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Physical review. B, Solid state

Exact Eigenvalues of the Constant-Coupled One-Band Model with a Finite Number of Sites

James D. Patterson

Published Aug 1, 1972• 11 citations
Publisher

Abstract

The exact eigenvalues (for ground and excited states) of the constant-coupled one-band model for ${N}_{f}$ fermions on $N$ sites were obtained. Both electrons ($T<0$, where $T$ is the coefficient of the term representing the "hopping energy") and holes ($T>0$) were considered, and results were obtained for $2\ensuremath{\le}{N}_{f}\ensuremath{\le}N$ and $N=2, 3, \mathrm{and} 4$. Only the ${N}_{f}=2$, $N=3$ and ${N}_{f}=3$, $N=4$ with $T>0$ cases gave a purely ferromagnetic ground state. These cases also showed a ferromagnetic state at higher energy. The eigenvalues were calculated as a function of $\frac{U}{T}$, where $U$ represents the Coulomb energy of repulsion for electrons on the same site. The total spins of the states were obtained from their splitting in a magnetic field. The hopping term was found to be dominant for $|\frac{U}{T}|$ as large as one. The calculations support the suggestion that the ground state is usually antiferromagnetic.

DOI
10.1103/physrevb.6.1041

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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.
We consider the question of ferromagnetism in the one-band Hubbard model on the square lattice with infinite on-site repulsion using exact diagonalization techniques. Our results show a nonmonotonic behavior as a function of the number of holes Nh{N}_{h}Nh​ away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.
We investigate two-site electronic correlations within generalized Hubbard model, which incorporates the conventional Hubbard model (parameters: t (hopping between nearest neighbours), U (Coulomb repulsion (attraction))) supplemented by the intersite Coulomb interactions (parameters: J (1) (parallel spins), J (2) (antiparallel spins)) and the hopping of the intrasite Cooper pairs (parameter: V ).As a first step we find the eigenvalues Eα and eigenvectors |E α of the dimer and we represent each partial Hamiltonian Eα|Eα Eα| (α = 1, 2, . . ., 16) in the second quantization with the use of the Hubbard and spin operators.Each dimer energy level possesses its own Hamiltonian describing different two-site interactions which can be active only in the case when the level will be occupied by the electrons.A typical feature is the appearance of two generalized t-J interactions ascribed to two different energy levels which do not vanish even for U = J (1) = J (2) = V = 0 and their coupling constants are equal to ±t in this case.In the large U -limit for J (1) = J (2) = V = 0 there is only one t-J interaction with coupling constant equal to 4t 2 /|U | as in the case of a real lattice.The competition between ferromagnetism, antiferromagnetism and superconductivity (intrasite and intersite pairings) is also a typical feature of the model because it persists in the case U = J (1) = J (2) = V = 0 and t = 0.The same types of the electronic, competitive interactions are scattered between different energy levels and therefore their thermodynamical activities are dependent on the occupation of these levels.It qualitatively explains the origin of the phase diagram of the model.We consider also a real lattice as a set of interacting dimers to show that the competition between magnetism and superconductivity seems to be universal for fermionic lattice models.
We study the multiband mechanism to favour a ferromagnetic insulating ground state within a two-band Hubbard model.Besides perturbation theory we use exact diagonalization studies to examine the ground state of chains and 2D-lattices.According to second order perturbation theory the exact diagonalization yields a fully polarized ferromagnetic ground state, if the hopping between two ground state orbitals of neighbouring atoms t gg is small and the hopping between a ground state orbital and an excited orbital tge dominates.However, in contrast to the suggestion from the second order perturbation theory this ferromagnetic state is stable only for very small hopping integrals U » tge > t gg .For larger tge quantum interference effects lead to complex magnetic structures.
We consider the one-orbital N -site repulsive Hubbard model on two kagome like chains, both of which yield a completely dispersionless (flat) one-electron band.Using exact many-electron ground states in the subspaces with n ≤ nmax (nmax ∝ N ) electrons, we calculate the square of the total spin in the ground state to discuss magnetic properties of the models.We have found that although for n < nmax the ground states contain fully polarized states, there is no finite region of electron densities n/N < 1 (N = N/3 or N = N/5) where ground-state ferromagnetism survives for N → ∞.
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.