Phase diagram of hole doped two-leg $\mathrm{Cu}\text{−}\mathrm{O}$ ladders

In the weak coupling limit, we establish the phase diagram of a two-leg ladder with a unit cell containing both Cu and O atoms, as a function of doping. We use bosonization and design a specific renormalization group procedure to handle the additional degrees of freedom. Significant differences are found with the single orbital case; for purely repulsive interactions, a completely massless quantum critical region is obtained at intermediate carrier concentrations (well inside the bands) where the ground state consists of an incommensurate pattern of orbital currents plus a spin density wave structure.

Figure 1

(Color online) (a) Molecular structure of the unit cell showing the hopping and interaction parameters included in the Hamiltonian. (b) Orbital current pattern and SDW modulation (bold arrows) in the $C2S2$ phase. The chain direction is vertical. $k_{F0}$ is the Fermi wave vector in the $0$ band.

Figure 2

(Color online) Knight shift on the outer O sites calculated for different phases: solid line $C1S0$ dashed line $C2S2$, and dash dotted line $C2S1$.