Persistent Ring Current: 2D Flux Sweep & Energy-Level Map

When a normal-metal or semiconductor ring is cooled below the point where its circumference is shorter than the electron's phase-coherence length, its ground state carries a real, non-decaying circulating current whenever a magnetic flux threads the loop. This 2D simulator builds the same physically grounded model as the 3D ring simulator: single-particle levels E_n(Φ) = (ħ²/2mR²)(n − Φ/Φ₀)² are filled with N electrons, the total energy is differentiated with respect to flux to get I(Φ), and the result is rendered across three linked flat panels — a top-down ring-and-flux view with flowing current markers, a live energy-level map that shows exactly which levels are occupied, and a flux-periodic I(Φ) strip chart. Sliders let you sweep the threaded flux, change the electron count to see the even/odd parity effect flip the current's sign, resize the ring to change the energy scale, and dial in dephasing to watch the effect wash out.