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Fermi Surface Volume of Interacting Systems

Three Fermion sumrules for interacting systems are derived at T=0, involving the number expectation $\bar{N}(μ)$, canonical chemical potentials $μ(m)$, a logarithmic time derivative of the Greens function $γ_{\vec{k} σ}$ and the static Greens function. In essence we establish at zero temperature the sumrules linking: $$ \bar{N}(μ) \leftrightarrow \sum_{m} Θ(μ- μ(m)) \leftrightarrow \sum_{\vec{k},σ} Θ\left(γ_{\vec{k} σ}\right) \leftrightarrow \sum_{\vec{k},σ} Θ\left(G_σ(\vec{k},0)\right). $$ 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 $μ(m)$ 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.