Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Note on the one-dimensional Hubbard model

William J. Camp

Published Oct 1, 1974• 2 citations
Publisher

Abstract

By use of a Jordan-Wigner transformation, the one-dimensional Hubbard model for an itinerant magnet is expressed as an anisotropic Heisenberg model on a double chain with two- and three-spin interactions. The limiting cases---(i) free electron, (ii) strong correlation, and (iii) zero-hopping rate---are briefly examined in the spin representation.

DOI
10.1103/physrevb.10.2903

Related papers

The one-dimensional Hubbard model with first-, second- and third-nearest-neighbour hopping in the strong-coupling limit

M W Long, C. W. M. Castleton, C A Hayward

Using an extension of the Jordan-Wigner transformation, we solve the one-dimensional Hubbard model at U = infinity in the limit of dominant nearest-neighbour hopping combined with infinitesimal hopping over slightly longer ranges. We find several possible phases at zero temperature, including ferromagnetism, paramagnetism (with and without a spin gap), and even long-range antiferromagnetism with a particular limiting procedure. Our solutions are always spin-charge separated, and we give evidence that the charge degrees of freedom are best described by bosonic statistics.

Journal of Physics Condensed Matter•Jan 10, 1994•10 citations

Spin-1 Heisenberg chain and the one-dimensional fermion gas

Jenő Sólyom, JUSSI TIMONEN

The composite-spin representation of the spin-1 Heisenberg chain is used to transform it through the Jordan-Wigner transformation to the one-dimensional fermion gas. To properly include the xy couplings between spins, we also consider the bosonized version of the fermion model. Phase diagrams deduced from the two versions of the fermion model are compared against numerical results for finite Heisenberg chains. One of the symmetries of the spin model is lost in the fermionization, and this leads to a topologically incorrect phase diagram in at least one part of the parameter space. There are clear indications of significant coupling of spin and charge degrees of freedom in the fermion model and of strong renormalization of the coupling constants in the continuum limit. The existence of the Haldane gap is discussed in connection with these results.

Physical review. B, Condensed matter•Oct 1, 1989•21 citations

Magnetization Curve for the Half-Filled Hubbard Model

Minoru Takahashi

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 physics•May 1, 1994•0 citations

Two-Dimensional Spin-1/2 J1 – J'1 – J2 Heisenberg Model within Jordan–Wigner Transformation

O. R. Baran, Taras Verkholyak

The Jordan–Wigner transformation is applied to the spatially anisotropic spin-1/2 Heisenberg model on a square lattice with the nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions. The transformed Hamiltonian describes the interacting spinless fermions that hop between neighbor sites in a gauge field. Using the mean-field-type approximation to both the direct interaction between fermions and the phase factors, which represent the gauge field, the problem is reduced to that concerning a free Fermi gas. Two types of antiferromagnetic ordering (the N´eel and collinear ones) are considered. By calculating the ground-state energies, the phase transitions induced by the interaction frustration were analyzed.

Ukrainian Journal of Physics•Jul 1, 2016•1 citation

Ferromagnetic Quantum critical behavior in three-dimensional Hubbard model with transverse anisotropy

Naoum Karchev

One-band Hubbard model with transverse anisotropy is considered at density of electrons $n=0.4$. It is shown that when the anisotropy is appropriately chosen, the ground state is ferromagnetic with magnetic order perpendicular to the anisotropy. The increasing of the ratio $\frac tU$, where $t$ is the hopping parameter and $U$ is the Coulomb repulsion, decreases the Curie temperature, and the system arrives at the quantum critical point $(T_C=0)$. The result is obtained introducing Schwinger bosons and slave Fermions representation of the electron operators. Integrating out the spin-singlet Fermi fields an effective Heisenberg model with ferromagnetic exchange constant is obtained for vectors which identifies the local orientation of the spin of the itinerant electrons. The amplitude of the spin vectors is an effective spin of the itinerant electrons accounting for the fact that some sites, in the ground state, are doubly occupied or empty. Owing to the anisotropy, the magnon fluctuations drive the system to quantum criticality and when the effective spin is critically small these fluctuations suppress the magnetic order.

arXiv:1207.1674•Jul 6, 2012•0 citations

Magnetization Curve for the Half-Filled Hubbard Model

Minoru Takahashi

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 Griffith's 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.

Progress of Theoretical Physics•Nov 1, 1969•166 citations
By use of a Jordan-Wigner transformation, the one-dimensional Hubbard model for an itinerant magnet is expressed as an anisotropic Heisenberg model on a double chain with two- and three-spin interactions. The limiting cases---(i) free electron, (ii) strong correlation, and (iii) zero-hopping rate---are briefly examined in the spin representation.
Using an extension of the Jordan-Wigner transformation, we solve the one-dimensional Hubbard model at U = infinity in the limit of dominant nearest-neighbour hopping combined with infinitesimal hopping over slightly longer ranges. We find several possible phases at zero temperature, including ferromagnetism, paramagnetism (with and without a spin gap), and even long-range antiferromagnetism with a particular limiting procedure. Our solutions are always spin-charge separated, and we give evidence that the charge degrees of freedom are best described by bosonic statistics.
The composite-spin representation of the spin-1 Heisenberg chain is used to transform it through the Jordan-Wigner transformation to the one-dimensional fermion gas. To properly include the xy couplings between spins, we also consider the bosonized version of the fermion model. Phase diagrams deduced from the two versions of the fermion model are compared against numerical results for finite Heisenberg chains. One of the symmetries of the spin model is lost in the fermionization, and this leads to a topologically incorrect phase diagram in at least one part of the parameter space. There are clear indications of significant coupling of spin and charge degrees of freedom in the fermion model and of strong renormalization of the coupling constants in the continuum limit. The existence of the Haldane gap is discussed in connection with these results.
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 Jordan–Wigner transformation is applied to the spatially anisotropic spin-1/2 Heisenberg model on a square lattice with the nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions. The transformed Hamiltonian describes the interacting spinless fermions that hop between neighbor sites in a gauge field. Using the mean-field-type approximation to both the direct interaction between fermions and the phase factors, which represent the gauge field, the problem is reduced to that concerning a free Fermi gas. Two types of antiferromagnetic ordering (the N´eel and collinear ones) are considered. By calculating the ground-state energies, the phase transitions induced by the interaction frustration were analyzed.
One-band Hubbard model with transverse anisotropy is considered at density of electrons n=0.4n=0.4n=0.4. It is shown that when the anisotropy is appropriately chosen, the ground state is ferromagnetic with magnetic order perpendicular to the anisotropy. The increasing of the ratio tU\frac tUUt​, where ttt is the hopping parameter and UUU is the Coulomb repulsion, decreases the Curie temperature, and the system arrives at the quantum critical point (TC=0)(T_C=0)(TC​=0). The result is obtained introducing Schwinger bosons and slave Fermions representation of the electron operators. Integrating out the spin-singlet Fermi fields an effective Heisenberg model with ferromagnetic exchange constant is obtained for vectors which identifies the local orientation of the spin of the itinerant electrons. The amplitude of the spin vectors is an effective spin of the itinerant electrons accounting for the fact that some sites, in the ground state, are doubly occupied or empty. Owing to the anisotropy, the magnon fluctuations drive the system to quantum criticality and when the effective spin is critically small these fluctuations suppress the magnetic order.
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 Griffith's 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.