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

Threaded Flux

Flux Φ / Φ₀ 0.00

Ring Population

Electrons N 12

Ring radius R (nm) 400

Dephasing (T / T_φ) 0.00

Live Readouts

Persistent current I

0.0 nA

Ground-state energy E

0.0 μeV

Direction

— none —

Parity

N even

Period

Φ₀

How it works

Thread a magnetic flux Φ through a metal or semiconductor ring whose circumference is shorter than the electron phase-coherence length ℓ_φ. The vector potential shifts the ring's periodic boundary condition, so the single-particle energy levels become

E_n(Φ) = (ħ²/2mR²) · (n − Φ/Φ₀)²,   Φ₀ = h/e,   n = 0, ±1, ±2, …

Filling the N lowest levels and summing gives a total ground-state energy E(Φ) that is periodic in Φ with period Φ₀ — the normal-metal h/e period, since single electrons (not Cooper pairs) carry the phase. Because the state is a true equilibrium ground state, it carries a current that never decays despite scattering:

I(Φ) = − dE(Φ)/dΦ   (sawtooth, period Φ₀)

Ring view (top) — a flat top-down map of the ring: drag left/right to tilt the view. Dot markers flow around the loop; blue = diamagnetic (opposes Φ), red = paramagnetic (aligns with Φ), and their speed tracks |I|.
Energy-level map (middle) — each curve is one single-particle level E_n(Φ)/E₀; the N lowest-lying curves at the current Φ are drawn bold and filled — those are the occupied levels. Watch the bold set re-sort as you sweep Φ or change N.
I(Φ) strip chart (bottom) — the sawtooth built by differentiating the summed energy; the moving dot marks the present Φ.
Electrons N — flipping N by one (odd ↔ even) shifts the sawtooth by half a period — the parity effect: an even-N ring is diamagnetic near Φ=0, an odd-N ring is paramagnetic.
Ring radius R — sets the level spacing ħ²/2mR² (E₀) and hence the current amplitude; smaller rings give a larger, more sharply-sawtoothed current.
Dephasing — models finite temperature / residual inelastic scattering by exponentially damping the current amplitude, reproducing why the effect vanishes above ~1 K in real devices.

Real-world relevance: exactly this h/e-periodic circulating current was measured in isolated gold and GaAs rings (Lévy et al. 1990; Bleszynski-Jayich et al. 2009, using a nano-SQUID cantilever) and is a direct, textbook demonstration that quantum phase coherence can drive a real, measurable, dissipationless equilibrium current at the nanoscale.

Ring & Flux (drag to tilt)

Energy levels Eₘ(Φ) / E₀ — bold = occupied

I(Φ) sawtooth

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