Superconducting Gap for a Two-Leg $t\mathrm{\text{−}}J$ Ladder

Single-particle diagonal and off-diagonal Green’s functions of a two-leg $t\mathrm{\text{−}}J$ ladder at $1/8$ doping are investigated by exact diagonalizations techniques. A numerically tractable expression for the superconducting gap is proposed and the frequency dependence of the real and imaginary parts of the gap are determined. The role of the low-energy gapped spin modes, whose energies are computed by a (one-step) contractor renormalization procedure, is discussed.

Figure 1

Low-energy magnetic excitations at $1/8$ doping and $J=0.4$ for $q_y=0$ (circles) and $q_y=\pi$ (diamonds) momenta. (a) Triplet collective mode obtained by CORE on $2\times 16$ (shaded symbols) and $2\times 24$ (solid symbols) ladders. The particle-hole continuum (open symbols) constructed from the data of Figs. 2, 3 is also shown; (b) $S=3/2$ excitations (measured from the chemical potential) calculated by ED of the $2\times 12$ cluster. The onsets of the three quasiparticle continua are shown by arrows.

Figure 2

Spectral function vs $\omega$ ($t=1$) on a $2\times 12$ ladder at $1/8$ doping and $J=0.4$ for two transverse momenta $0$ and $\pi$ in panels (a) and (b), respectively. Data for different chain momenta are shifted with respect to each other.

Figure 3

Momentum dispersion of the low-energy peaks seen in Fig. 2. Error bars give the widths of the peaks (if any) and the dotted lines correspond to the lowest excitation energies.

Figure 4

Spectral weight of the superconducting Green’s function. Same conventions as in Fig. 2. Note the change of sign between transverse momenta $0$ and $\pi$.

Figure 5

Superconducting amplitude as a function of the chain momentum $q_x$ for the bonding $q_y=0$ (solid circles) and antibonding $q_y=\pi$ (open circles) transfer momenta.

Figure 6

Real (a) and imaginary (b) parts of the superconducting gap vs $\omega$, normalization as discussed in the text, obtained by ED results for a $2\times 12$ $t\mathrm{\text{−}}J$ ladder at the two bonding and antibonding Fermi momenta $(2\pi /3,0)$ and $(\pi /3,\pi )$. An imaginary damping $ϵ=0.05$ is used and the irrelevant offset $F(0,0)\propto iϵ$ has been subtracted.