[Paper Review] A holographic view of topological stabilizer codes

Thomas Schuster, Nathanan Tantivasadakarn+2 more|arXiv (Cornell University)|Dec 7, 2023

Theoretical and Computational Physics4 citations

TL;DR

This paper establishes a general bulk-boundary correspondence for topological stabilizer codes by showing that the anyonic statistics of bulk topological order generate non-local constraints on the boundary Hilbert space, which cannot be realized with local degrees of freedom. Using a polynomial formalism over finite fields, the authors derive universal 'obstructor invariants' that classify these obstructions and generalize the framework to three-dimensional fracton models like the X-Cube and Haah’s code, revealing emergent subsystem symmetries at boundaries.

ABSTRACT

The bulk-boundary correspondence is a hallmark feature of topological phases of matter. Nonetheless, our understanding of the correspondence remains incomplete for phases with intrinsic topological order, and is nearly entirely lacking for more exotic phases, such as fractons. Intriguingly, for the former, recent work suggests that bulk topological order manifests in a non-local structure in the boundary Hilbert space; however, a concrete understanding of how and where this perspective applies remains limited. Here, we provide an explicit and general framework for understanding the bulk-boundary correspondence in Pauli topological stabilizer codes. We show -- for any boundary termination of any two-dimensional topological stabilizer code -- that the boundary Hilbert space cannot be realized via local degrees of freedom, in a manner precisely determined by the anyon data of the bulk topological order. We provide a simple method to compute this "obstruction" using a well-known mapping to polynomials over finite fields. Leveraging this mapping, we generalize our framework to fracton models in three-dimensions, including both the X-Cube model and Haah's code. An important consequence of our results is that the boundaries of topological phases can exhibit emergent symmetries that are impossible to otherwise achieve without an unrealistic degree of fine tuning. For instance, we show how linear and fractal subsystem symmetries naturally arise at the boundaries of fracton phases.

Motivation & Objective

• To establish a general bulk-boundary correspondence for topological stabilizer codes beyond symmetry-protected topological phases.
• To identify and compute universal invariants—called 'obstructors'—that obstruct the realization of boundary Hilbert spaces using local degrees of freedom.
• To extend the framework to three-dimensional fracton models, including the X-Cube and Haah’s code, where bulk topological order induces emergent subsystem symmetries at boundaries.
• To provide a systematic method for computing these obstructions using polynomial mappings over finite fields, applicable to both Abelian and non-Abelian anyon theories.
• To demonstrate that the boundary theory's non-local structure is directly inherited from the braiding and mobility properties of bulk excitations.

Proposed method

• The authors introduce 'obstructor invariants'—universal invariants of the boundary operator algebra that encode the obstruction to realizing the boundary Hilbert space via local degrees of freedom.
• They employ a polynomial formalism mapping stabilizer codes to polynomials over finite fields, enabling systematic computation of bulk conservation laws and boundary constraints.
• The framework distinguishes between self-obstructor and mutual-obstructor invariants, which correspond to the self-statistics and mutual statistics of anyons, respectively.
• For 2D stabilizer codes, the method computes obstructor invariants from the anyon data of the bulk, proving that non-trivial anyonic statistics imply non-local boundary structure.
• The approach is generalized to 3D fracton models by analyzing patch operators and their commutation relations, revealing fractal and subsystem symmetries.
• The method is validated through explicit calculations on the toric code, X-Cube model, and Haah’s code, with results confirmed via algebraic topology and finite field arithmetic.

Experimental results

Research questions

• RQ1 Can the bulk-boundary correspondence in topological stabilizer codes be formulated universally, independent of boundary termination?
• RQ2 What algebraic invariants in the boundary operator algebra encode the obstruction to local Hilbert space realization due to bulk topological order?
• RQ3 How can the polynomial formalism over finite fields be used to systematically compute these obstructor invariants in both 2D and 3D stabilizer codes?
• RQ4 Do fracton phases exhibit a bulk-boundary correspondence, and if so, how do their immobile excitations and restricted mobility affect boundary symmetries?
• RQ5 Can emergent subsystem symmetries at boundaries be derived directly from bulk anyon statistics and mobility?

Key findings

• The boundary Hilbert space of any 2D topological stabilizer code cannot be realized with local degrees of freedom due to obstructions directly inherited from bulk anyon statistics.
• The self-obstructor invariant is equal to the self-statistics of the anyons in the bulk, as demonstrated in the Fibonacci prism model where it matches the anyon's self-statistics.
• The mutual-obstructor invariant distinguishes the boundary of the X-Cube model from that of a stack of toric codes, reflecting different anyonic braiding statistics.
• For the X-Cube model, the boundary exhibits emergent linear and fractal subsystem symmetries, which are directly tied to the mobility and exchange statistics of bulk fractons.
• In Haah’s code, the framework identifies a non-trivial self-obstructor invariant that corresponds to the self-statistics of the fracton excitations, confirming the non-local nature of the boundary.
• The polynomial formalism enables a unified treatment of both 2D and 3D stabilizer codes, with obstructor invariants computable via algebraic operations on polynomials over F2.