Creation and Destruction of a Spin Gap in Weakly Coupled Quarter-Filled Ladders

We investigate weakly coupled quarter-filled ladders with model parameters relevant for ${\mathrm{NaV}}_2{\mathrm{O}}_5$ using density-matrix renormalization group calculations on an extended Hubbard model coupled to the lattice. ${\mathrm{NaV}}_2{\mathrm{O}}_5$ exhibits superantiferroelectric charge order with a zigzag pattern on each ladder. We show that this order causes a spin dimerization along the ladder and is accompanied by a spin gap of the same magnitude as that observed experimentally. The spin gap is destroyed again at large charge order due to a restructuring of the spins. An analysis of an effective spin model predicts a recreation of the gap by interladder singlets when the charge order increases further.

Figure 1

Quarter-filled coupled ladders in ${\mathrm{NaV}}_2{\mathrm{O}}_5$. The mapping of a rung (top) onto an effective spin site (bottom) is illustrated for (a) a charge-disordered and (b) a SAF-ordered regime. The shifted periodic boundary conditions in the $a$ direction are indicated by numbers which identify identical sites. The proposed singlet formation for ${\mathrm{NaV}}_2{\mathrm{O}}_5$ is depicted by dashed ellipses in the effective spin model [(b), bottom].

Figure 2

Examples of finite-size extrapolations for (a), (b) the spin gap ${\Delta }_{\mathrm{S}}$ and (c), (d) the dimerization $d$ of (a), (c) the Hubbard model and of (b), (d) the effective spin model. Linear and quadratic fits in $\frac{1}{L}$ are illustrated for charge-disordered states ($m_{\mathrm{CO}}=0$, circles) and SAF-ordered states ($m_{\mathrm{CO}}>0$).

Figure 3

Results for the EHM with coupling to the lattice, as a function of $V$. Extrapolated values of (a) the charge order parameter $m_{\mathrm{CO}}$, (b) the spin gap ${\Delta }_{\mathrm{S}}$, and (c) the dimerization $d$ are illustrated for two coupled ladders ($t_{\mathrm{ab}}=0.17t_{\mathrm{a}}$). The effective magnetic exchange constant $J_{\mathrm{eff}}$ [8] and the distortion $z_{\mathrm{opt}}$ [8] of an isolated ladder are also shown in (a). The three regimes discussed in the text are highlighted.

Figure 4

Effective spin model. Quadratically extrapolated values ($L=\infty$) for (a) the spin gap ${\Delta }_{\mathrm{S}}$ and (b) the dimerization $d$, as a function of the charge order parameter $m_{\mathrm{CO}}$. Calculations with OBC (black dots) and PBC (open circles) in the $b$ direction are shown. For comparison the spin gap ${\Delta }_{\mathrm{S}}$ for the EHM is shown, in units of $J_{\mathrm{eff}}$ [Fig. 3a].