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Bulk-boundary correspondence of (1+1)D symmetric gapped phases

  • Yizhou Ma
  • Gen Yue
  • Tian Lan
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Abstract

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 π’ž, an indecomposable semisimple right π’ž-module category β„³, a Q-system Qβˆˆπ’ž specifying the bulk phase, and a right Q-module Kβˆˆβ„³Q, regarded as an object of β„³Qop, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor β„³Qopβ†’BCond is an equivalence, so simple boundary conditions are classified by simple objects of β„³Q 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 (π’žβ„³βˆ¨)rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of Z1(π’žrev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of (π’žβ„³βˆ¨)rev on β„³Qop. 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.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026bulkboundary,
  title = {{Bulk-boundary correspondence of (1+1)D symmetric gapped phases}},
  author = {Yizhou Ma and Gen Yue and Tian Lan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.19137},
  eprint = {2606.19137},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.19137},
}
RIS
TY  - UNPB
TI  - Bulk-boundary correspondence of (1+1)D symmetric gapped phases
AU  - Yizhou Ma
AU  - Gen Yue
AU  - Tian Lan
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2606.19137
UR  - https://arxiv.org/abs/2606.19137
AB  - 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 π’ž, an indecomposable semisimple right π’ž-module category β„³, a Q-system Qβˆˆπ’ž specifying the bulk phase, and a right Q-module Kβˆˆβ„³_Q, regarded as an object of β„³_Q^op, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor β„³_Q^opβ†’BCond is an equivalence, so simple boundary conditions are classified by simple objects of β„³_Q 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 (π’ž_β„³^∨)^rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of Z_1(π’ž^rev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of (π’ž_β„³^∨)^rev on β„³_Q^op. 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.
ER  -