Single-particle spectral function of quarter-filled ladder systems

We study the single-particle properties of quarter-filled ladder systems such as ${\alpha }^{\prime }\text{−}\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ by means of a recently developed generalization of the variational cluster perturbation theory to extended Hubbard models. We find a homogeneous antiferromagnetic insulating phase for nearest-neighbor repulsions smaller than a critical value, without any metallic phase for small repulsions. Different from cluster dynamical mean-field theory and local density approximation considerations, the inclusion of diagonal hopping within a ladder has little effect on the bonding bands, while flattening and shifting the antibonding bands. In the low-temperature charge-ordered phase, the spectrum depends on whether the ordering is driven by the Coulomb repulsion or by the coupling to a static lattice distortion. The small change of the experimentally observed gap upon charge ordering implies that the lattice coupling plays an important role in this ordering. Interladder coupling is straightforward to include within our method. We show that it has only a minor effect on the spectral function. The numerically calculated spectra show good agreement with experimental angle-resolved photoemission data.

Figure 1

Clusters used for the V-CPT calculations. The diagonal hopping $t_d$ is indicated only once, but is present equivalently between other sites. Decoupled bonds treated perturbatively are marked by dashed lines, and the boxes show the clusters of finite size. Left: Single ladder with $6\times 2$ cluster. Right: Super cluster consisting of two 12 site clusters.

Figure 2

Grand potential $\Omega \left(\delta \right)$ as a function of the mean-field parameter $\delta$ with a $6\times 2$ cluster serving as reference system, and without coupling to the lattice. Upper panel: $V=1.5$. Lower panel: $V=1.7$.

Figure 3

Finite-size dependence of the critical Coulomb interaction $V_c$ without lattice coupling. Error bars are due to the finite step $\Delta V=0.01$ in the calculations. Dotted line: Linear extrapolation of the 8 and 6 rung cluster. Dashed line: Quadratic extrapolation of the 8, 6, and 4 rung cluster.

Figure 4

Single-particle spectral function $A(\mathit{k},\omega )$ calculated with a $6\times 2$ cluster in the disordered phase at $V_a=V_b=1.3$. Top panel: Momentum $k_a=0$ perpendicular to the ladder. Bottom panel: $k_a=\pi$. The dashed line marks the chemical potential calculated by Eq. (11), the dotted line marks the result obtained from Eq. (9).

Figure 5

Gap $\Delta$ in the spectral function as a function of $V_a$. Squares: $V_b=0$. Diamonds: $V_b=V_a$.

Figure 6

Magnetic properties of a single ladder at $V_a=V_b=1.3$. Upper panel: Spin correlation function $S_r=⟨S_1^z{S}_{1+r}^z⟩$ calculated on an isolated $6\times 2$ cluster. Lower panel: Grand potential $\Omega$ as a function of the strength of the fictitious field Eq. (13).

Figure 7

Spectral function $A(\mathit{k},\omega )$ when the diagonal hopping is included, $t_b=0.25,t_d=0.25$. The Coulomb interaction was $V_a=V_b=1.3$. The dotted line marks the chemical potential.

Figure 8

Spectral function $A(\mathit{k},\omega )$ in the ordered phase. Top: Transition driven by coupling to the lattice. Bottom: Transition driven by Coulomb interaction. For the interaction $V=V_a=V_b$ was used, with the values as given in the plots. The dotted line marks the chemical potential.

Figure 9

Spectral function $A(\mathit{k},\omega )$ in the disordered phase at $V_a=V_b=V_{xy}=1.3$ calculated on the $2\times 12$ super cluster. Top: Momentum $k$ along the ladder direction. Bottom: $k$ perpendicular to the ladder direction.