Accessing topological order in fractionalized liquids with gapped edges

Abstract

We consider manifestations of topological order in time-reversal-symmetric fractional topological liquids (TRS-FTLs), defined on planar surfaces with holes. We derive a formula for the topological ground state degeneracy of such a TRS-FTL, which applies to cases where the edge modes on each boundary are fully gapped by appropriate backscattering terms. The degeneracy is exact in the limit of infinite system size, and is given by $q^{N_{\mathrm{h}}^{}}$, where $N_{\mathrm{h}}^{}$ is the number of holes and $q$ is an integer that is determined by the topological field theory. When the degeneracy is lifted by finite-size effects, the holes realize a system of $N_{\mathrm{h}}^{}$ coupled spin-like $q$-state degrees of freedom. In particular, we provide examples where ${\mathbb{Z}}_q^{}$ quantum clock models are realized on the low-energy manifold of states. We also investigate the possibility of measuring the topological ground state degeneracy with calorimetry, and briefly revisit the notion of topological order in $s$-wave BCS superconductors.

arXiv comment16 pages, 5 figures, 1 appendix; expanded and refined discussion of gluing conditions and their connection to different ways of gapping the edge; published version

References in corpus (34)

• Quantum Spin Hall Effect in Graphene
• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Effect
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure
• Superconductors are topologically ordered
• Parafermionic edge zero modes in Z_n-invariant spin chains
• Fractional topological insulators
• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
• Protected edge modes without symmetry
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Classification and analysis of two dimensional abelian fractional topological insulators
• Fractional topological liquids with time-reversal symmetry and their lattice realization
• Theory of defects in Abelian topological states
• Stability of Chiral Luttinger Liquids and Abelian Quantum Hall States.
• Classification of Topological Defects in Abelian Topological States
• Superconducting analogue of the parafermion fractional quantum Hall states
• Thermopower as a Possible Probe of Non-Abelian Quasiparticle Statistics in Fractional Quantum Hall Liquids
• Time-reversal symmetric hierarchy of fractional incompressible liquids
• Topological phases in gapped edges of fractionalized systems
• Membrane-based nanocalorimeter for high-resolution measurements of low-temperature specific heat
• Observable Bulk Signatures of Non-Abelian Quantum Hall States
• Thermoelectric response of fractional quantized Hall and re-entrant insulating states in the N=1 Landau level
• Weak symmetry breaking in two dimensional topological insulators
• Thermopower of Quantum Hall States in Corbino Geometry as a Measure of Quasiparticle Entropy
• Fractional spin liquid hierarchy for spin S antiferromagnets
• Thermopower of Two-Dimensional Electrons at $\nu$ = 3/2 and 5/2
• Multiple Chern-Simons Fields on a Torus
• Thermal signatures of Little-Parks effect in the heat capacity of mesoscopic superconducting rings
• Topological Order in the (2+1)D Compact Lattice Superconductor
• Topological order and the deconfinement transition in the (2+1) dimensional compact Abelian Higgs model
• Absolute specific heat measurements of a microgram Pb crystal using ac nanocalorimetry

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• Non-Abelian topological spin liquids from arrays of quantum wires or spin chains
• Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Boundary Hamiltonian theory for gapped topological orders
• Twofold twist defect chains at criticality