Abstract
Prototypical models and their material incarnations are cornerstones to the understanding of quantum magnetism. Here we show theoretically that the recently synthesized magnetic compound Na_{2}BaCo(PO_{4})_{2} (NBCP) is a rare, nearly ideal material realization of the Sā=ā1/2 triangularlattice antiferromagnet with significant easyaxis spin exchange anisotropy. By combining the automatic parameter searching and tensornetwork simulations, we establish a microscopic model description of this material with realistic model parameters, which can not only fit well the experimental thermodynamic data but also reproduce the measured magnetization curves without further adjustment of parameters. According to the established model, the NBCP hosts a spin supersolid state that breaks both the lattice translation symmetry and the spin rotational symmetry. Such a state is a spin analog of the longsought supersolid state, thought to exist in solid Helium and optical lattice systems, and share similar traits. The NBCP therefore represents an ideal materialbased platform to explore the physics of supersolidity as well as its quantum and thermal melting.
Introduction
Quantum magnets are fertile ground for unconventional quantum phases and phase transitions. A prominent example is the Sā=ā1/2 triangularlattice antiferromagnet (TLAF). Crucial to the conception of the quantum spinliquid state^{1}, its inherent geometric frustration and strong quantum fluctuations give rise to exceedingly rich physics. In the presence of an external magnetic field, spin anisotropy, and/or spatial anisotropy, the system exhibits a cornucopia of magnetic orders and phase transitions^{2,3,4}. In particular, introducing an easyaxis spin exchange anisotropy to the TLAF results in the spin supersolid^{5,6,7,8,9,10,11} in zero magnetic field. Applying a magnetic field along the easyaxis drives the system through a sequence of quantum phase transitions by which the spin supersolidity disappears and then reemerges^{12}, whereas applying the field in the perpendicular direction yields distinct, even richer behaviors^{13}. The Sā=ā1/2 easyaxis TLAF therefore constitutes a special platform for exploring intriguing quantum phases and quantum phase transitions.
Lately, a cobaltbased compound Na_{2}BaCo(PO_{4})_{2} (NBCP) has been brought to light^{14,15,16,17}. This material features an ideal triangular lattice of Co^{2+} ions, each carrying an effective Sā=ā1/2 spin owing to the crystal field environment and the significant spinorbital coupling^{17,18} (Fig. 1a). Early thermodynamic measurements show that NBCP does not order down to ~300āmK with a large magnetic entropy (~2āJāmol^{ā1}āK^{ā1}) hidden below that temperature scale^{14}. A later thermodynamic measurement reveals a specific heat peak at ~150āmK, which accounts for the missing entropy and points to a possible magnetic ordering in zero magnetic field^{15}. However, the muon spin resonance (Ī¼SR) experiment finds strong dynamical fluctuation down to 80āmK^{16} which may suggest a spinliquid like state. The multitude of experimental results call for a theoretical assessment.
Previous works have attempted at establishing the spin exchange interactions in this compound. The authors in ref. ^{15} suggest an exchange coupling ~2āK based on an analysis of the magnetic susceptibility data, which is an order of magnitude smaller than an earlier estimate of 21.4 K in ref. ^{14}. Meanwhile, a firstprinciple calculation suggests potentially significant Kitaevtype exchange interaction^{17}. Despite these efforts, the precise spin Hamiltonian, its magnetic ground states, as well as the connection to experimental data, are yet to be established.
In this work, we show theoretically that NBCP can be welldescribed by a Sā=ā1/2 easyaxis TLAF with negligible perturbations. We establish the microscopic description of NBCP with realistic model parameters by fitting the model to intermediate and hightemperature experimental thermal data. We expedite the fitting process by using the Bayesian optimization^{19} equipped with an efficient quantum manybody thermodynamic solverāexponential tensor renormalization group (XTRG)^{20,21}. Our model is corroborated by reproducing quantitatively the experimental lowtemperature magnetization curves by density matrix renormalization group (DMRG)^{22} calculations. Furthermore, we are able to put the various experimental results into a coherent picture and connect them to the physics of the spin supersolid state. Therefore, the NBCP represents a rare material realization of this prototypical model system and thereby the spin supersolidity. The small exchange energy scale in this material (~1āK) implies that the phases of the NBCP can be readily tuned by weak or moderate magnetic fields. Our results also highlight the strength of the manybody computationbased, experimental datadriven approach as a methodology for studying quantum magnets.
Results
Crystal symmetry and the spin1/2 model
Figure 1a shows the lattice structure of NBCP and the crystallographic a, b, and caxes. Due to the octahedral crystal field environment and the spinorbital coupling, each Co^{2+} ion forms an effective Sā=ā1/2 doublet in the ground state, which is separated from higher energy multiplets by a gap of ~71āmeV (see Supplementary Note 1). Supersuperexchange path through two intermediate oxygen ions produces exchange interactions between two nearestneighbor (NN) spins, thereby connecting them into a triangular network (see density functional theory calculations in the Supplementary Note 2). Furtherneighbor spin exchange interactions are suppressed by the long distance. Meanwhile, the interlayer exchange interactions are expected to be much smaller than the intralayer couplings owing to the nonmagnetic BaO layer separating the adjacent cobalt layers. Therefore, we model the NBCP in the experimentally relevant temperature window as a Sā=ā1/2 TLAF with dominant NN exchange interactions. This hypothesis will be justified a posteriori.
The crystal symmetry constrains the NN exchange interactions as follows^{23}. The threefold symmetry axis ā„c passing through each lattice site relates the exchange interactions on the 6 bonds emanating from that site. On a given bond, there is a twofold symmetry axis passing through that bond and a center of inversion. The former symmetry forbids certain components of the offdiagonal symmetric exchange interaction, whereas the latter forbids DzyaloshinskiiMoriya interactions. We obtain
with
where i,āj are a pair of neighboring lattice sites, and Ī±,āĪ² label the spin x,āy,āz components^{24,25,26}. We choose the spin frame such that xā„a, zā„c. \(\varphi \,=\,\{0,\frac{2\pi }{3},\frac{2\pi }{3}\}\) for three different types of NN bonds parallel to a, b, and ā(aā+āb), respectively. J_{xy}, J_{z}, J_{Ī}, and J_{PD} are respectively the XY, Ising, offdiagonal symmetric, and pseudodipolar exchange couplings (see more details in Supplementary Note 3). The entire model parameter space is thus spanned by the four exchange constants, two LandĆ© factors (g_{ab} and g_{c}, for perpendicular and parallel to the caxis, respectively), as well as two van Vleck paramagnetic susceptibilities (\({\chi }_{ab}^{{{{\rm{vv}}}}},{\chi }_{c}^{{{{\rm{vv}}}}}\)), all of which are taken to be constants in the experimentally relevant temperature/magnetic field window.
Determination of the model parameters
We determine the model parameters in Eq. (1) by fitting the experimental magnetic specific heat (C_{m}) and magnetic susceptibility (Ļ) data at temperature Tāā„āT_{cut}, where T_{cut}ā=ā1āK for C_{m} and 3 K for Ļ. Note the magnetic susceptibilities are remeasured in this work with high quality samples. For each trial parameter set, we compute the same thermodynamic quantities from the model by using the XTRG solver^{21,27}. We search for the parameter set that minimizes the total loss function through an unbiased and efficient Bayesian optimization process^{19}. See Methods for more details.
We set the cutoff temperature T_{cut} based on the following considerations. The experimental data from independent measurements agree with each other at Tā>āT_{cut}, and our XTRG solver does not exhibit significant finitesize effects above T_{cut}. Meanwhile, the T_{cut} has to be less than or comparable with the characteristic energy scale of the material. These constraints fix T_{cut} to our present choices.
The searching process yields the following optimal parameter set: J_{xy}ā=ā0.88āK, J_{z}ā=ā1.48āK, and J_{Ī,PD} are negligible. The LandĆ© factors g_{ab}ā=ā4.24, and g_{c}ā=ā4.89. The van Vleck susceptibilities \({\chi }_{ab}^{{{{\rm{vv}}}}}\,=\,0.149\)ācm^{3}āmol^{ā1} and \({\chi }_{c}^{{{{\rm{vv}}}}}\,=\,0.186\)ācm^{3}āmol^{ā1}. To ensure that the algorithm does converge to the global minimum, we project the loss function onto the (J_{xy},āJ_{z}) plane in Fig. 1b, where, for fixing values of J_{xy},āJ_{z}, the loss function is minimized over the remaining parameters. The fitting landscape reveals a single minimum. The estimated value of exchange parameters and their bounds of uncertainty are shown in Fig. 2f.
The small uncertainties in J_{xy} and J_{z}, as well as the small loss, indicate that the experimental data are well captured by our parameters. Indeed, Fig. 2aāc show respectively the specific heat and the magnetic susceptibility as functions of temperature. We find excellent agreement between the model calculations and the experiments within the fitting temperature range Tāā„āT_{cut}. Reassuringly, the LandĆ© factors obtained by us are in excellent agreement with the latest electronspin resonance measurement (g_{ab}ā=ā4.24 and g_{c}ā=ā4.83)^{17}. We have also calculated the magnetic specific heat C_{m} in nonzero magnetic fields and find good agreement with the experiments whenever the two independent measurements^{14,15} mutually agree (Supplementary Note 4). Note in Fig. 2a the two experimental data sets of specific heat differ at Tā<āT_{cut}. Our modelās behavior below T_{cut} is in agreement with one of them. The discrepancy in the experimental data calls for further investigation.
Our model parameters pinpoint to an almost ideal Sā=ā1/2 TLAF with significant easyaxis anisotropy Īā=āJ_{z}/J_{xy}āāā1.68. In particular, the negligible offdiagonal exchange interactions imply that the NBCP features an approximate U(1) spin rotational symmetry with respect to c axis. As a result, the magnetization curve with field Bā„c in Fig. 2d has a couple of idiosyncratic features: It shows a 1/3magnetization plateau in an intermediate field range [B_{c1},āB_{c2}], and another fully magnetized plateau above the saturation field B_{c3}. As an independent estimate of Ī, we note the semiclassical analysis shows that \({B}_{c2}/{B}_{c1}\,=\,{{\Delta }}\,\,1/2\,+\,\sqrt{{{{\Delta }}}^{2}\,+\,{{\Delta }}\,\,7/4}\) and B_{c3}/B_{c1}ā=ā2Īā+ā1 (Supplementary Note 5). Using the experimental values B_{c1}āāā0.35āT and B_{c3}āāā1.62āT^{15}, we estimate Īāāā1.81, which is consistent with the Bayesian search result. Meanwhile, using these numbers, we can estimate B_{c2}āāā1.10āT, which is fairly close to the experimental value of 1.16āT^{15}.
As a corroboration of our model, we perform DMRG calculations of the zerotemperature magnetization curves and find quantitative agreement with the experimental results (Fig. 2d, e). With no further adjustment of the parameters, the model can not only produce the correct transition fields but also the details of the magnetization curve between the transitions. We find that the magnetization curve ā„c is a sensitive diagnostic for the anisotropy parameter Ī. The agreement between the model and the experiments is quickly lost when Ī deviates slightly from the optimal value in Fig. 2d.
Taken together, the broad agreements between the semiclassical estimates, the quantum manybody calculations, and experimental data strongly support the Sā=ā1/2 easyaxis TLAF as an effective model description for the NBCP.
Fieldtuning of the spin supersolid state
The Sā=ā1/2 easyaxis TLAF exhibits a sequence of magnetic phases and quantum phase transitions driven by magnetic fields^{12,28,29}. The experimentally measured differential susceptibilities (dM/dB) show a few anomalies when the field is ā„c and ā„a and are attributed to quantum phase transitions^{15}. Here, we clarify the nature of the magnetic orders of NBCP based on the Sā=ā1/2 easyaxis TLAF model.
Figure 3a shows the theoretical zerotemperature phase diagram of our model in field ā„c, obtained from DMRG calculations. The system goes through successively the Y, UpUpDown (UUD), V, and the polarized (P_{z}) phases with increasing field. These phases are separated by three critical fields B_{c1,c2,c3}ā=ā0.36āT, 1.14āT, 1.71āT discussed in the previous section, which are manifested as peaks in dM/dB. The Y phase, as well as the V phase, spontaneously break both the lattice translation symmetry and the U(1) symmetry, thereby constituting the supersolid state analogous to that of the Bose atoms. The UUD phase, on the other hand, restores the U(1) symmetry but breaks the lattice translation symmetry. This state is analogous to a Bose Mott insulator state. The magnetic plateau associated with the UUD state reflects the incompressibility of the Mott insulator.
The situation is yet more intricate when the field ā„a. DMRG calculations show that the system goes through the Ī», \(\tilde{{{{\rm{Y}}}}}\), \(\tilde{{{{\rm{V}}}}}\) (see Fig. 3c for an illustration), and the quasipolarized P_{x} phases, which are separated by three critical fields B_{a1}ā=ā0.075āT, B_{a2}ā=ā0.75āT, and B_{a3}ā=ā1.51āT. The presence of a field ā„a breaks the U(1)rotational symmetry with respect to the S^{z} axis but preserves the Ļrotational (Z_{2}) symmetry with respect to the S^{x} axis. The Ī» phase, where the spins sitting on three magnetic sublattices form the greek letter āĪ»ā, spontaneously breaks both the Z_{2} symmetry and the lattice translation symmetry. The Z_{2} symmetry is restored in the \(\tilde{{{{\rm{Y}}}}}\) phase, and spontaneously broken again in the \(\tilde{{{{\rm{V}}}}}\) phase.
Comparing to the fieldinduced transitions in Bā„c case, the transitions at B_{a1,a2} show much weaker anomalies in dM/dB when Bā„a. Numerically, we detect these two transitions using an order parameter \({{\Theta }}\,=\,\frac{1}{\pi } ({\theta }_{a}\,\,{\theta }_{b}\,\,{\theta }_{c})\), where Īø_{a,b,c} measure the angle between the spin moments on three sublattice and the S^{x} axis. Īā=ā0 when the Z_{2} symmetry is respected and Īāā ā0 when it is broken. Figure 3c shows Ī as a function of field, from which we can delineate the boundaries between Ī», \(\tilde{{{{\rm{Y}}}}}\), and \(\tilde{{{{\rm{V}}}}}\) phases, with the critical fields B_{a1}āāā0.07āT and B_{a2}āāā0.75āT. The small value of B_{a1} implies it could be easily missed in experiments. Meanwhile, the weak anomalies in dM/dB associated with B_{a1,a2} make them difficult to detect in thermodynamic measurements. We note that dM/dB shows a broad peak in the \(\tilde{{{{\rm{Y}}}}}\) phase at ~0.4ā0.5āT^{15}. This peak appears in the experimental data and was previously interpreted as a transition. The true transitions (B_{a1,a2}) are in fact above and/or below the said peak. We also note that, despite of the weak anomaly observed numerically at B_{a2}, the quantum phase transition there is likely of firstorder from symmetry analysis: the \(\tilde{{{{\rm{Y}}}}}\) and \(\tilde{{{{\rm{V}}}}}\) phases both have a sixfold groundstate degeneracy and they have incompatible symmetry breaking; thus the transition cannot be continuous according to Landauās paradigm.
Strong spin fluctuations and phase diagram at finite temperature
Having established the zerotemperature phase diagram of the NBCP, we now move on to its physics at finite temperature. Figure 4a shows the contour plot of C_{m}/T as a function of temperature and field Bā„c. In the temperature window accessible to the XTRG, we find a broad peak near zero field, which moves to higher temperature and becomes sharper as field increases. These features are in qualitative agreement with the experimental findings.
Figure 4b shows a crosssection of the contour plot at zero field. The model produces a peak in C_{m}/T atāāā150āmK, which is in excellent agreement with the experimental data from ref. ^{15}. Note this temperature is well below the temperature window (above T_{cut}) used for fitting, the difference between theory and experiment at very low temperatures may be ascribed to the finitesize effect inherent in the XTRG calculations (see Methods). The magnetic entropy also shows quantitative agreement with the experimental data. In particular, there is still a considerable amount of magnetic entropy to be released at 300 mK (and even down to 150 mK). The missing entropy at 300āmK reported in ref. ^{14} can be ascribed to the small spin interaction energy scale and high degrees of frustration in the NBCP.
To understand the finitetemperature phase diagram of the NBCP, we perform a Monte Carlo (MC) simulation of the classical TLAF model with appropriate rescaling of temperature and magnetic field^{30,31,32,33}. This approximation is amount to neglecting fluctuations in the imaginary time direction in the coherentstate path integral of the Sā=ā1/2 TLAF. As the finitetemperature phase transitions are driven by thermal fluctuations, we expect that the salient features produced by the classical MC simulations are robust. Meanwhile, the MC simulation allows for accessing much larger system sizes and lower temperatures comparing to the XTRG for quantum model simulations.
The physics of the classical TLAF model is well documented; here, we focus on the features that can be directly compared with available experimental data. Figure 4c shows the MCconstructed TB phase diagram with Bā„c. Figure 4d shows the specific heat as a function of temperature for various representative fields. The phase diagram shows a broad dome of UUD phase, beneath which lie the Y phase at low field and the V phase at high field. The UUD phase and the paramagnetic phase are separated by a transition of threestate Potts universality; the Y and V phases and the UUD phase are seperated by the BerezinskiiKosterlitzThoueless (BKT) transitions. Note the MC simulation may seem to suggest the onset of the V phase precedes that of the UUD phase at high field; this is an finitesize effect. On symmetry ground, we expect that the onset of the UUD phase either precedes that of the V phase through two continuous transitions, or the system enters the V phase directly through a firstorder transition.
At zero field, the specific heat shows two broad peaks, which are related to the two BKT transitions accompanying the onset of the algebraic longrange order in S^{z} and S^{x} components, respectively^{30,31}. The experimentally observed specific heat peak ~150āmK may be related to the highertemperature BKT transition; the lowertemperature BKT transition (around 54āmK as estimated by classical MC simulations) is yet to be detected as they lie below the temperature window probed by the previous experiments. The strong dynamical spin fluctuations found in Ī¼SR experiment at 80āmK is naturally attributed to the algebraic longrange order in the S^{z} component. Note there exists arguments for a third BKT transition^{32} although it is not observed in classical MC simulations^{31}.
When the magnetic field is switched on, the specific heat shows a sharp peak signaling the threestate Potts transition from the hightemperature paramagnetic phase to the UUD phase, corresponding to onset of the longrange order in the S^{z} component. This is consistent with the experimentally observed sharp specific heat peak at finite fields^{15}. At lower temperature, the specific heat shows a much weaker peak related to the BKT transition into either the Y phase or the V phase, corresponding to the onset of the algebraic longrange order in S^{x}. The lowertemperature BKT transitions are yet to be detected by experiments.
Discussion
The supersolid, a spatially ordered system that exhibits superfluid behavior, is a longpursued quantum state of matter. The question of whether such a fascinating phase of matter exists in nature has spurred intense research activity, and the search for supersolidity has become a multidisciplinary endeavor^{34,35,36,37,38,39}. The early claim of observation in He4^{34} turned out to be an experimental artifact^{35}. Nevertheless, it has inspired new lines of research in ultracold quantum gases^{36,37,38,39}. Meanwhile, it has been proposed theoretically that the ultracold Bose atoms in a triangular optical lattice can host a supersolid state^{5,6,7,8}. Yet, the realization of such a proposal has not been reported up to date.
An equivalent, yet microscopically different route to the triangularlattice supersolidity is via the easyaxis Sā=ā1/2 TLAF magnet. The spin up/down state of a magnetic ion can be viewed as the occupied/empty state of the lattice site by a Bose atom, and the spin rotational symmetry with respect to the easy axis is mapped to the U(1) phase rotation symmetry. By virtue of this mapping, the spin ground state in the easyaxis TLAF, which spontaneously breaks both lattice translation symmetry and spin rotational symmetry, is equivalent to the supersolid state of Bose atoms.
Despite its simple setting, ideal Sā=ā1/2 TLAF has rarely been found in real materials. Although TLAFs with higher spin (Sā>ā1/2) are known^{3}, Sā=ā1/2 systems with equilateral triangularlattice geometry, such as Ba_{3}CoSb_{2}O_{9}^{40,41,42,43,44,45,46,47} and Ba_{8}CoNb_{6}O_{24}^{48,49}, were synthesized and characterized not until recently. The former shows easyplane anisotropy^{28,47}, whereas the latter material is thought to be nearly spin isotropic^{48,49,50}. To the best of our knowledge, ideal Sā=ā1/2 easyaxis TLAFs are yet to be found. In this work, we show that the NBCP is an almost ideal material realization of such an Sā=ā1/2 easyaxis TLAF with the anisotropy parameter Īāāā1.7.
Our model arranges the various pieces of available experimental data into a coherent picture by connecting them to the rich physics of the TLAF model. It permits a quantitative fit of the thermodynamic data, including specific heat C_{m} and magnetic susceptibility Ļ down to intermediate and even low temperatures. In particular, we obtain the C_{m}/T peak at around 150āmK observed in experiments^{15}, which we associate to the BKT transition. Furthermore, we are able to accurately reproduce the spin state transition fields observed in previous AC susceptibility measurements along both a and c axes and clarify their nature.
The obtained spin exchange interactions are on similar orders of magnitude as previous estimation based on the CurieWeiss fitting of the magnetic susceptibility^{15} and firstprinciple calculation^{17}. However, the firstprinciple calculation suggests a significant Kitaevtype exchange interaction (in a rotated spin frame)^{17}, whereas our model, being directly fitted from the experimental data, possesses a nearly ideal U(1) symmetry and negligible Kitaevtype interaction (see more discussions in Supplementary Note 3). The nearly ideal U(1) symmetry in this material is indicated by the wellquantized magnetization plateau (Fig. 2d), which would be absent without the U(1) symmetry.
In our fitting procedure, we have omitted at the outset all furtherneighbor exchange interactions on the ground that their magnitude must be suppressed by the large distance between furtherneighbor Co^{2+} ions. This can be verified by including in the model a secondneighbor spinisotropic exchange interaction J_{2}. To verify it, we have performed additional 400 Bayesian iterations and find J_{2} with the median value ~0.1āK amongst the best 20 parameter sets, which are negligibly small. We thus conclude the obtained optimal parameters in the simulations are robust.
Despite the essential challenge in the firstprinciple calculations of the strongly correlated materials, we may nevertheless employ the density functional theory (DFT) + U approach to justify certain aspects of the microscopic spin model that are accessible to this approach. First of all, we find the charge density distributions of 3d electrons of Co^{2+} ions are well separated from one triangular plane to another (see Supplementary Note 2), which ensures twodimensionality of the compound. Moreover, the inplane charge density distribution reveals clearly a supersuperexchange path between the two NN Co^{2+} ions. We construct the Wannier functions of dorbitals of Co^{2+} ions and extract the hopping amplitude t between two NN Co^{2+} ions. From the secondorder perturbation theory in t/U, the NN exchange coupling can be estimated to be on the order of 2ā3āK for moderate and typical U_{eff}ā=ā4ā6āeV in this Cobased compound^{17}, which is consistent with the energy scales of the spin model.
The accurate model for the NBCP also points to future directions for the experimentalists to explore. The model hosts a very rich phase diagram in both temperature and magnetic field, which are yet to be fully uncovered by experiments. In particular, the model shows a second BKT transition at ~50āmK in zero field; in finite field Bā„a, the model shows two subtle transitions at B_{a1}āāā0.07āT and B_{a2}āāā0.75āT. These transitions may be detected by nuclear magnetic resonance^{51}, magnetotorque measurements^{52}, and magnetocaloric measurements^{53,54,55}. Neutron scattering experiments can also be employed to detect the simultaneous breaking of discrete lattice symmetry and spin U(1) rotational symmetry, as well as the behaviors of spin stiffness, so as to observe spin supersolidity in this triangular quantum magnet. On the theory front, while the Sā=ā1/2 easyaxis TLAF and its classical counterpart share similar features in their finitetemperature phase diagrams, it was realized early on that the quantum model also possess peculiar traits that are not fully captured by the classical model^{32}. Clarifying these subtleties in the context of NBCP would also presents an interesting problem for the future.
Methods
Exponential tensor renormalization group
The thermodynamic quantities including the magnetic specific heat C_{m}, and magnetic susceptibility Ļ can be computed with the exponential tensor renormalization group (XTRG) method^{21,27}. In practice, we perform XTRG calculations on the Ytype cylinders with width Wā=ā6 and length up to Lā=ā9 (denoted as YC6āĆā9, see Supplementary Note 4), and retain up to Dā=ā400 states with truncation errors Ļµāā²ā1āĆā10^{ā4} (down to 1āK) and ā²1āĆā10^{ā3} (down to about 100āmK). The XTRG truncation provides faithful estimate of error in the computed free energy, and the small Ļµāvalues thus guarantee high accuracy of computed thermal data down to low temperature.
The XTRG simulations start from the initial density matrix Ļ_{0}(Ļ) at a very hightemperature Tāā”ā1/Ļ (with the inverse temperature ĻāāŖā1), represented in a matrix product operator (MPO) form^{56}. The series of density matrices Ļ_{n}(2^{n}Ļ) (nāā„ā1) at lower temperatures are obtained by iteratively multiplying and compressing the MPOs Ļ_{n}ā=āĻ_{nāāā1}āā āĻ_{nāāā1}. As a powerful thermodynamic solver, XTRG has been successfully applied in solving triangularlattice spin models^{50} and related compounds^{51,57}, Kitaev model^{58} and materials^{59}, correlated fermions in ultracold quantum gas^{60}, and even moirĆ© quantum materials^{61}.
Automatic parameter searching
By combining the thermodynamic solver XTRG and efficient Bayesian optimization approach, the optimal model parameters can be determined automatically via minimizing the fitting loss function between the experimental and simulated data, i.e.,
\({O}_{\alpha }^{\exp }\) and \({O}_{\alpha }^{\mathrm{sim} ,{{{\bf{x}}}}}\) are respectively the experimental and simulated quantities with given model parameters \({{{\bf{x}}}}\,\equiv\, \{{J}_{xy},{J}_{z},{J}_{{{{\rm{PD}}}}},{J}_{{{\Gamma }}},{g}_{ab,c},{\chi }_{ab,c}^{{{{\rm{vv}}}}}\}\). The index Ī± labels different physical quantities, e.g., magnetic specific heat and susceptibilities, and N_{Ī±} counts the number of data points in O_{Ī±}. The optimization of \({{{\mathcal{L}}}}\) over the parameter space spanned by {J_{xy},āJ_{z},āJ_{PD},āJ_{Ī}} is conducted via the Bayesian optimization^{19}. The LandĆ© factor g_{ab,c} and the Van Vleck paramagnetic susceptibilities \({\chi }_{ab,c}^{{{{\rm{vv}}}}}\) are optimized via the NelderMead algorithm for each fixing {J_{xy},āJ_{z},āJ_{PD},āJ_{Ī}}. In practice, we perform the automatic parameter searching using the package QMagen developed by some of the authors^{19,62}, and the results shown in the main text are obtained via over 450 Bayesian iterations. After that, we introduce an additional parameter, the nextnearestneighbor Heisenberg term J_{2}, and perform another 400 searching iterations. We find J_{2} is indeed negligibly small and the obtained optimal parameters are robust.
Density matrix renormalization group
The groundstate magnetization curves of the easyaxis TLAF model for NBCP are computed by the density matrix renormalization group (DMRG) method^{22}, which is a powerful variational algorithm based on the matrix product state ansatz. The DMRG simulations are performed on YC6āĆā15 lattice, and we retain bond dimension up to Dā=ā1024 with truncation error Ļµā<ā3āĆā10^{ā5}, which guarantees well converged DMRG data.
Classical Monte Carlo simulations
We replace the Sā=ā1/2 operators by classical vectors, \({S}_{i}^{x,y,z}\,\to\, S{\hat{n}}_{i}\), where \({\hat{n}}_{i}\) is a unit vector, and Sā=ā1/2 is the spin quantum number. We use the standard Metropolis algorithm with single spin update. The largest system size is 48āĆā48. We compute the Binder ratio associated with the UUDphase order parameter \(\psi \,=\,{m}_{1}\,+\,{m}_{2}\exp (i2\pi /3)\,+\,{m}_{3}\exp (i2\pi /3)\), where m_{1,2,3} are respectively the S^{z}axis magnetization of the three sublattices, as well as the inplane spin stiffness Ļ^{31}. We locate the threestate Potts transition by examining the crossing of the Binder ratio, and the BKT transition by the criterion Ļ_{c}ā=ā(2/Ļ)T_{c}.
In the simulations, we use the natural unit in the calculation and thus the following process is required for comparing the model calculation results in the natural unit to experimental data in SI units:

(1)
The value of temperature T in natural unit should be multiplied by a factor of J_{xy}, and change it thus to the unit of Kelvin, where J_{xy}ā=ā0.88āK is taken as the energy scale in the calculation.

(2)
Multiply the value of specific heat C_{m} in natural unit by a factor of R, i.e., the ideal gas constant, and change it to the unit of Jāmol^{ā1}āK^{ā1}.

(3)
Multiply the magnetic field h in natural unit by a factor of J_{xy}k_{B}/(g_{z}Ī¼_{B}) and it is now in unit of Tesla, where g_{z} is the LandĆ© factor along S_{z} direction and Ī¼_{B} the Bohr magneton.
Sample preparation and susceptibility measurements
Single crystals of Na_{2}BaCo(PO_{4})_{2} were prepared by the flux method starting from Na_{2}CO_{3} (99.9%), BaCO_{3} (99.95%), CoO (99.9%), (NH_{4})_{2}HPO_{4} (99.5%), and NaCl (99%), mixed in the ratio 2:1:1:4:5. Details of the heating procedure were given in ref. ^{14}. The flux generated after the reaction is removed by ultrasonic washing. The anisotropic magnetic susceptibility measurements in this work were performed using a SQUID magnetometer (Quantum Design MPMS 3). The magnetic susceptibility as a function of temperature was measured in zero field cooled runs. During the measurements, magnetic field of 0.1 T was applied either parallel or perpendicular to the c axis. In the latter (inplane) measurements, no anisotropy is observed in the obtained susceptibility data.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
All numerical codes in this paper are available upon request to the authors.
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Acknowledgements
W.L. and Y.G. are indebted to Tao Shi for stimulating discussions, W.L. would also thank XueFeng Sun and Jie Ma for valuable discussions on the experiments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12222412, 11834014, 11874115, 11974036, 11974396, 12047503, and 12174068), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33020300), and CAS Project for Young Scientists in Basic Research (Grant Nos. YSBR057 and YSBR059). We thank the HPCITP for the technical support and generous allocation of CPU time.
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W.L., Y.Q., and Y.W. initiated this work. Y.G. and H.L. performed XTRG and DMRG calculations of the TLAF model. Y.W. conducted the symmetry and semiclassical analyses. X.T.Z., F.Y., and X.L.S. did the CEF point charge model analysis and DFT calculations. Y.C.F. undertook the MC simulations. R.Z. prepared the sample and performed the susceptibility measurements. All authors contributed to the analysis of the results and the preparation of the draft. Y.W. and W.L. supervised the project.
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Gao, Y., Fan, YC., Li, H. et al. Spin supersolidity in nearly ideal easyaxis triangular quantum antiferromagnet Na_{2}BaCo(PO_{4})_{2}. npj Quantum Mater. 7, 89 (2022). https://doi.org/10.1038/s41535022005003
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DOI: https://doi.org/10.1038/s41535022005003