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: $$\bar{N}(\mu) \leftrightarrow \sum_{m} \Theta(\mu- \mu(m)) \leftrightarrow \sum_{\vec{k},\sigma} \Theta\left(\gamma_{\vec{k} \sigma}\right) \leftrightarrow \sum_{\vec{k},\sigma} \Theta\left(G_\sigma(\vec{k},0)\right).$$ 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.