Compactification of quasi-local algebras on the lattice

We introduce a compactification construction for abstract quasi-local C*-algebras over countable metric spaces equipped with an isometric group action which is functorial with respect to bounded spread isomorphisms. In $1$D, the construction recovers Ocneanu’s Tube algebra for fusion spin chains, and provides a canonical bridge between infinite-volume observables and observables with periodic boundary conditions. We exploit this connection to derive an obstruction for the implementability of such topological symmetries as Kramers-Wannier type dualities on symmetric subalgebras.

💡 Research Summary

This paper introduces a novel, functorial framework for the “compactification” of abstract quasi-local C*-algebras defined over countable metric spaces equipped with an isometric group action. The construction is natural with respect to bounded-spread isomorphisms, which are the physically relevant locality-preserving maps such as quantum cellular automata (QCA).

The core motivation stems from two fundamental questions in the interplay between algebraic quantum field theory, topological order, and quantum lattice models: (1) How can one systematically relate the infinite-volume algebra of observables (with open boundary conditions) to algebras under periodic boundary conditions? (2) Which symmetries of the emergent bulk topological quantum field theory (TQFT) can be implemented by a QCA on the microscopic lattice model?

The authors begin by rigorously defining the category of quasi-local *-algebras over a space L, with morphisms being bounded-spread homomorphisms. Key physical examples include “fusion spin chains,” which are quasi-local algebras constructed from a unitary fusion category C and a generating object X, generalizing symmetric subalgebras of spin chains.

The main technical achievement is the definition of the compactified algebra. Under two key assumptions on the quasi-local algebra—“local generation” and “local presentation”—the authors construct a functor that “folds” the infinite algebra onto a fundamental domain with respect to a group action G. This process involves first defining a scale-dependent compactified algebra Comp_k(A) and then taking a categorical (inductive) limit over these scales to obtain the final, scale-independent compactified algebra Comp(A). Crucially, this construction is functorial: a bounded-spread isomorphism α: A → B induces an isomorphism Comp(α): Comp(A) → Comp(B).

The first major result (Theorem 3.20) establishes the “faithfulness” of this compactification. It shows that the original quasi-local algebra can be partially recovered from the infinite sequence of its compactified versions. In the specific case of 1D fusion spin chains, Corollary 4.5 proves that this recovery is complete; the isomorphism class of the original fusion spin chain algebra is fully captured by the sequence of its compactifications, which are shown to be precisely Ocneanu’s Tube algebras for increasing circumferences. This provides a canonical and rigorous bridge between the infinite-volume and periodic-boundary-condition viewpoints.

The second major result leverages this bridge to derive a powerful obstruction theorem for implementing topological symmetries as QCAs. When specializing to a fusion spin chain A(C,X) and a G-equivariant bounded-spread automorphism α, two induced actions are considered: 1) DHR(α), a braided autoequivalence of the Drinfeld center Z(C) (the bulk TQFT), and 2) Tube_k(α), an automorphism of the Tube algebra (the compactification). By analyzing the relationship between these, the authors prove a stabilization property (Corollary 4.9). It states that for a bounded-spread isomorphism α to be implementable on the symmetric subalgebra, a specific object I(X⊗k) in Z(C) must be fixed up to isomorphism by DHR(α) for all sufficiently large k. This condition is highly restrictive.

The practical power of this obstruction is demonstrated through concrete physical examples:

• Example 4.10 (Z/2Z gauge theory): For a lattice model built from a symmetry under a matrix H = diag(I_a, -I_b), the famous e-m swap symmetry (a Kramers-Wannier-type duality) in the emergent Z/2Z topological order is proven not to be inducible by any QCA on the model’s symmetric subalgebra.
• Example 4.11 (Non-abelian D4 theory): The obstruction criterion is applied to show the non-implementability of a specific non-trivial braided autoequivalence in a model derived from the D4 fusion category.

In conclusion, this work provides a robust mathematical framework for compactifying quasi-local algebras, firmly connecting infinite and finite perspectives. More importantly, it uses this framework to uncover fundamental constraints on realizing abstract topological symmetries as local quantum circuits on lattice models, thereby deepening our understanding of the relationship between microscopic physics and emergent topological phenomena.