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Judd Operator Methods in Superspace: Application to the Thermal Single Particle Green Function for the Hubbard Dimer.

Jay D. Mancini

Published Jan 1, 1982• 0 citations
OpenAlex

Abstract

The Fourier transforms of the thermal two-time single-particle Green functions may be expressed as matrix elements of the resolvent of the Liouville operator, in an abstract Hilbert space. This abstract Hilbert space (the "superspace") contains elements f, g, etc. which are products of an odd number of fermion creation and/or annihilation operators. These operators may also be expressed as linear combinations of a set of stepping operators Φ = fig, where I is the projection operator for the vacuum in the ordinary Fock space. The Judd operators are stepping operators which step between many-particle states which usually differ in particle number. In the calculation of the single-particle Green function, only those single-particle Judd operators which step between states differing by one electron are relevant. The Judd operators obey a Lie algebra analogous to the angular momentum stepping operators L±. The single-site and two-site Hubbard model for arvitrary electron density are solved exactly using the Judd operator formalism. The correlation functions are evaluated as functions of chemical potential, temperature and t/U, where t is the hopping energy and U is the intasite Coulomb energy.

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The Fourier transforms of the thermal two-time single-particle Green functions may be expressed as matrix elements of the resolvent of the Liouville operator, in an abstract Hilbert space. This abstract Hilbert space (the "superspace") contains elements f, g, etc. which are products of an odd number of fermion creation and/or annihilation operators. These operators may also be expressed as linear combinations of a set of stepping operators Φ = fig, where I is the projection operator for the vacuum in the ordinary Fock space. The Judd operators are stepping operators which step between many-particle states which usually differ in particle number. In the calculation of the single-particle Green function, only those single-particle Judd operators which step between states differing by one electron are relevant. The Judd operators obey a Lie algebra analogous to the angular momentum stepping operators L±. The single-site and two-site Hubbard model for arvitrary electron density are solved exactly using the Judd operator formalism. The correlation functions are evaluated as functions of chemical potential, temperature and t/U, where t is the hopping energy and U is the intasite Coulomb energy.
The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research.
The Thermal Matsubara Green Functions, and their zero temperature analogs, in many-fermion systems are shown to be built up from products, or sums of products, of factors each of which is determined by a Lie Algebra. These factors are: the structure constants of a mapping φ of the B2N+1 Fermion Algebra (consisting of singles and pairs of fermion operators) onto itself; and the structure constants of a mapping φ′ of the D2N+1 Dynamical Algebra of the Hamiltonian (consisting of pairs of fermion operators) onto itself; and eigenvalues of the generators of the Abelian Cartan subalgebra. Eigenstates of the many-fermion Hamiltonian typically relate to different types or numbers of elementary excitations (quasi-particles); these states are shwon to be bases for irreducible representations of the Dynamical Algebra. The contribution of each such irreducible set of states to the Green Functions is exhibited. These results are exact for mean-field Hamiltonians; generalization to include effects of fluctuations may be possible. The Green Functions for a model BCS singlet superconductor are determined (Dynamical Algebra SU (2)) to illustrate the various steps in the analysis.
Abstract We use the cumulant Green’s functions method (CGFM) to study the single-band Hubbard model. The starting point of the method is to diagonalize a cluster (‘seed’) containing N correlated sites and employ the cumulants calculated from the cluster solution to obtain the full Green’s functions for the lattice. All calculations are done directly; no variational or self-consistent process is needed. We benchmark the one-dimensional results for the gap, the double occupancy, and the ground-state energy as functions of the electronic correlation at half-filling and the occupation numbers as functions of the chemical potential obtained from the CGFM against the corresponding results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The particle-hole symmetry of the density of states is fulfilled, and the gap, occupation numbers, and ground-state energy tend systematically to the known results as the cluster size increases. We include a straightforward application of the CGFM to simulate the singles occupation of an optical lattice experiment with lithium-6 atoms in an eight-site Fermi-Hubbard chain near half-filling. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and extended to other strongly correlated models, like the Anderson Hamiltonian, the t − J, Kondo, and Coqblin-Schrieffer models.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
Abstract Describing spin ↑ ‐electrons the particle number operators ni↓ are replaced under the functional integral directly by c‐numbers vi↓(t) using a δ‐functional instead of the Hubbard‐Stratonovich transformation. The Green function of the Hubbard model is thus exactly represented as an averaged Green function of electrons moving in a stochastic potential Uvi↓(t). The probability distribution for the stochastic process v ↓ containing the many body aspect and the information about the ↓‐electrons is determined. The vi↓(t) are only allowed to jump between 0 and 1 reflecting the property of Fermion particle number operators. This exact scheme can be interpreted as a dynamically generalized alloy analogy.