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