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Effective functional for the supercoherent state of spinless algebra in the Hubbard model. (English. Russian original) Zbl 1259.82139
Russ. Phys. J. 54, No. 6, 658-667 (2011); translation from Izv. Vyssh. Uchebn. Zaved., Fiz., No. 6, 39-46 (2011).
The paper develops the superfield formulation based on the functional integral method on trajectories for the Hubbard model studied in earlier authors’ works. The effective functional is calculated for spinless fermions based on the nonlinear supergroup and corresponding matrix representation of the supercoherent state. For a comparison, the superfield Lagrangian is calculated for spinless fermions in the Hubbard model with the help of a simple quantum-field chiral model. First, the Hamiltonian in the Hubbard model in the representation of secondary quantization is written, which includes three parameters, namely a matrix element of the electron transition to the neighboring site related to the electron bandwidth, the parameter of the Coulomb repulsion for two electrons in a site, and the chemical potential or electron concentration. In the paper, it is considered a half-occupied band with strong repulsion. In this mode, the lowest states are described by the magnetic fields together with degrees of freedom for fermions. The repulsion parameter is in fact eliminated from the problem. Then the matrix elements of the evolution operator between two states are expressed through the functional integral with the given effective Lagrangian and measure. The last parameters are written in the superspace and the effective functional is presented via the state vectors written through the Grassmann variables. To calculate the functional integral of the strongly correlated electron system in the Hubbard model, the two-group supercoherent states are used. Transitions between states of the atomic basis are described by \(4\times 4\) matrices comprising 16 Hubbard operators. In the matrix representation, any state is assigned by the ket-vector. The production and annihilation operators are expressed through the Fermi-like Hubbard operators. By introducing spinless operators, a spinless algebra is formed describing the limit of the spinless Fermi gas. The matrix representation of the algebra comprises a product of a set of the dynamic Bose and Fermi fields by corresponding generators. In the work, the matrix of the supercoherent state of this spinless algebra is calculated. In the simplest case of a spinless algebra, it is derived an effective functional for the kinetic energy of the Hubbard Hamiltonian through effective temporal and lattice functionals. As a result, the authors obtain an explicit expression for the effective functional in the Hubbard model for the spinless case in the linear approximation in time and quadratic approximation in the coordinate. In the use of the Klein-Gordon equation instead of the Schrödinger equation, the authors obtain an effective nonlinear functional of the kinetic energy in the Hubbard model for the spinless case with quadratic approximation in time and coordinate. Finally, the Lagrangian of superfield action for spinless algebra in the Hubbard 2D model (1+1) is calculated. A comparison of the superfield Lagrangian with above-mentioned Lagrangian functional demonstrates their structure of the same type, coinciding terms in the second derivatives with respect to time and in the first derivatives with respect to the coordinate.
MSC:
82D55 Statistical mechanical studies of superconductors
17A70 Superalgebras
81R30 Coherent states
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