Doped two-leg ladder with ring exchange: Exact diagonalization and density matrix renormalization group computations

The effect of a ring exchange on doped two-leg ladders is investigated combining exact diagonalization (ED) and density matrix renormalization group (DMRG) computations. We focus on the nature and weights of the low-energy magnetic excitations and on superconducting pairing. The stability with respect to this cyclic term of a remarkable resonant mode originating from a hole pair-magnon bound state is examined. We also find that, near the zero-doping critical point separating rung-singlet and dimerized phases, doping reopens a spin gap.

Figure 1

(Color online) (a), (b) Spectra for undoped ladders from ED calculations on ladders of length $L=12$, 14, and 16. Grey areas are guide to the eyes. (c) Spectra for ladders doped with two holes and $L=9,11,13$. Note that $k_x=\pi$ does not belong to the Brillouin zone of systems with odd $L$. The dashed line shows ${\Delta }_p$ computed independently.

Figure 2

(Color online) A comparison between characteristic energies of elementary magnetic excitations in doped ladders. Data are extrapolated from DMRG computations on systems of length up to $L=48$. Proposed experimental values of $K∕J$ for undoped systems are in the Exp. window.

Figure 3

Hole-hole static correlation functions from ED computations on an $L=13$ ladder with two holes for different $K∕J$. The area of circles is proportional to the correlation value. The upper left circle indicates the mean expectation value.

Figure 4

Spin density around the hole pair in the first triplet state from ED data on an $L=13$ ladder with two holes for different $K∕J$. Weights w of the projected function are indicated. Lengths of arrows are proportional to the local density values.

Figure 5

(Color online) The spin gap for doped two-leg ladders as a function of hole doping $\delta$. Data extrapolated from DMRG computations on systems of length $L=16$ to 64 and doping $\delta =n∕16$, $n=0,\dots ,6$, with a number of kept states up to $m=1600$. The $\delta =0$ magnon energies are indicated by grey symbols (see the arrows). Half the pairing energy for a finite system is also given by dashed lines.

Figure 6

(Color online) (a) static structure factor $\mathcal{S}\left(\mathbf{k}\right)$ for a ladder of length 12 with 4 holes $(\delta =0.17)$. (b) $\mathcal{S}\left({\mathbf{k}}_{\delta }\right)∕⟨\mathcal{S}⟩$ for different doping as a function of $K∕J$, from systems with $L$ ranging from 8 to 16. $\mathcal{S}\left({\mathbf{k}}_{\delta }\right)$ is normalized wrt the average structure factor $⟨\mathcal{S}⟩={\left(2L\right)}^{-1}{\sum }_{\mathbf{k}}\mathcal{S}\left(\mathbf{k}\right)=\frac{1}{4}(1-\delta )$.

Figure 7

(Color online) (a) dynamic structure factor $\mathcal{S}(\mathbf{k},\omega )$, with $\mathbf{k}=(k_x,k_y)$, for a ladder of length 10 with 4 holes. Red dots indicate poles and their relative weight (see the text). (b) Relative weight of the resonant state at ${\mathbf{k}}_{\delta }$.