A Rigorous Treatment of the Perturbation Theory for Many-Electron Systems

Abstract

Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d (≥ 1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in 2 dimensional case.

Exponential Decay of Equal-Time Four-Point Correlation Functions in the Hubbard Model on the Copper-Oxide Lattice

“…For anyX m j ∈Ĩ m j L,h (j = 1, · · · , n), T ∈ T n , ξ ∈ S n (T ), s ∈ [0, 1] n−1 and l ∈ {N β , · · · , N h }, e n q,r=1 Mat(T,ξ,s)q,r∆q,r(C l (wep)) This can be proved by using (4.11) and the properties of M at (T, ξ, s) and by repeating the same argument as in[10, Lemma 4.5].…”

“…In this section we formulate the correlation function by using the notion of Grassmann integral and show that the Grassmann integral representation of the correlation function multiplied by the distance between the holes and the electrons is transformed into a contour integral of the Grassmann integral. This procedure is essentially the same as we did in [10], [11]. In order to avoid unnecessary repetition we present the proofs at a minimum.…”

Exponential decay of correlation functions in many-electron systems

“…The evaluation of the tree expansion presented above overcounts the combinatorial factor, which is defined as the number of monomials appearing in the expansion {p,q}∈T (∆ p,q + ∆ q,p ) · (the monomial (4.7)). If we consider the 4 point correlation functions for the on-site interaction V = U x∈Γ ψ * x↑ ψ * x↓ ψ x↓ ψ x↑ (U ∈ R), the exact calculation of the combinatorial factor without overcounting is possible as proved in [7,Lemma 4.7]. Consequently, the perturbative bound is improved in this case.…”

“…To show (iv), define an h-dependent finite set M h of Matsubara frequencies by Using the assumption that h ∈ 2N/β, the argument parallel to [7,Appendix C] demonstrates that…”

“…As in [7] we directly treat the perturbation theory of the original model without introducing any multi-scale technique. Though the results can be presented explicitly in a simple manner through the single scale analysis, it costs the temperature dependency of the interaction.…”