Crystalline Symmetry-Enriched Quantum Criticality

Gapped phases of quantum matter with crystalline symmetries – namely rotations and translations of the lattice – can be characterized by topological invariants which can only be defined in the presence of these symmetries. At critical points where the gap closes, these topological invariants are no longer well-defined. However, we show that the emergent Dirac fermion field theory at each critical point is determined by the values of the crystalline-symmetry protected topological invariants on either side of the transition. We also discuss how symmetries in the UV (free fermion lattice model) map onto symmetries in the IR (Dirac fermion field theory). More formally, we have a homomorphism from the symmetry group of the lattice model $G_{UV}$ to the symmetry group of the emergent Dirac fermion field theory $G_{IR}$. This homomorphism can be thought of as a topological invariant of the critical point. Finally, we discuss the observable signatures of the field theory and homomorphism. From the perspective of the field theory, we expect defects of the crystalline symmetries – lattice disclinations and dislocations – to contribute excess magnetic flux. We confirm this prediction numerically by measuring the microscopic current at critical points of several lattice models of topological insulators with defects and comparing the results to predictions from defect conformal field theory.