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arXiv:cond-mat/9406064 · Physical review. B, Condensed matter

Validity of the rigid-band picture for thet-Jmodel

R. Eder, Y. Ohta, Tomohiro Shimozato

Published Aug 1, 1994• 40 citations
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Abstract

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.

DOI
10.1103/physrevb.50.3350

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Physical review. B, Condensed matter•Mar 1, 1993•16 citations
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.
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.
We have calculated S(q) and the single particle distribution function 〈nq{\mathit{n}}_{\mathbf{q}}nq​〉 for N holes in the t-J model on a nonsquare \ensuremath{\surd}8 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}32 16-site lattice with periodic boundary conditions. We justify the use of this lattice by appealing to results obtained from the conventional 4\ifmmode\times\else\texttimes\fi{}4 16-site cluster, and an undoped 32-site system, each having the full square symmetry of the bulk. This new cluster has a high density of k points along the diagonal of reciprocal space, viz. along k=k(1,1). The results clearly demonstrate that when the single-hole problem has a ground state with a system momentum of k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), the resulting ground state for N holes involves a shift of the peak of the system's structure factor away from the antiferromagnetic state q=(\ensuremath{\pi},\ensuremath{\pi}). This shift effectively increases continuously with N. When the single hole problem has a ground state with a momentum that is not equal to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2), something that may easily be accomplished through the use of the t-t'-J model with t'/t small and positive, then the above-mentioned incommensurability for N holes is not found---the maximum of S(q) remains at q=(\ensuremath{\pi},\ensuremath{\pi}) for all N.The existence of the incommensurate ground states are presented in conjunction with the electron and hole momentum distribution functions. Our results may be interpreted as evidence for rigid-band filling of the dipolar spin distortion states of Shraiman and Siggia---by studying the hole-hole correlation function we demonstrate that the holes on our cluster tend to stay as far apart as is possible, and thus rigid-band filling is not unexpected. Thus, these results demonstrate that in some instances important results for moderately doped CuO2{\mathrm{CuO}}_{2}CuO2​ planes can be predicted from a knowledge of the properties of weakly doped planes. However, the electron Fermi surface that we obtain also satisfies Luttinger's theorem, and has dimples at the wave vectors of the single hole ground state, k=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2). Finally, we comment on the relation of all of these results to the incommensurability found in S(q).
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.
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.
We calculate the single-particle Green's function for a contact interaction with nearest-neighbor hopping on a square lattice as a function of chemical potential $\ensuremath{\mu}.$ This allows us to investigate the dependence of the leading Fermi-liquid dependencies on the band structure as the Fermi surface evolves from a circle at $\ensuremath{\mu}\ensuremath{\sim}\ensuremath{-}4t$ to a square at $\ensuremath{\mu}=0.$ The form of the single-particle self-energy $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ is determined by the density-density correlation function $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ which develops two peaks for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}2.5t$ unlike the parabolic band case. Near half-filling, $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ becomes independent of $\ensuremath{\omega},$ one-dimensional behavior, at intermediate values of $\ensuremath{\omega},$ which leads to the one-dimensional behavior in $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ However, with $\ensuremath{\mu}<~\ensuremath{-}0.1t$ there is no influence on the Fermi-liquid dependencies from the spin-density wave instability. We find that throughout the doping region $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ remains qualitatively the same as for the isotropic Fermi surface with quantitative differences. The strong $\stackrel{\ensuremath{\rightarrow}}{p}$ and E dependence of the off-shell self-energy $\ensuremath{\Sigma}(p,E)$ found earlier for the parabolic band is recovered for $\ensuremath{\mu}\ensuremath{\lesssim}\ensuremath{-}t$ but deviates from this develop for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}0.1t.$ The resonance peak width of the spectral function $A(\stackrel{\ensuremath{\rightarrow}}{p}E),$ has a linear dependence in ${\ensuremath{\xi}}_{\stackrel{\ensuremath{\rightarrow}}{p}}$ due to the E dependence of the imaginary part of $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ We point out that an accurate detailed form for $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ would be very difficult to recover from angle-resolved photoemission spectroscopy data for the spectral density. Since the leading corrections are determined by the long wavelength particle-hole excitations, the results found here for the Hubbard model carry over to Hamiltonians with finite range interactions.
We calculated the momentum distribution and one-particle spectral function for the ground state of two holes in a finite-cluster t-J model in the framework of the string picture. Previous numerical results for these quantities could be reproduced at least qualitatively. Our interpretation of these data, however, supports a rigid-band approximation starting from the single-hole case rather than a nearest-neighbor hopping band and large Fermi surface. We argue that in order to interpret the finite-cluster data properly, it is crucial to take into account the well-established fact that the ground state of two holes in a finite cluster is a bound state with nontrivial symmetry.