Non-Perturbative Resolution of Luttinger's Theorem Violations in Strongly Correlated Systems

Abstract

Many-body physics faces an apparent paradox concerning violations of Luttinger's theorem in strongly correlated systems. The theorem relates the electron number to the Luttinger count -- the monodromy of the logarithm of the imaginary-frequency Green's function $G(iε)$ -- and, at finite magnetic field, connects the Hall conductivity to the magnetic-field derivative of the Luttinger count via the Streda formula. Here, we provide a non-perturbative solution to this paradox. We demonstrate that the reported violations of Luttinger's theorem do not require a breakdown of the skeleton diagram representation of physical quantities, thereby resolving a long-standing point of contention in many-body theory. Furthermore, we show that the Luttinger integral, which accounts for the mismatch between the particle number and the Luttinger count, is itself a genuine topological invariant directly tied to the Luttinger-Ward functional $Φ[G]$ rather than to the Green's function alone. We demonstrate that this invariant, like the monodromy of $\ln G(iε)$ itself, stems exclusively from the zero-frequency discontinuity of the Matsubara trajectory.