Quantum mechanics predicts that a resistive (normal) metal ring threaded by magnetic flux carries an equilibrium persistent current, an Aharonov–Bohm-periodic dissipationless current. High-precision cantilever-torque measurements in arrays of aluminum rings (Bleszynski-Jayich et al., Science 326, 272 (2009)) found quantitative agreement between the typical persistent current and the theory of diffusive, non-interacting, fully phase-coherent electrons, but the source work explicitly flags what remains unresolved: (i) the quantitative effect of dephasing on the equilibrium typical (sample-specific) persistent current has, to the authors' knowledge, not been calculated, so the observed faster decay of the current above T≈2T\approx 2,K in the smallest rings — where the phase-coherence length approaches the ring circumference — could not be tested against theory; (ii) the roles of electron-electron interactions, coupling to the electromagnetic environment, and trace magnetic impurities remain experimentally undetermined in regimes those measurements did not access; and (iii) persistent currents in nonequilibrium, driven circuits (e.g., microwave irradiation) remain essentially unexplored for normal-metal rings. This dephasing calculation remains to be carried out, and the other strands remain experimentally unresolved.

Background

A mesoscopic metal ring threaded by a magnetic flux Φ\Phi carries an equilibrium circulating current — the persistent current (PC) — that is periodic in flux with period h/eh/e (one flux quantum Φ0=h/e\Phi_0=h/e), a direct thermodynamic manifestation of the Aharonov–Bohm effect (the phase acquired by an electron wave function encircling the flux). In a diffusive ring, where elastic disorder makes electron motion a random walk with diffusion constant DD, two current measures matter: the typical (root-mean-square, sample-specific) current, which is h/eh/e-periodic and differs from ring to ring with the microscopic disorder, and the ensemble-averaged current, which is h/2eh/2e-periodic and, in the standard theory, arises from electron-electron interactions. The energy scale governing thermal suppression is the Thouless energy Ec=ℏD/L2E_c=\hbar D/L^2 (with LL the ring circumference): thermal averaging over an energy window kBTk_B T suppresses the current when kBTk_B T exceeds EcE_c. Early measurements (Lévy et al., 1990; Chandrasekhar et al., 1991) found currents one to two orders of magnitude larger than the non-interacting theory of the typical current (Cheung, Riedel and Gefen, 1989; Altshuler, Gefen and Imry, 1991; Riedel and von Oppen, 1993), and candidate explanations invoked electron-electron interactions (Ambegaokar and Eckern, 1990), coupling of the ring to its electromagnetic environment (Kravtsov and Yudson, 1993), and trace magnetic impurities (Schwab and Eckern, 1997). Bleszynski-Jayich et al. (2009), using cantilever torque magnetometry — inferring the rings' magnetic moment from the frequency shift of an ultrasensitive mechanical resonator to which the rings are attached — measured the typical PC in single and arrays of normal-state aluminum rings across wide ranges of temperature, ring radius, and magnetic field, finding quantitative agreement with the coherent non-interacting theory (Riedel and von Oppen, 1993), with the diffusion constant DD as the only fitted parameter and DD independently confirmed by transport measurements. The compared calculation, however, assumes phase-coherent electron motion around the ring. The phase-coherence length Lϕ=DτϕL_\phi=\sqrt{D\tau_\phi} (the distance over which inelastic processes randomize the electron phase, with τϕ\tau_\phi the dephasing time, independently measurable in transport from weak localization — the quantum-interference correction that elastic disorder produces in the resistance of a diffusive metal) exceeded the ring circumference LL for nearly all measured temperatures, but in the smallest (308 nm radius) array Lϕ(3 K)=1.86 LL_\phi(3\,\mathrm{K})=1.86\,L, and the current there decayed faster than the coherent prediction above T≈2T\approx 2,K — precisely the regime where a quantitative theory of the dephasing-suppressed PC is needed and, as the authors state, had not been calculated. Related earlier work ties dephasing or decoherence to the persistent current, but for different quantities: the ensemble-averaged h/2eh/2e harmonic of the noise-induced current is related to the dephasing rate (Kravtsov and Altshuler, 1999); two-level-system environments are shown to enhance the ensemble-averaged equilibrium current (Schwab, 2000); and decoherence of the current is analyzed in a nonequilibrium, reservoir-coupled tight-binding ring (Sur and Giamarchi, 2025). None of these evaluates the equilibrium typical (sample-specific) current of a diffusive normal-metal ring in the Lϕ∼LL_\phi\sim L crossover. The same measurement technique also opens regimes the 2009 work did not access: low fields and ultrapure or deliberately doped rings where the interaction-dominated ensemble-averaged current, environmental coupling, and magnetic-impurity effects can be discriminated, and rings driven out of equilibrium (e.g., by microwave radiation), whose persistent-current properties have received little attention for normal metals.

Problem Statement

Bring the quantitative theory of persistent currents in normal (resistive, non-superconducting) metal rings beyond the idealized model of diffusive, non-interacting, fully phase-coherent electrons in equilibrium by closing the gap explicitly flagged by Bleszynski-Jayich et al. (2009): "To the best of our knowledge the effect of dephasing upon the PC has not been calculated." The gap is specific to the equilibrium typical (sample-specific) current in the regime Lϕ∼LL_\phi\sim L; the earlier dephasing- and decoherence-related calculations concern other currents (see Background).

Calculate the equilibrium persistent current of a diffusive metal ring including the physical dephasing rate τϕ(T)\tau_\phi(T) (the rate independently measurable, e.g., from weak-localization corrections to resistance) — for the typical (root-mean-square, sample-specific) current, including its h/eh/e- and h/2eh/2e-periodic Fourier components ⟨I(h/pe)2⟩1/2\langle I^2_{(h/pe)}\rangle^{1/2} — valid in the crossover regime where the phase-coherence length Lϕ=DτϕL_\phi=\sqrt{D\tau_\phi} becomes comparable to the ring circumference LL; the calculation must reduce to the known coherent results (Riedel and von Oppen, 1993) for Lϕ≫LL_\phi\gg L and yield quantitative predictions for the temperature dependence that can be tested against the published data, in particular the faster decay above T≈2T\approx 2,K in the smallest (308 nm radius) ring arrays.

Related programmatic directions noted by the source, which frame the significance of this problem but are not required deliverables of it: (a) using persistent-current measurements in regimes the 2009 experiment did not access (e.g., low magnetic fields, ultrapure or deliberately doped rings) to study many-body and environmental effects — the roles of electron-electron interactions, coupling to the electromagnetic environment, and trace magnetic impurities, especially in the interaction-sensitive ensemble-averaged h/2eh/2e-periodic current, which the source calculates to be unobservably small in the 2009 experiment — discriminating among the proposed theoretical scenarios (Ambegaokar-Eckern; Kravtsov-Yudson; Schwab-Eckern; Bary-Soroker et al.); and (b) studying the persistent current of normal-metal rings in circuits driven out of equilibrium (e.g., by the controlled introduction of microwave radiation), whose properties, in the source's words, "have received relatively little attention to date but could offer new insights into the behavior of isolated nanoelectronic systems."

References

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