Search for quantum dimer phases and transitions in a frustrated spin ladder

A two-leg spin-1/2 ladder with diagonal interactions is investigated numerically. We focus our attention on the possibility of columnar dimer phase, which was recently predicted based on a reformulated bosonization theory. By using density matrix renormalization group technique and exact diagonalization method, we calculate columnar dimer order parameter, spin correlation on a rung, string order parameters, and scaled excitation gaps. Carefully using various finite-size scaling techniques, our results show no support for the existence of columnar dimer phase in the spin ladder under consideration.

Figure 1

(Color online) (a) Size dependence of the DC order parameter $D$ for various $J_{\perp }$ with $J_{\times }=0.2$. (b) $\ln D$ versus $L$ for various values of $J_{\perp }$ with $J_{\times }=0.2$. Labels for various $J_{\perp }$ are the same as those in (a). The inset shows the inverse of the correlation length $\xi$ for various values of $J_{\perp }$.

Figure 2

(Color online) Spin correlation on the $(N∕2)$-th rung as a function of $J_{\perp }$ for various sizes $L$ with $J_{\times }=0.2$. The inset shows the $L^{-1}$ scaling behavior of the crossing points.

Figure 3

(Color online) String order parameters for ladders with various sizes $L$ as functions of $J_{\perp }$ with $J_{\times }=0.2$. The inset shows the $L^{-1}$ scaling behavior of the crossing points.

Figure 4

(Color online) Scaled gaps (a) $L\mathrm{\Delta }{E}_s$ for spin-singlet excitation and (b) $L\mathrm{\Delta }{E}_t$ for spin-triplet excitation as functions of $J_{\perp }$ for various sizes $L$ with $J_{\times }=0.2$. Insets show the $L^{-2}$ scaling behavior of the left and the right crossing points represented by open and solid circles, respectively. Dotted lines are guides to the eye. They are straight lines fitted to the last few points.