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arXiv:2308.08087

Boundary states of a bulk gapped ground state in 222-d quantum spin systems

Yoshiko Ogata

Published Aug 16, 2023• 1 citation
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Abstract

We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of $2$-d quantum spin systems. With approximate Haag duality at the boundary, we derive a $C^*$-tensor category $\tilde{\mathcal{M}}$ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of $\tilde{\mathcal{M}}$ is equivalent to the bulk braided $C^*$-tensor category derived in [14].

DOI
10.48550/arxiv.2308.08087

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We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of 222-d quantum spin systems. With approximate Haag duality at the boundary, we derive a C∗C^*C∗-tensor category M~\tilde{\mathcal{M}}M~ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of M~\tilde{\mathcal{M}}M~ is equivalent to the bulk braided C∗C^*C∗-tensor category derived in [14].
We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category C\mathcal{C}C, an indecomposable semisimple right C\mathcal{C}C-module category M\mathcal{M}M, a Q-system Q∈CQ\in\mathcal{C}Q∈C specifying the bulk phase, and a right QQQ-module K∈MQK\in\mathcal{M}_{Q}K∈MQ​, regarded as an object of MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor MQop→BCond\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}MQop​→BCond is an equivalence, so simple boundary conditions are classified by simple objects of MQ\mathcal{M}_{Q}MQ​ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of Z1(Crev)Z_1(\mathcal{C}^{\mathrm{rev}})Z1​(Crev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev on MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.
Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on Zk\mathbb {Z}^{k}Zk , we define a local net of boundary algebras on Zk−1\mathbb {Z}^{k-1}Zk−1 , which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type II\mathrm {II}II , and we show that Levin-Wen models can have cone algebras of type III\mathrm {III}III . Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states.
This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.
In this work, we give a mathematical description of a chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk rational conformal field theories naturally emerge on certain gapless walls between two trivial phases.