Renormalization-group exponents for superconducting phases in two-leg ladders

In previous studies, we proposed a scaling ansatz for electron-electron interactions under renormalization group transformation. With the inclusion of phonon-mediated interactions, we show that the scaling ansatz, characterized by the divergent logarithmic length $l_d$ and a set of renormalization-group exponents, also works rather well. The superconducting phases in a doped two-leg ladder are studied and classified by these renormalization-group exponents as demonstration. Finally, nontrivial constraints among the exponents are derived and explained.

Figure 1

(a) Band structure for a two-leg ladder at filling $0<n<1$. The hopping amplitudes along ($t$) and across ($t_{\perp }$) the chains are chosen to be equal, $t_{\perp }=t$. The low-energy excitations are described by four pairs of chiral fields ${\psi }_{Ri\alpha },{\psi }_{Li\alpha }$, with $i=1,2$ and $\alpha =\uparrow ,\downarrow$. (b) Phase diagram for a doped two-leg ladder. In the filling regime $0.5<n\lesssim 0.54$, the interband Cooper scattering becomes irrelevant and the superconducting phase decomposes into the two-channel Luttinger liquid.

Figure 2

Phase diagram for a doped two-leg ladder with both electron-electron and electron-phonon interactions. The critical filling $n_c\approx 0.57$ separates two types of superconducting instabilities driven by electron-electron interactions and phonon-mediated ones.

Figure 3

RG exponents of intraband Cooper scattering in different superconducting phases for the (a) antibonding band and (b) bonding band. The pattern of the RG exponents provides a simple yet unique indicator to classify different ground states.