This dissertation discusses some properties of topologically ordered states as they appear in the setting of infinite quantum spin systems. We begin by studying the set of infinite volume ground states for Kitaev's abelian quantum double models. We show that states describing a single excitation in the bulk are infinite volume ground states, that is, local perturbations cannot remove the charge. The single excitations states, which are inequivalent for distinct charges, give a complete characterization of the sector theory for the set of ground states. Furthermore, any pure ground state is equivalent to some single excitation ground state. We proceed to study the stability of charges that are classified by certain representations of the algebra of observables. We introduce a new superselection criterion selecting almost localized and transportable
∗-endomorphisms with respect to a vacuum state. Equivalence classes of representations satisfying the superselection criterion form different charged superselection sectors of the system. We show that if the vacuum state satisfies certain locality conditions then the superselection structure will be a braided tensor
C∗-category. Further, this superselection structure is stable up to deformations by a quasi-local dynamics. This result is then applied to show that the anyon structure of the abelian quantum double models is stable under local perturbations.
We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if the theory does not display conformal symmetry.
We rigorously define superselection sectors in the 3d (spatial dimensions) Toric Code Model on the infinite lattice
Z3. We begin by constructing automorphisms that correspond to infinite flux strings, a phenomenon that's only possible in open manifolds. We then classify all ground state superselection sectors containing infinite flux strings, and find a rich structure that depends on the geometry and number of strings in the configuration. In particular, for a single infinite flux string configuration to be a ground state, it must be monotonic. For configurations containing multiple infinite flux strings, we define "infinity directions" and use that to establish a necessary and sufficient condition for a state to be in a ground state superselection sector. Notably, we also find that if a state contains more than 3 infinite flux strings, then it is not in a ground state superselection sector.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
Can excitation structure classify topological quantum codes up to finite-depth local circuits? For translation-invariant topological Calderbank--Shor--Steane (CSS) codes in arbitrary dimensions, we introduce excitation superselection profiles
Eσ (
σ=X,Z). Their cohomology layers
Eσℓ:=Hℓ(Eσ) encode excitation types, fusion rules, and translation action, while
Eσ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees
ℓ=1,2,…, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if
Eσ is supported in a single degree
ℓ, then
Eσℓ and
ℓ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If
Eσℓ is also finite, prime-qudit codes decouple into toric-code stacks by Clifford circuits after coarse graining, with the exact minimal threshold set by the translation kernel. We develop a unified topological theory of Koszul-complex models (encompassing toric codes, bivariate-bicycle codes, and Haah's cubic code): for regular models, we compute
Eσℓ, prove
Eσ is single-degree, express finite-torus logical modules as
Tor groups, and obtain coarse-grained toric-stack normal forms at full length. Two constructions extend this framework to realize any admissible single-degree profile and build multi-degree extensions. The latter yields eight 3D qubit toric-code extensions with identical layers in both Pauli sectors and trivial translation action, yet distinct size-dependent ground-state degeneracies. The seven nonsplit models require blocking to decouple. Thus, cross-degree gluing carries translation-symmetry-enriched topological information invisible to individual layers.