In solid state physics, the Luttinger–Ward functional,[1] proposed by Joaquin Mazdak Luttinger and John Clive Ward in 1960,[2] is a scalar functional of the bare electron-electron interaction and the renormalized one-particle propagator.
In terms of Feynman diagrams, the Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible diagrams, i.e., all diagrams without particles going in or out that do not fall apart if one removes two propagator lines.
is the one-particle Green's function and
The Luttinger–Ward functional has no direct physical meaning, but it is useful in proving conservation laws.
The functional is closely related to the Baym–Kadanoff functional constructed independently by Gordon Baym and Leo Kadanoff in 1961.
[3] Some authors use the terms interchangeably;[4] if a distinction is made, then the Baym–Kadanoff functional is identical to the two-particle irreducible effective action
, which differs from the Luttinger–Ward functional by a trivial term.
, the partition function can be expressed as the path integral: where
By expansion in the Dyson series, one finds that
is the sum of all (possibly disconnected), closed Feynman diagrams.
in turn is the generating functional of the N-particle Green's function: The linked-cluster theorem asserts that the effective action
is the sum of all closed, connected, bare diagrams.
As an example, the two particle connected Green's function reads: To pass to the two-particle irreducible (2PI) effective action, one performs a Legendre transform of
One chooses an, at this point arbitrary, convex
as the source and obtains the 2PI functional, also known as Baym–Kadanoff functional: Unlike the connected case, one more step is required to obtain a generating functional from the two-particle irreducible effective action
By subtracting it, one obtains the Luttinger–Ward functional:[5] where
Along the lines of the proof of the linked-cluster theorem, one can show that this is the generating functional for the two-particle irreducible propagators.
Diagrammatically, the Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible Feynman diagrams (also known as “skeleton” diagrams): The diagrams are closed as they do not have any external legs, i.e., no particles going in or out of the diagram.
They are two-particle irreducible since they do not become disconnected if we sever up to two fermionic lines.
The Luttinger–Ward functional is related to the grand potential
is a generating functional for irreducible vertex quantities: the first functional derivative with respect to
gives the self-energy, while the second derivative gives the partially two-particle irreducible four-point vertex: While the Luttinger–Ward functional exists, it can be shown to be not unique for Hubbard-like models.
[6] In particular, the irreducible vertex functions show a set of divergencies, which causes the self-energy to bifurcate into a physical and an unphysical solution.
[7] Baym and Kadanoff showed that we can satisfy the conservation law for any functional
responding to one-body external fields apparently satisfies the space- and time- translational symmetries as well as the abelian gauge symmetry (phase symmetry), as long as the equation of motion is given with the derivative of
Based on the diagramatic analysis, what Baym found is that
is needed to satisfy the conservation law.
This is nothing but the completely-integrable condition, implying the existence of