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arXiv:2003.13136

Systematic Resummation of the Large N expansion of Vector Models: Application to the Hubbard model and $2 + 1 $ dimensional QED

Tom Banks

Published Mar 29, 2020• 0 citations
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Abstract

We introduce a hierarchy of closed equations for charge density correlation functions in the Hubbard model and $2 + 1$ dimensional QED. Each step in the hierarchy can be considered a large $N$ truncation of an exact, but infinite set of equations relating all $k-$point charge correlators. $N$ is the number of fermion spin components. Each step in the hierarchy sums up an infinite number of large $N$ diagrams, including all diagrams up to some fixed order, for $k$ point functions with $k \leq K$. Higher point functions are replaced with their leading large $N$ behavior. The simplest truncation gives a closed nonlinear equation for the $2$ point function.

DOI
10.48550/arxiv.2003.13136

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We introduce a hierarchy of closed equations for charge density correlation functions in the Hubbard model and 2+12 + 12+1 dimensional QED. Each step in the hierarchy can be considered a large NNN truncation of an exact, but infinite set of equations relating all k−k-k−point charge correlators. NNN is the number of fermion spin components. Each step in the hierarchy sums up an infinite number of large NNN diagrams, including all diagrams up to some fixed order, for kkk point functions with k≤Kk \leq Kk≤K. Higher point functions are replaced with their leading large NNN behavior. The simplest truncation gives a closed nonlinear equation for the 222 point function.
We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.
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We develop a diagrammatic approach for calculating the high temperature expansion of dynamic correlation functions, such as the electron Green's function and the time-dependent density-density and spin-spin correlation functions, for the infinite-U Hubbard Model with any number of spin species. The formalism relies on the use of restricted lattice sums, in which distinct vertices of the diagram represent distinct sites on the lattice. We derive a new formula for the restricted lattice sum of a disconnected diagram consisting of several connected components, and use it to prove the linked cluster theorem with respect to "generalized connected diagrams", formed by overlapping the original connected components on the lattice. This enables us to express all quantities as a sum over these generalized connected diagrams. We compute the Green's function to 4th order in βt for the case of m spin species on a d-dimensional hypercube by hand. We take the m\to\infty limit, enabling us to obtain expressions for the Dyson-Mori self-energy to 4th order in βt for the case of an infinite number of spin species. This may have connections to slave boson techniques used for the study of this model. Our approach is computationally more efficient than any used previously for the calculation of the high temperature expansion of dynamic correlation functions, and high order results for both the Green's function and the time-dependent density-density and spin-spin correlation functions shall be presented in a separate paper [25].
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In this paper, we show that a system of localized particles, satisfying the Fermi statistics and subject to finite-range interactions, can be exactly solved in any dimension. In fact, in this case it is always possible to find a finite closed set of eigenoperators of the Hamiltonian. Then, the hierarchy of the equations of motion for the Green's functions eventually closes and exact expressions for them are obtained in terms of a finite number of parameters. For example, the method is applied to the two-state model (equivalent to the spin-(1/2) Ising model) and to the three-state model (equivalent to the extended Hubbard model in the ionic limit or to the spin-1 Ising model). The models are exactly solved for any dimension d of the lattice. The parameters are self-consistently determined in the case of d = 1.