Quantitative vortex tunneling with pinning, dissipation, and the Magnus force: full rates, crossover regimes, and experimental tests beyond the Ao-Thouless limiting cases

Ao and Thouless (Phys. Rev. Lett. 72, 132 (1994)) reduced the quantum tunneling of a vortex out of a metastable pinning well in a superconducting film - in the presence of the Magnus force, a dissipative environment, and a harmonic pinning potential transverse to the escape direction - to an effective one-dimensional Euclidean action containing a normal damping kernel (from dissipation) and an anomalous, nonlocal kernel generated by the vortex-velocity-dependent part (VVDP) of the Magnus force. They computed the semiclassical tunneling exponent only in limiting cases (no pinning, no dissipation, or one of the two), predicted a dynamical localization threshold (in the absence of pinning and dissipation) above which the tunneling rate vanishes at zero temperature, and argued qualitatively that pinning and dissipation both suppress Magnus-force effects, plausibly explaining their observed absence in dirty superconductors. What remains open is the quantitative completion of this program: full tunneling rates - exponent and fluctuation-determinant prefactor - for realistic pinning potentials in the general regime where both kernels contribute, a systematic map of the crossover regimes including the vicinity of the localization threshold, and material-specific experimental tests of the localization criterion and the VVDP-suppression prediction. Within the source framework the prefactor with both kernels, the general case, and a direct experimental test of the localization criterion have not been worked out.

Background

In a type-II superconductor, magnetic field penetrates as vortices - flux tubes each carrying one flux quantum - and the decay of supercurrents is governed by the motion of these vortices across energy barriers produced by defects (pinning centers). At high temperature vortices escape thermally (flux creep); at sufficiently low temperature the dominant escape channel is macroscopic quantum tunneling (MQT), the tunneling of a mesoscopic coordinate through a barrier. The standard tool is the Caldeira-Leggett path-integral formalism, in which the tunneling object couples to a bath of harmonic oscillators whose spectral density $J(\omega)\propto\eta\omega^{s}$, with cutoff frequency $\omega_c$, classifies the dissipation as subohmic ($s<1$), ohmic ($s=1$), or superohmic ($s>1$); subohmic dissipation strongly suppresses tunneling, superohmic dissipation barely affects it. The tunneling rate is $\Gamma\propto\exp(-S_c/\hbar)$, where $S_c$ is the Euclidean action of the bounce - the periodic saddle-point solution in imaginary time. Computing this exponent is the standard step; the full rate additionally requires the prefactor, a fluctuation determinant around the bounce.

A vortex is not an ordinary particle. Besides an inertial mass $M$ (whose magnitude was itself long debated), it experiences the Magnus force, a transverse force proportional to the relative velocity between the vortex and the background superflow. Ao and Thouless split the Magnus force into a superfluid-velocity-dependent part (SFVDP), which contributes to the effective potential, and a vortex-velocity-dependent part (VVDP), $F_m=q_v(h\rho_s d/2)(v_s-\dot r)\times\hat z$, which acts on the vortex like a Lorentz force on a charged particle, i.e. like a magnetic field coupling the two spatial directions of the vortex coordinate $r=(x,y)$. Here, in the source's notation, $h$ is Planck's constant (so $h\rho_s d/2=\pi\hbar\rho_s d$), $\rho_s$ is the superfluid electron density, $d$ the film thickness, and $q_v=\pm 1$ the vorticity sign (the sense of the vortex circulation relative to the film normal $\hat z$). Using the effective vortex Hamiltonian derived by Niu, Ao and Thouless, with vector potential $\mathbf{A}=(h\rho_s d/2)(y,0,0)$, Ao and Thouless studied a vortex in a semi-infinite superconducting film escaping from a metastable point near the film edge, with a harmonic pinning potential $\frac{1}{2}k_x x^2$ along the edge, a general metastable potential $V_1(y)$ in the escape direction (their Eq. (4) gives one explicit example, $V_1(y)=\frac{h}{2}\rho_s d\left(-v_0 y+\frac{\hbar}{2m_c}\ln y\right)$, the sum of the potential due to the external supercurrent, of superflow velocity $v_0$, and the edge image potential, with $m_c$ the Cooper-pair mass), and a dissipative bath of the above type. Integrating out the $x$ motion and the bath yields an effective one-dimensional Euclidean action for $y(\tau)$ containing two nonlocal-in-time kernels: a normal damping kernel $k(\tau)$ from the bath, and an anomalous damping kernel $g(\tau)$ generated by the VVDP Magnus force through the pinned transverse motion.

Analytic results were obtained only in limiting cases. With neither pinning nor dissipation, the anomalous kernel acts like a subohmic ($s=0$) bath: for weak Magnus force the action acquires a correction proportional to $\rho_s^2$ with a linear temperature dependence, while for strong Magnus force the rate vanishes at zero temperature - dynamical localization - once $\frac{1}{M}\left(\frac{h\rho_s d}{2}\right)^2>\left|\frac{d^2V_1}{dy^2}\right|_{\mathrm{barrier\ top}}$; a large vortex mass favors tunneling, opposite to ordinary tunneling. With dissipation but no pinning, the anomalous kernel maps a bath exponent $s$ to $2-s$ (ohmic dissipation is unchanged and combines with the Magnus effect into an effective damping $\eta_{\mathrm{eff}}=\eta+(h\rho_s d/2)^2/\eta$). With pinning but no dissipation the anomalous kernel is superohmic and the rate is always finite; in the strong-Magnus limit $x$ and $y$ become canonically conjugate and $S_c=h\rho_s d\int_0^{y_t}dy\,\sqrt{2V_1(y)/k_x}$, with $y_t$ the turning point where $V_1(y_t)=0$. With both pinning and dissipation present - the generic experimental situation - only a qualitative conclusion was drawn (both suppress the VVDP effect, explaining its observed absence in dirty superconductors), without a detailed study, and throughout the paper only the exponential $\exp(-S_c/\hbar)$ was computed, with no fluctuation determinants.

Problem Statement

Within the Ao-Thouless effective-action framework - a vortex tunneling out of a metastable well in a superconducting film, with finite vortex mass $M$, Magnus (VVDP) coupling strength $h\rho_s d/2$ ($h$ being Planck's constant), harmonic transverse pinning $k_x$, escape-direction potential $V_1(y)$, and dissipative bath $(\eta,s,\omega_c)$ - determine the quantitative tunneling rate in the general regime where both the normal kernel $k$ and the anomalous kernel $g$ contribute, and test the predictions against experiment. Concretely, the source-derived program develops along four linked directions: (1) semiclassical actions and tunneling rates for realistic forms of $V_1(y)$ beyond the limiting-case evaluations (the source's Eq. (4) being only one example potential); (2) a systematic treatment of the crossover regimes between the analytic limits, in particular intermediate pinning strength combined with intermediate dissipation where neither kernel is parametrically dominant, together with the behavior near the dynamical localization threshold $\frac{1}{M}\left(\frac{h\rho_s d}{2}\right)^2=\left|\frac{d^2V_1}{dy^2}\right|_{\mathrm{barrier\ top}}$; (3) the full tunneling rate including the prefactor - the fluctuation determinant of the nonlocal effective action with both kernels - beyond the leading semiclassical exponent; and (4) a quantitative comparison with experiment using material-specific values of $\rho_s$, $M$, $\eta$, $k_x$, and $\omega_c$, testing in particular the localization criterion and the predicted suppression of VVDP (Magnus-force) effects in dirty superconductors. These items serve one objective: converting the 1994 limiting-case analytics into a quantitatively complete and experimentally testable theory of vortex tunneling with pinning, dissipation, and the Magnus force. The four directions are linked parts of that objective rather than alternative routes, and directions (1) and (3) are the exponent-level and full-rate levels of the same rate computation, with (1) not subsumed by (3).

References

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