Nondiagonal anisotropic quantum Hall states

We propose a family of Abelian quantum Hall states termed the nondiagonal states, which arise at filling factors ν = p/2q for bosonic systems and ν = p/(p+2q) for fermionic systems, with p and q being two coprime integers. Nondiagonal quantum Hall states are constructed in a coupled wire model, which shows an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for bulk quasiparticles, featuring a nontrivial interplay between charge symmetry and translation symmetry. The nondiagonal state is established as a distinctive symmetry-enriched topological order. Aside from the usual U(1) charge sector, there is an additional symmetry-enriched neutral sector described by the quantum double model D(Zp) , which relies on the presence of both the U (1) charge symmetry and the Z translation symmetry of the wire model. Translation symmetry distinguishes nondiagonal states from Laughlin states, in a way similar to how it distinguishes weak topological insulators from trivial band insulators. Moreover, the translation symmetry in nondiagonal states can be associated with the e ↔ m anyonic symmetry in D(Zp) , implying the role of dislocations as twofold twist defects. The boundary theory of nondiagonal states is derived microscopically. For the edge perpendicular to the direction of wires, the effective Hamiltonian has two components: a chiral Luttinger liquid and a generalized p -state clock model. Importantly, translation symmetry in the bulk is realized as self-duality on the edge. The symmetric edge is thus either gapless or gapped with spontaneously broken symmetry. For p = 2 , 3 , the respective electron tunneling exponents are predicted for experimental probes.