Scale-Dependent Competing Interactions: Sign Reversal of the Average Persistent Current

The interaction-induced orbital magnetic response of a nanoscale system, modeled by the persistent current in a ring geometry, is evaluated for a system which is a superconductor in the bulk. The interplay of the renormalized Coulomb and Fröhlich interactions is crucial. The diamagnetic response of the large superconductor may become paramagnetic when the finite-size-determined Thouless energy is larger than or on the order of the Debye energy.

Figure 1

Upper panel: the renormalized interactions, Eq. (1), as functions of the energy, for a metal that is a superconductor in the bulk. When only the attraction exists, its absolute value is renormalized up (with ${\omega }_{>}={\omega }_{\mathrm{D}}$) until it diverges at $T_c^a$ (the dash-dotted curve). The repulsion (dashed curve) is renormalized down from its bare value at ${\omega }_{>}=E_{\mathrm{F}}$. The total interaction starts at ${\omega }_{\mathrm{D}}$ from the bare value $g_a({\omega }_{\mathrm{D}})+g_r({\omega }_{\mathrm{D}})$ and then its absolute value is renormalized up until it diverges at the true transition temperature $T_c$ (solid curve). In a mesoscopic system the renormalization stops at the Thouless energy and therefore the relative location of $E_c$ vs ${\omega }_{\mathrm{D}}$ and the ratio $|g_a|/g_r(\max \{E_c,{\omega }_{\mathrm{D}}\})$ will determine the sign of the effective interaction at ${\omega }_{\mathrm{D}}$. The figure is drawn for demonstration purposes with the unrealistic parameter $E_{\mathrm{F}}/{\omega }_{\mathrm{D}}=2.5$. Lower panel: the lines in the plane $g_r-E_c/{\omega }_{\mathrm{D}}$ where the first harmonic of the current changes sign from paramagnetic (above the curve) to diamagnetic (below), for three values of the attraction $g_a$. Here and in all figures below, $T={\omega }_{\mathrm{D}}/125\simeq 2–4\mathrm{K}$ and $E_{\mathrm{F}}/{\omega }_{\mathrm{D}}=25$.

Figure 2

$I_m(E_c)$ for $m=1$, 2, and 3 (see the legend), with $|g_a|=0.2$ and $g_r=0.1$.

Figure 3

$I_{m=1}$ for three values of the repulsion and $|g_a|=0.2$.

Figure 4

The current as a function of $\phi$. Solid (dashed) curve is for $g_r=0.17$ ($g_r=0.1$), with $|g_a|=0.2$ and $E_c/{\omega }_{\mathrm{D}}=0.76$.