Introduction

In the last decade, the thermal Hall effect has become a useful probe of insulators1, because it can reveal whether the carriers of heat in a material have chirality. (Here we use the term “chirality” to mean handedness in the presence of a magnetic field.) In insulators, the carriers of heat are not charged, but neutral, and so the electrical Hall effect is zero. The thermal Hall conductivity κxy is measured by sending a heat current along the x-axis and detecting a transverse temperature gradient along the y-axis, in the presence of a perpendicular magnetic field (along the z-axis). It has been shown that in certain conditions, spins can produce such chirality2. For example, magnons give rise to a thermal Hall signal in the antiferromagnet Lu2V2O7 (ref. 3). As a result, a measurement of the thermal Hall effect can in principle provide access to various topological excitations in insulating quantum materials, such as Majorana edge modes in chiral spin liquids4. Recently, the thermal Hall conductivity κxy seen in α-RuCl3 below T ≃ 80 K (refs. 5,6) has been attributed to the excitations of a Kitaev spin liquid7. Similarly, the κxy signal observed in some frustrated magnets—in which there is no magnetic order down to the lowest temperatures—has been attributed to spin-related heat carriers8.

However, phonons can also generate a nonzero thermal Hall conductivity if some mechanism confers chirality to them. For instance, an intrinsic mechanism is the Berry curvature of phonon bands acquired from a magnetic environment9. In the ferrimagnetic insulator Fe2Mo3O8, the large κxy signal is attributed to the strong spin–lattice coupling characteristic of multiferroic materials10. An extrinsic mechanism is the skew scattering of phonons by rare-earth impurities11, as in the rare-earth garnet Tb3Ga5O12 (refs. 12,13). Recently, a large phononic κxy has been observed in the nonmagnetic insulator SrTiO3 (ref. 14). A proposed explanation involves the large ferroelectric susceptibility of this oxide insulator, together with an extrinsic mechanism whereby phonons are scattered by the polar boundaries from the antiferrodistortive structural transition at 105 K (ref. 15). This interpretation is supported by the fact that κxy is negligible in the closely related material KTaO3 (ref. 14), which remains cubic and free of structural domains.

In cuprates, a large negative κxy signal was observed at low temperature inside the pseudogap phase16, i.e., for dopings p < p*, where p* is the pseudogap critical doping17. Because it persists down to p = 0, in the Mott insulator state, this negative κxy cannot come from charge carriers, which are not mobile at p = 0. Therefore, it must come either from spin-related excitations (possibly topological, as in refs. 18,19) or from phonons (as in ref. 15). To distinguish between these two types of heat carriers, a simple approach was recently adopted: the thermal Hall conductivity was measured for a heat current along the c-axis, normal to the CuO2 planes, a direction in which only phonons move easily20. In La2CuO4, at p = 0, the thermal Hall signal was found to be just as large as for an in-plane heat current, i.e., κzy(T) ≈ κxy(T) (ref. 21). This is compelling evidence that phonons are the heat carriers involved in the thermal Hall signal of this insulator. Moreover, it was found that this phonon Hall effect vanishes entirely immediately outside the pseudogap phase, i.e., κzy(T) = 0 at p > p*, revealing that phonons only become chiral upon entering the pseudogap phase21.

The question is: what makes the phonons in cuprates become chiral? In order to provide answers to this question, we have investigated two other cuprate Mott insulators, Nd2CuO4 and Sr2CuO2Cl2, and find in both a large negative thermal Hall conductivity similar to that of La2CuO4. While the three materials share the same fundamental characteristic of cuprates, namely they are a stack of single CuO2 planes, there are significant differences between them (see “Methods” and Fig. 1). Our comparative study allows us to conclude that none of the distinguishing features—orthorhombicity, structural domain boundaries, apical oxygens, spin canting, noncollinear alignment of spins, and nature of the cation—play a key role in causing the chirality. This points to a chiral mechanism associated with the coupling of phonons to the CuO2 planes themselves.

Fig. 1: Crystal structure of La2CuO4, Nd2CuO4, and Sr2CuO2Cl2.
figure 1

Sketch of the crystal structure of the three single-layer cuprate Mott insulators compared in the present study: a La2CuO4, b Nd2CuO4, and c Sr2CuO2Cl2. Note that the small orthorhombic distortion in La2CuO4 below 530 K is not shown here, nor is the tilt in the oxygen octahedra surrounding the Cu atoms.

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Results

Thermal Hall conductivity

In Fig. 2, we show our data for κxx and κxy in Sr2CuO2Cl2 (sample A) and Nd2CuO4. We see that as in La2CuO4, both materials show a large negative thermal Hall signal. We also observe a certain field dependence of κxx, larger than the small one observed in La2CuO4 (ref. 16). In Sr2CuO2Cl2, the field increases κxx slightly, below T ≈ 20 K. In Nd2CuO4, the field decreases κxx, below T ≈ 40 K.

Fig. 2: Thermal transport in Sr2CuO2Cl2 and Nd2CuO4.
figure 2

a Thermal conductivity of Sr2CuO2Cl2 (sample A) in zero field (H = 0, black) and in a field of 15 T applied parallel to the c-axis (light red), plotted as κxx/T vs T. The field is seen to increase κxx slightly at low temperature. b Thermal Hall conductivity of Sr2CuO2Cl2 (same sample) in a field of 15 T applied parallel to the c-axis, plotted as κxy/T vs T. c Thermal conductivity of Nd2CuO4, plotted as κxx/T vs T, for three values of the magnetic field applied parallel to the c-axis: H = 0 (black), H = 10.6 T (blue), and H = 15 T (light red). In this case, the field is seen to decrease κxx at low temperature. d Thermal Hall conductivity of Nd2CuO4, plotted as κxy/(TH) vs T, for two values of the magnetic field applied parallel to the c-axis: H = 10.6 T (blue); H = 15 T (light red). The Hall conductivity κxy is seen to be sublinear in H at low T and linear in H at high T (T > 50 K). All lines are a guide to the eye.

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In Fig. 3, we compare the three cuprate Mott insulators. We observe that the curves of −κxy vs. T (right panels) are similar in shape, peaking at T ≈ 25 K, a temperature close to that where κxx vs. T peaks (left panels). At low temperature, κxx is dominated by phonons. Indeed, because there is a gap in the magnon spectrum of these antiferromagnets22, their contribution to κxx becomes negligible at low T compared to the phonon contribution. At T = 35 K, the magnon conductivity is only 2% of the measured κxx (ref. 20), and it rapidly becomes vanishingly small below that temperature. At T = 20 K, the magnitude of κxx is 8 times larger in Nd2CuO4 compared to Sr2CuO2Cl2 (Fig. 3). So, phonons are a lot more conductive in Nd2CuO4. We see from Fig. 3e, f that κxy is correspondingly (ten times) larger in Nd2CuO4. This is strong evidence that phonons are the heat carriers responsible for the Hall response.

Fig. 3: Thermal Hall conductivity in the three Mott insulators.
figure 3

Left panels: thermal conductivity of the three cuprate Mott insulators, plotted as κxx vs T: a La2CuO4, b Nd2CuO4, and c Sr2CuO2Cl2. Right panels: Corresponding thermal Hall conductivity, plotted as −κxy vs T: d La2CuO4, e Nd2CuO4, and f Sr2CuO2Cl2. All data shown in this figure are taken in a field of 15 T (along the c-axis).

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In Fig. 4, we plot the ratio κxy/κxx vs T for the three materials. We see that not only is this ratio of similar magnitude in the three cuprates, but its temperature dependence is also very similar, growing with decreasing T to reach a maximal (negative) value at T ≈ 10–15 K, where |κxy/κxx| ≈ 0.3–0.4% (at H = 15 T).

Fig. 4: Ratio of κxy over κxx.
figure 4

Ratio of κxy over κxx in the three cuprate Mott insulators (expressed in %), measured in a field of 15 T applied parallel to the c-axis: Sr2CuO2Cl2 (green), La2CuO4 (blue), and Nd2CuO4 (red). All lines are a guide to the eye. We see that despite a factor 10 in the magnitude of κxy between Sr2CuO2Cl2 and Nd2CuO4 (Fig. 2), the ratio κxy/κxx is very similar in magnitude for all three cuprates.

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Having observed a large negative thermal Hall conductivity κxy in both Nd2CuO4 and Sr2CuO2Cl2 that is very similar to that previously reported for La2CuO4 (i.e., of comparable magnitude when measured relative to κxx) allows us to draw several conclusions about the underlying mechanism for chirality in the cuprate Mott insulators.

Spin canting

It has been shown theoretically that in ferromagnetic or antiferromagnetic insulators, under certain conditions, magnons can have chirality and should give rise to a thermal Hall effect1. In the collinear Néel antiferromagnetic order of La2CuO4, no thermal Hall effect is expected theoretically, because of the so-called “no-go” theorem, which states that Néel order on a square lattice has zero chirality1. However, if the spins of the Néel order cant out of the plane, as they do in La2CuO4, due to some Dzyaloshinskii–Moriya (DM) interaction, then some chirality becomes possible. In this case, one could get a nonzero κxy signal, but it is expected to be much smaller than the measured κxy signal in La2CuO4 (ref. 23). Our data on Nd2CuO4 and Sr2CuO2Cl2 completely eliminate this possibility, because a κxy signal of similar or larger magnitude is found in these materials for which there is no canting of spins out of the plane (see “Methods”), and so no DM interaction. We conclude that magnons are not responsible for the thermal Hall effect in cuprate Mott insulators. This conclusion is consistent with the fact that in La2CuO4 a large κxy signal persists down to temperatures well below the smallest magnon gap, of magnitude 26 K (ref. 22), and up in doping well above the critical doping for the suppression of Néel order, i.e., p ≈ 0.02 in La2−xSrxCuO4 (ref. 16).

As for a phonon scenario whereby phonons would acquire chirality through their coupling to spins, spin canting also appears to be unimportant.

Let us now consider two mechanisms known to confer chirality to phonons in other materials—skew scattering off rare-earth impurities and scattering off structural domain boundaries—and show that neither is relevant to cuprates.

Nature of cation

The initial observation of a phonon thermal Hall effect, in the garnet Tb3Ga5O12 (refs. 12,13), has been attributed to the skew scattering of phonons by superstoichiometric Tb3+ ions11. This extrinsic mechanism depends crucially on the details of the crystal-field levels of the rare-earth ion. A different rare-earth ion will in general produce skew scattering of a very different strength. The fact that the ratio κxy/κxx is the same in all three cuprates considered here is compelling evidence that the underlying mechanism does not depend on the nature of the particular cation, whether La, Sr, or Nd.

Note also that strong skew scattering by rare-earth impurities shows up as a major reduction in κxx (ref. 24). In Tb3Ga5O12, 2% of Tb3+ impurities gives rise to both a finite κxy signal from phonons (whose magnitude is given in Table 1) and a fivefold reduction in κxx (ref. 13), whose value at T = 15 K is then only κxx = 1.2 W/Km (ref. 11). In the pyrochlore oxide Tb2Ti2O7, a frustrated magnet with a sizable thermal Hall effect8 (Table 1), κxx is massively reduced compared to Y2Ti2O7, by a factor 15 at T = 15 K (H = 0) (ref. 25), pointing again to strong scattering of phonons by Tb3+ ions. (Note that the thermal Hall effect in Tb2Ti2O7 has recently been attributed to phonons26.) By comparison, the thermal conductivity in the cuprate Mott insulators is an order of magnitude larger (Table 1), evidence that no strong skew scattering is at play: κxx = 10 W/Km in La2CuO4, 45 W/Km in Nd2CuO4, and 6 W/Km in Sr2CuO2Cl2, at T = 15 K (H = 15 T, Fig. 3). We conclude that skew scattering of phonons by superstoichiometric cation atoms is not the mechanism that confers chirality to phonons in cuprates.

Table 1 Thermal Hall conductivity in various oxide insulators.
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Structural domains

In the nonmagnetic insulator SrTiO3, a negative thermal Hall conductivity was recently observed14, with a magnitude comparable to that of the three cuprate Mott insulators (Table 1). There is little doubt that the thermal Hall effect in SrTiO3 is due to phonons. Importantly, the κxy signal in the closely related oxide KTaO3 is 30 times smaller (and of opposite sign)14 (Table 1). The key difference between the two materials is that SrTiO3 undergoes an antiferrodistortive structural transition at 105 K, whereas KTaO3 remains cubic down to T ≈ 0 K. The authors of the study on those two materials conclude that the large signal in SrTiO3 is linked to the structural domain boundaries that exist below 105 K (ref. 14), although the precise mechanism whereby these confer chirality to phonons is still unclear. Our comparative study of the three cuprates allows us to rule out a similar role for structural domains. Indeed, whereas La2CuO4 undergoes a structural transition to an orthorhombic phase below 530 K, both Nd2CuO4 and Sr2CuO2Cl2 remain tetragonal down to T ≈ 0 K, and yet all three have a similar thermal Hall effect, in both T dependence (Fig. 3) and magnitude—relative to κxx (Fig. 4 and Table 1).

Magnetostructural domains

Because the collinear spin order in Sr2CuO2Cl2 breaks the fourfold symmetry of the lattice, there will be antiferromagnetic domains below TN and these will in principle be accompanied by an orthorhombic distortion of the tetragonal lattice aligned with the moment direction in each domain. To investigate the possible effect of these putative structural distortions, we have measured κxy in the same sample of Sr2CuO2Cl2 (sample B) under three different conditions: (1) for a field H = 10.6 T applied along the c-axis, (2) for a field H = 15 T applied at an angle of 45° from the c-axis (whose components normal and parallel to the CuO2 planes are both 10.6 T), applied at T = 2 K (zero-field cooling), and (3) same as for (2), but applied at T = 300 K > TN (in-field cooling). In the latter in-field cooling condition, the in-plane component of the field (of magnitude 10.6 T) applied at T > TN will ensure that a single antiferromagnetic domain is present below TN. (We expect the in-plane field needed to create a monodomain to be approximately 5 T, as verified in YBa2Cu3O6 (ref. 27).) Comparing conditions (2) and (3) amounts to comparing a multidomain sample vs a monodomain sample.

The results of this comparative study are displayed in Fig. 5. We see that κxy is identical in the three situations, within error bars. So, magnetic domains in Sr2CuO2Cl2, and any associated structural distortions, do not influence the thermal Hall response. Note that the noncollinear order in Nd2CuO4 does not break the fourfold symmetry of the lattice, so here no magnetic domains are expected.

Fig. 5: Effect of magnetostructural domains in Sr2CuO2Cl2.
figure 5

Thermal transport in Sr2CuO2Cl2 (sample B) for a heat current J//a, measured as a function of increasing temperature from T = 2 K up to 80 K, in three different conditions: (1) for a field H = 10.6 T along the c-axis, applied at T = 300 K (blue circles), (2) for a field H = 15 T at 45° from the c-axis (meaning equal in-plane and out-of-plane fields, i.e., H ⊥ c = H // c = 10.6 T), applied at T = 2 K (zero-field cooling, red squares), and (3) for a field H = 15 T at 45° from the c-axis, applied at T = 300 K (in-field cooling, green circles). All lines are a guide to the eye. In conditions (1) and (2), we expect that multiple orthorhombic magnetostructural domains and associated boundaries exist below TN = 270 K. Condition (3)—in-field cooling in the presence of an in-plane field of 10.6 T—ensures that a single antiferromagnetic domain exists when magnetic order sets in below TN, and so there should be no, or very few, structural domain boundaries in that case. a Thermal conductivity κxx vs T. There is no detectable difference between the three curves, showing that the presence of magnetostructural domains has a negligible impact on the phonon thermal conductivity. b Thermal Hall conductivity κxy vs. T. Within error bars, there is no significant difference between the three curves, demonstrating that magnetostructural domains do not play a significant role in generating the thermal Hall signal in Sr2CuO2Cl2.

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We conclude that structural (or magnetostructural) domains are not the mechanism that confers chirality to phonons in cuprates. Moreover, the thermal Hall conductivity of cuprates is independent of whether the system has orthorhombic or tetragonal symmetry, or whether there are apical oxygens in the structure or not.

In summary, our results show that the cuprate Mott insulators Nd2CuO4 and Sr2CuO2Cl2 exhibit a large negative thermal Hall conductivity κxy very similar to that found in La2CuO4. The fact that the magnitude of κxy scales with the magnitude of the phonon-dominated κxx as the latter varies by a factor 10 between Sr2CuO2Cl2 and Nd2CuO4 is further evidence in favor of phonons as the carriers of heat responsible for the thermal Hall effect in these materials. Given the different crystal structures and cations involved in those three materials, the similarity in κxy/κxx allows us to rule out two extrinsic mechanisms of phonon chirality proposed for other oxides, namely the scattering off rare-earth impurities—invoked for Tb3Gd5O12—and the scattering off structural domain boundaries—invoked for SrTiO3. This suggests that phonon chirality in the cuprates comes from an intrinsic coupling of phonons to their environment.

Discussion

Phonons can acquire chirality through a coupling to their intrinsic environment (see, e.g., ref. 9). This could involve a coupling to charge or a coupling to spin, for example. In ref. 15, a flexoelectric coupling of phonons to their charge environment was shown to generate a Hall response. However, even in the nearly ferroelectric insulator SrTiO3, where the electric polarizability is exceptionally large, this intrinsic mechanism is estimated to be much too small. The inclusion of some additional, extrinsic, scattering mechanism—possibly structural domain boundaries—is deemed necessary. Applied to cuprates, the intrinsic flexoelectric coupling is certainly much too small. It is not clear what extrinsic mechanism could be added to make this mechanism strong enough to account for the observed data in the cuprate Mott insulators.

In multiferroic materials like Fe2Mo3O8, a large κxy signal is observed even in the paramagnetic phase10, where κxy/κxx ≃ 0.5% (at T = 65 K and H = 14 T) (Table 1). This is attributed to a strong spin–lattice coupling. In cuprates, a coupling of phonons to spins in their environment should be investigated as a possible source of chirality.

Another avenue of investigation for cuprates is the possibility that they harbour exotic chiral excitations, like spinons18,19 that could couple to phonons. Such a coupling has recently been considered for the case of Majorana fermions in a Kitaev spin liquid28.

In a scenario of phonons coupled to their environment, there could be two relevant regimes of temperature, namely above and below the peak in κxy vs. T, so roughly above 25 K and below 15 K, respectively (Fig. 3). At temperatures above the peak, it has been shown that if the heat carriers have Berry curvature, they would be expected to exhibit a characteristic exponential dependence, namely κxy/T ∝ exp(−T/T0) (ref. 29). In Fig. 6, we fit our data on Sr2CuO2Cl2 and Nd2CuO4 to that form and find a good fit over the intermediate temperature range from 30 to 100 K. An equally good fit is found for La2CuO4 (ref. 21). Whether this implies that phonons acquire a Berry curvature through their coupling to the environment remains to be determined. At low temperature, we would expect phonons to eventually decouple from their environment, whether that be spins or other excitations of electronic origin. The temperature below which they do so would shed light on the nature of that coupling. In Fig. 4, we see that upon cooling below 10 K, |κxy| in Sr2CuO2Cl2 falls more rapidly to zero than κxx does. (Our current data on Nd2CuO4 and La2CuO4 do not allow us to explore their regime below 10 K.) In Fig. 7, we zoom on the low-T regime in Sr2CuO2Cl2. We see that whereas κxx/T3 rises monotonically as T → 0, κxy/T3 drops rapidly toward zero, starting roughly at 5 K. We identify 5 K as the approximate decoupling temperature between acoustic phonons and their chiral environment.

Fig. 6: Phenomenological fit to the phonon thermal Hall conductivity.
figure 6

Thermal Hall conductivity, plotted as κxy/T vs T in a Sr2CuO2Cl2 (sample A) and b Nd2CuO4. The data are fit to the phenomenological expression κxy/T = A exp(−T/T0) + C from ref. 29. The fit interval is 30–100 K. The resulting fit parameters are a A = − 5 mW/K2m, C = − 0.03 mW/K2m, T0 = 16 K; b A = − 67 mW/K2m, C = − 0.3 mW/K2m, T0 = 12 K.

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Fig. 7: Low-temperature regime in Sr2CuO2Cl2.
figure 7

Thermal conductivity κxx and thermal Hall conductivity κxy of Sr2CuO2Cl2 (sample A), plotted as κxx/T3 (blue) and −κxy/T3 (red) vs T. The dashed lines are a guide to the eye. Note that the κxy data are multiplied by a factor 325, to compare them more easily with κxx, using only one y-axis scale.

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It is instructive to compare our data on undoped cuprates to prior data on hole-doped cuprates. At a doping p = 0.24, in both Nd-LSCO and Eu-LSCO, the thermal Hall signal coming from phonons—as opposed to charged carriers—is zero21 (Table 1). It only becomes nonzero when the doping is reduced below the critical doping for the pseudogap phase, i.e., when p < p* (p* = 0.23). The magnitude of κxy is relatively constant from p* down to p = 0 when measured relative to κxx (see Table 1), and the sign is negative throughout. This continuity suggests that the same chiral mechanism is at play in the Mott insulator and within the pseudogap phase.

Moreover, because the phononic κxy signal in Nd-LSCO goes from zero at p = 0.24 to its full value at p = 0.21, rising abruptly upon crossing below p*, this chiral mechanism must be an intrinsic property of the pseudogap phase—since there is no change in the crystal structure21,30 and little change in the amount of impurity scattering between p = 0.24 and p = 0.21. This is consistent with our finding that structural domains and cation impurities are unimportant, and it extends the argument to all other defects and impurities, e.g., oxygen vacancies, all of which are essentially unchanged between p = 0.24 and p = 0.21. (In a scenario of oxygen vacancies screened by mobile charge carriers, we would expect zero screening at p = 0, in the insulator, so we should see a much larger κzy signal in La2CuO4 than in Nd-LSCO at p = 0.21, for example. This is not the case, on the contrary. As seen from Table 1, the κzy signal is larger at p = 0.21, relative to κzz: |κzy/κzz| = 0.48% in Nd-LSCO p = 0.21 vs. 0.2% in La2CuO4.)

One intrinsic mechanism has recently been proposed whereby phonons couple to an electronic state that breaks time-reversal and inversion symmetries, which would be realized in the pseudogap phase of cuprates31.

Another possible mechanism is the coupling of phonons to short-range antiferromagnetic correlations. Experimental evidence for such correlations includes the Fermi-surface transformation across p* observed by angle-dependent magnetoresistance32 and the drop in carrier density across p* observed in the electrical Hall effect33,34,35, both consistent with spin modulations with a wavevector Q = (π, π). Solutions of the Hubbard model in the paramagnetic state find that, in doped Mott insulators such as the cuprates, local moments36,37 persist all the way from half-filling up to a critical doping where the pseudogap disappears38,39,40. In calculations within the pseudogap phase, superexchange between local moments naturally favors short-range antiferromagnetic41,42,43 or singlet correlations36,37. In such a scenario, the question becomes: how can the coupling of phonons to spins make these phonons chiral (in the presence of a magnetic field)?

Methods

Crystal structures

La2CuO4: La2CuO4 is the parent compound of the most widely studied family of single-layer cuprates, La2–xSrxCuO4. In La2CuO4, there is an (apical) oxygen atom above the Cu atom, thereby forming an octahedron of O atoms around Cu (Fig. 1). Upon cooling from high temperature, La2CuO4 goes from a tetragonal (I4/mmm) structure to an orthorhombic (Cmca) structure at 530 K (ref. 44), wherein the octahedra are tilted (the orthorhombic distortion and associated tilt are not shown in Fig. 1). This means that unless they are deliberately detwinned by application of uniaxial stress, crystals of La2CuO4 will be full of orthorhombic structural domains (twins) whose boundaries can in principle scatter phonons. (The samples of La2CuO4 studied in refs. 16,21 were twinned.) Below TN = 270 K, the Cu spins order into a collinear antiferromagnetic arrangement, whereby all alternating moments point along the same direction ([110]) within every CuO2 plane inside a given orthorhombic domain. The tilting of the oxygen octahedra causes a slight canting of the spins out of the CuO2 plane (by 0.17°) (ref. 45), thereby producing a Dzyaloshinskii–Moriya (DM) interaction that could, in principle, be a source of chirality.

Nd2CuO4: Nd2CuO4 is the parent compound of the electron-doped family of cuprates Nd2−xCexCuO4. Unlike La2CuO4, it does not undergo any structural transition and remains tetragonal down to T = 0. A significant difference from La2CuO4 is the absence of apical oxygens, so that Cu atoms in Nd2CuO4 are not surrounded by oxygen octahedra (Fig. 1). So, in Nd2CuO4, there are no structural domain boundaries and no spin canting.

Magnetically, Nd2CuO4 differs from La2CuO4 in two ways: there is a large moment on the Nd3+ ions and the Cu spins adopt a noncollinear antiferromagnetic order46. Below TN = 255 K, the spins of the Cu2+ ions order antiferromagnetically along the Cu–O bond ([100]). This breaks the fourfold symmetry within a single CuO2 plane. However, in the next CuO2 plane along the c-axis, the same spin configuration is rotated by 90°, thereby restoring the fourfold symmetry of the entire system. This noncollinear magnetic structure therefore preserves the tetragonal symmetry of the crystal.

Sr2CuO2Cl2: Sr2CuO2Cl2 has the same crystal structure as tetragonal La2CuO4 (T > 530 K), with La replaced by Sr and the apical O replaced by Cl (Fig. 1). Unlike La2CuO4, it remains tetragonal down to low temperature and its octahedra show no sign of tilting47,48,49. So here, again, there are no structural domain boundaries and no spin canting. Sr2CuO2Cl2 develops collinear antiferromagnetic order below TN = 250 K, with a magnetic structure similar to that of La2CuO4 (moments along [110]), except with no spin canting out of the plane48. It remains in the same magnetic phase down to at least T = 10 K (ref. 48).

Samples

Our single crystal of Nd2CuO4 was grown at the University of Science and Technology of China by a standard flux method, annealed in helium for 10 h at 900 °C, and cut in the shape of rectangular platelets with dimensions 0.50 × 0.69 × 0.066 mm3 (length between contacts × width × thickness in the c direction). Contacts were made with silver epoxy, diffused at 500 °C for 1 h. The thermal conductivity κxx of similar samples was studied in detail at low temperature (T < 20 K) (refs. 50,51). Single crystals of Sr2CuO2Cl2 were grown at the University of British Columbia using a flux-growth method. Here, we report data on two samples (labeled A and B), cut in the shape of rectangular platelets with dimensions 0.6 × 0.11 × 0.03 mm3. Contacts were made using silver paint. In all cases, the heat current was made to flow along the a-axis of the tetragonal structure.

Measurements

The thermal conductivity κxx was measured applying a heat current Jx within the CuO2 plane, generating a longitudinal temperature difference ΔTx = T + − T −. The thermal conductivity along the x-axis is given by κxx = (Jx/ΔTx) (L/wt), where L is the distance between T + and T −, w is the width of the sample, and t its thickness. By applying a magnetic field along the c-axis of the crystal, normal to the CuO2 planes, a transverse temperature gradient, ΔTy, was generated (see inset of Supplementary Fig. 1). The thermal Hall conductivity is defined as

$$\kappa _{{xy}} = - \,\kappa _{{yy}}(\Delta T_{x}/\Delta T_{y})\,(L/w)$$

where κyy is the longitudinal thermal conductivity along the y-axis. Due to the tetragonal structure of our samples, we can take κxx = κyy.

The measurements were made with a steady-state method as a function of temperature, using differential type-E thermocouples for ΔTx and ΔTy (see inset of Supplementary Fig. 1). This method consists in keeping the sample in a fixed magnetic field H and changing its temperature in discrete steps, typically of 2–3 K. At each fixed temperature, the background value of the thermocouple that measures ΔTy is recorded before sending heat J to the sample. Once the sample is entirely in equilibrium, we measure ΔTy(H). Here, the voltage (heat-off) background in the thermocouple is carefully subtracted from the heat-on signal of the thermocouple to give the correct ΔTy(H). Once the entire temperature range is covered, say from 10 to 100 K, the field direction is reversed to –H. The same procedure is now applied for these negative values of the field. We then define ΔTy(H) = [ΔTy(T,H) − ΔTy(T, −H)]/2, thereby removing any symmetric contamination of the signal coming from the longitudinal thermal gradient and the possible misalignment of transverse contacts.

The heat current along the x-axis is generated by a heater stuck at one end of the sample. The other end is glued to a block that serves as a heat sink (see inset of Supplementary Fig. 1). For the data reported here (and in refs. 16,21), this block was made of copper. To confirm that the Hall response of copper in a field does not contaminate the Hall response coming from the sample, we performed the same measurement twice, once with the copper block (using metallic contacts made with Ag paint) and then with a block made of the insulator LiF (using insulating contacts made with GE varnish), using the same sample of Nd2CuO4 in both cases. The results are shown in Supplementary Fig. 1; we see that the same κxy curve is obtained with the two setups. We conclude that using copper for the heat sink does not lead to any detectable contamination of the thermal Hall signal.