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arXiv:2102.07707 · Communications in Mathematical Physics

The Split and Approximate Split Property in 2D Systems: Stability and Absence of Superselection Sectors

Pieter Naaijkens, Yoshiko Ogata

Published Mar 28, 2022• 10 citations
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Abstract

Abstract The split property of a pure state for a certain cut of a quantum spin system can be understood as the entanglement between the two subsystems being weak. From this point of view, we may say that if it is not possible to transform a state $$\omega $$ ω via sufficiently local automorphisms (in a sense that we will make precise) into a state satisfying the split property, then the state $$\omega $$ ω has a long-range entanglement. It is well known that in 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev’s toric code. Here we show that this failure is the reason that anyons can exist in that model. There is a folklore saying that the existence of anyons, like in the toric code model, implies long-range entanglement of the state. In this paper, we prove this folklore in an infinite dimensional setting. More precisely, we show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors. Anyons in particular give rise to such non-trivial sectors. States with the split property for cones, on the other hand, do not admit non-trivial sectors. A key technical ingredient of our proof is that under suitable assumptions on locality, the automorphisms generated by local interactions can be “approximately factorized.” That is, they can be written as the tensor product of automorphisms localized in a cone and its complement respectively, followed by an automorphism acting near the “boundary” of $$\Lambda $$ Λ , and conjugation with a unitary. This result may be of independent interest. This technique also allows us to prove that theapproximatesplit property, a weaker version of the split property thatissatisfied in e.g. the toric code, is stable under applying such automorphisms.

DOI
10.1007/s00220-022-04356-3

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Abstract The split property of a pure state for a certain cut of a quantum spin system can be understood as the entanglement between the two subsystems being weak. From this point of view, we may say that if it is not possible to transform a state ω\omega ω ω via sufficiently local automorphisms (in a sense that we will make precise) into a state satisfying the split property, then the state ω\omega ω ω has a long-range entanglement. It is well known that in 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev’s toric code. Here we show that this failure is the reason that anyons can exist in that model. There is a folklore saying that the existence of anyons, like in the toric code model, implies long-range entanglement of the state. In this paper, we prove this folklore in an infinite dimensional setting. More precisely, we show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors. Anyons in particular give rise to such non-trivial sectors. States with the split property for cones, on the other hand, do not admit non-trivial sectors. A key technical ingredient of our proof is that under suitable assumptions on locality, the automorphisms generated by local interactions can be “approximately factorized.” That is, they can be written as the tensor product of automorphisms localized in a cone and its complement respectively, followed by an automorphism acting near the “boundary” of Λ\Lambda Λ Λ , and conjugation with a unitary. This result may be of independent interest. This technique also allows us to prove that theapproximatesplit property, a weaker version of the split property thatissatisfied in e.g. the toric code, is stable under applying such automorphisms.
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∗C^*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 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 rigorously define superselection sectors in the 3d (spatial dimensions) Toric Code Model on the infinite lattice Z3\mathbb{Z}^3Z3. 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 consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.