2020/05/31 by Naren Manjunath, Maissam Barkeshli · 1 voice · 49 citations
Materials Science · Physics and Astronomy · #Abelian group #Chemical and Physical Properties of Materials #Gauge theory #Hamiltonian lattice gauge theory #Lattice (music) #Lattice gauge theory #Quantum and electron transport phenomena #Symmetry (geometry) #Topological Materials and Phenomena #Topological defect #Topological quantum computer #Topology (electrical circuits) #cond-mat.str-el #hep-th #quant-ph
paper · pdf · open access · doi:10.1103/physrevresearch.3.013040
published in Physical Review Research 3(1) (American Physical Society) · 25 + 10 pages; revised Appendix C and D; Added a new Table V in main text with other minor edits; added some additional references
openalex created_date 2020/05/29 · arxiv created 2020/12/17 · openalex publication_date 2021/01/13 · arxiv updated 2021/01/15 · openalex updated_date 2026/08/06
Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected quantized invariants, such as the Hall conductivity, shift and Hall viscosity. Here we develop a theory of symmetry-protected quantized invariants for topological phases defined on a lattice, where quantized invariants with no continuum analog can arise. We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders with symmetry group G = U(1) × Gspace, where Gspace consists of orientation-preserving space group symmetries on the lattice. We show how discrete rotational and translational symmetry fractionalization can be characterized by a discrete spin vector, a discrete torsion vector which has no analog in the continuum or in the absence of lattice rotation symmetry, and an area vector, which also has no analog in the continuum. The discrete torsion vector implies a type of crystal momentum fractionalization that is only non-trivial for 2, 3, and 4-fold rotation symmetry. The quantized topological response theory includes a discrete version of the shift, which binds fractional charge to disclinations and corners, a fractionally quantized angular momentum of disclinations, rotationally symmetric fractional charge polarization and its angular momentum counterpart, constraints on charge and angular momentum per unit cell, and quantized momentum bound to dislocations and units of area. The fractionally quantized charge polarization, which is non-trivial only on a lattice with 2, 3, and 4-fold rotation symmetry, implies a fractional charge bound to lattice dislocations and a fractional charge per unit length along the boundary. An important role is played by a finite group grading on Burgers vectors, which depends on the point group symmetry of the lattice.