a magnetic transition alters both $\kappa _{xx}^{{\rm sp}}$ and $\kappa _{xx}^{{\rm ph}}$ as observed in Nd₂CuO₄ [53]. In volborthite, NMR measurement [47] has reported that a broad single peak in phase I becomes a double-horn shape in phase II, showing a change of the magnetic structure. Therefore, the different field dependence and the sharp drop of ${{\kappa}_{xx}}$ between phase I and phase II should reflect this change of the magnetic structure. Recently, a phase with an orthogonal spin order in the ac plane has been suggested as phase I by a theory based on the coupled trimer model with DM interactions [54]. The magnetic field perpendicular to the ab plane frustrates this order, which is expected to lead to a drastic change in elementary excitations. In contrast, the elementary excitations of the spin density wave state, which is a candidate for phase II, are phasons [55], whose velocity depends only slowly on the magnetic field. These considerations may be consistent with the different field dependences in these phases (figures 4(c) and (d)).

The difference between ${{\kappa}_{xx}}$ along the $a$ axis and that along the $b$ axis is smaller than the error in the estimation of the geometrical factor. This isotropic thermal conduction is more consistent with the coupled trimer model [51] rather than the anisotropic coupled-chain model [50].

3.2. ${{{\kappa}_{xy}}}$ measurements in volborthite

The thermal Hall effect was investigated in sample #1 with the smallest ${{\kappa}_{xx}}$ (figure 3(a)). Figure 5(a) shows the field dependence of the transverse temperature difference $\Delta {{T}_{y}}$ measured by different heat current $Q$, where the magnitude of the longitudinal temperature difference $\Delta {{T}_{x}}/T$ at the corresponding $Q$ is also shown. As shown in figure 5(a), the transverse temperature difference $\Delta {{T}_{y}}$ is dominated by the symmetric longitudinal response mixed by the misalignment effect. Figure 5(b) shows the field dependence of antisymmetrized transverse temperature difference $\Delta T_{y}^{{\rm Asym}}$. We find that a large heat current $Q$ for $\Delta {{T}_{x}}/T>10\%$ was necessary to increase $\Delta T_{y}^{{\rm Asym}}$ above the resolution limit (the error bar in figure 5(b)). Because the small $\Delta T_{y}^{{\rm Asym}}$, ${{\kappa}_{xy}}$ measurement was successful only in sample #1 with the smallest ${{\kappa}_{xx}}$, but not in other samples with larger ${{\kappa}_{xx}}$. The temperature dependence of ${{\kappa}_{xy}}$ is obtained by measuring $\Delta T_{y}^{{\rm Asym}}$ with a heat current to set $\Delta {{T}_{x}}/T$ as 10%–40% (the horizontal error bar in figure 9). As shown in figure 9, a negative ${{\kappa}_{xy}}/TB$ starts to appear at $T\lesssim J/{{k}_{{\rm B}}}$. At lower temperatures, $\left| {{\kappa}_{xy}}/TB \right|$ shows a broad peak ~10 K, which is followed by a sharp decrease and a sign inversion just above ${{T}_{{\rm N}}}$. This temperature dependence is discussed in detail later together with ${{\kappa}_{xy}}/TB$ of Ca kapellasite.

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Ca kapellasite, CaCu₃(OH)₆Cl₂·0.6H₂O, is a newly synthesized kagomé compound, in which Cu²⁺ ions form a non-distorted, ideal kagomé lattice (figure 6) [39]. Kapellasite is a structural polymorph of herbertsmithite, ZnCu₃(OH)₆Cl₂ [56]. Compared to herbertsmithite where the non-magnetic Zn²⁺ ions are located between the kagomé layers, the non-magnetic Ca²⁺ ions in Ca kapellasite are in the same kagomé layer, resulting in a better two dimensionality by the smaller coupling between the kagomé layers. Moreover, owing to the bigger ionic radii of Ca²⁺ ion than that of Cu²⁺, there is no site mixings in Ca kapellasite, in contrast to that in herbertsmithite [57].

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The spin interaction energy $J$ is estimated as ~60 K from the temperature dependence of the magnetic susceptibility and the first-principles calculation [16, 39]. Both the next-nearest neighbor spin interaction ${{J}^{\prime}}$ and the diagonal interaction ${{J}_{{\rm d}}}$ are estimated as negligile compared to $J$. Therefore, the spin Hamiltonian of Ca kapellasite is well approximated to an ideal KHA. The temperature dependence of $\chi$ and $C$ feature a peak at ${{T}_{{\rm N}}}\sim 7.4$ K, which is shown later as an onset of a long-ranged ordered state by NMR measurement [58].

4.1. ${{{\kappa}_{xx}}}$ measurements in Ca kapellasite

Figure 7 shows the temperature dependence of ${{\kappa}_{xx}}/T$ of Ca kapellasite with that of volborthite (sample #1). As shown in figure 7, ${{\kappa}_{xx}}/T$ of Ca kapellasite is about one order of magnitude smaller than that of sample #1 which shows the smallest ${{\kappa}_{xx}}/T$ measured in the crystals of volborthite (figure 3(a)). Given that the heat capacity of Ca kapellasite [39] is about two times larger than that of volborthite [43], this very small ${{\kappa}_{xx}}/T$ of Ca kapellasite can be attributed to a very short mean free path of phonons by the structural disorders in Ca kapellasite. In Ca kapellasite, Ca²⁺ ions are randomly located in the two equivalent Wyckoff positions. Moreover, there are vacancies for H₂O molecules in Ca kapellasite. These defects are likely to scatter phonons, giving rise to the small ${{\kappa}_{xx}}$.

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As lowering temperatures, only a small increase is observed in ${{\kappa}_{xx}}$ upon entering the magnetic ordered phase below ${{T}_{{\rm N}}}$. Also, the field effect in ${{\kappa}_{xx}}$ is smaller in Ca kapellasite than that in volborthite. These results show that $\kappa _{xx}^{{\rm sp}}$ has a negligible contribution in ${{\kappa}_{xx}}$ in Ca kapellasite owing to the structural disorders, which is consistent with the smaller contribution of $\kappa _{xx}^{{\rm sp}}$ in the crystals of volborthite with smaller ${{\kappa}_{xx}}$ (figure 3(a)). These systematic relations between the magnitude of $\kappa _{xx}^{{\rm sp}}$ and that of ${{\kappa}_{xx}}$ observed in volborthite and Ca kapellasite demonstrate that a magnetic thermal conduction is more easily scattered by defects in crystals than that by phonons.

4.2. ${{{\kappa}_{xy}}}$ measurements in Ca kapellasite

Although this small ${{\kappa}_{xx}}$ in Ca kapellasite is inconvenient to measure $\kappa _{xx}^{{\rm sp}}$, it turned out to work better to measure ${{\kappa}_{xy}}$. Remarkably, as shown in figure 8(a), the field dependence of $\Delta {{T}_{y}}$ observed in Ca kapellasite clearly shows an antisymmetric field dependence, which is in sharp contrast to that of volborthite in which antisymmetrizing $\Delta {{T}_{y}}$ is necessary to resolve the thermal Hall signal (figure 5). This large $\Delta T_{y}^{{\rm Asym}}$ observed in Ca kapellasite is not owing to a large ${{\kappa}_{xy}}$ in Ca kapellasite, but is simply due to small ${{\kappa}_{xx}}$ as expected in equation (3). Because of the large $\Delta T_{y}^{{\rm Asym}}$, a heat current $Q$ for $\Delta {{T}_{x}}/T\sim 5$% was good enough to resolve the thermal Hall signal, enabling us to measure the temperature dependence of ${{\kappa}_{xy}}$ of Ca kapellasite with fixed $\Delta {{T}_{x}}/T=5$% in the whole temperature range. As shown in figure 8(b), the field dependence of ${{\kappa}_{xy}}$ obtained by antisymmetrizing $\Delta {{T}_{y}}(H)$ shows a linear field dependence above ${{T}_{{\rm N}}}$. Below ${{T}_{{\rm N}}}$, ${{\kappa}_{xy}}$ shows a non-linear field dependence with a peak around 4 T, which becomes negligibly small above ~6 T. This drastic change in the field dependence of ${{\kappa}_{xy}}$ below ${{T}_{{\rm N}}}$ is attributed to the change of the elementary excitations by the magnetic order, demonstrating a better sensitivity of ${{\kappa}_{xy}}$ to a change of the elementary excitation in sharp contrast to the small change in ${{\kappa}_{xx}}$ at ${{T}_{{\rm N}}}$.

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Figure 9 shows the temperature dependence of ${{\kappa}_{xy}}/TB$ of Ca kapellasite above ${{T}_{{\rm N}}}$, which is determined by the linear fit of ${{\kappa}_{xy}}$ as a function of the field (the solid lines in figure 8(b)). Quite unexpectedly, we find that both the temperature dependence of ${{\kappa}_{xy}}/TB$ and the magnitude $\left| {{\kappa}_{xy}}/TB \right|$ of Ca kapellasite are similar to that of volborthite. This is in sharp contrast to ${{\kappa}_{xx}}$ which differ by one order of magnitude between the two kagomé materials (see figure 7). The very similar behaviors of $\left| {{\kappa}_{xy}}/TB \right|$ in the two kagomé materials suggest that the origin of the thermal Hall effect does not come from phonons, but rather from spins of which the magnitude of the interaction energy is similar in the two compounds.

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For further investigation of a thermal Hall effect by kagomé spins, we calculate ${{\kappa}_{xy}}$ by using Schwinger-boson mean field theory (SBMFT) [59]. In SBMFT, the spins are described by the pair of bosons $\left({{b}_{i\uparrow}},~{{b}_{i\downarrow}} \right)$ as

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle {{\boldsymbol{S}}_{i}}=\frac{1}{2}\sum\limits_{\alpha ,\beta =\uparrow ,\downarrow}{b_{i\alpha}^{\dagger}{{\boldsymbol{\sigma}}_{\alpha \beta}}{{b}_{i\beta}}},\nonumber \end{align}$$

where $\alpha ,\beta =\uparrow ,\downarrow$ and ${{\boldsymbol{\sigma}}_{\alpha \beta}}$ is the Pauli matrices.

We take the mean-field value of the bond operator $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}{{\chi}_{ij}}\,=$$\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger} \left\langle b_{i\sigma}^{\dagger}{{b}_{j\sigma}} \right\rangle$ to diagonalize the kagomé Heisenberg Hamiltonian

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H=\frac{1}{2}\sum\limits_{\left\langle i,j \right\rangle}{\left(J{{\boldsymbol{S}}_{i}}\cdot {{\boldsymbol{S}}_{j}}+D{{\boldsymbol{S}}_{i}}\times {{\boldsymbol{S}}_{j}}\cdot \hat{z} \right)}-g{{\mu}_{B}}\sum\limits_{i}{\boldsymbol{B}\cdot {{\boldsymbol{S}}_{i}}},\nonumber \end{align}$$

where $D$ is the $c$ axis component of the DM interaction.

Figures 10(a) and (d) show the energy bands calculated by the SBMFT method for kagomé Heisenberg antiferromagnet ($J>0$) and ferromagnet ($J<0$), respectively. From these energy bands, $\kappa _{xy}^{{\rm SBMF}}$ is calculated by the relation [17, 18]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\kappa _{xy}^{{\rm SBMF}}}{T}=-\frac{k_{{\rm B}}^{2}}{\hbar {{N}_{t}}}\sum\limits_{n,\boldsymbol{k},\sigma}{{{c}_{2}}\left[{{n}_{{\rm B}}}\left(\frac{{{E}_{n\boldsymbol{k}\sigma}}}{{{k}_{{\rm B}}}T} \right) \right]{{\Omega}_{n\boldsymbol{k}\sigma}}},\nonumber \end{align}$$

where ${{c}_{2}}\left(x \right)=\left(1+x \right){{\left(\ln \frac{1+x}{x} \right)}^{2}}-{{\left(\ln x \right)}^{2}}-2{\rm L}{{{\rm i}}_{2}}(-x)$ is a distribution function of the Schwinger bosons, ${\rm L}{{{\rm i}}_{2}}(x)$ polylogarithm function, ${{n}_{{\rm B}}}\left(x \right)={{\left({{e}^{x}}-1 \right)}^{-1}}$ the Bose–Einstein distribution function, ${{E}_{n\boldsymbol{k}\sigma}}$ the energy eigenvalue of $n$-th energy band, and ${{\Omega}_{n\boldsymbol{k}\sigma}}=i{{\partial}_{{{k}_{x}}}}{{u}_{n\boldsymbol{k}\sigma}}|{{\partial}_{{{k}_{y}}}}{{u}_{n\boldsymbol{k}\sigma}}+{\rm c}.{\rm c}.$ the Berry curvature (see [16] for details). The bands of up-spin Schwinger bosons and those of down-spin have exactly opposite ${{\Omega}_{n\boldsymbol{k}\sigma}}$ to each other. This degeneracy is lifted by a magnetic field, giving rise to a finite ${{\kappa}_{xy}}$ in the sum of the whole Brillouin zone.

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From the definition of the Berry curvature, ${{\Omega}_{n\boldsymbol{k}\sigma}}$ becomes larger in the momentum space where the energy gap between the bands is smaller, i.e. around the $\Gamma$ and the $K$ points (figures 10(a) and (d)). The contribution of these high ${{\Omega}_{n\boldsymbol{k}\sigma}}$ points to ${{\kappa}_{xy}}$ is determined by the energy dependence of the distribution function ${{c}_{2}}$. Owing to the monotonically-decreasing energy dependence of ${{c}_{2}}$, the high ${{\Omega}_{n\boldsymbol{k}\sigma}}$ point at lower energy has a larger contribution to ${{\kappa}_{xy}}$. As shown in figures 10(b) and (e), ${{\Omega}_{nk\sigma}}$ near $\Gamma$ point has the largest contribution for the kagomé antiferromagnet, whereas it is at $K$ point for the ferromagnetic case. Non-trivial temperature dependence of ${{E}_{n\boldsymbol{k}\sigma}}$, and thus that of ${{c}_{2}}$, determine the temperature dependence of ${{\kappa}_{xy}}$. As a result, the peak temperature of ${{\kappa}_{xy}}$ sensitively reflects the energy where ${{\Omega}_{n\boldsymbol{k}\sigma}}$ is large. This is demonstrated by the difference in the temperature dependence and the magnitude of ${{\kappa}_{xy}}$ between the antiferromagnet (figure 10(c)) and the ferromagnet (figure 10(f)), which is caused by the different energy position of the high-${{\Omega}_{n\boldsymbol{k}\sigma}}$.

To compare the calculated $\kappa _{xy}^{{\rm SBMF}}$ with the measurements of ${{\kappa}_{xy}}$ in volborthite and Ca kapellasite, the thermal Hall conductivity for one two-dimensional (2D) kagomé layer is estimated by $\kappa _{xy}^{2{\rm D}}={{\kappa}_{xy}}\times d$, where $d$ is the inter-layer distance (7.22 Å and 5.76 Å for volborthite and Ca kapellasite, respectively). Figure 11(a) shows $\kappa _{xy}^{2{\rm D}}$ normalized by $k_{{\rm B}}^{2}T/\hbar$ at 1 T as a function of ${{k}_{{\rm B}}}T/J$ where $J$ is assumed as 60 K for both compounds.

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Most remarkably, both the temperature dependence and the magnitude of $\kappa _{xy}^{2{\rm D}}$ are close to the calculated $\kappa _{xy}^{{\rm SBMF}}$ for kagomé antiferromagnet. We further normalize both the experiment and the numerical results by one-variable scaling function ${{\tilde{f}}_{{\rm SBMF}}}\left(x \right)$ and $\newcommand{\e}{{\rm e}} {{\tilde{f}}_{\exp}}\left(x \right)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\kappa _{xy}^{{\rm SBMF}}}{T}=\frac{k_{{\rm B}}^{2}}{\hbar}\cdot \frac{D}{J}\cdot \frac{g{{\mu}_{{\rm B}}}B}{J}{{\tilde{f}}_{{\rm SBMF}}}\left(\frac{{{k}_{{\rm B}}}T}{J} \right)\nonumber \end{align}$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\kappa _{xy}^{2{\rm D}}}{T}=\frac{k_{{\rm B}}^{2}}{\hbar}\cdot \frac{D}{J}\cdot \frac{g{{\mu}_{{\rm B}}}B}{J}{{\tilde{f}}_{\exp}}\left(\frac{{{k}_{{\rm B}}}T}{J} \right),\nonumber \end{align}$$

where $D$ and $J$ in $\newcommand{\e}{{\rm e}} {{\tilde{f}}_{\exp}}\left({{k}_{{\rm B}}}T/J \right)$ are fitting parameters.

As shown in figure 11(b), the numerical and the experimental results show a perfect agreement by choosing ($J/{{k}_{{\rm B}}}$, $D/J$)  =  (60, −0.07) and (66, 0.12) for volborthite and Ca kapellasite, respectively. The fitting results of $J$ are close to the value estimated by the temperature dependence of the magnetic susceptibility and those of $D/J$ are in good agreement with $({{g}_{{\rm c}}}-2)/{{g}_{{\rm c}}}$, further supporting the agreement between the experiment and the theory. This excellent agreement, both qualitative and quantitative, demonstrates not only that the thermal Hall effect in these kagomé antiferromagnets is caused by spins in the spin liquid phase, but also that the elementary excitations of this spin liquid phase are well described by the bosonic spin excitations. These spin excitations can be detetcted by neutron scattering experiments if a large single crystal becomes available, which remains as an important future work.

As discussed in section 3, the effective spin Hamiltonian in volborthite is not a kagomé Heisenberg antiferromagnet at low temperatures. However, the difference from the observed ${{\kappa}_{xy}}$ and that of SBMFT is not clearly observed, implying that a KHA still gives a good approximation in the temperature range where ${{\kappa}_{xy}}$ is observed. Numerical calculations of ${{\kappa}_{xy}}$ for the spins in the trimer model remain a future issue.

It is also an open question if the observed ${{\kappa}_{xy}}$ is reproduced by calculations for other spin liquid states, in particular spin liquids with fermionic spinons. We note that the results of the ferromagnetic kagomé (figure 10(f)) clearly fail to reproduce the observed ${{\kappa}_{xy}}$ both in the temperature dependence and the magnitude. Also, the thermal Hall effect of the 120${}^\circ$-ordered state in a kagomé antiferromagnet is shown to have a different ${{\kappa}_{xy}}$ [60]. Therefore, short-ranged magnons are excluded as the origin of ${{\kappa}_{xy}}$ observed in the two compounds.

We investigated both longitudinal (${{\kappa}_{xx}}$) and transverse (${{\kappa}_{xy}}$) thermal conductivities of kagomé material volborthite and Ca kapellasite. From ${{\kappa}_{xx}}$ measurements in volborthie, a spin contribution is found to be larger in crystals with larger ${{\kappa}_{xx}}$. In the highest-${{\kappa}_{xx}}$ crystal, the dominant magnetic contribution is observed in the ordered state, which shows ${{\kappa}_{xx}}\propto {{T}^{2}}$ at the lowest temperatures. A different field dependence in ${{\kappa}_{xx}}$ is observed in phase I and II in volborthite, and ${{\kappa}_{xx}}$ is isotropic in the ab plane. These results put strong constraints on the unknown magnetic interactions in volborthite. In Ca kapellasite, ${{\kappa}_{xx}}$ is so small that a magnetic contribution is not observed. On the other hand, a spin thermal Hall effect is observed in both volborthite and Ca kapellasite. We find that κ_xy in both compounds can be well reproduced by Schwinger-boson mean field theory, demonstrating that the elementary excitations in these kagomé material are well described by bosonic spin excitations.

We thank fruitful discussions with S Furukawa and T Momoi. This work was supported by Yamada Science Foundation, Toray Science Foundation, KAKENHI (Grants-in-Aid for Scientific Research) Grant No. 18H05227, No. 19H01809 and No. 19H01848, and Grants-in-Aid for Scientific Research on Innovative Areas 'Topological Material Science' (No. 15H05852) from Japan Society for the Promotion of Science (JSPS). HL and NK were supported by MEXT as 'Exploratory Challenge on Post-K computer' (Frontiers of Basic Science: Challenging the Limits).