Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of
G-crossed braided tensor categories (
G×BTCs) for SETs. We consider systems with
U(1) charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all
5 orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains
9 terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the
G×BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the
G×BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.
The permutation symmetry of Newtonian stationary-state functions – such as the total momentum and the Hamiltonian – is crucial for correct state counting in classical statistical mechanics, but plays virtually no role for classical many-body systems. In contrast, it yields novel effects for quantum many-body systems of equal/identical particles. In order to avoid the misunderstandings Jaynes (1992) has rightly stressed, basic notions, such as equal and identical particles, (in)distinguishability and exchangeability (the really relevant notion) are discussed first within classical mechanics. Then, they are applied to the weight (limiting) functions. The permutation symmetry of the wave functions follow largely from that of the weight functions. The different effects for bosons and fermions are mentioned and illustrated. For the generalization towards anyons and the fractional quantum Hall effect, some ideas for further exploration are proposed. Entanglement is treated as a consequence of the conservation laws, in particular, for the angular momentum.
We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.