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 \({{{{{\rm{P}}}}}}\bar{3}{{{{{\rm{m}}}}}}\) 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, d temperature stacks of differential PDFs are shown for data in a, 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 \({{{{{\rm{P}}}}}}\bar{3}{{{{{\rm{m}}}}}}\) lattice spacing in each \({{{{{\rm{P}}}}}}\bar{3}{{{{{\rm{m}}}}}}\) 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 \({{{{{\rm{P}}}}}}\bar{3}{{{{{\rm{m}}}}}}\)).