Gianguido Conradson

P Baldinozzi

D Sutar

Mihailovic

E S Bozin

M Abeykoon

S Conradson

G Baldinozzi

P Sutar

D Mihailovic

Crystallization of polarons through charge and spin ordering transitions in 1T-TaS2

INTRODUCTION

Quasi-two dimensional quantum-ordered materials display very diverse and often exotic spin [START_REF] Law | 1T-TaS2 as a quantum spin liquid[END_REF][START_REF] Klanjsek | A high-temperature quantum spin liquid with polaron spins[END_REF][START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] and charge ordering behaviour [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF] , including exciton condensation [START_REF] Kogar | Signatures of exciton condensation in a transition metal dichalcogenide[END_REF] , chiral spin liquids [START_REF] Persky | Magnetic memory and spontaneous vortices in a van der Waals superconductor[END_REF] and superconductivity [START_REF] Hamill | Two-fold symmetric superconductivity in few-layer NbSe2[END_REF][START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF] . They can also display hidden phases which can be manipulated by pressure [START_REF] Sipos | From Mott state to superconductivity in 1T-TaS2[END_REF][START_REF] Dong | Structural phase transition and superconductivity hierarchy in 1T-TaS2 under pressure up to 100 GPa[END_REF] , light [START_REF] Zhang | All-optical switching of magnetization in atomically thin CrI3[END_REF][START_REF] Stojchevska | Ultrafast Switching to a Stable Hidden Quantum State in an Electronic Crystal[END_REF][START_REF] Ravnik | A time-domain phase diagram of metastable states in a charge ordered quantum material[END_REF][START_REF] Dabrowski | All-optical control of spin in a 2D van der Waals magnet[END_REF][START_REF] Kogar | Light-induced charge density wave in LaTe3[END_REF] or doping [START_REF] Tedstone | Properties, and Applications of Transition Metal-Doped Layered Transition Metal Dichalcogenides[END_REF][START_REF] Qiao | Mottness Collapse in 1T-TaS2-xSex Transition-Metal Dichalcogenide: An Interplay between Localized and Itinerant Orbitals[END_REF][START_REF] Li | Fe-doping-induced superconductivity in the charge-density-wave system 1T-TaS2[END_REF] . An important aspect that determines spin ordering behaviour is the competition between Coulomb repulsion and interlayer interactions [START_REF] Wang | Band insulator to Mott insulator transition in 1T-TaS2[END_REF] . A common feature of these systems is that the strength of the electron phonon interaction (EPI) and electron density control the electron kinetic energy. Increasing the EPI leads to bandwidth narrowing, and lattice dressed electron quasiparticles -as originally introduced by Landau in 1933 [START_REF] Landau | Electron motion in crystal lattices[END_REF] . A further consequence of electron localization is an enhanced Coulomb interaction that leads to correlations, ubiquitous mesoscopic textures [START_REF] Camjayi | Coulomb correlations and the Wigner-Mott transition[END_REF][START_REF] Mertelj | Self-organization of charged particles on a two-dimensional lattice subject to anisotropic Jahn-Teller-type interaction and three-dimensional Coulomb repulsion[END_REF] , charge-density wave (CDW) formation [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF] and superconductivity at high critical temperatures in the fluctuation-dominated intermediate coupling regime of layered cuprate superconductors [START_REF] Mihailovic | Why Does Maximum T c Occur at the Crossover From Weak to Strong Electronphonon Coupling in High-temperature Superconductors[END_REF] . Recent model calculations [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF] applied to wide range of transition metal dichalcogenide (TMD) materials and comparisons with scanning tunneling microsopy experiments highlighted the importance of the Coulomb interaction that lead to ubiquitous charge ordering at magic filling fractions [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF] . Specific features of the band structure may lead to further effects, such as exciton condensation [START_REF] Kogar | Signatures of exciton condensation in a transition metal dichalcogenide[END_REF] and CDW ordering aided by Fermi surface nesting [START_REF] Rossnagel | On the origin of charge-density waves in select layered transition-metal dichalcogenides[END_REF] on top of the seemingly universal charge ordering [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF] . Experimental evidence for polarons comes from many different techniques in a very wide variety of materials [START_REF] Franchini | Polarons in materials[END_REF][START_REF] Mihailovic | Application of the polaron-transport theory to sigma ( omega ) in Tl2Ba2Ca1-xGdxCu2O8, YBa2Cu3O7-d, and La2-xSrxCuO4[END_REF] , but their role in forming quantum states, such as Wigner crystals, superconductivity and quantum spin liquids is not experimentally clear, particularly on the transition from the low to high density limit. Direct observation of symmetry-specific polaronic lattice distortions on the low-to high-density limit are crucial for progress in understanding a wide class of quantum materials.

As a prototypical example, the layered dichalcogenide 1T-TaS2 in equilibrium is of interest as a Mott insulator [START_REF] Fazekas | Charge Carrier Localization in Pure and Doped 1t-Tas2[END_REF] , CDW system [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF] , polaronic superconductor [START_REF] Sipos | From Mott state to superconductivity in 1T-TaS2[END_REF] and quantum spin liquid [START_REF] Klanjsek | A high-temperature quantum spin liquid with polaron spins[END_REF] . Additional recent interest in the material comes from the discovery of hidden light-or charge injection-induced mesoscopic metastable phases [START_REF] Stojchevska | Ultrafast Switching to a Stable Hidden Quantum State in an Electronic Crystal[END_REF][START_REF] Gerasimenko | Quantum jamming transition to a correlated electron glass in 1T-TaS 2[END_REF] that are important both fundamentally and for applications [START_REF] Mraz | Charge Configuration Memory Devices: Energy Efficiency and Switching Speed[END_REF][START_REF] Mihailovic | Ultrafast non-thermal and thermal switching in charge configuration memory devices based on 1T-TaS[END_REF][START_REF] Liu | A charge-density-wave oscillator based on an integrated tantalum disulfide-boron nitride-graphene device operating at room temperature[END_REF] . Structurally, it is a kinetically trapped 1T polymorph of TaS2, with octahedral stacking (Fig 1a) [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF][START_REF] Landuyt | Electron diffraction study of inter-and intrapolytypic phase transitions in transition metal dichalcogenides. IV Cinematogeraphic Observa ons of the 1Tβ → 1Tγ-Phase Transi on in TaS2[END_REF][START_REF] Revelli | Studies of the system TaS2-xSex[END_REF][START_REF] Givens | Thermal expansion of NbSe2 and TaS2[END_REF] . It's deceptively simple electronic band structure in the undistorted phase has a single Ta metal band crossing the Fermi level. [START_REF] Stojchevska | Ultrafast Switching to a Stable Hidden Quantum State in an Electronic Crystal[END_REF][START_REF] Revelli | Studies of the system TaS2-xSex[END_REF] While nominally metallic at high temperatures, it shows an incommensurate (IC) CDW (ICCDW) along the principal crystal axes already at high temperatures (between 550 and 350 K), with an associated Kohn anomaly at a wavevector ≃ (0.283, 0,0) in units of the reciprocal lattice vector * 36,37 which implies a strong coupling between electrons and the lattice. Below 350 K, the IC state breaks up into a patchy periodic network of commensurate islands separated by discommensurations [START_REF] Nakanishi | Domain-Like Incommensurate Charge-Density-Wave States and Collective Modes[END_REF][START_REF] Nakanishi | The nearly commensurate phase and effect of harmonics on the successive phase transition in 1 T-TaS 2[END_REF][START_REF] Mcmillan | Theory of discommensurations and the commensurate-incommensurate chargedensity-wave phase transition[END_REF] . Upon further cooling, this nearly commensurate (NC) CDW (NCCDW) state becomes fully commensurate (C) CDW (CCDW) after a first order transition at ~ 180 K, with the appearance of a Mott gap in the charge excitation spectrum observed by single-particle tunneling [START_REF] Butler | Mottness versus unit-cell doubling as the driver of the insulating state in 1T-TaS2[END_REF] and angle-resolved photoemission [START_REF] Rossnagel | On the origin of charge-density waves in select layered transition-metal dichalcogenides[END_REF][START_REF] Hellmann | Time-domain classification of charge-density-wave insulators[END_REF] . In contrast, gapless spin relaxation in the range 50-200 K is attributed to a possible quantum spin liquid (QSL) phase, arising from the spin frustration on the triangular superlattice. While band structure calculations [START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] suggest that the QSL phase may be suppressed by inter-layer coupling [START_REF] Ritschel | Orbital textures and charge density waves in transition metal dichalcogenides[END_REF] , this cannot explain the marked difference of spin and charge excitation spectra [START_REF] Klanjsek | A high-temperature quantum spin liquid with polaron spins[END_REF] . A further intriguing feature is a discontinuity in the magnetic susceptibility and an anomalous peak in the spin-spin relaxation time concurrent with a crossover around ~ 50 K of the spin-lattice relaxation time from a QSL-like, to a gapped T-dependence [START_REF] Spijkerman | X-ray crystal-structure refinement of the nearly commensurate phase of 1T-TaS2 in (3+ 2)-dimensional superspace[END_REF][START_REF] Jellinek | The system tantalum-sulfur[END_REF] . Near 50 K, a resistivity upturn from nearly T-independent resistivity to strongly T-dependent variable-range hopping behavior has been commonly reported, suggesting an onset of low-temperature charge localization [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF][START_REF] Svetin | Three-dimensional resistivity and switching between correlated electronic states in 1T-TaS2[END_REF] . The common picture of the 1T-TaS2 lattice structure in the C state assumes simple distortions from the average lattice structure (trigonal P3 m, Fig. 1a), whereby 12 Ta atoms are attracted symmetrically towards the central Ta atom that carries an extra electronic charge, forming a star of David (SoD) pattern (Fig. 1c). The SoDs are arranged in a √13 × √13 unit cell superlattice structure (P 3 symmetry [START_REF] Brouwer | Superstructure of 1s-TaS2[END_REF] ). At such an electron density of 1/13, the dimensionless ratio of the Coulomb energy to the kinetic (Fermi) energy

= = ℏ √ ≃ 70~100,
where is the elementary charge, is the (2D) density, and ~3 is the effective electron mass [START_REF] Rossnagel | On the origin of charge-density waves in select layered transition-metal dichalcogenides[END_REF] . Considering that the Wigner crystal stability limit is = 31~38 [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF][START_REF] Drummond | Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas[END_REF][START_REF] Pankov | Self-doping instability of the Wigner-Mott insulator[END_REF][START_REF] Radonjić | Wigner-Mott scaling of transport near the two-dimensional metal-insulator transition[END_REF] , 1T-TaS2 is comfortably in the Wigner crystal regime, which means that electronic superlattice ordering on the basis of dominant Coulomb interactions and tell-tale lattice deformations around localized carriers may be anticipated. Room temperature extended X-ray absorption fine structure (EXAFS) [START_REF] Kochubey | Charge density waves in 1T-TaS2: an EXAFS study[END_REF] , high-resolution transmission electron microscopy (HRTEM) [START_REF] Ishiguro | Electron microscopy of phase transformations in 1T-TaS2[END_REF][START_REF] Ishiguro | High-resolution electron microscopy of discommensuration in the nearly commensurate phase on warming of 1 T-TaS2[END_REF][START_REF] Ishiguro | Layer Number Dependence of Charge Density Wave Phase Transition Between Nearly-Commensurate and Incommensurate Phases in 1T-TaS 2[END_REF] and X-ray structural measurements [START_REF] Spijkerman | X-ray crystal-structure refinement of the nearly commensurate phase of 1T-TaS2 in (3+ 2)-dimensional superspace[END_REF] indeed suggest the existence of symmetry-breaking atomic displacements, but studies over a wide range of ordering temperatures have not yet been performed.

Here we present the first systematic measurements of the local lattice structure that covers temperatures from 15 to 915 K. The data reveal local symmetry breaking from the established P3 m symmetry at all measured temperatures (Fig. 1a, inset), and unambiguous evidence for individual polarons in the dilute high-temperature limit, revising our current notions about the origin of charge and spin ordering, with clear implications for other materials of current interest.

RESULTS

The local structure of 1T-TaS2 is investigated using X-ray pair distribution function (PDF) analysis [START_REF] Billinge | The Problem with Determining Atomic Structure at the Nanoscale[END_REF][START_REF] Billinge | Underneath the Bragg Peaks[END_REF] , giving picosecond-timescale structural snapshots for 15 K ≤ T ≤ 915 K. This encompasses the known electronically ordered states below 600 K portrayed by the resistivity vs. temperature plot shown in Fig. 1a 9 , as well as irreversible polytype transformations above 600 K.

The anisotropic atomic displacement parameters, Uij, of Ta atoms were retrieved from a fit to the established P3 m structure over a range of 20-60 Å (Fig. 1b). For T < 600 K, the Uij, reveal an anomalously large in-plane component, U11, approximately a factor of 2 larger than the component along the stacking axis, U33 (e.g. at 500 K U11 is 0.025 Å 2 whereas U33 is 0.01 Å 2 ). This is unusual for a layered system in which a larger out-of-plane component is expected to arise from interlayer disorder. At high T, in the metallic (M) phase, the large U11 signifies an apparent intralayer 'disorder', unaccounted for by the P3 m structural model. At lower temperature, the P3 m model is inadequate at any temperature, revealed by the difference PDFs shown in Fig. 1c. The spatial extent and character of this symmetry breaking changes across the cascade of CDW transitions, tracking complex correlations in different electronic regimes. Intriguingly, as we describe below, nanoscale symmetry breaking is also evident in the polymorphs above 630 K (Fig. 2b,c).

As an aid to understanding, pair-specific contributions to the total PDF within the P3 m model are shown in Fig. 1e, where Ta-Ta pair contributions are dominant (see Methods). In the P3 m structure, the undistorted Ta sublattice features a single-valued nearest-neighbor Ta-Ta distance, seen as a sharp peak at ~3.4 Å, Fig. 1e. However, the data reveal a distinct additional shoulder, (e.g. Figs. 1e and 2b for 600 K), which is unexplained by the P3 m model (Figs. 1c and 2c (top)). At high temperature this distortion is most likely dynamic, as expected for symmetry-specific lattice fluctuations associated with carrier localization and polaron formation. Very specific distributions of Ta-Ta and Ta-S pairs in the 3.3-5.9 Å range are expected for SoD polaronic distortions [START_REF] Brouwer | Superstructure of 1s-TaS2[END_REF] .

In the C phase with a single layer √13 × √13 supercell P3 model (Methods and Supplementary Table 1) the SoD polaron structure is adequately described (Fig. 2f) [START_REF] Brouwer | Superstructure of 1s-TaS2[END_REF][START_REF] Brouwer | The low-temperature superstructures of 1T-TaSe2 and 2H-TaSe2[END_REF] . However, for all other electronic regimes it was necessary to add P3 m as a secondary phase to achieve good fits (see Fig. 3c, inset). The temperature dependence of the interatomic distances and bond angles, and deductions resulting from fits to the data are addressed sequentially below, starting from high temperatures, well above any known charge or spin ordered phase.

Polymorphic regime -Heating 1T-TaS2 above 600 K results in irreversible polymorphic transformations altering average symmetry and stacking, [START_REF] Franchini | Polarons in materials[END_REF][START_REF] Kochubey | Charge density waves in 1T-TaS2: an EXAFS study[END_REF] where nominally 50% of octahedrally coordinated 1T layers convert into trigonal prismatically coordinated 1H layers [START_REF] Givens | Thermal expansion of NbSe2 and TaS2[END_REF][START_REF] Tidman | Resistivity of thin TaS2 crystals[END_REF] (Fig. 2a inset). This is accompanied by polymorph-specific stacking of the two layer-types [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF][START_REF] Jellinek | The system tantalum-sulfur[END_REF] and changes of the local Ta environment in the prismatic layers. Two such transformations are seen in the PDF data (Fig. 2b), at 630 and 880 K (see Supplementary Figure 1 for more details). Remarkably, a shoulder-signal (~3.5 Å, Fig. 2b) associated with local lattice distortions (polarons) is observable for 730 K < T < 915 K. The distorted fraction does not vanish in the polymorphs but instead drops to half its value observed in the 1T regime (T < 630 K), with no significant reduction through the second transformation (Fig. 2b). Notably, PDF data above 730 K conform to the 6R-TaS2-like model (R3m symmetry) featuring equally abundant alternating 1T and 1H layers (Fig. 2c, bottom) [START_REF] Achari | Alternating Superconducting and Charge Density Wave Monolayers within Bulk 6R-TaS2[END_REF] . The model features undistorted 1T and 1H layers, but a clear distortion signal, albeit weaker, such as is seen at 600 K, persists in the fit differential, implying the existence of polaron-specific deformations in the 1T layers well above 600K (Fig. 2c, top), with ~10 wt% distorted fraction (see below).

High-temperature metallic regime -the data at 600 K show multiple components at the position where the P3 m model predicts a single Ta-Ta peak (Fig. 1c and red trace fit in Fig. 2c,Rw=15%). The distortion spans ~5 Å with weaker signal extending up to ~12 Å, depicted by the difference PDFs. Such signatures, also seen in the IC phase, are weaker than in the NC and C regimes (Fig. 1c). However, their similarity and continuity imply a common origin and unambiguously reveals the presence of high temperature electron localization, detectable by charge-lattice coupling [START_REF] Koch | Dual Orbital Degeneracy Lifting in a Strongly Correlated Electron System[END_REF][START_REF] Bozin | Local orbital degeneracy lifting as a precursor to an orbital-selective Peierls transition[END_REF] . Despite similar P3 m model misfits shown in Fig. 1c, explicit data differences, Fig. 1d, reveal subtle changes across the M-IC transition for r > 5 Å. The electron localization results in incomplete breakup of the P3 m matrix, but without the formation of the full SoD motif. The light red profile in Fig. 2d and the intensity band in Fig. 1e at ~3.3 Å remain broad, consistent with incoherent distortions from weakly developed fluctuating charge correlations. In this diluted limit local distortions were approximated as P 3 nanoscale inclusions, Fig. 2g, within the undistorted P3 m matrix. This model (cyan traces in Fig. 2c (top), Rw=8%) accounts for the misfit, resulting in ~20 wt % of P3 distortions involving heavily puckered hexagonal Ta discs surrounding central tantalum. (This is consistent with ~10 wt% of locally distorted environments in the 1T-layers in the polymorphic regime above 630 K discussed in the previous section).

Incommensurate regime -The growth of charge correlations and structural coherence of associated distortions results in their signature extending over a longer length-scale (Fig. 1c). Stacks of experimental PDFs for 15-500 K (Fig. 2d,e) show that short length-scale peaks (r < 5 Å) sharpen gradually on cooling. However, the data still does not show a clear SoD motif (Fig. 2d). The data at ~3.5 Å are broad and unresolved, and at ~3.75 Å are rather featureless. Conversely, peaks at higher distances show the opposite temperature trend. In the light red stack, Fig. 2e, the signal at ~23.5 Å is the sharpest just below the M-IC transition, portraying high symmetry, and systematically broadens with intensity reduction as temperature decreases, evidencing growth of brokensymmetry correlations. The change in slope of Te U11(T) at 550 K (Fig. 1b) indicates an increased localized charge density. Model-derived Ta-Ta distances for T > 350 K shown in Fig. 3a portray substantially distorted hexagonal SoD cluster cores comprised of short Ta-Ta contacts (Fig. 3d,f). The distorted fraction increases to ~30 wt% compared to the M state (Fig. 3c). The PDF peaks which are sensitive to interlayer stacking are very broad, as shown in Fig. 4 a,c, implying inhomogeneous stacking. The distribution centroids of the closest interlayer separation, Fig. 4a, are shifted to the left, well below 5.9 Å, indicating local compression of layers which are, on average, closer together than they would be in the undistorted 1T structure. The observation of negative thermal expansion jumps along the stacking axis is relevant in this context [START_REF] Sezerman | Thermal expansion of lT-TaS2 and 2H-NbSe2[END_REF] (see Supplementary Figures 2 and3 for discussion).

Nearly commensurate regime -Significant local structure changes occur at the IC-NC transition. The distortion fingerprint, Fig. 1c, grows dramatically and PDF intensities vividly redistribute below 5 Å (Figs. 1f and2d). The Ta-Ta peak at 3.25 Å, which is broad in the IC phase, splits into an incompletely resolved triplet of unequal intensity peaks at ~3.15 Å, ~3.3 Å and ~3.8 Å, marking the formation of well-defined SoD clusters. The feature at ~3.15 Å corresponds to the Ta13 SoD intra-cluster configuration, whereas the other two features depict the inter-cluster correlations (Fig. 3). Residual intensity at ~3.4 Å is associated with discommensuration regions of approximately P 3 m character, separating hexagonal domains of well-ordered SoD clusters within the Ta layers [START_REF] Ma | A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2[END_REF][START_REF] Cho | Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2[END_REF] , well observable by STM. Multiphase modeling with coexisting P3 and P3 m phases confirms this picture (Fig. 3c inset, see Methods). As SoD clusters emerge, the distortion magnitude increases (Fig. 3a,b), consistent with increasingly resolved signal seen for 3 Å < r < 4 Å shown in Fig 2d . Importantly, the star vertices exhibit sudden asymmetric contraction, accompanied by substantial puckering (Fig. 3d,f) and SoD twisting (Fig. 3e). The distorted P3 phase becomes the dominant fraction below 350 K, at ~60 wt% (Fig. 3c). This is consistent with the patchy nature of the discommensurate NCCDW phase. Narrow thermal hysteresis, shown in Figs. Commensurate regime -At the NC-C transition, the PDF intensity at ~3.4 Å diminishes, implying the gradual disappearance of the ordered domain walls, while the intra-SoD-cluster peak sharpens (Fig. 2d), consistent with increased SoD density and additional charge localization. Similar changes are observed in the far-r limit (Fig. 2e), as a homogeneous crystal structure forms in the commensurate state. Closer inspection of the data unveils an additional restructuring below ~50 K, embodied in subtle peak shifts and sharpening observed in the dark blue traces in Figs. 2d and4a. Modeling shows that the distribution of Ta-Ta distances depicting intra-and inter-SoD Ta-Ta contacts changes character in the C phase, with tightly bound SoDs structurally further regularizing below 50 K (Fig. 3a Notably, there are 6 choices of such offsets, consistent with 13x stacking period and helical stacking arrangement suggested in some works [START_REF] Spijkerman | X-ray crystal-structure refinement of the nearly commensurate phase of 1T-TaS2 in (3+ 2)-dimensional superspace[END_REF] . Further qualitative changes are seen below 50 K in the PDF peak which includes the nearest neighbor intra-bilayer stacking peak just below 6 Å, whose intensity distinctly and abruptly shifts to lower distance (Figs. 4a,c). In contrast, the inter-bilayer peak at ~13.1 Å sharpens, but does not shift (Figs. 4b,d). This could imply enhanced intra-bilayer coupling, with neighboring bilayers presumably adjusting to preserve the inter-bilayer spacings. [START_REF] Ziebeck | Kohn anomaly in the 1T 2phase of TaS 2[END_REF] Symmetry of the order parameter. On the basis of a symmetry analysis of the observed PDF displacements we can identify the symmetry of order parameters which transform as the and irreducible representations in the parent P3 m structure space group. The displacements are described by two modes with wavevectors of the form = ( , , 0) and = (-, + , 0) at 60° to . In the C phase, = * + * , and = - * + * , where * and * are the reciprocal lattice vectors of the undistorted lattice. It is possible to decompose the atomic displacements of the Ta atoms at any temperature in terms of the independent amplitudes of the and modes described by and + . The displacements at 15 K are shown in the insert to Figure 3f. In the NC phase at room temperature [START_REF] Spijkerman | X-ray crystal-structure refinement of the nearly commensurate phase of 1T-TaS2 in (3+ 2)-dimensional superspace[END_REF] , aNC = 0.2448(2) and bNC = 0.0681 (2), which is close to the C values ( = 3/13 and = 1/13), while in the IC phase, aIC = 0.283 (2), and b = 0, where aIC is close to 2/7=0.2857. Note that our PDF analysis does not reveal any information about long range order, and 1 and 2 should not be interpreted to imply its existence. Rather, they describe the symmetry of the lattice distortions associated with the twists and displacements of the Ta atoms in the distorted SoD polaron structures. (See Supplementary Fig. 4).

DISCUSSION

The local structure data in 1T-TaS2 reveals symmetry-specific local structural fingerprints of polaron formation whose ordering evolves with temperature as shown schematically in Fig. 4f. Surprisingly, the polaron fingerprints are already identifiable in the polymorphic 6R-TaS2 regime (where individual 1T-monolayers appear between 1H layers), in the metallic phase (> 600 K), then successively through the IC, NC and C ordering transitions, and eventually in the non-QSL-like regime below 50 K, where -remarkably -SoD symmetry is restored in a layerdimerized undistorted in-plane SoD structure. An important fundamental question arises in the context of conventional CDW viewpoint regarding the lattice distortions in the high-temperature M phase: Should the deformations be discussed in terms of a uniform reduced modulation amplitude in the IC-like phase, or in terms of dynamical polaron fluctuations? Figure 1d indicates the range of the structural correlations. The black trace shows a difference across the IC-M transition, while the red trace is a difference corresponding to two PDFs in the same (M) phase separated by the same Δ . This comparison shows that dominant changes across IC-M occur at > 5 Å, with smaller changes for < 5 Å (Supplementary Fig. 5). This implies that at the M to IC transition both the number of distorted sites and the extent of IC spatial correlations grow. The M phase thus looks like a sparse polaron gas, whereas the IC phase is a sparse polaron crystal (incommensurate with the lattice). We can now consider the three possible scenarios for the IC melting transition based on the PDF data: In the first one, in the M phase, the long-range IC-CDW becomes dynamic, with amplitude and phase fluctuations on a timescale faster than the lattice can respond. In the second scenario, all relevant charges become fully itinerant on all length scales in the M phase without lattice distortions. In the third, polaronic, scenario the IC-CDW breaks up in the M regime but the lattice follows the localized charge fluctuations, which is equivalent to thermally activated polaron hopping. In all three scenarios, we would see a disappearance of the reflections at the IC modulation wave vector on going from IC to M in Bragg reflections, with the third scenario leaving some diffuse scattering in the M phase. In the PDF data, the first and second scenario would result in no characteristic polaronic distortions in the M phase. The fact that we do see them implies that only the polaron fluctuation scenario is relevant in 1T-TaS2. The persistence of polarons in the polymorphic regime (Fig. 2 b,c) and the change from 20 wt% to 10 wt% on the transition from a uniform 1T to the 6R multilayer structure confirms this notion and implies that polarons exist only in individual 1T monolayers. These may be expected to appear also in freestanding monolayers or monolayers on substrates, provided that strain and interaction with the substrate do not interfere.

Magnetic ordering.

It is interesting to consider the symmetry of the magnetic polaron structures that are compatible with the two possible in-plane magnetic space groups P 3 (#147.13) and P 3 ' (147.15). Two (ferro)magnetic P3 m subgroups can be derived from the analysis of the symmetric irreducible representation at the Γ point of the trigonal Brillouin zone of the 1T stack. For P3 , inversion does not flip the magnetic moment, allowing a magnetic moment component along for Ta1 and no constraints for the magnetic moments of the Ta2 and Ta3 atoms. In contrast, for the P3 'm magnetic space group, inversion reverses time and thus flips the magnetic moment, the magnetic moments of the Ta atoms of the same kind are not flipped by the symmetry operators. By choosing different components for Ta2 and Ta3 magnetic moments it is possible to create different motifs maintaining the same symmetry. Figs. 4 g,h show the motifs compatible with the symmetry operations of the two magnetic groups for in-plane chiral and non-chiral arrangements of magnetic moments respectively. An in-plane QSL-like state is consistent with fluctuating in-plane moments, and c-axis antiparallel moments in the P3 ' magnetic group, consistent with dimerized stacking of CDW orders in the C state.

A remarkable feature of the data is that local in-plane mirror symmetries appear to be restored below 50 K, with ≃ 120° and ≃ 150°. The data below 50 K are strongly suggestive of dimerization along the axis, consistent with inter-plane spin singlet formation and spin gap formation, that could explain the apparent cross-over from a ~ QSL-like spin relaxation above 50 K to a gapped, exponential T dependence below this temperature [START_REF] Klanjsek | A high-temperature quantum spin liquid with polaron spins[END_REF] . A plausible origin of the apparent symmetry restoration at low temperature comes from the disappearance of domain walls at low temperature. In-plane domain walls have been observed to persist well below the NC-C transition temperature, which topologically prevent interlayer stacking and spin dimerization. As long as domain walls exist, un-dimerized layers support an intra-layer QSL composed of spin structures shown in Figs. 4g andh, while regions with dimerized lattice layers are expected to exhibit inter-layer spin pairing [START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] . The restoration of SoD symmetry at lowest temperatures may thus be considered to be associated with the disappearance of DWs. A slightly different situation occurs in the metastable 'hidden' domain state [START_REF] Gerasimenko | Intertwined chiral charge orders and topological stabilization of the light-induced state of a prototypical transition metal dichalcogenide[END_REF] , where a QSL is allowed to form within domains whenever inter-layer stacking is indirect.

The polaron crystallization paradigm [START_REF] Fratini | Polaron Crystallization and Melting: Effects of the Long-Range Coulomb Forces[END_REF] emerging on the basis of these data has the benefit of providing a common framework for the understanding the polaronic physics in both the equilibrium and metastable phases of 1T-TaS2 and numerous related materials that display similar charge and spin ordering [START_REF] Vodeb | Configurational electronic states in layered transition metal dichalcogenides[END_REF] . The local structure symmetry analysis is particularly relevant for revealing the origin of the still poorly understood spin polaron structure and reconciles the observation of QSL-like behavior with theoretical band structure modelling, which commonly suggests a spin-paired dimerized ground state in a perfect crystal [START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] . Finally, we note that it was shown theoretically that Wigner-crystal ordering is not limited to systems with a half-filled electronic band, but may be also present at other fillings [START_REF] Camjayi | Coulomb correlations and the Wigner-Mott transition[END_REF][START_REF] Kapcia | Doping-driven metal-insulator transitions and charge orderings in the extended Hubbard model[END_REF] , which suggest that the crystallization phenomenon reported here may be more general than previously thought.

METHODS

Crystal growth and analysis. The 1T-TaS2 samples were grown using iodine vapor transport reaction, grown at 850 ° , and quenched to room temperature from the growth temperature. Single crystal XRD shows a pure single 1T phase is retained after the quench, with Ta:S composition determined by EDS to be 33: 66 ± 1 %.

Synchrotron X-ray atomic pair distribution function (PDF) measurements:

Temperature dependent X-ray total scattering data were collected at 28-ID-1 beamline of the National Synchrotron Light Source II (NSLS II) at Brookhaven National Laboratory. Finely ground powders of 1T-TaS2 were sealed in 1mm (outer diameter) polyimide capillary (referred to as experiment 1 or exp 1) and 1.5 mm (outer diameter) quartz capillary (experiment 2 or exp 2) within a glovebox under light vacuum. Measurements were carried out in capillary transmission geometry using a 2D PerkinElmer amorphous silicon area detector (2048 × 2048 pixels with 200 μm 2 pixel size) placed ~204 mm downstream of the sample. The setup utilized a monochromatic X-ray beam with 74.5 keV energy (λ = 0.1665 Å). Sample temperature control was achieved using a Cryo Industries of America cryostat (exp 1, 15 K ≤ T ≤ 500 K) and a FMB Oxford Hot Air Blower model GSB1300 (exp 2, 300 K ≤ T ≤ 915 K). Data for each experiment were collected in 5 K steps using 5 K/minute temperature ramp, 2 minutes thermalization, and 2 minutes data collection at each temperature. In exp1 data in temperature range 15 K ≤ T ≤ 300 K were collected on cooling, while these in temperature range 300 K ≤ T ≤ 500 K were collected on warming and cooling. In exp 2 data were collected on warming.

PDF data processing and analysis: Calibrations of the experimental geometry, momentum transfer range, and detector orientation were carried out by utilizing nickel standard measurements performed under the same conditions. Appropriate masking of the beam-stop shadow, inactive and outlier pixels, and subsequent azimuthal integration of the 2D images to obtain 1D diffraction patterns of intensity versus Q data were done using pyFAI software package [START_REF] Ashiotis | The fast azimuthal integration Python library: pyFAI[END_REF][START_REF] Kieffer | New tools for calibrating diffraction setups[END_REF] Standardized corrections to the data for experimental effects to obtain the reduced total scattering structure function, F(Q), and the subsequent sine Fourier transforms to obtain experimental PDFs, G(r), with Qmax=25 Å -1 (exp 1) and Qmax=20 Å -1 (exp 2) were carried out using the PDFgetX3 program within the xPDFsuite software package [START_REF] Juhás | PDFgetX3: a rapid and highly automatable program for processing powder diffraction data into total scattering pair distribution functions[END_REF] The PDF analysis was carried out using the PDFgui 71 modeling platform. The PDF, derived from powder diffraction-based Bragg and diffuse scattering data collected over a broad range of momentum transfer, describes direction-averaged distribution of atom pairs in a material as a function of interatomic distance, r, and provides structural information across different length scales. Uncertainties in reported model-derived parameters were estimated from their maximum variations observed by changing either the r-range of modelling (Fig. 1b, Fig. 3a-f) or integration (Fig. 2b, inset) by 10%. A double of the maximal values of such uncertainties was adopted as the error bars shown. Due to large X-ray scattering contrast (Z(Ta)=73, Z(S)=16) the dominant contribution to total PDF (Fig. 1e) originates from Ta-Ta pairs (red trace), followed by the Ta-S pairs' contribution (blue trace). The S-S (green trace) contribution is about 20 times weaker than the Ta-Ta contribution and 4 times weaker than that of Ta-S pairs. Estimate of distorted fraction and structure modeling in polymorphic regime: By using a calculated PDF based on undistorted P3 m model as a reference, we form difference PDFs for all T>600 K data. Such differences are then integrated over a 2.6-4 Å range (highlighted region on abscissa in Fig. 2b) and normalized to the 600 K value to quantify the relative change with temperature of the fraction of distorted sites across the polymorphic transformations, as shown in inset to Fig. 2b (for illustration see Supplementary Fig. 1). Above 730 K the shortrange PDF data can be explained by a 6R-TaS2-like model (R3m symmetry) featuring alternating 1T and 1H layers in equal abundance, Fig. 2c (bottom). Although the exact stacking and long-range character are likely more complex, as the model does not fully explain the data over longer length-scales, local analysis is rather robust. Importantly, the 6R-TaS2 model features undistorted 1T & 1H layers, hence not accounting for local distortions responsible for nontrivial nearest neighbor Ta-Ta distance distribution.

Local structure model with P symmetry: To map out temperature evolution of local interatomic distances, first data fits were carried out over 1 nm range and utilized a simple single S-Ta-S layer 1T-TaS2 model conforming to P3 symmetry. The model includes 19 independent structure parameters, constrained by symmetry to describe fractional coordinates of Ta (6 parameters) and S (13 parameters) atoms in the unit cell, two lattice parameters, two thermal parameters, and an overall scale factor. This explains the data features in the C state very well. The refinement range was established as a compromise between the model over-parametrization (data window too narrow) and the model deficiency (data window too wide) regimes. The 1 nm fitting range encompasses intrabilayer correlations but excludes inter-bilayer correlations, which were subsequently explored in a modelindependent manner as described below. (See Supplementary Table 1, Supplementary Figures 6 and7, and Section 2 for additional model details.)

Multiphase treatment: The fits using only the P3 phase in the NC regime were noticeably worse compared to these in the C state, with observable increase of fit residual, as shown in inset to Fig. 3c. This is consistent with the presence of discommensuration domain walls and lower SoD density in the NC state whose atomic structure is less distorted and not accounted for in our simple distorted P3 model. We approximated domain wall contribution by adding an undistorted P3 m structure as a secondary (minority) phase to account for local environment of the discommensuration regions. We utilized this approach to fit the PDF data in the IC and M regimes as well, where the distorted and undistorted phases swap roles and the distorted phase becomes a minority phase.

Assignment and consideration of inter-layer PDF peaks:

The inter-layer PDF peaks considered in this study were assigned based on undistorted P3 m model as reported for 300 K and its supercell variants. In this, interlayer spacings along the stacking direction are defined by multiples of c-axis lattice parameter, including ~5.9 Å which corresponds to intra-bilayer separation. The ~13.1 Å peak in undistorted P3 m structure corresponds to a welldefined off-axis Ta-Ta contribution connecting two Ta atoms that are 2 layers apart. These Ta atoms belong to next nearest neighbor Ta-layers and are offset within planes away from the stacking axis. In the high temperature regime, if structural distortions are incoherent, this peak in the data would be smeared, as is observed. As order establishes, re-appearance (with respect to putative P3 m structure that never materializes locally) of a sharp signal at ~13.1 Å implies re-emergence of structural coherence along this interatomic vector. The signal strength likely implies alignment of SoDs belonging to neighboring bilayers along this direction.

  3 be, reflects the 1 st order character of the IC-NC transition. As SoD local order is established in the NC state, they simultaneously form well defined bilayer correlations, Figs.4 a,c: The SoDs in adjacent layers couple and align along the c-axis. In contrast, inter-bilayer correlations are still fuzzy: different bilayers exhibit appreciable stacking disorder, evident from inter-bilayer sensitive peak, Figs.4 b,d, which probes off-axis correlations whose distribution is still considerably broad in the NC state.

  ,b). The intra-SoD angle θ reduces, approaching 120 o at base temperature, Fig 3e. The intra-star twist angle α increases, approaching 90 o below 50 K (Fig. 3d), while the local inter-star angle β reaches 150 o as SoDs align and form fully commensurate order (Fig. 3f) and the undistorted phase diminishes, consistent with vanishing domain walls (Fig 3c). Simultaneously, well defined inter-bilayer correlations develop (Fig 4 b,d). The most dramatic changes are observed at ~13.1 Å where PDF intensity redistributes and sharpens, possibly indicating an offset of pairs of SoDs, going from one bilayer to another, by one P3 m lattice spacing along both planar lattice vectors, Fig 4e.

Fig. 1

 1 Fig. 1 Electronic transitions & lattice symmetry breaking in 1T-TaS2. (a) Temperature dependent resistivity, adopted from Sipos et al. 9 , highlighting electronic phases in 1T-TaS2. The insert shows the undistorted P3 m lattice structure. (b) Temperature evolution of in-plane ADP of Ta from P3 m model fit to PDF data over 20-60 Å range. Red color indicates heating, blue color indicates cooling. Sloping black dotted line is a reference. Inset: Fit residual, Rw, implicates IC-M and IC-NC transitions. (c) P3 m model fit to 600 K data. Green difference trace (offset for clarity) reveals short range deviations. Stack of difference curves for the same model against data at different temperatures (as indicated) is shown underneath. Red arrow indicates a length-scale coarsely corresponding to the SoD polaron and the in-plane nearest neighbor inter-star center-to-center distance (inset). (d) Comparison of difference PDF between experimental data that are 10 K apart within metallic phase (T > 550 K) and across the IC-M transition, implicating existence of local distortions in the M phase over at least a radius r. (e) Calculated total PDF and partial pair-contributions in the P 3 m structure. (f) False color plot of PDF intensity versus temperature over a short r-range.

Fig. 2

 2 Fig. 2 Temperature progression of atomic structure in 1T-TaS2. (a) Stack of diffraction patterns revealing a polymporphic transformation on warming at ~630 K. Inset: intensity collapse of the (012) P3 m Bragg reflection during the restructuring from pure 1T to a hybrid 1T & 1H coordination. Further restructuring above 880 K results in additional Bragg reflections, marked by asterisk. (b) Corresponding PDFs over a narrow r-range. Data feature isosbestic points (circled) -nodes of temperature-invariant PDF intensity. Label 1T* indicates presence of local distortions in the 1T-type layers. Vertical blue arrow (1T*) marks polaronic distortion intensity. Inset: Change of fraction of distorted 1T* environments, relative to 600 K, estimated over the shaded region on the abscissa. (c) Polaron signature, seen in the M phase (top), persists in the restacked polymorph structure (bottom), and is associated with the 1T-type layers. Adding distortions, described in (g), as nanoscale inclusions results in improved fit (cyan trace). In (d) and (e) a stack of PDF data on different length scales is shown over the 15 K -500 K range, color-coded according to the electronic phase they belong to. At high temperature longer lengthscale peaks sharpen (indicated by vertical arrows) due to apparent average symmetry increase, despite expected broadening due to ADP increase at elevated temperature and in contrast to the shorter length-scale behavior. Features exhibiting negative thermal expansion (NTE) are indicated by block arrows. (f) At 15 K clear distortions associated with the star of David (SoD) are seen in the local structure, incompatible with the P3 m symmetry (top) but explained well (bottom) by a simple P3 broken symmetry model, depicted in (g) with focus on Ta atoms. Colored arrows show the Ta displacements. Out-of-plane degrees of freedom (exaggerated in the illustration for clarity) provide for puckering distortions.

Fig. 3

 3 Fig. 3 Local structure quantification from distorted local SoD model. (a) Local Ta-Ta distances from a P3 model fits to PDF data, obtained on cooling, over 1 nm length-scale in 15-500 K range. For reference, dashed horizontal gray line marks the undistorted P3 m value at 300 K. Intra-and inter-star distances are color coded as indicated in the legend. (b) Closeup of C phase behavior (left) and hysteretic response across the IC-NC transition (right). (c) In NC and IC phases secondary undistorted P3 m phase was necessary to reduce fit residual, Rw (inset). Distorted phase fraction, displaying thermal hysteresis consistent with one seen in resistivity, is calculated from atomic weight-based phase content obtained from the fits. (d)-(f) show selected intra-SoD and inter-SoD bond angles, as indicated in the sketch. In the IC phase the SoD motif is heavily distorted, with local distortions resembling discs. Upon entering the NC phase, the SoD distortion becomes better differentiated, with stars exhibit twists, and with interatomic angles beginning to evolve towards ideal values. In the C phase twists start to diminish. Below ~50 K a sharp regularization of stars is observed, with intra-star Ta-Ta distances becoming equal, intra-star angles approaching ideal values, and no twisting. Colored arrows in (b)-(f) indicate data collection on warming (red) and on cooling (blue) cycles. Inset to (f): displacements of the Ta atoms for the q1 (red) and q1+q2 (blue) modes from symmetry analysis at 15 K, compared to the parent P3 m phase (gray).

Fig. 4

 4 Fig. 4 Model independent considerations of stacking correlations. (a) Temperature stack of experimental PDFs around ~5.9 Å peak containing intra-bilayer correlations of SoDs, indicated in (e) by blue and red double arrows. (b) Temperature stack of PDFs around ~13.1 Å peak depicting inter-bilayer correlations of SoDs, indicated in (e) by purple double arrows. The contribution assignment is made based on features in P3 m model, Fig. 1e. There, the 5.9 Å feature is the c-axis lattice repeat distance involving strong signal from the nearest neighbor interlayer Ta-Ta correlations, whereas the 13.1 Å feature is well isolated and dominated by the next nearest neighbor off-axis interlayer Ta-Ta correlations. In (c) and (d) temperature stacks of differential PDFs are shown for data in (a) and (b), respectively, with differentials calculated using 500 K reference. Vertical dashed lines mark positions of apparent maxima in the C phase. (e) Sketch of two bilayers highlighting correlations discussed in text. Inter-bilayer SoDs are separated by ~13.1 Å at low temperature, a peak forming on cooling from a broad distribution at high temperature. This separation results from a SoD-shift by approximately one P3 m lattice spacing in each P3 m lattice direction. Note 6 possible choices of relative positioning of nearest bilayers, indicated by the purple shaded hexagon. (f) Schematics of distortions across different electronic phases in 1T-TaS2. (g, h) magnetic moments consistent with the distortions of the SoDs as determined by the PDF analysis for two different settings. The arrows at the center Ta1 atom pointing along the z axis are shown in green (only in P3 m).

ACKNOWLEDGMENTS

We wish to acknowledge Igor Vaskivskyi, Jaka Vodeb and Viktor Kabanov for valuable discussions. Work at Brookhaven National Laboratory is supported by the Office of Basic Energy Sciences, Materials Sciences, and Engineering Division, U.S. Department of Energy (DOE) under Contract No. DE-SC0012704. This research used beamline 28-ID-1 of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. DM and PS wish to acknowledge funding from ARRS. The work at the Jozef Stefan Institute was supported by the Slovenian Research Agency (P1-0040).

DATA AVAILABILITY

All of the data supporting the conclusions are available within the article and the Supplementary Information. Additional data are available from the corresponding authors upon request.

AUTHOR CONTRIBUTIONS

EB and MA designed and conducted the X-ray experiments, EB performed PDF analysis, GB and SC performed group theoretical analysis. DM and EB conceptually proposed the experiments. EB and DM wrote the paper. PS synthesized and characterized the crystals.

COMPETING INTERESTS

The authors declare no competing interests in this work.

Supplementary Information

Crystallization of polarons through charge and spin ordering transitions in 1T-TaS 2 E.S. Bozin 1 , M. Abeykoon 2 , S. Conradson 3 , G. Baldinozzi 4 , P. Sutar 3 and D. Mihailovic [START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] 1 Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, NY 11973, USA [START_REF] Klanjsek | A high-temperature quantum spin liquid with polaron spins[END_REF] Photon Sciences Division, Brookhaven National Laboratory, Upton, NY 11973, USA [START_REF] Darancet | Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides[END_REF] Dept. of Complex Matter, Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia [START_REF] Wilson | Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides[END_REF] Centralesupélec, CNRS, SPMS, Université Paris-Saclay, bât Eiffel, Gif-sur-Yvette, Île-de-France, 91190, FRANCE

SUPPLEMENTARY FIGURES AND TABLES

Supplementary Figure 1. Polymorphic transformations in 1T-TaS2 on warming in the M phase.

Supplementary Figure 2. Evolution of stacking correlations in 1T-TaS2.

Origin of c-axis negative thermal expansion (NTE) jumps Supplementary Figure 3. Possible origin of the c-axis NTE jumps.

Supplementary Figure 4. Comparison of PDF data of 1T-TaS2 in the low temperature regime.

Supplementary Figure 5. Local structure changes across M-IC transition in 1T-TaS2.

Supplementary Table 1. Details of the single layer 𝟏𝟑 × 𝟏𝟑 supercell P𝟑 model. Supplementary Figure 6. Features of the P𝟑 structure fit to the 15 K PDF data of 1T-TaS2 in the C state. Supplementary Figure 7. Considerations of the fitting range and the applicability of the P𝟑 approximant model.

ORDER PARAMETERS DESCRIBING THE SYMMETRY CHANGE LEADING TO THE COMMENSURATE CDW LOW TEMPERATURE PHASE OF 1T-TAS2.

Revisiting the 1T (single layer) C-CDW: atomic vibrations

The low T variants Symmetry adapted coordinates of Ta displacements at q1 and at q1+q2. Cartesian components of the displacements of each independent Ta atom in the daughter phase Supplementary table 2. Fractional displacement contributions originating from the amplitudes transported at q1 and q1+q2.

SUPPLEMENTARY FIGURES AND TABLES

Supplementary Figure 1. Polymorphic transformations in 1T-TaS2 on warming in the M phase.

a Stack of diffraction patterns revealing two polymorphic transformations on warming, T > 600 K. Red stack data, T > 870 K, reveal additional transformation to that focused on in the main text. This additional restructuring likely involves a distorted mixture of 4Hb (P63/mmc) and 6R (R3m) polymorphs with significant stacking faults. Low resolution of our diffraction data did not allow unambiguous characterization. Tentative PDF assessment suggested a complex mixture of alternating 1T and 1H layers (inset). b Transformation temperatures charted by fit residuals of P3 m and R3m models, as indicated in the panel. Inset: sketch of the free energy of TaS2, depicting metastable state (1T) and more stable configurations (1T+1H) achieved as temperature is elevated. c Stack of narrow r-range difference PDFs, ΔG, obtained by subtracting simulated PDF from a literature reference based on undistorted P3 m at 300 K, from experimental PDFs (lower r-resolution, Qmax=20 Å -1 ) collected at T > 600 K. The difference PDF is acquired for all M-phase data. To estimate the change in fraction across the polymorphic transformations, the difference PDF, ΔG, was integrated between 2.6 -4 Å (isosbestic points encompassing PDF feature of interest, between vertical arrows) for each temperature, and the final integral was normalized to the value obtained at 600 K. The uncertainties in the distorted fraction were estimated by extending the integration range over the grey window (nearest low-r zero crossings) and repeating the process. A double of the maximal value of such uncertainty was adopted as the error bars shown in the inset of Fig. 2b Coloured traces mark 500 K (solid red line), 350 K (dashed red line), 200 K (dashed dark red line), 50 K (solid cyan line) and 15 K (solid blue line). b. In P3 m structure layers are equidistant, as sketched. c. In distorted heavily disordered structure, such as that observed at higher temperatures, puckering distortions result in heavily distorted layers, resulting in interlayer stacking disorder and observed broad distribution of nearest neighbour interlayer distances, such as seen in a. In proposed NTE model this results in layer-shears and monoclinic compressions yielding shorter-than-expected average c-lattice parameter. d. As interlayer orders set in on cooling, the puff-pastry-like stacking disorder gets lifted, and the c-axis expands resulting in observed NTE jumps, as described in text.

Origin of c-axis negative thermal expansion (NTE) jumps

Dilatometry measurements in 1T-TaS2 [1] revealed anomalous NTE jumps in c-axis parameter, most significantly at the NC-C transition, but also at the IC-NC transition, origin of which has not been understood to date. Our data also reveal NTE behavior of certain PDF features, as marked in Figs. 2b,d,e and 4a of the Main Text, as well as in Supplementary Figure 3, and offer a simple explanation for this phenomenon. Based on these observations we propose a disorder-based rationale for NTE jumps. In putative undistorted P3 m crystal structure 1T layers stacked along c-axis are equidistant, Suppl. Fig. 4b. However, as our PDF analysis demonstrates, 1T-TaS2 is locally distorted at all temperatures at which this metastable polymorph exists, including the highest temperature in the M phase, resulting in 1T*-TaS2. There, the distortions involve local puckering of Ta layers, such as schematically shown the top panel of Fig. 4f in the Main Text as well as in Supp. Fig. 4c. As a consequence, disordered layers exhibit a range of interlayer distances and concomitant sheer displacements due to Coulomb repulsion of polarons in neighbouring layers, evident from broad interlayer distance distributions that are, importantly, centered at nominal distance shorter than simple integer multiples of undistorted P3 m c-axis lattice parameter (Main Text Fig. 4a,c at high temperature, and also Supplementary Fig. 3c and4a) resulting in, on average, closer packing of the layers. Due to this, the experimentally measured reference c-axis lattice parameter at high temperature is anomalously short, resembling puff-pastry configuration as depicted in Fig. 4 Text. c The r-region sensitive to the nearest neighbour intra-bilayer SoD correlations. The peak also exhibits observable shifting, consistent with enhanced inter-bilayer binding. Vertical black line in b and c marks the centroid of feature of interest at 50 K. One possible origin of this behaviour, as discussed in the Main Text, is a formation of spin-singlet dimers associated with the stray spins, presumably from the dz 2 manifolds, in two adjacent SoDs within the bilayers. Notably, the amount of change in PDF signal seen in c is quite large. It is unlikely than only one Ta-Ta contact per SoD (1:13) contracts, implying that the mechanism is more complex than simple pairing of spins in dz 2 orbitals portrayed in d. Importantly, the features in the differential in a are well structured, stronger under some PDF peaks and weaker under others. Given that the differential is of two data sets, it is unlikely that they originate from artifacts such as termination ripples or noise in the data. Even if present, such effects would effectively cancel out in the subtraction, leaving only signature of subtle structural differences. We further note that as part of the experiment we carried out a counting statistics test to assess the signal to noise ratio. In this, data statistics was found to be satisfactory after counting for 10 seconds. The actual counting time was 120 seconds, an order of magnitude longer. In order to establish suitable range for the local structure study, the model was tested against the 15 K dataset. In this, 19 structural parameters, described in Suppl. Fig. 6, were varied, while the lattice parameters, thermal parameters, and the overall scale were kept fixed. The fitting test was conducted over a variable range of data by incrementing the upper limit of refinement, rMAX, by 1 Å, from 5 Å to 15 Å, as shown in the figure. In this process we monitored two quantities: the fit quality captured by the fit residual, RW, and the number of parameter-pairs whose correlation is larger than |0.8| as reported by the fitting program. In the former, large value of RW indicates inadequacy of the model used. In the latter, parameters exhibiting correlation >|0.8| are typically considered to be unreliable since they are not independent when too narrow a data range is considered. Their report by the fitting platform indicates an over-parametrization of the model or, conversely, insufficient information in the data to constrain the model. In our test, fits of exceptionally good quality were obtained for rMAX 8 Å, as shown by the RW(T) dependence (red symbols in the bottom panel, right ordinate) and illustrated in the top left panel. However, fits over these ranges result in heavily over-parametrized fits, as indicated by the temperature dependence of the number of correlated parameter pairs (blue symbols in the bottom panel, left ordinate). Increasing the fitting range results in parameters becoming uncorrelated, as sufficient amount of data is added to the fitting. While this initially helps, with a moderate jump in RW, for rMAX  11 Å the fits begin to deteriorate, as illustrated in the upper right panel.

The reason behind this is that wider range of PDF data provides access to significant contributions of the NNN interlayer correlations which are beyond the scope of our present model. We identified rMAX = 10 Å as a sweet spot offering a reasonable compromise between the over-parametrization (model too complex for the data range) and underfitting (model inadequacy). The result is a reliable determination of the local structure, which is the focus of the present work.

ORDER PARAMETERS DESCRIBING THE SYMMETRY CHANGE LEADING TO THE COMMENSURATE CDW LOW TEMPERATURE PHASE OF 1T-TAS 2 .

The analysis of the symmetry of the high temperature structure of 1T-TaS2 can help understanding the structural changes of the low temperature daughter structures in terms of the symmetry adapted coordinates belonging to the irreducible representations of the parent space-group symmetry. The physical order parameters describing the change (in this case, the atomic displacements) can be classified as basis functions of the irreducible representations of the parent symmetry group. In the current case, multiple order parameters contribute to the final observed distortion: the individual degrees of freedom provided by an instance of these multi-dimensional order parameters are physically meaningful because they provide a natural description of the distortions of the parent structure.

Lowering the temperature, some of the symmetry elements of the symmetry group of the parent phase are lost (e.g the mirror planes), but the surviving symmetries that describe the daughter phase are a subgroup of the parent phase that is uniquely identified by its combination with the new superlattice basis (and eventual new supercell origin choice). Using a frozen-phonon picture, the structural parameters describing the low temperature daughter structures can be organized as a superposition of polarization vectors containing the cartesian components of the displacements of each one of the independent atoms of the daughter phase.

Revisiting the 1T (single layer) C-CDW: atomic vibrations

In the following, we totally neglect any c-stack rule and consider the symmetry changes occurring in a single layer of the TaS2 structure. We start from the standard description of the prototype phase that has space group symmetry P-3m1 (#164, D 3 3d, aP=bP≈3.34 Å, cP≈5.894 Å, =120°). In this structure, there are two independent atoms with positions described by the coordinates summarized in the table below:

The low T variants

All the low temperature phases have axes rotated when compared to the original frame of the high temperature prototype structure. The lattice changes produced by rotation around the z-axis (which has A2g representation) can maintain the -3 symmetry operations, but they systematically remove the mirrors and 2-fold axes of the parent group. The low temperature phases also involve superlattice reflections and therefore require the definition of a superlattice cell in real space. The superlattice reflections characteristic of the low temperature variants (IC, NC, and C-CDW phases) involve a condensation of Bragg intensity at some q-points in the high temperature Brillouin zone.

In the simplest description of the low temperature variants (incommensurate or commensurate), two additional independent vectors of the high temperature Brillouin zone are required. The low temperature phase (C-CDW) is obtained by the following matrix transformation of the prototype lattice

In this reference cell, there are 13 replicas of the TaS2 motif. The atomic positions of the prototype structure can be described in this new metric that has symmetry P-3 (#147, a=b≈12.04254 Å, c≈5.894 Å, =120°) by a new set of independent atomic positions summarized in the table below: For the q1 star we can simply take the spots indexed by the elements (E, -3 + and -3 -) with explicit components (3/13 1/13 0) (-1/13 4/13 0) (4/13 -3/13 0): therefore, the OP describing the superpositions of the atomic displacements has components (a0a0a0000000). The same choice can be made for the q1+q2 star, or explicitly (2/13 5/13 0) (-5/13 7/13 0) (7/13 -2/13 0). The OP describing the transformation from the parent phase to the C-CDW lock-in phase has overall symmetry components (a0a0a0000000;b0b0b0000000). It is then possible to decompose the atomic displacements of the Ta atoms at any temperature in terms of the amplitudes of the Eu and A2u modes transported at q1 and at q1+q2.

Symmetry adapted coordinates of Ta displacements at q1 and at q1+q2. This analysis can be also extended to S atoms, but:

-more components are involved, reducing the value of the decomposition procedure -optic phonons do not seem particularly relevant for this phase transition mechanism

Cartesian components of the displacements of each independent Ta atom in the daughter phase

The supplementary table 2 summarizes the fractional displacement contributions originating from the amplitudes transported at q1 and q1+q2. These displacements add to the undistorted positions of the Ta2 and Ta3 atoms inherited from the high temperature parent phase. The displacements of the Ta1 atom at the origin of the lattice are fixed by the symmetry element of the daughter phase, so they are not reported in the table. Some trends can be remarked:

-The out-of-plane z component decreases with decreasing T, resulting in puckering of the Ta atoms