Abstract
The authors develop a topological field theory of crystalline gauge fields, which they use to classify the fractional charge, angular momentum, and linear momentum of anyons and lattice defects in (2+1)D topologically ordered phases of matter.
Highlights
One of the most striking discoveries in physics is the quantized Hall conductivity of integer and fractional quantum Hall (FQH) systems [1,2]
We find that in general symmetry fractionalization for G = U (1) × Gspace is determined by four invariants, which are specified by a charge vector q, a discrete spin vector s, a discrete torsion vector, and an area vector m
The discrete torsion vector t has no analog in the continuum and can only be nontrivial for M = 2, 3, 4-fold lattice rotational symmetry; it specifies a fractional linear momentum for the anyons that does not appear to have been discussed in previous studies of topological phases of matter
Summary
One of the most striking discoveries in physics is the quantized Hall conductivity of integer and fractional quantum Hall (FQH) systems [1,2]. Clean isotropic quantum Hall systems possess additional symmetry-protected invariants, such as a quantized Hall viscosity [4,5,6,7,8,9], the shift, and fractional orbital spin of quasiparticles [10]. The crystalline gauge fields include gauge fields associated with the discrete translation and rotation symmetries, which keep track of certain geometric properties of the lattice, such as the presence of dislocations and disclinations, and areas and lengths of closed cycles in lattice units As such, they form a discrete analog of the coframe field and spin connection used in continuum geometry. We note that recently crystalline gauge fields have been used in the study of quantum phases of matter, see, e.g., Refs. [17,18], effective actions involving both translation and rotation gauge fields have not to our knowledge been discussed previously
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