Fermi Surface Volume of Interacting Systems

Three Fermion sumrules for interacting systems are derived at T=0, involving the number expectation $bar{N}(mu)$, canonical chemical potentials $mu(m)$, a logarithmic time derivative of the Greens function $gamma_{vec{k} sigma}$ and the static Greens function. In essence we establish at zero temperature the sumrules linking: $$ bar{N}(mu) leftrightarrow sum_{m} Theta(mu- mu(m)) leftrightarrow sum_{vec{k},sigma} Thetaleft(gamma_{vec{k} sigma}right) leftrightarrow sum_{vec{k},sigma} Thetaleft(G_sigma(vec{k},0)right). $$ Connecting them across leads to the Luttinger and Ward sumrule, originally proved perturbatively for Fermi liquids. Our sumrules are nonperturbative in character and valid in a considerably broader setting that additionally includes non-canonical Fermions and Tomonaga-Luttinger models. Generalizations are given for singlet-paired superconductors, where one of the sumrules requires a testable assumption of particle-hole symmetry at all couplings. The sumrules are found by requiring a continuous evolution from the Fermi gas, and by assuming a monotonic increase of $mu(m)$ with particle number m. At finite T a pseudo-Fermi surface, accessible to angle resolved photoemission, is defined using the zero crossings of the first frequency moment of a weighted spectral function.