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Physical review. B, Condensed matter

One- and two-hole excitation spectra of the Hubbard model in the non-half-filled case

Andrzej M. Oleś, Guy Tréglia, D. Spanjaard, Rémi Jullien

Published Oct 15, 1986• 17 citations
Publisher

Abstract

Exact results for the one- and two-hole excitation spectra of the one- dimensional Hubbard model for filling larger than one electron per site are presented. We consider short chains of up to eight atoms and use the so-called modified periodic boundary conditions, with which we are able to extrapolate the results up to infinite chains. The one-hole excitation spectra demonstrate a gradual transition from the band picture to the spectra characterized by new satellite structures, with increasing electron-electron interaction U. The weight of the satellite part of these spectra increases with increasing U, approaching the atomic-limit value. In the ferromagnetic case, a substantial reduction of the exchange splitting is found as compared with the independent-electron approximation. The two-hole excitation spectra exhibit characteristic atomic lines at large values of U, while at intermediate values of U they are much broader in the non-completely-filled chain than the respective ones of the filled chain. The distance between the satellite and the band part is found to be larger than the value of U for small and intermediate values of the interaction. The results obtained support those found with second-order perturbation theory and the t-matrix approximation.

DOI
10.1103/physrevb.34.5101

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Physical review. B, Condensed matter•Apr 1, 1993•30 citations
Exact results for the one- and two-hole excitation spectra of the one- dimensional Hubbard model for filling larger than one electron per site are presented. We consider short chains of up to eight atoms and use the so-called modified periodic boundary conditions, with which we are able to extrapolate the results up to infinite chains. The one-hole excitation spectra demonstrate a gradual transition from the band picture to the spectra characterized by new satellite structures, with increasing electron-electron interaction U. The weight of the satellite part of these spectra increases with increasing U, approaching the atomic-limit value. In the ferromagnetic case, a substantial reduction of the exchange splitting is found as compared with the independent-electron approximation. The two-hole excitation spectra exhibit characteristic atomic lines at large values of U, while at intermediate values of U they are much broader in the non-completely-filled chain than the respective ones of the filled chain. The distance between the satellite and the band part is found to be larger than the value of U for small and intermediate values of the interaction. The results obtained support those found with second-order perturbation theory and the t-matrix approximation.
Exact results for the one-hole excitation spectra of the one-dimensional half-filled Hubbard Hamiltonian are presented. We consider short chains of N=2, 4, 6 atoms and introduce modified periodic boundary conditions which allow us to make a comparison between different N and to extrapolate the results up to infinite chains. By the increasing weight of the scattering states in the one-hole spectrum, we demonstrate a gradual disappearance of the free-electron (band) picture with increasing local electron-electron interaction U. The exact spectra of the N=6 chain verify the results of second-order perturbation theory for U/W1, where W is the bandwidth. The extrapolation to the infinite system, made with the finite-cell-scaling method, gives the gap which agrees rather well with the exact result for U/W>0.2. The average energies and weights of the band and scattering states obtained with this method for an infinite chain do not differ significantly from those of the N=6 chain.
We investigate the dynamical properties of the one-dimensional Hubbard model using the quantum Monte Carlo method and the numerical analytic continuation method based on the maximum entropy principle. Results for the spin, charge and current excitation spectrum are presented for various values of the coupling and the band filling for the system of 24 sites with the periodic boundary condition. We find that the total spectral density of the spin excitation exhibits a gapless structure irrespective of the coupling or the band filling. The charge excitation spectrum, on the other hand, clearly shows the gap formation at the half filled band and shifts its weight toward the higher energy region with increase of the coupling. This charge excitation gap, however, disappears with the hole doping. We also discuss the momentum dependence of the spectral functions.
Using the density-matrix renormalization group in combination with the Chebyshev polynomial expansion technique, we study the two-hole excitation spectrum of the one-dimensional Hubbard model in the entire filling range from the completely occupied band ( n = 2) down to half-filling ( n = 1). For strong interactions, the spectra reveal multiplon physics, i.e., relevant final states are characterized by two (doublon), three (triplon), four (quadruplon) and more holes, potentially forming stable compound objects or resonances with finite lifetime. These give rise to several satellites in the spectra with largely different spectral weights as well as to different two-hole, doublon–hole, two-doublon etc continua. The complex multiplon phenomenology is analyzed by interpreting not only local and k -resolved two-hole spectra but also three- and four-hole spectra for the Hubbard model and by referring to effective low-energy models. In addition, a filter-operator technique is presented and applied which allows to extract specific information on the final states at a given excitation energy. While multiplons composed of an odd number of holes do neither form stable compounds nor well-defined resonances unless a nearest-neighbor density interaction V is added to the Hamiltonian, the doublon and the quadruplon are well-defined resonances. The k -resolved four-hole spectrum at n = 2 represents an interesting special case where a completely stable quadruplon turns into a resonance by merging with the doublon–doublon continuum at a critical wave vector. For all fillings with , the doublon lifetime is strongly k -dependent and is even infinite at the Brillouin zone edges as demonstrated by k -resolved two-hole spectra. This can be traced back to the 'hidden' charge-SU(2) symmetry of the model which is explicitly broken off half-filling and gives rise to a massive collective excitation, even for arbitrary higher-dimensional but bipartite lattices.
We report numerical results for the two-dimensional Hubbard model in the strong-coupling approximation, including the three-site terms omitted in the t-J model, for a 4\ifmmode\times\else\texttimes\fi{}4 system with periodic boundary conditions. We study the one- and two-hole band structures, the root-mean-square hole-hole separation, and the hole-hole binding energy for both the t-J and the strong-coupling models. In the physical parameter region (8U/t40) believed appropriate for the high-temperature superconductors, the three-site terms enhance the hole mobility, increasing the one- and two-hole bandwidths and decreasing the binding energy, but changing the hole-hole separation only slightly; quantitative differences between the models range up to 40%. The transition (at large U/t) to the ferromagnetic Nagaoka state is sharper in the strong-coupling model; our results do not support a transition through all intermediate S values in the bulk.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
We calculate the photoemission and inverse photoemission spectra of holes in copper-oxygen planes, the characteristic structural unit of high-temperature superconductors. The computations are based on the extended Hubbard or Emery model in the limit of infinitely strong Coulomb repulsion Ud{\mathit{U}}_{\mathit{d}}Ud​ on copper sites. In order to evaluate the corresponding one-particle correlation functions the projection technique is applied, which is especially suitable for strongly correlated systems. Thereby the hole dynamics is restricted to a subspace of relevant operators, within which it can be treated exactly. In contrast to independent-particle approximations the excitation spectra show an energy gap at half-filling, which disappears when the system is doped with additional holes. Besides the insulator-to-metal transition the strong correlations lead also to a significant shift of spectral weight to states close to the Fermi energy when the hole concentration increases. Our results are in good agreement with the ones of exact diagonalization studies of the (CuO2{\mathrm{CuO}}_{2}CuO2​)4{)}_{4})4​ cluster with periodic boundary conditions.