Hubbard Hamiltonian equation

The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is comprised of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is ... 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [ 128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V, computed for each microscopic model by band-structure calculations for Uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement... 

The F . in Eq. (20) is a many-body force because the electronic states sample the environment of the interacting atom / and / in the system. The formula above for the Hellmann-Feynman force calculation is also applicable to the tight-binding Hamiltonians when the Hubbard-like u term is present. Inclusion of the overlap matrix 5 in a nonorthogonal tight-binding scheme can also be handled by replacing the Hamiltonian matrix H by H - e 5 in the equations above. 

Here, the first and second terms define a Hubbard-like Hamiltonian for the extraparticles of the form of Equation 12.8, where the operators Cj(cJ) are destruction... 

Applying an optical lattice provides a periodic structure for the polar molecules described by the Hamiltonian of Equation 12.1, with yjj) given by Equation 12.32. In the limit of a deep lattice, a standard expansion of the field operators i] (r) = w(r - Ri)b] in the second-quantized expression of Equation 12.1 in terms of lowest-band Wannier functions w(r) and particle creation operators b] [107] leads to the realization of the Hubbard model of Equation 12.9, characterized by strong nearest-neighbor interactions [85]. We notice that the particles are treated as hardcore because of the constraint Rq. The interaction parameters Uy and Vyk in Equation 12.9derive from theeffective interaction V ( ri ), and in the limit of well-localized Wannier functions reduce to... 

The PPP model is the most complete model in this hierarchy (of Huckel, Hubbard and PPP), and it begins with the HF equations for the one electron ir-molecular orbitals of a conjugated molecule. Only IT orbitals are considered so all the a electrons are lumped together with the nuclear potential to form a "core" Hamiltonian, h ore carbon nucleus has one p atomic orbital... 

The vibrational consequences of 7r-electron fluctuations discussed in Sections II and III drew on both molecular spectroscopy and solid-state physics. The analysis of NLO and EA spectra in Section IV combined PPP models for molecules with quantum cell models of alternating chains. We proposed at the outset to relate the conjugated polymers in Fig. 6.2 to alternating Fliickel or PPP chains and have so far discussed vibrational and optical implications of 7r-electron models rather than the Hamiltonian, Eq. (7), or its mathematical properties. The analysis holds for any H(8) with appropriate vibrational or optical susceptibilities. Equation (7) is sufficiently general to encompass Hiickel, Hubbard, extended Hubbard, PPP, and other models with suitable choices of U and Vp,. This generality is an extremely useful feature of solid-state models. 

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