Ground-state degeneracy for abelian anyons in the presence of gapped boundaries

TL;DR

The paper presents a general method to compute the ground-state degeneracy of abelian fractional quantum Hall states in the presence of gapped boundaries, including configurations with boundary segments and domain walls. By reducing 3d abelian Chern-Simons theory to a 2d TQFT with discrete gauge group $G={\mathsf D}^*=\mathrm{Hom}({\mathsf D},U(1))$ and encoding boundary data with Lagrangian subgroups ${\mathsf L}\subset{\mathsf D}$ (via subgroups $H\subset G$) and domain walls with elements of $G/(H_i+H_{i+1})$, the authors derive an explicit disk correlator formula $Z(x;g)$ that yields the GSD for arbitrary boundary segmentation. The key result is a concrete, normalization-adjusted partition function on disks with multiple boundary segments and domain walls, together with several illustrative examples showing how geometry and boundary data control degeneracy. This framework clarifies how ground-state degeneracy can arise even in planar geometries and provides a practical tool for analyzing Abelian FQH systems, with potential extensions to nonabelian phases via 2d reductions of lattice models. Overall, the work links bulk anyon data, boundary condensates, and 2d TQFT techniques to deliver a versatile GSD computation method for abelian topological orders.

Abstract

Gapped phases with long-range entanglement may admit gapped boundaries. If the boundary is gapped, the ground-state degeneracy is well-defined and can be computed using methods of Topological Quantum Field Theory. We derive a general formula for the ground-state degeneracy for abelian Fractional Quantum Hall phases, including the cases when connected components of the boundary are subdivided into an arbitrary number of segments, with a different boundary condition on each segment, and in the presence of an arbitrary number of boundary domain walls.

Figures (3)

• Figure 1: (a) A cellular decomposition of a disk, with an insertion of a local operator $e_g$ in the interior of the 2-cell. (b) A cellular decomposition of a disk, with an insertion of a local operator $e_g$ in the interior of the 2-cell and two boundary-changing operators.
• Figure 2: In the lattice model, a boundary-changing operator $\psi_x$ corresponds to a special 1-cell separating two segments of the boundary. One sums over all values of the variable $g\in G$ living on this 1-cell such that $g$ corresponds to a fixed $x\in G/(H_1+H_2)$.
• Figure 3: A nontrivial degeneracy on a triangle can be obtained by starting with a square with an alternating pattern of boundary conditions and folding along the diagonal.