Fig. 2: Demonstration of universal scaling for tunnelling into a Tomonaga-Luttinger Liquid (TLL).

a, b Temperature dependent point spectra of the edge state’s local density of states (LDOS) up to T = 25 K, measured at the Y-edges off WTe₂/HOPG and WTe₂/BLG, respectively. The tunnelling spectra are offset by 0.6 on the y-axis with the zero position highlighted by horizontal dashed lines. Solid black lines are fits to TLL theory, taking into account thermal broadening (Section S3 of the Supplementary Information). The corresponding bare power-law fits are shown as dotted curves. The shaded bands indicate the conduction band tail. c, d The data in a and b can be collapsed onto a single universal curve with common scaling exponents α = 0.45 (WTe₂/HOPG) and α = 0.98 (WTe₂/BLG). Shaded bands indicate statistical confidence defined by the mean absolute error across the individual fits at different measurement temperature. The shaded band in d indicates the conduction band tail. e, f Temperature dependence of the measured zero-bias (E = E_F) conductance in a–d (blue circles), compared with the TLL model (solid black lines) and Coulomb gap (CG) model (solid red lines). Fitting parameters of the CG model, such as the disorder parameter Ω, have been determined from a best fit to each data at T = 4.5 K (see Supplementary Fig. 4). The error bars are obtained from the Gaussian width of the TLL fitting residual. Shaded bands indicate statistical confidence as defined for (c, d).