Realizing the 2+1D Parity anomaly on a Lattice

Given a quantum field theory, realising its global symmetries and anomalies on a lattice has been a fruitful approach to gain new insights of these symmetries. In this talk, we present an exact lattice model in 2+1D which hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. We first show that our lattice model has a Lieb-Schultz-Mattis (LSM) anomaly of the “Onsager symmetries” in the UV, which indeed enforces that every Hamiltonian which is symmetric has to be gapless. We then show that the SU(2) Parity anomaly on the IR can be exactly matched by this LSM anomaly. Finally, we briefly discuss our results in relation to similar anomaly matching schemes in 1+1D and 3+1D.