Sum Rules and Ward Identities in the Kondo Lattice

TL;DR

This paper derives a generalized sum rule for the Kondo lattice, revealing how the Fermi surface volume changes at quantum critical points and suggesting the possibility of a new intermediate phase with a distinct Fermi surface.

Contribution

It introduces a generalized Luttinger sum rule for the Kondo lattice that accounts for spinless fermions and their impact on Fermi surface volume changes.

Findings

The sum rule includes two components: heavy electron Fermi surface and spinless fermion sea.

The Fermi surface contracts from large to small at a quantum critical point.

A potential new phase with a true Fermi surface of spinless fermions is proposed.

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 explicity 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.