Extended Data Fig. 8: Power-law scalings (theory) and diffusion coefficients.

a, b, Decay time constants τ for different anisotropies Δ ranging from negative (a) to positive (b). Numerical results are shown in a for Δ = −1 (blue), −1.5 (green) and −2 (purple) and in b for Δ = 0 (red), 0.5 (orange), 0.85 (yellow), 1 (blue) and 1.5 (green). Solid lines are power-law fits (to the filled symbols). Open symbols are excluded from the fit owing to finite-size effects. Crossed symbols are results from \(\tilde{t}\)–J model simulations including 5% hole fraction. Fitted power-law exponents are shown in Fig. 3c. For positive anisotropies Δ ≥ 0 the decay time τ is defined as τ = τ′/ln(1/0.60) with c(τ′) = 0.60. For negative anisotropies Δ < 0, the decay time τ_I for short times (I) is defined as \({\tau }_{{\rm{I}}}=10{\tau }_{{\rm{I}}}^{{\prime} }\) with \(c({\tau }_{{\rm{I}}}^{{\prime} })=0.90\). For longer times (II), the decay time τ_II is obtained from exponential fits \({{\rm{e}}}^{-t/{\tau }_{{\rm{II}}}}\) to the diffusive long-time tail (see dotted curves in Extended Data Fig. 7a–c). c, Diffusion coefficients for the diffusive long-time regime (II) obtained from theory (open symbols) and experiment (filled symbols). For negative anisotropies Δ < 0, values were determined from quadratic power-law fits 1/τ = DQ² to the data points in a (theory) and Fig. 3a (experiment) for the diffusive regime (II). Note that for Δ ≥ 0 the system is only diffusive for Δ = +1, as shown in b (theory) and Fig. 3b (experiment). From the experimental diffusion coefficients, we estimate mean free paths δx using the velocity v = 0.76(1)v_F from the ballistic short-time regime (I), and obtain δx = 3.35(15)a, 1.07(7)a and 0.66(4)a for Δ = −1.02, −1.43 and −1.79, respectively. d, Short-time (t ≪ ħ/J_xy) decay constant τ = |Γ|⁻¹ obtained from Taylor expansion of the contrast c(t) = 1 + Γ²t² + … as a function of Q, for Δ = 0 (red), 0.55 (orange), 0.85 (yellow), 0.95 (purple) and 1 (blue). For Δ < 1 all curves in the log–log plot asymptote to the same slope as Q → 0 (continuum limit), whereas there are deviations for larger wavevectors Q. For Δ = 1 the slope is instead different. This indicates that the power-law exponent α in τ ∝ Q^−α depends on the range of wavevectors Q used to determine it. e, Power-law exponents α determined for the short-wavelength regime between λ = 6a and 20a (filled symbols) as in experiments and numerics, and for the long-wavelength regime between λ = 150a and 200a (open symbols) approaching the continuum limit. In the former case, the exponents show a smooth crossover from superballistic to diffusive as Δ → 1 similar to that in the experiments and numerics, whereas in the latter case the exponents show a sharp jump from ballistic to diffusive occurring exactly at Δ = 1.