Note on Derivation of the Exact Double-counting
ثبت نشده
چکیده
Most of the double-counting formulas were historically derived by approximating the Hubbard interaction term (defined with the help of matrix elements Eq. 33) by some static approximation, either in the atomic limit, or, in Hartree-Fock limit. Such static approximations were argued to be a good substitute for the LDA treatment of the Hubbard interaction. Hence, the problem arrose, because it is not clear how to solve the Hubbard model (or any lattice model) by LDA, so that the LDA approximation for the Hubbard term could be subtracted from dynamic self-energy, computed by many body method. Here we show that, if Luttinger-Ward functionals for the two approximate methods are written side-by side in the same form, the intersection of the two is evident. In other words, we can either perform the DMFT approximation on the LDA functional, or, the LDA approximation on the DMFT functional, and in both cases we arrive at the same term, which is counted twice. Let’s start with the lowest order term in the interaction, the Hartree term, because it can be explicitely written down. The exact Hartree term takes the form
منابع مشابه
Double derivations of n-Lie algebras
After introducing double derivations of $n$-Lie algebra $L$ we describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual derivation Lie algebra $mathcal Der(L)$. In particular, we prove that the inner derivation algebra $ad(L)$ is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra wit...
متن کاملDouble Derivations, Higher Double Derivations and Automatic Continuity
Let be a Banach algebra. Let be linear mappings on . First we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple Banach algebras, in certain case. Afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...
متن کاملCharacterization of $(delta, varepsilon)$-double derivation on rings and algebras
This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, delta, varepsilon$ are additive mappings satisfyingbegin{equation}d(x^n) = sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+sum^{n-1}_{j=1}sum^{j}_{i=1}x^{n-1-j}Big(delta(x)x^{j-i}varepsilon(x)+varepsilon(x)x^{j-i}delta(x)Big)x^{i-1}quadend{e...
متن کاملCharacterization of $delta$-double derivations on rings and algebras
The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...
متن کامل