In this paper, we discuss relativistic hydrodynamics for a massless Dirac fermion in $(2+1)$ dimensions, which has the parity anomaly -- a global t Hooft anomaly between $mathrm{U}(1)$ and parity symmetries. We investigate how hydrodynamics implement
s the party anomaly, particularly focusing on the transport phenomena at the boundary. Based on the parity anomaly matching and the second law of local thermodynamics, we find $mathrm{U}(1)$ and entropy currents localized at the boundary as well as the bulk anomalous current with vanishing divergence. These edge currents are similar to the $(1+1)$-dimensional chiral transports, but the coefficients are given by half of theirs. We also generalize our discussion to more general anomalies among multiple $mathrm{U}(1)$ symmetries and single $mathbb{Z}_2$ symmetry.
We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin
Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $mathbb{Z}_2$ t Hooft anomaly of this system, which involves the $mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrains the possible phase diagram at any temperatures and quark chemical potentials at the isospin RW point, and we compare it with previous results obtained by chiral effective field theory and lattice simulations. We also point out an interesting similarity of two-flavor massless QC$_2$D with $(2+1)$d quantum anti-ferromagnetic systems.
