The Hubbard model is studied for the Mott-transition region in the limit of high dimension d →+∞. Leading-order effects with respect to 1/ d are local spin fluctuations, all of which are fully included in the single-site approximation (SSA). Because the SSA is reduced to solving the Anderson model, it is argued that a normal heavy-electron liquid forms through the Kondo effect. The Fermi-liquid relations are proven within the SSA. The other leading-order effects are mean fields of magnetism caused by exchange interaction. Therefore it depends on the competition between the Kondo effect and the exchange interaction which the ground state is, magnetic or paramagnetic. However, any other effect is a higher-order effect. The 1/ d expansion from the heavy-electron liquid in the SSA is one of the most useful methods in examining lower-temperature phases in the real dimension.
The Hubbard model is studied for the Mott-transition region in the limit of high dimension d →+∞. Leading-order effects with respect to 1/ d are local spin fluctuations, all of which are fully included in the single-site approximation (SSA). Because the SSA is reduced to solving the Anderson model, it is argued that a normal heavy-electron liquid forms through the Kondo effect. The Fermi-liquid relations are proven within the SSA. The other leading-order effects are mean fields of magnetism caused by exchange interaction. Therefore it depends on the competition between the Kondo effect and the exchange interaction which the ground state is, magnetic or paramagnetic. However, any other effect is a higher-order effect. The 1/ d expansion from the heavy-electron liquid in the SSA is one of the most useful methods in examining lower-temperature phases in the real dimension.
The Hubbard model with strong correlation is examined. The mean-field approximation (MFA) based on the best single-site approximation (SSA) is rigorous in the limit of high dimension of d =+∞. The best SSA is reduced to solving the Anderson model. The ground state is a heavy-electron liquid, presumably superconducting, in the weak-coupling regime, which is approximately defined by g ≈ J ( Q )/4 T K <1 with J ( Q ) being the maximum value of exchange interactions and T K the Kondo temperature to show the energy scale of heavy electrons. In the strong-coupling regime defined by g >1, on the other hand, the ground state is a MFA-type antiferromagnetic state.
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.
The Hubbard model is studied in the vicinity of the Mott transition. Local and quantal spin fluctuations are completely included through the mapping to the Anderson model; a local Kondo temperature TK is defined as their characteristic temperature or energy scale. Antiferromagnetic spin fluctuations are included perturbatively in terms of d1 and to leading order in UkBTK with d being the spatial dimensionality, kB the Boltzmann constant, and U the intrasite repulsion. The two different kinds of spin fluctuations are responsible for the formation of heavy quasiparticles. Two intersite exchange interactions are responsible for both the development of the antiferromagnetic spin fluctuations and $d\ensuremath{\gamma}$-wave Cooper pairing between the heavy quasiparticles: the superexchange interaction and an exchange interaction due to the virtual exchange of spin excitations within the heavy quasiparticle band. An experimental specific-heat coefficient of about 14 mJ/K2 CuO2 mol and a Wilson ratio of about 0.4 imply that the interplay between the local and the antiferromagnetic spin fluctuations plays a crucial role in both the normal and the superconducting states of $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ with critical temperatures of ${T}_{c}\ensuremath{\simeq}90$ K. A pairing interaction deduced from the normal-state properties is strong enough to give critical temperatures as high as ${T}_{c0}\ensuremath{\simeq}200$ K in the absence of any pair breaking; there is experimental evidence that the pair-breaking effect of the antiferromagnetic spin fluctuations reduces ${T}_{c0}\ensuremath{\simeq}200$ K down to ${T}_{c0}\ensuremath{\simeq}90$ K.
We investigate the magnetic order and related strongly correlated effects in a one-dimensional Ising-Kondo lattice with transverse field. This model is the anisotropic limit of the conventional isotropic Kondo lattice model, in the sense that the itinerant electrons interact with the localized magnetic moments via only longitudinal Kondo exchange. Adopting the numerical density-matrix renormalization group method, we map out the ground-state phase diagram in various parameter spaces. Depending on the Kondo coupling and filling number, three distinct phases, including a metallic paramagnetic, a metallic ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-density wave is characterized by an ordering wave vector, which coincides with the nesting wave vector of the Fermi surface. This makes the corresponding magnetic transition a spin analog of the Peierls transition occurring in the one-dimensional metal. Moreover, by analyzing the momentum distribution function and charge correlation function, the conduction electrons are shown to behave like free spinless fermions in the ferromagnetic phase. We finally discuss the effect of the repulsive Hubbard interaction between conduction electrons. Our paper enriches the Kondo physics and deepens the current understanding of the heavy fermion compounds.
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.