Sum rules and Ward identities in the Kondo lattice

Abstract

We derive a generalized Luttinger-Ward expression for the Free energy of a many body system involving a constrained Hilbert space. In the large N limit, we are able to explicitly write the entropy as a functional of the Green's functions. Using this method we obtain a Luttinger sum rule for the Kondo lattice. One of the fascinating aspects of the sum rule, is that it contains two components, one describing the heavy electron Fermi surface, the other, a sea of oppositely charged, spinless fermions. In the heavy electron state, this sea of spinless fermions is completely filled and the electron Fermi surface expands by one electron per unit cell to compensate the positively charged background, forming a "large" Fermi surface. Arbitrarily weak magnetism causes the spinless Fermi sea to annihilate with part of the Fermi sea of the conduction electrons, leading to a small Fermi surface. Our results thus enable us to show that the Fermi surface volume contracts from a large, to a small volume at a quantum critical point. However, the sum rules also permit the possible formation of a new phase, sandwiched between the antiferromagnet and the heavy electron phase, where the charged spinless fermions develop a true Fermi surface.

A comment on holographic Luttinger theorem

“…The proof of [3] is based on a U(1) gauge symmetry, Fermi-liquids description near the Fermisurface and a mild assumption for dynamical degrees of freedom, namely, all the momentum and charge carrying degrees of freedom are quasi-particles near the Fermi-surfaces. See also [4,5] for further developments concerning the proof of the Luttinger theorem.…”

Fermi surfaces and gauge-gravity duality

“…There is a repulsive interaction u > 0 between the bosons necessary to stabilize the theory, and an attractive interaction g between the bosons and fermions. Now it is useful to introduce a fermionic 'molecular' field c σ by a Hubbard-Stratonovich decoupling [26] of the two-body interaction:…”

“…A more recent discussion of the Luttinger theorem appeared in the works of Powell et al [24,25] and Coleman et al, [26] who applied it to arbitrary interacting systems of fermions, bosons, and gauge fields. They pointed out the key role played by continuous symmetries and associated global conservation laws, and we will now review their presentation.…”

“…24 (see Section V) and Ref. 26 (see Section IV) for further details, and proceed to the final result:…”

“…(1.4) involves all fermions, whether canonical or not. [26] • Some of the Fermi surfaces could be of fermions which carry additional charges of fluctuating gauge fields, or are coupled to other gapless scalars associated with a symmetry-breaking transition. In these cases, the singularities near the Fermi surface can differ from those in a Landau Fermi liquid.…”