Extended Data Fig. 9: Effective models.
From: Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system
a, Illustration of the effective potential between chain y with its neighbouring chain y + 1. Grey lines illustrate energy contributions proportional to \({J}_{\mu }{C}_{1}^{\mu }\), μ = x, y; green line denotes diagonal correlation with energy contributions of order JyC2 starting at |Σ| ≥ 2. Intrachain energy corrections from the Néel state of strength \({J}_{x}{C}_{1}^{x}\) are constant and not written down explicitly in the potential. b, Thermally averaged two-point correlations of the undoped Heisenberg model \({C}_{1}^{\mu }(T)=\langle {\widehat{{\bf{S}}}}_{{\bf{i}}}\cdot {\widehat{{\bf{S}}}}_{{\bf{i}}+{{\bf{e}}}_{\mu }}{\rangle }_{T}\), μ = x (grey), y (red) and \({C}_{2}(T)=\langle {\widehat{{\bf{S}}}}_{{\bf{i}}}\cdot {\widehat{{\bf{S}}}}_{{\bf{i}}+{{\bf{e}}}_{x}+{{\bf{e}}}_{y}}{\rangle }_{T}\) (blue) calculated from DMRG calculations for Jx/Jy = 0.3 on a 12 × 4 lattice with periodic boundary conditions applied along the short (y) direction. c, Thermally averaged string-length distribution \(\langle \varSigma | {\widehat{\rho }}_{{\rm{MF}}}^{(0)}| \varSigma \rangle \) for temperatures kBT/tx = [0.2, 0.625] and tx/Jy = 2 using the thermal correlations in the Heisenberg model in b. d, Hole distance distributions in the MHZ approach (equation (29)) for various temperatures kBT/Jy = 0.4–0.9. e, Mean length of excess stripes as calculated from the MHZ approach as a function of temperature. The dashed line marks the experimental temperature. f, Difference in stripe lengths from mean-field theory to random distribution for temperatures kBT/tx ∈ [0.2, 0.625] and Jx/Jy = 0.3 and experimental data for δ = 0.111 (markers) as in the inset of Fig. 4a. g, Stripe-length histograms using the classical MHZ estimate for temperatures kBT/Jy ∈ [0.4, 1], which shows qualitatively similar results.
