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Haag Duality in the Thermodynamic Limit of the Toric Code

Quintin James Trayling

Published Jan 1, 2023• 0 citations
Publisher

Abstract

In 1997, Kitaev introduced the toric code as an error-tolerant model for quantum computations, which has become a prominent model in recent years. This thesis is dedicated to providing a detailed exploration of Pieter Naaijkens’ work on the toric code in the thermodynamic limit. The primary focus of this thesis is providing an exposition on local Hamiltonians for the toric code and their ground states, showing that Haag duality holds for cones in the corresponding ground state representation. That is, given a cone Λ, the von Neumann algebras R_Λ and R_Λ_c of observables localized in Λ and Λ_c respectively are each other’s commutant. We also show that given Haag duality, we also have the distal split property, which states that if Λ_1 ⊂ Λ_2 are two cones with sufficiently separated boundaries, then there is a Type I factor M such that R_Λ_1 ⊂ M ⊂ R_Λ_2.

DOI
10.22215/etd/2023-15677

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In 1997, Kitaev introduced the toric code as an error-tolerant model for quantum computations, which has become a prominent model in recent years. This thesis is dedicated to providing a detailed exploration of Pieter Naaijkens’ work on the toric code in the thermodynamic limit. The primary focus of this thesis is providing an exposition on local Hamiltonians for the toric code and their ground states, showing that Haag duality holds for cones in the corresponding ground state representation. That is, given a cone Λ, the von Neumann algebras R_Λ and R_Λ_c of observables localized in Λ and Λ_c respectively are each other’s commutant. We also show that given Haag duality, we also have the distal split property, which states that if Λ_1 ⊂ Λ_2 are two cones with sufficiently separated boundaries, then there is a Type I factor M such that R_Λ_1 ⊂ M ⊂ R_Λ_2.
We prove that Haag duality holds for cones in the toric code model. That is, for a cone Λ, the algebra RΛ{\mathcal{R}_{\Lambda}}RΛ​ of observables localized in Λ and the algebra RΛc{\mathcal{R}_{\Lambda^c}}RΛc​ of observables localized in the complement Λ c generate each other’s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if Λ1⊂Λ2{\Lambda_1 \subset \Lambda_2}Λ1​⊂Λ2​ are two cones whose boundaries are well separated, there is a Type I factor N{\mathcal{N}}N such that RΛ1⊂N⊂RΛ2{\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}RΛ1​​⊂N⊂RΛ2​​ . We demonstrate this by explicitly constructing N{\mathcal{N}}N .
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.
Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation. One of the simplest examples of such a spin system is Kitaev’s toric code. Naaijkens made mathematically rigorous the treatment of toric code on an infinite planar lattice (the thermodynamic limit), using an operator algebraic approach via algebraic quantum field theory. We adapt his methods to study the case of toric code with gapped boundary. In particular, we recover the condensation results described in Kitaev and Kong and show that the boundary theory is a module tensor category over the bulk, as expected.
The uniqueness of purifications of quantum states on a system A up to local unitary transformations on a purifying system B is central to quantum information theory. We show that, if the two systems are modeled by commuting von Neumann algebras M_{A} and M_{B} on a Hilbert space H, then uniqueness of purifications is equivalent to Haag duality M_{A}=M_{B}^{'}. In particular, the uniqueness of purifications can fail in systems with infinitely many degrees of freedom-even when M_{A} and M_{B} are commuting factors that jointly generate B(H) and hence allow for local tomography of all density matrices on H. We present a simple argument showing that the uniqueness of purifications, and, hence, Haag duality, fail in ground state sectors of topologically ordered models at renormalization group fixed points on infinite two-dimensional lattices, partitioned into the union of two spatially separated cones and its complement.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for abstract quantum spin systems which allow one to access topological order via a boundary algebra construction. Using the LTO axioms, we produced a canonical pure state on the quasi-local algebra, which gives a net of von Neumann algebras associated to a poset of cones in Rn\mathbb{R}^nRn. In this article, motivated by [arXiv:2509.23734], we introduce an axiom for LTOs which ensures Haag duality for cone-like regions using Tomita-Takesaki theory. We prove this axiom is satisfied for all known topologically ordered commuting projector models. We thus get an independent proof of Haag duality for the Levin-Wen string net models originally proved in [arXiv:2509.23734]. We also give a reflection positivity axiom for LTOs, connecting to the recent article [arXiv:2510.20662]. We again prove this axiom is satisfied for all known topologically ordered commuting projector models about some Z/2\mathbb{Z}/2Z/2-reflection symmetry.
We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.