Resonant magnetic mode in superconducting two-leg ladders

The spin dynamics of a doped two-leg spin ladder is investigated by numerical techniques. We show that a hole pair-magnon boundstate evolves at finite hole doping into a sharp magnetic excitation below the two-particle continuum. This is supported by a field theory argument based on a $\mathrm{SO}\left(6\right)$-symmetric ladder. Similarities and differences with the resonant mode of the high-$T_C$ cuprates are discussed.

Figure 1

Schematic image of the low-energy triplet spectrum of a spin ladder doped with two holes. The left and right panels correspond to the two transverse momenta (as written on the plots). The two-hole continuum (h-h) and the pair-magnon (P-M) scattering continuum are shown as hashed regions. The thick line corresponds to the pair-magnon boundstate (P-M BS). Note that the single magnon branch is not shown (hidden by the P-M states).

Figure 2

(Color online) Dynamic spin structure factor computed on a $2\times 12$ ladder at $1∕6$ doping and $J=0.4$. Data for 0 and $\pi$ transverse momenta are shown on the left and right panels, respectively, and $q_x$ varies from 0 (top) to $\pi$ (bottom). The weights of the largest peaks (truncated and symbolized as thick arrows) are indicated on the plot in units where the “local” $\omega$-integrated weight is $(1∕N){\sum }_{\mathbf{q}}S\left(\mathbf{q}\right)=1-n_h\simeq 0.833$. Down arrows indicate (almost) silent excitations.

Figure 3

(Color online) Contribution of the two-kink (or two-antikink) bound states to the dynamic spin structure factor in the $\mathrm{SO}\left(6\right)$-symmetric ladder at momentum $\mathbf{q}=(2k_F,\pi )$ (shown for two values of $K_{\rho +}$). The frequency is normalized to the bound state energy $M^{\prime }=\sqrt{2}M$. The arrow indicates the onset of a small kink-kink continuum starting at $\omega =2M$ (not shown). Note that the ED data of the $t–J$ model shown on the same scale (setting $M^{\prime }$ as the finite size spin gap) exhibit only a small number of poles ($\delta$-functions).

Figure 4

(Color online) Spin gap of the two-leg ladder at $J=0.4$; (a) finite size scaling analysis for a fixed hole number (of 2) or a fixed hole density computed using bosonic or boson-fermion range-2 CORE Hamiltonians or DMRG methods. Dashed lines are tentative fits of the data with algebraic finite size corrections for DMRG ($\sim 1∕L^2$) and exponential for CORE; (b) variation of the spin gap as a function of doping.

Figure 5

Tunneling density of state of the two-leg ladder computed on a $2\times 12$ cluster. The integrated weights (sum rules) for electrons and holes are shown on the plot. Inset: close up of the low energy region.