Introduction

The two-dimensional (2D) triangular lattice antiferromagnet (TAF) with spin-1/2 is one of the simplest geometrically frustrated systems with strong quantum spin fluctuations, which recently has caught attention due to its exotic quantum magnetism. One celebrated example is the 2D gapless quantum spin liquid (QSL)1,2,3,4 that can host non-abelian quasiparticle5 and fractional excitations6,7 known as spinons8,9, which allows quantum mechanical encryption and transportation of information with potential for creating a qubit that is protected against environmental influences10. Three experimental hallmarks have been widely accepted as evidence for spinon, including (i) a broad continuous magnetic intensity in the inelastic neutron scattering (INS) spectrum6,11,12; (ii) a large magnetic specific heat with power law (C ~ Tα) temperature dependence13,14,15, and (iii) a non-zero residual thermal conductivity κ0/T in the zero-temperature limit16,17,18,19. While most of the suggested 2D gapless QSLs exhibit the first two hallmarks, they do not exhibit the third one. In reality, so far only the organic EtMe3Sb[Pd(dmit)2]2 reported by Yamashita et al.18,19 and the inorganic 1T-TaS2 reported by Murayama et al.20 exhibit a non-zero κ0/T term, both of which are spin-1/2 TAFs. However, some other groups also reported a zero κ0/T term in these two materials, raising a controversy21,22,23. For another QSL candidate pyrochlore Tb2Ti2O7, a saturated value of κ/T at 0.3 K was reported which resembles that of a dirty metal24. For other oxides, such as YbMgGaO425, another TAF with effective spin-1/2 Yb3+ ions, and Ca10Cr7O28 with bilayer kagome lattice26, the reported κ0/T term tends to be zero upon approaching zero temperature. This behavior could be closely related to the chemical disorder in both cases. For instance, YbMgGaO4 has Mg2+/Ga3+ site mixture27 and Ca10Cr7O28 has disorder among the two different Cr3+ positions28,29.

Another example of exotic magnetism in spin-1/2 TAFs is the quantum spin state transition. The theoretical studies have proposed that in a spin-1/2 TAF, the quantum spin fluctuations (QSFs) stabilize a novel up up down (UUD) phase while approaching zero temperature with the applied field parallel to either easy plane or easy axis30,31. This UUD phase exhibits itself as a magnetization plateau within a certain magnetic field regime and with one-third of the saturation moment. Experimentally, it is very rare to observe such a UUD phase in TAFs while approaching zero temperature. One example is Ba3CoSb2O9, another TAF with effective spin-1/2 Co2+ ions, which orders around 3.5 K and exhibits a UUD phase at ultralow temperatures32,33,34. More recently, the UUD phase also has been proposed for AYbCh2 (A = Na and Cs, Ch = O, S, Se), one TAF family with effective spin-1/2 Yb3+ ions35,36,37,38. Further detailed experimental and theoretical studies on Ba3CoSb2O9 revealed more complex quantum spin state transitions (QSSTs)39,40,41,42,43,44,45. Specifically, with increasing field along the ab plane, its 120° spin structure at zero field is followed by a canted 120° spin structure, the UUD phase, a coplanar phase (the V phase), and another coplanar phase (the V′ phase) before entering the fully polarized state. While for B // c, the 120° spin structure will be followed by an umbrella spin structure and a V phase.

While searching for new spin-1/2 TAFs without chemical disorder to explore the novel physics of QSL and QSSTs, the new Co-based triangular lattice antiferromagnet Na2BaCo(PO4)246 caught our attention. This system has a trigonal crystal structure with lattice parameter a = 5.3185 Å and c = 7.0081 Å. The magnetic CoO6 octahedra form a triangular network in the ab plane, separated by a layer of nonmagnetic BaO12 polyhedra. Meanwhile, the Na+ ions fill the gaps in the CoO6 layers (Fig. 1a, b). Overall, no site mixture among the ions has been observed. Due to its Kramers ion nature, the Co2+ ions can be treated as effective spin-1/2 at low temperatures. The magnetic susceptibility, INS spectrum, and specific heat data show no magnetic ordering down to 50 mK but with large magnetic specific heat and localized low-energy spin fluctuations. Then, as discussed above, to confirm whether this system is a truly gapless QSL or not, it is crucial to look for the possible existence of itinerant spinons. Moreover, until now, no magnetic phase diagram has been reported for this new TAF and its possibility for QSSTs is awaiting exploration.

Fig. 1: Structure and magnetic susceptibility of Na2BaCo(PO4)2.
figure 1

a The crystallographic structure. b The triangular lattice of Co2+ ions in the ab plane. c The inverse of the DC susceptibility measured with 0.1 T magnetic field along the ab plane. The solid lines are the Curie–Weiss fittings to high-temperature or low-temperature data.

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Results

Magnetic susceptibility

By following the recipe in ref. 46, we grew single crystals of Na2BaCo(PO4)2. Figure 1c shows the inverse of the DC magnetic susceptibility (1/χ) with applied field B // ab. A change of slope is observed around 50 K. The effective moment is estimated to be 5.37 µB for 150 < T < 300 K and 4.0 µB for 2 < T < 20 K by using the linear Curie–Weiss fittings. This decrease of effective moment indicates a crossover of spin state for Co2+ ions from S = 3/2 at high temperatures to an effective spin-1/2 at low temperatures. For Co2+ ions in an octahedral environment, as for Na2BaCo(PO4)2, the crystal field and spin-orbital coupling can lead to a Kramers doublet with the effective spin-1/2 as the ground state. For other triangular lattice antiferromagnets with octahedral Co sites, such as Ba3CoSb2O932 and ACoB3 (A = Cs, Rb; B = Cl, Br)47, the ground state also has effective spin-1/2. Therefore, the θCW = −2.5 K from the low-temperatures fitting represents its intrinsic antiferromagnetic exchange energy. According to the mean field theory, θCW is given as (−zJS(S + 1))/3kB, where J is the exchange interaction of the Heisenberg Hamiltonian \(J{\boldsymbol{{\Sigma}}}_{({\mathrm{i,j}})}S_iS_j\), and z is the number of nearest neighbors. For the effective S = 1/ 2 triangular lattice with z = 6, we obtained J/kB = − 2/3θCW = 1.7 K.

Thermal conductivity

Figure 2a shows the zero-field thermal conductivity of Na2BaCo(PO4)2 in the temperature range of 70 mK to 30 K. At higher temperature, it behaves like a usual insulating crystal. The peak at 12 K with a large value of 90 WK−1 m−1 can be understood as the so-called phonon peak. It is notable that such a large phonon peak indicates high quality of the single crystal sample. Also shown are the thermal conductivity in 14 T field, either along the c or the a axis, which can increase the κ at most temperatures up to 7 K.

Fig. 2: Ultralow-temperature thermal conductivity of Na2BaCo(PO4)2.
figure 2

a The zero-field thermal conductivity of Na2BaCo(PO4)2 in temperature range of 70 mK – 30 K. The heat current is along the a axis. The peak at 12 K is the so-called phonon peak. Also shown are the thermal conductivity in 14 T magnetic field along the c or the a axis. In most of this temperature region, high magnetic field enhances the thermal conductivity. b Data in zero field plotted in κ/T vs T2. The solid line is a linear fit of the data at T < 700 mK. A nonzero residual thermal conductivity κ0/T = 0.0062 WK−2 m−1 is resolved. The inset shows a magnified view of the lowest-temperature data. There is a very weak anomaly at T < 100 mK. c, d Thermal conductivity in 14 T magnetic field plotted as κ/T vs T2. The solid lines are a linear fits for data at T < 550 mK (for B // c) and at T < 500 mK (for B // a). There is no residual term (κ0/T = 0).

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Figure 2b–d show the ultralow-temperature thermal conductivity at 0 T and 14 T fields. Several features are noteworthy. First, all these data are well fitted by κ/T = κ0/T + bT2 with b as a constant in a very broad temperature range (from several tens to 500 mK or more, particularly up to 700 mK for zero field), while the fitting parameters κ0/T and b are clearly different for them. Second, in zero field the fitting gives κ0/T = 0.0062 WK−2 m−1, that is, the presence of a residual value in κ/T while extending to zero temperature is clearly resolved. Third, the fitting curves in Fig. 2 yield intercepts of 0 ± 0.0005 WK−2 m−1 for data with 14 T // c and 14 T // a. The error is at least one order of magnitude smaller than the zero-field κ0/T value, which indicates zero κ0/T for the 14 T data.

Specific heat

As shown in the inset to Fig. 2b, the zero-field κ(T) data also shows a very weak anomaly around 100 mK. To learn the nature of this anomaly, the specific heat (Cp) was measured at very low temperatures down to 50 mK, as shown in Fig. 3. At zero field, the Cp exhibits a broad peak around 630 mK followed by a small and sharp peak at 148 mK. This sharp peak should represent an antiferromagnetic ordering, which is likely related to the anomaly observed from the zero field κ(T). With increasing field along the c axis, this peak’s position shifts to ~310 mK for B = 0.5 and 1 T; meanwhile, its intensity abruptly increases for B = 0.5 and 1 T and then this peak disappears for B = 1.5 T, as shown in Fig. 3a. By assuming that the lattice contribution can be described by Cph = βT3 + β5T5 + β7T7 with β = 8.83 × 10−4 JK−4mol−1, β5 = −3.32 × 10−7 JK−6 mol−1, and β7 = 6.67 × 10−11 JK−8 mol−1 (see Supplementary Fig. 1), the change of magnetic entropy below 4 K, ΔSmag, was calculated by integrating (Cp–Cph)/T (Fig. 3b). The obtained values are 5.1 and 5.4 JK−1 mol−1 for B = 0 and 1 T, respectively, which are approaching the value of Rln2. This is another strong evidence that Na2BaCo(PO4)2 can be treated as an effective spin-1/2 system. At zero field, the recovered entropy below 200 mK (where the peak starts) is 1.6 JK−1mol−1. This is only 28% of Rln2, which indicates the strong spin fluctuations above TN. A small upturn of the specific heat is observed at the lowest temperatures, which could be attributed to a contribution from the nuclear entropy. Figure 3c shows the specific heat data for B // a. Similar to the results for B // c, the low magnetic fields along the a axis can also change the position of the peak at 148 mK but with weaker field dependence.

Fig. 3: Ultralow-temperature specific heat of Na2BaCo(PO4)2.
figure 3

a, c The zero-field data and those in different magnetic fields along the c axis or the a axis. The temperature range is 50 mK to 4 K. b, d The magnetic entropy for B // c and B // a, obtained by integrating the magnetic specific heat.

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Residual thermal conductivity

It is abnormal for Na2BaCo(PO4)2 to exhibit a non-zero κ0/T term extrapolated from the data above TN. One possible scenario is that it behaves as a QSL above TN with gapless magnetic excitations, which give rise to power-law temperature dependences of the low temperature physical properties. Indeed, the reported INS spectrum46 and specific heat data reported here support the presence of strong spin fluctuations above TN. While the 2D QSL is stable at zero temperature in the strict sense, it is also known that QSL behavior, such as spinon excitations can survive at a finite temperature regime if the temperature scale is smaller than the exchange interaction, J. To our knowledge, a couple of quantum magnets exhibit quantum spin disordered states including QSL in a temperature range TN < T < J due to the combination of strong geometrical frustration with enhanced quantum spin fluctuations for spin-1/2, as present in Na2BaCo(PO4)2 with TN = 148 mK and J ~ 1.7 K. One good example is the Volborthite, Cu3V2O7(OH)2·2H2O, with a 2D distorted kagome lattice of Cu2+ (S = 1/2) ions, which antiferromagnetically orders at TN ~ 1 K with exchange constant J ~ 60 K48,49. A finite linear T-dependent contribution of specific heat extrapolated to T = 0 K48 and a negative thermal Hall conductivity observed above TN49 both strongly support the presence of a QSL state with gapless spin excitations above TN for Volborthite. While no clear non-zero κ0/T term was observed for Volborthite due to its relatively high ordering temperature, the estimated mean free path of the spin excitations from the 8 K magnetic thermal conductivity is about 80 times its inter-spin distance, which indicates its spin excitations are highly mobile49. The related theoretical work also proposes the existence of spinon Fermi surface in Volborthite above TN50,51. Another relevant example is pyrochlore Yb2Ti2O7 with effective spin-1/2 Yb3+ ions, which ferromagnetically orders at TC ~ 0.2 K52. For Yb2Ti2O7, the XY and off diagonal components of the interactions, J⊥ ~ 0.58 K and Jz± ~ 1.7 K, respectively, produce quantum spin fluctuations53,54. Its reported specific data suggests strong quantum fluctuations above TC55,56. Its observed diffuse scattering and pinch point structure of the INS spectrum and related model calculation further suggest the presence of a quantum spin ice state above TC57. Lately, the unusual behavior of the magneto-thermal conductivity58 and thermal Hall conductivity59 suggests the emergence of highly itinerant quantum magnetic monopoles in this quantum spin ice state.

By following ref. 18’s method, we estimate the mean free path (ls) and life time of the spin excitation (τs) of the quasiparticles responsible for the excitations in Na2BaCo(PO4)2 by calculating \(\frac{{k_0}}{T} = \frac{{\pi \kappa _B^2}}{{9\,h}}\frac{{l_s}}{{ad}} = \frac{\pi }{9}\left( {\frac{{\kappa _B}}{h}} \right)^2\frac{J}{d}\tau _{_s}\). Here, a (~5.32 Å) and d (~7.01 Å) are nearest-neighbor and interlayer spin distance, respectively. From the observed κ0/T = 0.0062 WK−2 m−1, the ls is obtained as 36.6 Å, indicating that the excitations are mobile to a distance seven times as long as the inter-spin distance without being scattered. Third, in high magnetic field of 14 T, although the κ is much larger than the zero-field data, the fitting gives a negligibly small value of, or vanishing κ0/T. This is reasonable since 14 T is strong enough to polarize all spins and completely suppress the spinon excitations of the QSL state. Thus, the 14 T data should be a result of pure phonon heat transport. From the specific heat data (see Supplementary Fig. 1), it is found that the phonon specific heat can be approximated as Cph = βT3 at very low temperatures with the coefficient β = 8.83 × 10−4 JK−4 mol−1. The phonon velocity can be calculated from the β value as vph = 2430 ms−1. The phonon thermal conductivity in the ballistic scattering limit is κph = (1/3)Cphvphlph, where the phonon mean free path is determined by the averaged sample width of \(l_{{\mathrm{ph}}} = 2\sqrt {A/\pi } = 0.32\) mm for this sample. Thus, the phonon thermal conductivity at low temperature is expected as κph = 2.21 × T3 WK−1 m−1. Note that this estimation is different from the 14 T data by only a factor of 2, which is acceptable. If one assumes that the 14 T data is purely due to the phonon term, much smaller signal in zero field indicates that the phonon ballistic scattering limit is not achieved, although κ/T nicely follows κ0/T + bT2. Therefore, in zero field the phonons are always suffering some scattering effect besides the boundary. Apparently, at very low temperatures only the magnetic excitations can take the role of phonon scattering.

Another possible scenario is that this non-zero κ0/T term is related to other abnormal quasiparticles besides spinon, which means the high-T (>TN) phase may not be ascribed to the QSL. Either way, future studies are desirable to learn the exact origin for this interesting residual thermal conductivity in Na2BaCo(PO4)2.

Field dependence of thermal conductivity and AC susceptibility

The dramatic change of the Cp peak with B // c suggests the possibility of spin state transitions. For further investigation, more detailed κ and AC susceptibility in magnetic fields were measured. For B // c, the κ(B) curve at 92 mK exhibits four minima at Bc1, Bc2, Bc3, and Bc4 (Fig. 4a). With increasing temperatures, κ(B) only exhibits two minima at 151 mK and no minimum at T > 300 mK. The κ(T) measured at 0.5 and 1.0 T (Fig. 4b) clearly shows a slope change around 310 mK, which is consistent with the Cp peaks’ position measured at the same fields. The AC susceptibility, χ′, measured at 22 mK (Fig. 4c) shows three peaks at Bc1, Bc2, and Bc3. The values of these three critical fields are consistent with the Bc1, Bc2, and Bc3 observed from κ(B). With increasing temperatures, the Bc1 peak shifts to higher fields and the Bc2 and Bc3 peaks shifts to lower fields. At T > 280 mK, the peaks almost disappear. Since the measured χ′ shows no frequency dependence (not shown here), it could be approximately treated as the derivative of the DC magnetization M(B). We calculated M(B) by integrating χ′. The obtained M(B) at 22 mK (Fig. 4d) clearly shows a plateau regime between Bc1 and Bc2 and a slope change at Bc3 followed by saturation around 2.5 T. Although we cannot infer the absolute value of M(B) here, it is obvious that the magnetization of the plateau (around 0.29, here we scaled the M value to the 3 T value) is around 1/3 of the saturation value (around 0.84 after we subtract the Van Vleck paramagnetic background, which is the upper dashed line in Fig. 4d).

Fig. 4: Thermal conductivity and AC susceptibility of Na2BaCo(PO4)2 for B // c.
figure 4

a Magnetic-field dependence of thermal conductivity at different temperatures. For clarifying, the 151 mK, 252 mK and 380 mK curves are shifted upward by 0.5, 1.0 and 1.5, respectively. At 92 mK, there are four minima indicated by arrows. With increasing temperature, the minima become weaker and disappear above 380 mK. b Temperature dependence of κ in different fields. At low fields of 0.5 and 1 T, there is a clear slope change of κ(T) curves around 310 mK, which has good correspondence to the specific-heat anomaly. c AC magnetic susceptibility at different temperatures. There are three peaks in the low-temperature curves. d Magnetization curves obtained by integrating the AC susceptibility data and renormalized with the 3 T value. The lower dashed line indicates a ~1/3 plateau and the upper dashed line indicates the Van Vleck paramagnetic background.

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For comparison, the above measurements were also performed for B // a. The κ(B) curve (Fig. 5a) at 92 mK shows two minima at Ba1 and Ba2, while the κ(T) (Fig. 5b) measured at different fields shows no obvious slope change. The χ′ (Fig. 5c) measured at 25 mK shows a broad peak around Ba1 and a sharp peak around Ba2. With increasing temperature, Ba1 and Ba2 shift to higher and lower fields, respectively, and both disappear at T > 290 mK. The calculated M(B) at 25 mK shows a slope change around the 1/3Ms position. These results are clearly different from those for B // c.

Fig. 5: Thermal conductivity and AC susceptibility of Na2BaCo(PO4)2 for B // a.
figure 5

a Magnetic-field dependence of thermal conductivity at different temperatures. For clarifying, the 151 mK, 252 mK and 380 mK curves are shifted upward by 0.5, 1.0 and 1.5, respectively. At very low temperatures, there are two minima on the κ(B) curves which disappear above 252 mK. b Temperature dependence of κ in different fields. No clear anomaly is observed. c AC magnetic susceptibility at different temperatures. There are two peaks in the low-temperature curves. d Magnetization curves obtained by integrating the AC susceptibility data and renormalized with the 3 T value. The dashed line indicates the Van Vleck paramagnetic background.

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Phase diagram

Based on the critical fields and ordering temperatures presented above, the magnetic phase diagrams for B // c and B // a are constructed in Fig. 6. For B // a, since both the κ(B) and χ′ data consistently show two critical fields and Ba2 is very close to the field where the magnetization becomes flat (or saturated), it is natural for us to assign Ba2 as the saturation field and Ba1 as a critical field for a spin state transition. On the other hand, for B // c, the κ(B) exhibits four critical fields while the χ′ shows three. Here we assign the Bc4 as the saturation field for two reasons: (i) if we assign Bc3 as the saturation field, it will be difficult to understand why there is still a possible spin state transition at Bc4 > Bc3 after all spins have been polarized; (ii) a close look of the calculated M(B) curve shows that Bc3 represents a slope change before the magnetization becomes flat, which most likely represents a spin state transition. One possible situation is that since this Bc3 peak of χ′ data is so close to the saturation field position, it may smear out the expected χ′ peak at Bc4. Accordingly, besides the paramagnetic phase at high temperatures and fully polarized phase at high fields, with increasing field, there are four phases for B // c (Fig. 6a) and two phases for B // a (Fig. 6b).

Fig. 6: Phase diagram of Na2BaCo(PO4)2.
figure 6

The magnetic phase diagrams of Na2BaCo(PO4)2 for B // c (a) and B // a (b). The data points are obtained from the AC susceptibility (AC), specific heat (Cp) and temperature or field dependence of thermal conductivity (κ(T) and κ(B)) measurements. The dashed lines are phase boundaries. For B // c, there are four phases (I, II, III, and IV) in the low-temperature and low-field region. Whereas, there are two phase (I and II) at low-temperature and low-field for B // a. The dashed lines are a guide to the eye.

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Now we compare the phase diagrams of Na2BaCo(PO4)2 to those of Ba3CoSb2O9 listed in the introduction. For B //c we are confident that the phase II is the UUD phase based on the 1/3Ms plateau observed at 22 mK. Since the 120° spin structure is a pre-required phase for the appearance of UUD phase, we ascribe the phase I as the canted 120° spin structure. Whether the phase III and IV are the V and V’ phase or the phase I and II for B // a are the umbrella and V phases or not cannot be said at this stage. Further studies such as neutron diffraction are needed to address this question.

We emphasize that in Na2BaCo(PO4)2, the UUD phase only survives for B // c, which strongly suggests its easy axis anisotropy as the theory predicted31. Ba3CoSb2O9 and AYbCh2 are both TAFs with easy plane anisotropy. To our knowledge, Na2BaCo(PO4)2 is a very rare example of spin-1/2 TAF with single crystalline form to exhibit series of QSSTs along the easy axis. Another two examples for spin-1/2 TAFs with easy axis anisotropy to show UUD phase are Ba3CoNb2O960,61 and Ba2La2CoTe2O1262, but both of them are polycrystalline form.

Discussion

In summary, we clearly observed a nonzero residual thermal conductivity, κ0/T, extrapolated from the data above TN (~148 mK) in an effective spin-1/2 triangular antiferromagnet Na2BaCo(PO4)2. This abnormal feature indicates that Na2BaCo(PO4)2 possibly behaves as a gapless QSL with itinerant spin excitations above TN. Moreover, its strong quantum spin fluctuations persist below TN and help to stabilize a series of spin state transitions while approaching zero temperature. With applied field along the c axis, this includes the UUD phase with a 1/3Ms magnetization plateau. This makes Na2BaCo(PO4)2 a unique TAF with easy axis anisotropy to exhibit a UUD phase.

Methods

Sample preparation and characterization

The single crystals were grown by flux method as reported in ref. 46. One adjustment made is that Platinum crucibles instead of alumina crucibles were used in our growth. The powder X-ray diffraction measurement on the ground single crystals confirmed its lattice structure is the same as reported in ref. 46. Laue back diffraction confirmed the flat surface of the as grown crystals is the ab plane. DC magnetic susceptibility was measured with a Quantum Design superconducting quantum interference device (SQUID) magnetometer. The applied field B = 0.1 T is parallel to the ab plane. Specific heat was measured with a Quantum Design Physical Property Measurements System (PPMS), equipped with a dilution refrigerator insert.

AC susceptibility measurements

The AC susceptibility measurements were conducted with a voltage controlled current source (Stanford Research, CS580) and lock-in amplifier (Stanford Research, SR830). The phase of the lock-in amplifier is set to measure the first harmonic signal. Single crystal samples of Na2BaCo(PO4)2 were prepared to allow the AC and DC magnetic fields to be perpendicular and parallel to the c axis separately in the measurements. The rms amplitude of the ac excitation field is set to be 0.6 Oe with the frequency fixed to be 220 Hz. The measurements were performed at SCM1 of the National High Magnetic Field Laboratory, Tallahassee, by using a dilution refrigerator. The data was obtained by the zero field cooling process and we increased the magnetic field during the ramping process.

Thermal conductivity measurements

Thermal conductivity was measured by using a “one heater, two thermometers” technique in a 3He/4He dilution refrigerator and in a 3He refrigerator, equipped with a 14 T superconducting magnet63,64,65,66,67,68,69,70,71,72,73. The sample was cut precisely along the crystallographic axes with dimensions of 3.0 × 0.63 × 0.14 mm3, where the longest and the shortest dimensions are along the a and the c axes, respectively. The heat currents were applied along the a axis while the magnetic fields were applied along either the a or c axis. Since the AC susceptibility clearly showed no hysteresis with sweeping field, we did not perform all the specific heat and thermal conductivity measurements with the zero-field cooling process. However, we carefully checked the first κ(B) measurement at 92 mK for both B // c and B // a. The sample was zero-field cooled to 92 mK and the κ was measured with increasing field to 14 T and then decreasing field to 0 T. No hysteresis was observed in κ(B).

For low-temperature thermal conductivity measurements, calibrating the magnetoresistance effect of resistor thermometers is a basic requirement. The thermometers (RuO2) used at 300 mK to 30 K in the 3He refrigerator are pre-calibrated by using a capacitance sensor (Lakeshore Cryotronics) as the reference65,67,68,69; the thermometers (RuO2) used at 50 mK to 1 K in the dilution refrigerator are pre-calibrated by using a RuO2 reference sensor (Lakeshore Cryotronics) mounted at the mixture chamber (the superconducting magnet was equipped with a cancellation coil at the height of mixture chamber)70,71,72. The resolution of the κ measurements is typically better than 3% (better at higher temperature). The sample size was determined by using microscopy and has uncertainty of <5%. Therefore, the total error bar of is κ always < 8%. The uncertainty of κ0/T caused by the fitting is ~2%. The κ0/T value of Na2BaCo(PO4)2 is ~30 times smaller than that of EtMe3Sb[Pd(dmit)2]2 and ~10 times smaller than that of 1T-TaS218,19,20. One may ask whether this value is too small to be resolved by the κ measurement at ultralow temperatures. We would like to mention that this residual thermal conductivity is actually comparable to those in high-Tc cuprate superconductors (HTSC). For HTSC, the κ0/T is contributed by the nodal quasiparticles from the d-wave superconducting state and has been experimentally observed by us in many materials, including La2−xSrxCuO4, YBaCu3Oy, Bi2Sr2-xLaxCuO6+δ, and Bi2Sr2CaCu2O8+δ64,65,66,67. In these materials, it is well resolved that the κ0/T varies from 0.0015 to 0.06 WK−2m−1 and shows systematic changes with the carrier concentration. Similar experimental results have also been reported by other groups for both the cuprate and the iron-based superconductors74,75,76,77. Therefore, a κ0/T value of 0.0062 WK−2m−1 is big enough to be correctly detected by a high-level measurement.

Demagnetization effect

Here we list the used samples’ dimensions and weights for various measurements. AC susceptibility: for B // a, 1.44 × 1.15 × 4.80 mm3, 33.2 mg; for B // c, 1.30 × 1.30 × 5.20 mm3, 36.7 mg. For both cases, the field is along the longest dimension. Specific heat: for B // a, 1.96 × 0.45 × 0.38 mm3, 1.40 mg; for B // c, 1.87 × 1.16 × 0.18 mm3, 1.63 mg. For both cases, the field is along the shortest dimension. Thermal conductivity: 3.0 × 0.63 × 0.14 mm3, 1.11 mg. For B // a, the field is along the longest dimension, for B // c, the field is along the shortest dimension. The estimated upper limit of the modification of B by the demagnetization effects for AC susceptibility is <1% for both directions, for specific heat is <4% for B // a and <8% for B // c, and for thermal conductivity is <1% for B // a and <8% for B // c. Such kind of small modification was neglected.