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Physical review. B, Solid state

Excitation spectrum of the one-dimensional Hubbard model

Cornelius F. Coll

Published Mar 1, 1974• 166 citations
Publisher

Abstract

We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio $\frac{U}{t}$, where $U$ represents the on-site Coulomb repulsion and $t$ is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.

DOI
10.1103/physrevb.9.2150

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We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio Ut\frac{U}{t}tU​, where UUU represents the on-site Coulomb repulsion and ttt is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.
With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.
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 consider a one-dimensional two-band model of electrons on a lattice with equal nearest-neighbour hopping, an interband splitting Delta and a Hubbard-like repulsion U. The model is defined via the SU(4) generalization of Lieb and Wu's Bethe ansatz solution of the one-dimensional Hubbard model. At T=0 the model has a Mott metal-insulator transition at a critical value U c ( Delta ) for a band filling of exactly one electron per site. U c decreases with Delta , being zero if the excited-electron band is empty, and U c =2.981 when the bands are degenerate. We discuss the ground-state properties, the spectrum of elemental excitations, the specific heat and the magnetic susceptibility, in both the metallic and insulating phases, as a function of the crystal-field splitting for exactly one electron per site. There are four branches of elemental excitations: (i) charge excitations, (ii) crystalline-field excitations and (iii) two branches of spin waves. The Fermi velocity is finite in the metallic phase, diverges as the metal-insulator transition is approached from the metallic side and vanishes for the insulator. Each band contributes to the susceptibility with a term that is inversely proportional to the spin-wave velocity for that band. The low-temperature specific heat is proportional to T and to the sum of the inverses of the velocities of the four branches.
The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​. This change is probably associated with a Mott metal-insulator transition (Uc{\mathit{U}}_{\mathit{c}}Uc​=0 for N=2). The Fermi velocity is finite for UUc{\mathit{U}}_{\mathit{c}}Uc​ but vanishes for U>Uc{\mathit{U}}_{\mathit{c}}Uc​ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.
The magnetic properties of a Hubbard chain with on-site attraction are studied as a function of the band filling. Magnetization curves are calculated numerically for several values of the interaction and band filling. An unexpected feature of these curves is that the susceptibility is always finite except for the empty and half-filled band. The gap in the magnetic excitation spectrum is also calculated as a function of the band filling. The numerical results are compared with analytical results, which are obtained for the nearly empty and nearly-half-filled band.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.