The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffiths' theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffiths' theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.

Advanced series in mathematical physicsMay 1, 19940 citations

Phase transitions and exact ground-state properties of the one-dimensional Hubbard model in a magnetic field

Chia‐Wei Yang, ARMEN N. KOCHARIAN, Y. L. Chiang

The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.

Journal of Physics Condensed MatterAug 2, 200024 citations

Gap in the spin excitations and magnetization curve of the one-dimensional attractive Hubbard model

Thomas B. Bahder, Ferenc Woynarovich

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.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.

Physical review. B, Condensed matterFeb 15, 198668 citations

Magnetic Susceptibility at Zero Temperature for the One-Dimensional Hubbard Model

Hiroyuki Shiba

The magnetic susceptibility at absolute-zero temperature for the one-dimensional Hubbard model is studied exactly as a function of the concentration of electrons by using Lieb and Wu's theory for this system. Our analysis essentially follows Griffiths's method for the magnetic susceptibility of the one-dimensional Heisenberg antiferromagnet, and is a generalization of Takahashi's calculation to the systems with an arbitrary concentration of electrons. The ground-state energy and the magnitude of local moments at each site are also studied. Combined with the results on the susceptibility, they should suggest how the effect of the Coulomb interaction on the properties of the system at low temperatures changes with the concentration of electrons.The magnetic susceptibility at absolute-zero temperature for the one-dimensional Hubbard model is studied exactly as a function of the concentration of electrons by using Lieb and Wu's theory for this system. Our analysis essentially follows Griffiths's method for the magnetic susceptibility of the one-dimensional Heisenberg antiferromagnet, and is a generalization of Takahashi's calculation to the systems with an arbitrary concentration of electrons. The ground-state energy and the magnitude of local moments at each site are also studied. Combined with the results on the susceptibility, they should suggest how the effect of the Coulomb interaction on the properties of the system at low temperatures changes with the concentration of electrons.

Physical review. B, Solid stateAug 1, 1972310 citations

Magnetization Curve at Zero Temperature for the Antiferromagnetic Heisenberg Linear Chain

Robert B. Griffiths

The highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton, $g$ the electron $g$ factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.The highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton, $g$ the electron $g$ factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.

Physical ReviewFeb 3, 1964402 citations

Excitation spectrum of the one-dimensional Hubbard model

Cornelius F. Coll

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.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.

Physical review. B, Solid stateMar 1, 1974166 citations

g

the electron

g

factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.

\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.