Topological holography, quantum criticality, and boundary states

TL;DR

<3-5 sentences>Topological holography reinterprets (1+1)d quantum phases through a (2+1)d bulk topological order with a sandwich boundary structure, separating topological symmetry data from local dynamics. The framework unifies gapped and gapless phases, encodes dualities and 't Hooft anomalies via anyon permutations and domain walls, and provides a concrete method to compute partition functions and boundary RCFT data from bulk topological data. It yields explicit correspondences between gapped phases and boundary states, reveals how twisted partition functions capture SPT anomalies, and extends to conformal boundary states in RCFTs with rich examples (Ising, tricritical Ising, Potts, SU(2)k). The approach offers new insights into symmetry-enriched critical points, DQCPs, and igSPT phenomena, and points to broader applications in higher dimensions and non-invertible fusion-category symmetries.

Abstract

Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1)d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1)d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.

Figures (10)

• Figure 1: The sandwich picture in the topological holography. A (1+1)$d$ theory can be viewed as a sandwich built from a topological order $\mathcal{T}$ in the bulk with a topological gapped boundary conditions on the left and a potentially non-topological boundary condition on the right.
• Figure 2: When $M$ is a torus $T^{2}$, we can shrink the inner topological boundary of the sandwich $T^{2} \times I$, and it becomes a Lagrangian algebra given by $\mathcal{A}$. The figure shows a spatial slice $S^{1} \times I$ of this process.
• Figure 3: Topological defect lines on the left boundary implement the fusion category $\mathcal{C}$ symmetry.
• Figure 4: Symmetry actions on the local and defect operators. (a) A local operator $\mathcal{O}_{\alpha}$ corresponds to a bulk anyon line $\alpha$ that condenses on the left topological boundary, and stretches across the sandwich. The symmetry $\mathcal{C}$ acts on the local operator as the mutual braiding between the bulk anyon line $\alpha$ and the boundary topological line $m \in \mathcal{C}$. The boundary line $m$ can be lifted to a bulk line $\mu$ and the symmetry transformation is given by the mutual braiding between the bulk anyons $\alpha$ and $\mu$. (b) A defect operator corresponds to a bulk anyon line $\beta$ stretched across the bulk and connecting to a topological line $a \in \mathcal{C}$ on the left boundary. The action of the symmetries on a defect operator can be formulated in terms of a lasso diagram (shown on the left boundary), which can also be lifted and became a mutual braiding between the bulk anyon $\mu$ and $\beta$.
• Figure 5: A (1+1)$d$ RCFTs corresponds to a (2+1)$d$ chiral topological orders $\mathcal{B}$ on a strip with an insertion of a gapped domain wall $\mathcal{S}$ in the middle. The sandwich picture is obtained by folding along the gapped domain wall, which results in a double topological order $\mathcal{Z}({\mathcal{B}}) = \mathcal{B} \boxtimes \overline{\mathcal{B}}$ in the bulk with a gapped boundary condition $\mathcal{A}_{\mathcal{S}}$.