Low-energy effective field theories of fermion liquids and the mixed U(1)×Rd anomaly

In this paper, we study gapless fermionic and bosonic systems in $d$-dimensional continuum space with $U(1)$ particle-number-conservation ${\mathbb{R}}^{d}$ translation symmetry. We present low-energy effective field theories for several phases the $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{d}$ The symmetry has a property that twist will induce nonzero momentum proportional to charge density $\overline{\ensuremath{\rho}}$, which be referred as mixed anomaly. different of same system must have As result, all fields momenta order ${\overline{\ensuremath{\rho}}}^{1/d}$. particular, theory infinite number Fermi liquid. also Fermi-liquid presence real magnetic $\mathbit{k}$-space ``magnetic'' field, well interaction described by Landau parameters. Our correctly captures anomaly, constrains dynamics, such determines volume surface terms anomaly [i.e., density]. This is another formulation Luttinger-Ward-Oshikawa theorem.