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<p>Symmetry topological field theory (SymTFT), or topological holography, posits a correspondence between symmetries in a <i>d</i>-dimensional theory and topological order in a (<i>d</i> + 1)-dimensional theory. In this work, we extend this framework to subsystem symmetries and develop subsystem SymTFT as a systematic tool to characterize and classify subsystem symmetry-protected topological (SSPT) phases. For (2 + 1)D gapped phases, we introduce a 2-foliated (3 + 1)D exotic tensor gauge theory (which is equivalent to 2-foliated (3 + 1)D BF theory via exotic duality) as the subsystem SymTFT and systematically analyze its topological boundary conditions and linearly rigid subsystem symmetries. Taking subsystem symmetry groups <i>G</i> = <i>ℤ</i><sub><i>N</i></sub> and <i>G</i> = <i>ℤ</i><sub><i>N</i></sub> × <i>ℤ</i><sub><i>M</i></sub> as examples, we demonstrate how to recover the classification scheme <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_27250_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">C</mi> <mfenced close="]" open="["> <mi>G</mi> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{C}\left[G\right] \)</EquationSource> </InlineEquation> = <i>H</i><sup>2</sup>(<i>G</i><sup>×2</sup>, U(1))/ <i>H</i><sup>2</sup>(<i>G</i>, U(1)))<sup>3</sup>, which was previously derived by examining topological invariant under linear subsystem-symmetric local unitary transformations in the lattice Hamiltonian formalism. To illustrate the correspondence between field-theoretic and lattice descriptions, we further analyze <i>ℤ</i><sub>2</sub> × <i>ℤ</i><sub>2</sub> and <i>ℤ</i><sub><i>N</i></sub> × <i>ℤ</i><sub><i>M</i></sub> cluster state models as concrete examples.</p>

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Subsystem symmetry-protected topological phases from subsystem SymTFT of 2-foliated exotic tensor gauge theory

  • 期刊论文

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  • 发表日期：

    2025年9月19日

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  • Qiang Jia,
  • Zhian Jia

摘要

Symmetry topological field theory (SymTFT), or topological holography, posits a correspondence between symmetries in a d-dimensional theory and topological order in a (d + 1)-dimensional theory. In this work, we extend this framework to subsystem symmetries and develop subsystem SymTFT as a systematic tool to characterize and classify subsystem symmetry-protected topological (SSPT) phases. For (2 + 1)D gapped phases, we introduce a 2-foliated (3 + 1)D exotic tensor gauge theory (which is equivalent to 2-foliated (3 + 1)D BF theory via exotic duality) as the subsystem SymTFT and systematically analyze its topological boundary conditions and linearly rigid subsystem symmetries. Taking subsystem symmetry groups G = ℤN and G = ℤN × ℤM as examples, we demonstrate how to recover the classification scheme C G \( \mathcal{C}\left[G\right] \) = H2(G×2, U(1))/ H2(G, U(1)))3, which was previously derived by examining topological invariant under linear subsystem-symmetric local unitary transformations in the lattice Hamiltonian formalism. To illustrate the correspondence between field-theoretic and lattice descriptions, we further analyze ℤ2 × ℤ2 and ℤN × ℤM cluster state models as concrete examples.