Same disordered Heisenberg model as the 3D version, exactly diagonalized, but shown two-dimensionally instead of as a rotating 3D spectrum:
H = J Σᵢ [Sᶻᵢ Sᶻᵢ₊₁ + ½(S⁺ᵢS⁻ᵢ₊₁ + S⁻ᵢS⁺ᵢ₊₁)] + Σᵢ hᵢ Sᶻᵢ
hᵢ ~ Uniform[−W, W] (open chain, J = 1)
Clicking "Resample & diagonalize sweep" builds the full Hamiltonian in the zero-magnetization sector (dimension C(L, L/2)) at nine disorder strengths W/J = 0…8 and exactly diagonalizes each one with the same cyclic Jacobi eigenvalue solver used by the 3D version — for every one of the chosen number of disorder realizations. That gives two genuinely 2D-native views for free, at no extra diagonalization cost:
- Level-flow diagram (top) — every eigenvalue plotted directly as energy (y) against disorder strength W (x), connected level-by-sorted-level across the nine samples. This is not a camera view of a 3D object: energy vs. a swept control parameter is a plane by construction. Where lines swerve apart before touching (avoided crossings) the chain is thermal; where they cross freely and clump, it is localized — colored red→green by each level's local gap ratio, exactly like the 3D spiral's bead colors.
- ⟨r⟩ vs. W phase curve (middle) — the mean gap ratio at each sampled W, connected into a curve against the two flat reference lines (Poisson 0.386, GOE 0.531). This is the MBL crossover itself drawn as an order-parameter-vs-control-parameter phase diagram — the most natural 2D representation of a phase transition, and something the 3D snapshot view never shows directly since it only renders one W at a time.
- r-statistic histogram (bottom) — flat 2D bars for the W value highlighted by the slider, over all sampled realizations, against the same P_Poisson(r) = 2/(1+r)² and P_GOE(r) = (27/8)(r+r²)/(1+r+r²)^(5/2) reference curves as the 3D version's 3D bar chart.
Moving the W slider after a sweep is free — it just re-reads the already-computed column and repaints the flow-diagram highlight and histogram; only "Resample & diagonalize sweep" runs new diagonalizations (9 disorder strengths × the chosen number of realizations, so keep realizations modest at L=10 if you want a snappy sweep).
Real-world relevance: this level-statistics test is the standard exact-diagonalization fingerprint physicists use to map the MBL phase diagram in cold-atom and trapped-ion quantum simulators, since it needs only the static spectrum, not a time-resolved measurement.