Hubbard model is a fundamental model in solid state physics; it describes the itinerant electrons interacting with each other through the on-site repulsive interaction. It is assumed that the intra-atomic Coulomb interaction is more important than the long-ranged 1/r Coulomb interaction, so that it can be approximated by the on-site repulsion. The model may exhibit fruitful possibilities of electronic states in the correlated electron systems. They are, for example, ferromagnetism, antiferromagnetism and superconductivity. Inspite of its simplicity the phase diagram has never been clarified except the one-dimensional system.
The Hubbard model has attracted much attention recently, stimulated by studies of high-temperature superconductivity.
One-dimensional Hubbard model has been understood in a very elegant way by means of the Bethe ansatz.
The solution revealed that the weak-coupling bosonization theory can well describe the ground state of the one-dimensional
Hubbard model and established a novel concept of the Luttinger liquid. In spite of the success for the one-dimensional
models, correlated electron systems in two- or three-dimensional space are still far from a complete understanding.
In the study of the Hubbard model, main topics are
・ Metal-insulator transition (Mott transition)
・ Magnetism of insulators and itinerant electrons
・ Possibility of high temperature superconductivty
These significant properties arise from the quantum many-body effects due to the Coulomb repulsive interaction.
References
[1] J. H. Hubbard: Proc. Roy. Soc. A276, 238 (1963). (Hubbard I)
[2] J. H. Hubbard: Proc. Roy. Soc. A277, 237 (1964). (Hubbard II)
[3] J. H. Hubbard: Proc. Roy. Soc. A281, 401 (1964). (Hubbard III)
[4] M. C. Gutzwiller: Phys. Rev. Lett. 10, 159 (1963).
[5] J. Kanamori: Prog. Theor. Phys. 30, 275 (1963).
[6] A. B. Harris and R.V. Lange: Phys. Rev. 157, 296 (1967).
One-dimensional model:
[7] H. Bethe: Z. Phys. 71 (1931) 205.
[8] C. N. Yang: Phys. Rev. Lett. 19 (1967) 1312.
[9] E. H. Lieb and F. Y. Wu: Phys. Rev. Lett. 20 (1968) 1445.
[10] J. Solyom: Adv. Phys. 28 (1979) 201.
[11] V. J. Emery: Highly Conducting One-Dimensional Solids (Plenum, 1979).
Two-dimensional model:
[12] E. Dagotto: Rev. Mod. Phys. 66 (1994) 763.
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