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Physica C Superconductivity

Variational cluster treatment of the three-band Hubbard model: Electron vs. hole doping

Enrico Arrigoni, Maria Daghofer, Markus Aichhorn, Werner R. Hanke

Published Apr 18, 2007• 1 citation
Publisher
DOI
10.1016/j.physc.2007.03.191

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A detailed numerical study of the three-band Hubbard (3BH) model in the underdoped region is carried out, both in the hole- as well as in the electron-doped case, by means of the variational cluster approach. The phase diagram and the low-energy single-particle spectrum are found to be very similar to recent results for the single-band Hubbard model with next-nearest-neighbor hoppings t '. In particular, we obtain a mixed antiferromagnetic+superconducting phase at low doping with a first-order transition to a pure superconducting phase accompanied by phase separation. Despite the fact that the physically relevant and generally accepted parameters ( t pp / t pd , t pd /Δ pd ) of our 3BH-model are not in the small regime, required for a 'downward' projection onto a single-band model, in the single-particle spectrum a clear Zhang–Rice (ZR) singlet band (with a projection on d- and p-orbitals consistent with the phase factor of the ZR singlet wave function) can be seen. It displays an incoherent and a coherent part, in which holes enter upon doping around (π/2, π/2). The latter is very similar to the coherent quasi-particle band crossing the Fermi surface in the t – t '– U single-band model, with generally accepted values of the parameters. Doped electrons go instead into the upper Hubbard band, first filling the regions of the Brillouin zone around (π, 0). This fact can be related to the enhanced robustness of the antiferromagnetic phase as a function of electron doping compared to hole doping.
Using the variational cluster approach, we study the transition from the antiferromagnetic to the superconducting phase of the two-dimensional Hubbard model at zero temperature. Our calculations are based on a method to evaluate the VCA grand potential which employs a modified Lanczos algorithm and avoids integrations over the real or imaginary frequency axis. Thereby, very accurate results are possible for cluster sizes not accessible to full diagonalization. This is important for an improved treatment of short-range correlations, including correlations between Cooper pairs in particular. We apply this improved method in order to investigate the cluster-size dependence of the phase-separation tendency that has been proposed recently on the basis of calculations for smaller clusters. While the energy barrier associated with phase separation rapidly decreases with increasing cluster size for both hole and electron doping, the extension of the phase-separation region behaves differently in the two cases. More specifically, our results suggest that phase separation remains persistent in the hole-doped case and disappears in the electron-doped case. We also study the evolution of the single-particle spectrum as a function of doping and point out the relevance of our results for experimental findings in electron and hole-doped materials.
The concentration dependences of the band structure, spectral weight, density of states, and Fermi surface in the paramagnetic state are studied in the Hubbard model within cluster pertubation theory with 2 × 2 clusters. Representation of the Hubbard X operators makes it possible to control conservation of the spectral weight in constructing cluster perturbation theory. The calculated value of the ground-state energy is in good agreement with the results obtained using nonperturbative methods such as the quantum Monte Carlo method, exact diagonalization of a 4 × 4 cluster, and the variational Monte Carlo method. It is shown that in the case of hole doping, the states in the band gap (in-gap states) lie near the top of the lower Hubbard band for large values of U and near the bottom of the upper band for small U . The concentration dependence of the Fermi surface strongly depends on hopping to second ( t ′) and third ( t ″) neighbors. For parameter values typical of HTSC cuprates, the existence of three concentration regions with different Fermi surfaces is demonstrated. It is shown that broadening of the spectral electron density with an energy resolution typical of contemporary ARPES leads to a pattern of arcs with a length depending on the concentration. Only an order-of-magnitude decrease in the linewidth makes it possible to obtain the true Fermi surface from the spectral density. The kinks associated with strong electron correlations are detected in the dispersion relation below the Fermi level.