Data for "Diagnosing phase transitions through time-scale entanglement"

This data set contains the raw numerical data, the Julia plot scripts and the resulting figure files for all figures of the main text (Figs. 1-6) and of the Supplemental Material (Figs. 1-27) of the publication "Diagnosing phase transitions through time-scale entanglement" by S. Rohshap, H. Ishida, F. Bippus, L. M. Verhoff, A. Kauch, K. Held, H. Shinaoka and M. Wallerberger. A preprint is available at arXiv:2507.11276. The README.txt in this record describes the folder structure, every data file and how to run the scripts.

Background

Spatial entanglement of quantum states is a central paradigm of many-body physics. This work studies a different kind of entanglement, the entanglement between imaginary time scales, which is accessible through quantics tensor train diagnostics (QTTD): an n-particle correlator is written on a quantics grid of its imaginary-time (or Matsubara-frequency) arguments and compressed into a quantics tensor train, whose bond dimension D_l at bond l encodes the coupling between the temporal scales 2^l. The central result is that this time-scale entanglement, measured by the maximum bond dimension D_max or the sum of bond dimensions D_sum, is generically enhanced near phase transitions and crossovers, becomes scale invariant at quantum critical points, and shows distinct signatures at first-order transitions. The data set covers all systems of the paper: the Hubbard dimer and four-site Hubbard rings (ground-state crossings, compared with entanglement measures and susceptibilities), the transverse-field Ising model (quantum critical point), the single-impurity Anderson model (Kondo crossover) and the Mott transition in the dynamical mean-field theory (DMFT) solution of the Hubbard model on the Bethe lattice and in DFT+DMFT for NdNiO2.

Structure

data_publication.zip contains six self-contained folders, one per physical system. Each holds the raw data, the plot script(s) and the figure PDFs:

• hubbard_dimer/: main-text Figs. 1 and 5, SM Figs. 1-6. One Julia script per figure (main_fig1.jl, main_fig5.jl, sm_fig1.jl, ..., sm_fig6.jl), a helper file common.jl, make_all.jl, Project.toml, a README.md, file_manifest.csv (one row per raw file with its parameters), the raw HDF5 output files of the QTT runs in data/<dataset>/ and the figures in plots/.
• four_site_ring_nn/: four-site Hubbard ring with nearest-neighbour hopping, SM Figs. 7-12; same layout (sm_fig7.jl, ..., sm_fig12.jl). SM Fig. 12 is computed by exact diagonalisation in compute_double_occupancy.jl.
• four_site_ring_tprime/: four-site Hubbard ring with next-nearest-neighbour hopping t', SM Figs. 13-17; same layout (sm_fig13.jl, ..., sm_fig17.jl).
• Mott/: main-text Figs. 3 and 6, SM Figs. 18-20. QTT bond dimensions of the particle-hole bubble G(iν)G(iν+iω) from DMFT solutions of the Bethe lattice (insulating and metallic starting points) and from DFT+DMFT calculations for NdNiO2, the double occupancies and the analytically continued spectral function of NdNiO2 (8 HDF5 files), the script mott_plot_script.jl and the figures fig_3.pdf, fig_6.pdf, fig_supp_18.pdf, fig_supp_19.pdf and fig_supp_20.pdf.
• SIAM/: main-text Fig. 4, SM Figs. 21-24. All data in plot_data.jld2, the script plot_all.jl, Project.toml and Manifest.toml with the exact Julia environment, a README.md and the figures in figures/.
• TFIM/: main-text Fig. 2, SM Figs. 25-27. QTT bond dimensions of the Green's function G(k,iν) of the transverse-field Ising model on a 30 x 30 grid of J and h for several QTT cutoffs (2 HDF5 files), the script tfim_plot_script.jl, which also recomputes the QTT of the local Green's function G(iν) shown in Fig. 2(c,d) and SM Fig. 27, and the figures fig_2.pdf, fig_sup_25.pdf, fig_sup_26.pdf and fig_sup_27.pdf.

A figure index in README.txt lists, for every figure of the paper, the folder, the script and the output file.

Technical Details

• All scripts are written in Julia (julialang.org) and use CairoMakie for plotting and HDF5 or JLD2 for reading the data. SIAM/ was run with Julia 1.10.5 (CairoMakie 0.12.18, Makie 0.21.18, JLD2 0.4.51, LaTeXStrings 1.3.1; exact versions in SIAM/Manifest.toml). The other folders were run with Julia 1.12.4, CairoMakie 0.15.15 and HDF5 0.17.4; TFIM/ additionally needs ITensors 0.7.13, ITensorMPS 0.3.6, QuanticsGrids 0.7.2, QuanticsTCI 0.7.3 and TensorCrossInterpolation 0.9.19. The folders hubbard_dimer/, four_site_ring_nn/, four_site_ring_tprime/ and SIAM/ contain a Project.toml and are run with "julia --project=. make_all.jl" (SIAM: plot_all.jl) after "julia --project=. -e 'using Pkg; Pkg.instantiate()'"; the scripts in Mott/ and TFIM/ are run from inside their folder with the required packages in the active environment.
• Data generation: the correlators of the Hubbard dimer and rings and of the single-impurity Anderson model were obtained by exact diagonalisation and compressed into quantics tensor trains with the ITensors (SVD), QuanticsGrids and TensorCrossInterpolation (TCI) libraries; the Green's functions of the transverse-field Ising model are the exact Jordan-Wigner results; the DMFT and DFT+DMFT data were produced with w2dynamics (DFT input from Wien2k, Wien2wannier and Wannier90), and the spectral functions were analytically continued with ana_cont.
• All data files are HDF5 (.h5, .hdf5, .jld, .jld2); the figures are PDF files.

Licensing

The data and the PDF files are published under the Creative Commons Attribution 4.0 International licence (CC BY 4.0); the distributed code (all *.jl files) is published under the MIT licence.