The momentum range about the Fermi momentum in which the concept of ``adiabatic continuity'' is valid is argued to decrease continuously to zero as the Mott transition is approached. This implies that the momentum distribution of occupied single-particle states will not decrease monotonically with momentum if the spectral function for single-particle excitations has no feature other than the quasiparticle peak. The value of the single-particle density of states at the Fermi energy is argued to drop to zero discontinuously at the Mott transition.
The momentum range about the Fermi momentum in which the concept of ``adiabatic continuity'' is valid is argued to decrease continuously to zero as the Mott transition is approached. This implies that the momentum distribution of occupied single-particle states will not decrease monotonically with momentum if the spectral function for single-particle excitations has no feature other than the quasiparticle peak. The value of the single-particle density of states at the Fermi energy is argued to drop to zero discontinuously at the Mott transition.
The Mott transition is usually considered as resulting from the divergence of the effective mass of the quasiparticle in the Fermi-liquid theory; the dispersion relation around the Fermi level is considered to become flat toward the Mott transition. Here, to clarify the characterization of the Mott transition under the assumption of a Fermi-liquid-like ground state, the electron-addition excitation from the Gutzwiller wavefunction in the $t\text{\ensuremath{-}}J$ model is investigated on a chain, ladder, square lattice, and bilayer square lattice in the single-mode approximation using a Monte Carlo method. The numerical results demonstrate that an electronic mode that is continuously deformed from a noninteracting band at zero electron density loses its spectral weight and gradually disappears toward the Mott transition. It exhibits essentially the magnetic dispersion relation shifted by the Fermi momentum in the small-doping limit as indicated by recent studies for the Hubbard and $t\text{\ensuremath{-}}J$ models, even if the ground state is assumed to be a Fermi-liquid-like state exhibiting gradual disappearance of the quasiparticle weight. This implies that, rather than as the divergence of the effective mass or disappearance of the carrier density that is expected in conventional single-particle pictures, the Mott transition can be better understood as freezing of the charge degrees of freedom while the spin degrees of freedom remain active, even if the ground state is like a Fermi liquid.
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
An intuitive interpretation of the relationship between the dispersion relation of the single-particle excitation in a metal and that of the spin excitation in a Mott insulator is presented, based on the results for the one- and two-dimensional Hubbard models obtained by using the Bethe ansatz, dynamical density-matrix renormalization group method, and cluster perturbation theory. The dispersion relation of the spin excitation in the Mott insulator is naturally constructed from that of the single-particle excitation in the zero-doping limit in both one- and two-dimensional Hubbard models, which allows us to interpret the doping-induced states as the states that lose charge character toward the Mott transition. The characteristic feature of the Mott transition is contrasted with the feature of a Fermi liquid and that of the transition between a band insulator and a metal.
An intuitive interpretation of the relationship between the dispersion relation of the single-particle excitation in a metal and that of the spin excitation in a Mott insulator is presented, based on the results for the one- and two-dimensional Hubbard models obtained by using the Bethe ansatz, dynamical density-matrix renormalization group method, and cluster perturbation theory. The dispersion relation of the spin excitation in the Mott insulator is naturally constructed from that of the single-particle excitation in the zero-doping limit in both one- and two-dimensional Hubbard models, which allows us to interpret the doping-induced states as the states that lose charge character toward the Mott transition. The characteristic feature of the Mott transition is contrasted with the feature of a Fermi liquid and that of the transition between a band insulator and a metal.
The zero temperature Mott-Hubbard transition as a function of the Coulomb repulsion U is investigated in the limit of large dimensions. The behavior of the density of states near the transition at U=Uc is analyzed in all orders of the skeleton expansion. It is shown that only two transition scenarios are consistent with the skeleton expansion for U<Uc: (i) The Mott-Hubbard transition is ``discontinuous'' in the sense that in the density of states finite spectral weight is redistributed at Uc. (ii) The transition occurs via a point at U=Uc where the system is neither a Fermi liquid nor an insulator.