Fig. 3: Dielectric screening of the TLL many-body interactions.

a, b Histograms of the Luttinger parameter K, recorded across 69 locations along the X- and Y-edges of WTe₂/HOPG and WTe₂/BLG islands. Solid lines are fits to Gaussian normal distributions characterised to extract statistical mean and standard deviation. Arrows indicate data points from literature^{26,33 -- 35}. c Luttinger parameter K, extracted across different samples, and literature, plotted as a function of the substrate dielectric constant ϵ_r. Red (blue) circles are the mean and standard error of the mean extracted for the distributions in a, b. Shaded bands and dashed lines are numerical calculations of K based on the electrostatic energy W(q = 0) stored in a screened edge charge distribution of varying density (see text), as well as the calculated edge dependent Fermi velocity v_F extracted from d and e. The bands have been modeled to coincide in width with the extracted standard deviation of the distributions in a and b. The solid red line represents tight-binding calculations of the Luttinger parameter. Insert: Representative spectra measured at the Y- and X-edge, respectively, on HOPG, with extracted Luttinger parameters K ≈ 0.35 (Y-edge) and K ≈ 0.24 (X-edge) from TLL fits shown as black lines. For the dielectric constants, we assume ϵ_r = 6 (BLG/SiC)⁴⁰ and ϵ_r → ∞ (HOPG), approximating the perfect metal limit). d, e Tight-binding band structure of the Γ-Y and Γ-X directions, respectively, assuming the atomic terminations shown by coloured arrows in (insert to d). Grey shading indicates the 2D bulk band edges with a band gap comparable to the data in Fig. 1e, f. Calculations are based on the edge terminations highlighted in the insert to (d), showing the position of W atoms (blue) and Te atoms (orange). Insert to (e), Corresponding WTe₂ Brillouin zone.