Fig. 4 : Verifying the generalized eigenstate thermalization hypothesis.

a Fidelity \({{{\mathcal{F}}}}(\rho _{{{{\mathrm{st}}}}}|\rho _{{{{\mathrm{GETH}}}}})\) between the measured steady state \(\rho _{{{{\mathrm{st}}}}}\) and GME prediction \(\rho _{{{{\mathrm{GETH}}}}}\) against the standard deviation Δ_k for a walk with (blue circles) \({{{\mathcal{H}}}}_{{{{\mathrm{eff}}}}}(\pi /9,6\pi /9)\) and (purple diamonds) \({{{\mathcal{H}}}}_{{{{\mathrm{eff}}}}}(\pi /9,4\pi /9)\). The colored dashed and solid lines represent the theoretical simulations for the steady state estimated by \(\bar \rho _{t = 10}\) and \(\bar \rho _{t \to \infty }\), respectively. The dashed black line represents the upper bound of the initial Δ_k; below this line, the fidelity always approaches 1. The insets illustrate the effective band structures (colored solid lines with left-hand scale) and occupations of the initial state (with \({\Delta}_k = 0.3\) as an example) on the two bands (gray curves with right-hand scale). The expectation of the energy and momentum of the initial state is E₀ and k₀, respectively. In the insets, the vertical gray regions and horizontal colored regions show the chosen momentum window with \(\delta k = 0.4\) and associated energy window. b Measured expectations of the three spin observables of the reconstructed spinor eigenvectors \({{{\mathbf{n}}}}_{{{\mathcal{H}}}}(k)\) within the chosen energy-momentum window. The upper and lower panels show the results for \({{{\mathcal{H}}}}_{{{{\mathrm{eff}}}}}(\pi /9,6\pi /9)\) and \({{{\mathcal{H}}}}_{{{{\mathrm{eff}}}}}(\pi /9,4\pi /9)\), respectively. The colored dashed lines give the theoretical results for \(\rho _{{{{\mathrm{GETH}}}}}(E_{k_0}^{{{\mathrm{d}}}},k_0)\), and the translucent shadings illustrate the statistical errors