Pairing and charge distribution in Emery ladders preserving the ratio of Cu to O atoms

Abstract

We study the Emery model (three-band Hubbard model) for superconducting cuprates on three distinct ladder-like lattices that are supercell of the CuO$_2$ plane and thus preserve the ratio of copper to oxygen atoms. Using the density-matrix renormalization group method we confirm that these Emery ladders are charge-transfer insulators for the hole concentration corresponding to undoped cuprates but become Luther-Emery liquids with enhanced pairing correlations upon doping. The preservation of the Cu to O ratio allows us to study the distribution of charges between these atoms in the Luther-Emery phase. We show that these Emery ladders can describe the relations between charge distribution, pairing strength, and interactions that have been observed in the Emery model on two-dimensional clusters and in experiments.

Figures (17)

• Figure 1: Schematic of various two-leg ladder lattice structures. Squares and circles represent copper $d$ orbitals and oxygen $p$ orbitals, respectively. Solid black circles and squares are included in all lattice structures. White circles denote the outer oxygen $p$ orbitals that are included in specific ladders only. Dashed-dotted lines indicate the unit cells. Solid lines represent the hopping terms between orbitals in the Emery model (\ref{['eq:hamiltonian']}). (a) Three-chain ladder made of the solid black symbols, five-chain ladder made of all symbols, and four-chain tube, where the upper and lower white circles represent the same oxygen atoms, (b) t-ladder with translation symmetry, (c) g-ladder with glide symmetry, and (d) r-ladder with reflection symmetry.
• Figure 2: Charge gaps $E_c$ (\ref{['eq:charge_gap']}) as a function of the Hamiltonian parameters $U_d$ (left panel) and $\varepsilon$ (right panel) for undoped (t,g,r)-ladders with length $L=24$. The other Hamiltonian parameters are given in Table \ref{['table2']}. Vertical dashed lines indicate the selected values $U_d=8$ and $\varepsilon=2$, respectively.
• Figure 3: Normalized hole density variation (\ref{['eq:local_density']}) of the r-ladder with length $L=24$ and the parameter set in Table \ref{['table2']}: (a) for two added holes and (b) for two removed holes ($=$ two added electrons) with respect to the undoped ladder.
• Figure 4: Spin gaps $E_s$ (\ref{['eq:spin_gap']}) as a function of the Hamiltonian parameters $U_d$ (left panel) and $\varepsilon$ (right panel) for undoped (t, g, r)-ladders with a length of $L=24$. The other Hamiltonian parameters are given in Table \ref{['table2']}. Vertical dashed lines indicate the selected values $U_d=8$ and $\varepsilon=2$, respectively.
• Figure 5: Convergence analysis of (a) spin gaps $E_s$ and (b) charge gaps $E_c$ as a function of the inverse of the ladder length $L$ in undoped (t,g,r)-ladders with the parameter sets from Table \ref{['table2']}. Lines represent quadratic fits of the data for $40 \geq L \geq 16$.