The Hubbard model in the strong-coupling regime is studied by the Kondo-lattice theory. If no symmetry is broken at a sufficiently low temperature in sufficiently low dimensions, an electron liquid is stabilized by the Fock-type exchange effect of the superexchange interaction. Since the stabilization mechanism is none other than the resonating-valence-bond (RVB) mechanism, the electron liquid is none other than an RVB electron liquid. The RVB electron liquid in the Hubbard model is adiabatically connected to the RVB spin liquid in the Heisenberg model.
The Hubbard model in the strong-coupling regime is studied by the Kondo-lattice theory. If no symmetry is broken at a sufficiently low temperature in sufficiently low dimensions, an electron liquid is stabilized by the Fock-type exchange effect of the superexchange interaction. Since the stabilization mechanism is none other than the resonating-valence-bond (RVB) mechanism, the electron liquid is none other than an RVB electron liquid. The RVB electron liquid in the Hubbard model is adiabatically connected to the RVB spin liquid in the Heisenberg model.
The Hubbard model in D dimensions, with the on-site repulsion U and the transfer integral between nearest neighbors −t/D, is studied on the basis of the Kondo-lattice theory. If U/∣t∣≫1, ∣n−1∣≲∣t∣/(DU), where n is the number of electrons per unit cell, and D is so small that ∣J∣/D≫kBTc, where J=−4t2/U and Tc is 0K for D=1 and is the highest critical temperature among possible ones for D≥2, a low-T phase where Tc<T≪∣J∣/(kBD) is a frustrated electron liquid. Since the liquid is stabilized by the Kondo effect in conjunction with the resonating-valence-bond (RVB) mechanism, it is simply the RVB electron liquid; in one dimension, it is also the Tomonaga-Luttinger liquid. The Kondo energy of the RVB liquid is kBTK=O(∣J∣/D); its effective Fermi energy is O(kBTK). A midband appears on the chemical potential between the upper and lower Hubbard bands; the Hubbard gap is a pseudogap. As regards the density of states per unit cell of the midband, its bandwidth is O(kBTK) or O(∣J∣/D), its peak height is O(1/U), and its spectral weight is O[t2/(DU2)]. Since the midband almost disappears in the Heisenberg limit, the RVB electron liquid in the Heisenberg limit is simply the RVB spin liquid. The RVB electron and spin liquids adiabatically continue to each other. Since local moments form in a high-T phase where T≳TK, the high-T phase is simply the Mott insulator.
The ground state of the Hubbard model is studied within the constrained Hilbert space where no order parameter exists.The self-energy of electrons is decomposed into the single-site and multisite self-energies.The calculation of the single-site self-energy is mapped to a problem of self-consistently determining and solving the Anderson model.When an electron reservoir is explicitly considered, it is proved that the single-site self-energy is that of a normal Fermi liquid even if the multisite self-energy is anomalous.Thus, the ground state is a normal Fermi liquid in the supreme single-site approximation (S 3 A).In the strong-coupling regime, the Fermi liquid is stabilized by the Kondo effect in the S 3 A and is further stabilized by the Fock-type term of the superexchange interaction or the resonating-valence-bond (RVB) mechanism beyond the S 3 A. The stabilized Fermi liquid is frustrated as much as an RVB spin liquid in the Heisenberg model.It is a relevant unperturbed state that can be used to study a normal or anomalous Fermi liquid and an ordered state in the whole Hilbert space by Kondo lattice theory.Even if higher-order multisite terms than the Fock-type term are considered, the ground state cannot be a Mott insulator.It can be merely a gapless semiconductor even if the multisite self-energy is so anomalous that it is divergent at the chemical potential.A Mott insulator is only possible as a high temperature phase.
The ground state of the Hubbard model is studied within the constrained Hilbert space where no order parameter exists. The self-energy of electrons is decomposed into the single-site and multisite self-energies. The calculation of the single-site self-energy is mapped to a problem of self-consistently determining and solving the Anderson model. When an electron reservoir is explicitly considered, it is proved that the single-site self-energy is that of a normal Fermi liquid even if the multisite self-energy is anomalous. Thus, the ground state is a normal Fermi liquid in the supreme single-site approximation (S3A). In the strong-coupling regime, the Fermi liquid is stabilized by the Kondo effect in the S3A and is further stabilized by the Fock-type term of the superexchange interaction or the resonating-valence-bond (RVB) mechanism beyond the S3A. The stabilized Fermi liquid is frustrated as much as an RVB spin liquid in the Heisenberg model. It is a relevant unperturbed state that can be used to study a normal or anomalous Fermi liquid and an ordered state in the whole Hilbert space by Kondo lattice theory. Even if higher-order multisite terms than the Fock-type term are considered, the ground state cannot be a Mott insulator. It can be merely a gapless semiconductor even if the multisite self-energy is so anomalous that it is divergent at the chemical potential. A Mott insulator is only possible as a high temperature phase.
The Hubbard model in the strong-coupling regime is mainly studied by Kondo-lattice theory or 1/d expansion theory, with d the spatial dimensionality. In two dimensions and higher, the ground state within the Hilbert subspace with no order parameter is a normal Fermi liquid except for n=1 and U/W=+infinity, with n the electron density per unit cell, U the on-site repulsion, and W the bandwidth; the cooperation between the Kondo effect, which favors a local singlet on each unit cell, and a resonating-valence-bond effect, which favors a local singlet on each pair of nearest-neighbor unit cells, stabilizes the Fermi liquid, whose ground state is a singlet as a whole, in the strong-coupling regime. In the whole Hilbert space with no restriction, the normal Fermi liquid is unstable at least against a magnetic or superconducting state. This analysis confirms an early Fermi-liquid theory of high-temperature superconductivity, F. J. Ohkawa, Jpn. J. Appl. Phys. 26, L652 (1987). The ground state for n=1 and U/W=+infinity is a Mott insulator. Actual metal-insulator transitions cannot be explained within the Hubbard model. In order to explain them, the electron-phonon interaction, multi-band or multi-orbital effects, and effects of disorder should be considered beyond the Hubbard model. The crossover between local-moment magnetism and itinerant-electron magnetism corresponds to that between a localized spin and a normal Fermi liquid in the Kondo effect and it is simply a Mott metal-insulator crossover.
It is proved that the ground state under the supreme single-site approximation (S^3A), the dynamical mean-field theory (DMFT), or the dynamical coherent potential approximation (DCPA) is the normal Fermi liquid in the presence of an infinitesimally weak hybridization with an electron reservoir, except for the just half filling of electrons and the infinite on-site repulsion. In the strong-coupling regime, in particular, the Fermi liquid is stabilized under S^3A, DMFT, or DCPA by the Kondo effect, which stabilizes a local singlet on each unit cell, and is further stabilized beyond it by the Fock-type term of the superexchange interaction or a resonating valence bond (RVB) mechanism, which stabilizes a local singlet on each pair of nearest neighbors. The Fermi liquid is a relevant normal state to study possible lower-temperature phases or the true ground state. It is proposed that the Fermi liquid stabilized by the Kondo effect and the RVB mechanism is the normal state of cuprate high-temperature superconductors.
Recent numerical simulations find a possible chiral spin-liquid state in the intermediate coupling regime of the triangular lattice Hubbard model. Here we provide a simple picture for its origin in terms of a bosonic resonating valence bond (RVB) description. More specifically, we show that such a chiral spin-liquid state can be understood as a quantum disordered tetrahedral spin state stabilized by the four spin ring exchange coupling, which suppresses the order-by-disorder effect toward a stripy spin state. Such a chiral spin-liquid state features a spin Berry phase of $\frac{\ensuremath{\pi}}{2}$ per triangle. However, we show that the topological property of such a state is totally missed in the Schwinger boson mean-field description as a result of the lack of boson rigidity caused by the no double occupancy constraint.