Ward identities and Dyson equation for $d=1$ interacting Fermi systems

G. Benfatto, Vieri Mastropietro

G. Morandi, Francesco Napoli

Ioan Bâldea

A Ward identity is derived for the one-dimensional system with both electron-electron and electron-phonon forward scattering interactions. This identity is used to solve exactly the Dyson equations of the generalized effective interaction, phonon and electron Green functions for both T = 0 and T ≠ 0. The absence of both the quasi-fermion excitations and the jump in electron momentum distribution at the Fermi "surface" points out the failure of the Landau Fermi liquid theory for this system. The long-wavelength dielectric response function and the thermodynamic potential are also computed exactly.

Takeru Nakashima, Hannes Raebiger, Kaoru Ohno

It has been a long-standing problem in $N$-electron systems as to how to solve the Dyson equation with the quasiparticle wave functions defined as $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{0}^{N}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\ensuremath{\psi}}_{s}(\mathbit{r})\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{\ensuremath{\nu}}^{N+1}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$ and $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{\ensuremath{\mu}}^{N\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\ensuremath{\psi}}_{s}(\mathbit{r})\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Psi}}}_{0}^{N}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, which are not mutually orthogonal and have a norm less than 1. We show that quasiparticle wave functions without multiple excitations can exactly be normalized to unity owing to the Ward identity and the vertex function in $(\mathbit{q},{\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{-}\ensuremath{\omega})\ensuremath{\rightarrow}0$, although they are not necessarily mutually orthogonal. Since they satisfy an eigenvalue equation with the hermitized self-energy due to Baym-Kadanoff's conservation laws, the present theory can be regarded as an extension of the Kohn-Sham density functional theory, with correlated, interacting, one-electron orbitals.

R. F. Bishop

Andrey V. Chubukov

We discuss Ward identities for strongly interacting fermion systems described by Eliashberg-type theories. We show that Ward identities are not in conflict with the Migdal theorem and derive diagrammatically a Ward identity for a charge vertex both in a Fermi liquid and when a Fermi liquid is destroyed at quantum criticality. We argue that a Ward identity for the spin vertex cannot be obtained within the Eliashberg theory.