Symmetries in Quantum Field Theory may have 't Hooft anomalies. If the symmetry is unbroken in the vacuum, the anomaly implies a nontrivial low-energy limit, such as gapless modes or a topological field theory. If the symmetry is spontaneously broken, for the continuous case, the anomaly implies low-energy theorems about certain couplings of the Goldstone modes. Here we study the case of spontaneously broken discrete symmetries, such as Z2 and T . Symmetry breaking leads to domain walls, and the physics of the domain walls is constrained by the anomaly. We investigate how the physics of the domain walls leads to a matching of the original discrete anomaly. We analyze the symmetry structure on the domain wall, which requires a careful analysis of some properties of the unbreakable CP T symmetry. We demonstrate the general results on some examples and we explain in detail the mod 4 periodic structure that arises in the Z2 and T case. This gives a physical interpretation for the Smith isomorphism, which we also extend to more general abelian groups. We show that via symmetry breaking and the analysis of the physics on the wall, the computations of certain discrete anomalies are greatly simplified. Using these results we perform new consistency checks on the infrared phases of 2 + 1 dimensional QCD. * which by construction leaves the boundary conditions invariant and therefore indeed acts on the wall! But observe that this is an anti-unitary symmetry, since CP ⊥ T is anti-unitary. Thus, the correct answer to the question about the fate of the spontaneously broken, unitary, Z 2 symmetry is that it becomes some anti-unitary symmetry acting on the wall.By the same argument, if we began with a spontaneously broken anti-unitary Z 2 symmetry, the domain wall would inherit a unitary Z 2 symmetry. In fact, we will derive a 4-periodic dimensional

Abstract:We analyze the mechanism of spontaneous symmetry breaking of scale invariance in Galilean invariant field theories. We show that the existence of a dynamic gapless dilaton mode depends on whether the U(1) particle number or the Galilean boost symmetry are spontaneously broken. When both scale and particle number symmetries are spontaneously broken there is one propagating gapless Nambu-Goldstone mode. Its dispersion relation is linear if the chemical potential is nonzero and quadratic otherwise. We discuss the reversibility of RG flows in such theories.

We study the entanglement entropy of the vacuum in non-relativistic local theories with Galilean or Schrödinger symmetry. We clear some confusion in the literature on the free Schrödinger case. We find that with only positive U (1) charge particles (states) and a unique zero U (1) charge state (the vacuum) the entanglement entropy must vanish in that state. *

We analyze the effect of a gravitational field on the sound modes of superfluids. We derive an instability condition that generalizes the well-known Jeans instability of the sound mode in normal fluids. We discuss potential experimental implications.

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