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Boundary states of a bulk gapped ground state in $2$-d quantum spin systems

Yoshiko Ogata

TL;DR

This work provides an operator-algebraic formulation of boundary states for bulk gapped ground states in two-dimensional quantum spin systems. By defining a boundary state and, under approximate Haag duality at the boundary, constructing a boundary $C^*$-tensor category $\tilde{\mathcal{M}}$ and its Drinfeld center ${\mathcal Z}_a(\tilde{\mathcal{M}})$ with an asymptotic constraint, the authors prove a bulk–center correspondence: ${\mathcal Z}_a(\tilde{\mathcal{M}})$ is braided equivalent to the bulk braided $C^*$-tensor category (under a nontrivial bulk braiding condition). They develop bulk-to-boundary and boundary-to-bulk functors, and show a bulk copy of the bulk theory resides at the boundary, culminating in a full bulk-boundary correspondence via idempotent completion and a Drinfeld center construction. A key technical feature is the asymptotic half-braiding, which, together with a nondegeneracy (raichi) assumption, ensures the braided center recovers the bulk category. The framework is exemplified by the toric code and connects with Kitaev–Kong type conjectures, providing a robust operator-algebraic route to boundary theories in topologically ordered systems.

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].

Boundary states of a bulk gapped ground state in $2$-d quantum spin systems

TL;DR

This work provides an operator-algebraic formulation of boundary states for bulk gapped ground states in two-dimensional quantum spin systems. By defining a boundary state and, under approximate Haag duality at the boundary, constructing a boundary -tensor category and its Drinfeld center with an asymptotic constraint, the authors prove a bulk–center correspondence: is braided equivalent to the bulk braided -tensor category (under a nontrivial bulk braiding condition). They develop bulk-to-boundary and boundary-to-bulk functors, and show a bulk copy of the bulk theory resides at the boundary, culminating in a full bulk-boundary correspondence via idempotent completion and a Drinfeld center construction. A key technical feature is the asymptotic half-braiding, which, together with a nondegeneracy (raichi) assumption, ensures the braided center recovers the bulk category. The framework is exemplified by the toric code and connects with Kitaev–Kong type conjectures, providing a robust operator-algebraic route to boundary theories in topologically ordered systems.

Abstract

We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of -d quantum spin systems. With approximate Haag duality at the boundary, we derive a -tensor category 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 is equivalent to the bulk braided -tensor category derived in [14].
Paper Structure (14 sections, 58 theorems, 207 equations, 10 figures)

This paper contains 14 sections, 58 theorems, 207 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Assumptions
  4. Results
  5. Comparisons with former works
  6. Boundary theories
  7. Braiding on $O_{\mathop{\mathrm{bd}},\Lambda_0}^{\mathop{\mathrm{bu}}}$
  8. A bulk-to-boundary map
  9. A boundary to bulk map
  10. Bulk-boundary correspondence
  11. Assumption \ref{['assump7']}
  12. Basic Lemmas
  13. Conditional expectations and approximations
  14. $O_{\mathop{\mathrm{bd}}}^{\mathop{\mathrm{bu}}}$ in $O^{rU}_{\mathop{\mathrm{bd}}}$

Key Result

Lemma 1.5

Consider the setting in subsection setting2 and assume Assumption assum3. Then for any $\Lambda\in {\mathcal{C}}_{{\mathop{\mathrm{bk}}}}^U$, we have $\pi_{\mathop{\mathrm{bk}}}\left ({\mathcal{A}}_{\Lambda^c}\right )'\subset{\mathcal{G}}$.

Figures (10)

  • Figure 1:
  • Figure 5: $\pi_{\gamma'_1}^Z=\mathop{\mathrm{Ad}}\nolimits(V)\pi_{\gamma'}^Z$
  • Figure 6: $\pi_{\gamma}^X=\mathop{\mathrm{Ad}}\nolimits(V)\pi_{\mathop{\mathrm{bd}}}$
  • Figure 7:
  • Figure 8:
  • ...and 5 more figures

Theorems & Definitions (130)

  • Definition 1.1
  • Example 1.2
  • Lemma 1.5
  • Remark 1.8
  • Lemma 1.11
  • proof
  • Lemma 1.12
  • proof
  • Lemma 1.13
  • Example 1.15
  • ...and 120 more