Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States

TL;DR

This work develops a framework in which bulk anyons define emergent edge symmetries that organize the phase structure of boundaries of 2D topological orders. By mapping bulk anyon content to edge symmetry actions and dualities, the authors classify gapped and gapless boundary phases, including symmetry-breaking, SPT, and fusion-category symmetry phenomena. They illustrate the approach with concrete models: toric-code boundaries (both single and bilayer) and domain walls in Kitaev spin liquids, revealing Ising-, Z2 × Z2-, and Ising × Ising-category structures and a spectrum of critical points such as Ising, tricritical Ising, and orbifold CFTs. The results provide a bulk-boundary diagnostic tool and a path to analyze higher-dimensional boundary phenomena via fusion-category symmetries and dualities, with potential implications for diagnosing bulk topological order from boundary data.

Abstract

We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry breaking transitions or transitions between symmetry protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality-breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.

Figures (13)

• Figure 1: A topological order is placed on a cylinder geometry, where the bottom edge (dashed) is fixed to be in some reference gapped boundary condition, and the top edge (solid) is tuned. We explore the phase diagram of the top edge using emergent symmetries which arise from anyon lines (red) encircling the cylinder, $\mathcal{L}_a^X$. Any anyon which is not condensed at the bottom edge defines a global symmetry of the top edge. Meanwhile, anyon tunneling operators between the two edges act as local operators $\mathcal{L}_a^Y(x)$ in this quasi-1d system. Such operators are charged under the emergent symmetries, according to the braiding rules. A tunneling operator for an anyon which is condensed at both edges corresponds to a local operator with a nonzero VEV, and implies spontaneous symmetry breaking of the emergent symmetries. This way, we can understand transitions between gapped phases at the top edge as symmetry-breaking transitions in this quasi-1d system.
• Figure 2: In the study of domain walls in topological order, a natural quasi-1d geometry is given by a thin torus with the domain wall (solid black) running along the top cycle. We can relate this geometry to the cylinder geometry by squashing the cross-section of the torus. If one begins with a chiral theory, the squashed geometry is non-chiral, being a product of the original theory with its anti-chiral partner. The domain wall is placed at the top of the cylinder, while the bottom boundary (dashed) is in the "fold" gapped boundary condition. Then, we may invoke the symmetry methods described in Fig. \ref{['figcylinder']}.
• Figure 3: (a) The Kitaev honeycomb model, with letters indicating the direction of the ferromagnetic couplings on each edge. We gradually turn off some of the bonds, which results in a "decoupling transition". (b) The tricritical Ising model phase diagram. The c=7/10 point has 2 relevant operators: $\epsilon$ is duality odd, and tunes between the 2 gapped phases, while $\epsilon'$ is duality even and tunes between the c=1/2 Ising CFT and a threefold degenerate gapped coexistence line. (c) The phase diagram of the 1 dimensional KSL-KSL boundary is the self dual part of the tricritical Ising model phase diagram. This is so since KW duality is enforced as a symmetry of the quasi 1d system.
• Figure 4: (a) A two-parameter phase diagram of a boundary of a toric code bilayer near a $c = 1$ multicritical point. The multicritical point corresponds to a free fermion theory. The phase diagram of the boundary maps onto that of a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric one-dimensional system, where the correspondence between the gapped phases is summarized in Table \ref{['tab:tcbilayer']}. The four $c = 1/2$ lines in black are all distinguished by the emergent $\mathbb{Z}_2 \times \mathbb{Z}_2$ charges of their order and disorder operators verresen_gapless_2019. Additional phase diagrams can be obtained by applying dualities associated with anyon permuting symmetries in the bulk. (b) A dual phase diagram obtained by applying $S_1 :e_{1} \leftrightarrow m_{1}$. The $c=1$ theory in the middle is a product of two Ising critical theories.
• Figure 5: (a) The three domain walls in the Kitaev spin liquid bilayer, inserted along the top cycle of a torus (compare Fig. \ref{['figtorus']}). The view is from the direction parallel to the domain wall. (1) is the trivial domain wall with GSD 9, (2) is the genon domain wall with GSD 3, and (3) has the two Ising layers glued with a toric code (shown in yellow) with GSD 6. The gluing of each single KSL to a toric code may be understood when the gluing line is regarded as a gapped boundary between a KSL$\times \overline{\text{KSL}}$ and a toric code burnell_anyon_2018. (b) In the toric code gluing domain wall, the state $|\sigma,\sigma\rangle$ is split into $|\sigma_1\sigma_2\rangle$ (left) and $|\sigma_1\sigma_2*\rangle$ (right). The red lines denote $\sigma$ lines wrapping around the tori. The state are distinguished by a $\psi$ line going through the domain wall, ending on the $\sigma$ lines.