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Physical Review Letters

Fermi surface and dynamics of thet-Jmodel at moderate doping

W. Stephan, Peter Horsch

Published Apr 29, 1991• 182 citations
Publisher

Abstract

The 2D t-J model is studied with exact-diagonalization techniques applied to clusters of sixteen to twenty sites with various hole doping levels. Already for two holes (\ensuremath{\ge}10% doping), a large electronic Fermi surface, compatible with Luttinger's theorem, is observed. The full single-particle Green's function is examined, and is shown to exhibit quasiparticlelike behavior, with dispersion consistent with the band structure of the noninteracting limit, and bandwidth scaling approximately as J for J smaller than t.

DOI
10.1103/physrevlett.66.2258

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The 2D t-J model is studied with exact-diagonalization techniques applied to clusters of sixteen to twenty sites with various hole doping levels. Already for two holes (\ensuremath{\ge}10% doping), a large electronic Fermi surface, compatible with Luttinger's theorem, is observed. The full single-particle Green's function is examined, and is shown to exhibit quasiparticlelike behavior, with dispersion consistent with the band structure of the noninteracting limit, and bandwidth scaling approximately as J for J smaller than t.
The evolution of the Fermi surface upon hole doping is studied in the t-J model of correlated electrons by exact diagonalization of chains and planes. In one dimension at low hole doping, the momentum distribution function n(k) indicates the presence of pocketlike features at the (noninteracting) Fermi momentum, while increasing the density of holes a large Fermi surface is observed. Although the results in two dimensions are consistent with this picture, conclusive evidence for the existence of hole pockets (at intermediate doping levels) cannot be provided in the present study of 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 square lattices. In order to improve the resolution in momentum space, twisted boundary conditions are used for the two-dimensional clusters.
Recent numerical results for the single-particle spectral function and optical conductivity of the two-dimensional Hubbard and t−J models are reviewed. Already for two holes in systems of sixteen to twenty sites (≥ 10% doping) a large electronic Fermi surface, compatible with Luttinger’s theorem, is observed. The full single-particle Green’s function is examined, and is shown to exhibit quasiparticle-like behavior, with dispersion consistent with the band structure of the non-interacting limit, and band width scaling approximately as J for J smaller than t. The optical conductivity of the Hubbard and t−J models is shown to have many features in common with recent experiments on copper oxide superconductors. The importance of the often neglected 3-site terms which arise in the derivation of the t−J model from the Hubbard model for optical properties is discussed.
We present an exact diagonalization study of the doping dependence of the single-particle Green's function in 16-, 18-, and 20-site clusters of the t-J model. We find evidence for the validity of the rigid-band picture starting from the half-filled case: upon doping, the topmost states of the quasiparticle band observed in the photoemission spectrum at half-filling cross the chemical potential and reappear as the lowermost states of the inverse photoemission spectrum. Features in the inverse photoemission spectra, which are inconsistent with the rigid-band picture, are shown to originate from the nontrivial point-group symmetry of the ground state with two holes, which enforces different selection rules than at half-filling. Deviations from rigid-band behavior which lead to the formation of a large Fermi surface in the momentum distribution are found to occur at energies far from the chemical potential. A Luttinger Fermi surface and a nearest-neighbor hopping band do not exist.
We investigate the effect of doping in the hole spectral function by exact diagonalization of a 16-site cluster with two holes. It is shown that for more than one hole the spectral function differs qualitatively from the one in the single-hole limit. In particular, we find no evidence for the formation of a Fermi surface having hole ``pockets'' localized around (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) in momentum space. On the contrary, a large low-energy peak appears in the spectral function of the hole propagator at k=(\ensuremath{\pi},0) or (0,\ensuremath{\pi}) which is then the best candidate for the existence of those pockets. A (\ensuremath{\pi}/2,\ensuremath{\pi}/2) quasihole state is obtained as a small energy excitation of the two-hole system. We discuss some consequences for photoemission and optical-conductivity experiments in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials at intermediate doping.
We present an exact diagonalization study of the self-energy of the two-dimensional Hubbard model. To increase the range of available cluster sizes we use a corrected ttt-JJJ model to compute approximate Green's functions for the Hubbard model. This allows to obtain spectra for clusters with 18 and 20 sites. The self-energy has several ``bands'' of poles with strong dispersion and extended incoherent continua with kkk-dependent intensity. We fit the self-energy by a minimal model and use this to extrapolate the cluster results to the infinite lattice. The resulting Fermi surface shows a transition from hole pockets in the underdoped regime to a large Fermi surface in the overdoped regime. We demonstrate that hole pockets can be completely consistent with the Luttinger theorem. Introduction of next-nearest-neighbor hopping changes the self-energy strongly and the spectral function with nonvanishing next-nearest-neighbor hopping in the underdoped region is in good agreement with angle-resolved photoelectron spectroscopy in the cuprates.
The effect of doping in the two-dimensional Hubbard model is studied within finite-temperature exact diagonalization combined with cluster dynamical mean-field theory. By employing a mixed basis involving cluster sites and bath molecular orbitals for the projection of the lattice Green's function onto $2\ifmmode\times\else\texttimes\fi{}2$ clusters, a considerably more accurate description of the low-frequency properties of the self-energy is achieved than in a pure site picture. To evaluate the phase diagram, the transition from Fermi-liquid to non-Fermi-liquid behavior for decreasing hole doping is studied as a function of Coulomb energy, next-nearest-neighbor hopping, and temperature. The self-energy component ${\ensuremath{\Sigma}}_{X}$ associated with $X=(\ensuremath{\pi},0)$ is shown to develop a collective mode above EF{E}_{F}EF​, whose energy and strength exhibits a distinct dispersion with doping. This low-energy excitation gives rise to non-Fermi-liquid behavior as the hole doping decreases below a critical value ${\ensuremath{\delta}}_{c}$, and to an increasing particle-hole asymmetry, in agreement with recent photoemission data. This behavior is consistent with the removal of spectral weight from electron states above EF{E}_{F}EF​ and the opening of a pseudogap, which increases with decreasing doping. The phase diagram reveals that ${\ensuremath{\delta}}_{c}\ensuremath{\approx}0.15\dots{}0.20$ for various system parameters. For electron doping, the collective mode of ${\ensuremath{\Sigma}}_{X}(\ensuremath{\omega})$ and the concomitant pseudogap are located below the Fermi energy, which is consistent with the removal of spectral weight from the hole states just below EF{E}_{F}EF​. The critical doping, which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.