Non-Invertible Interfaces Between Symmetry-Enriched Critical Phases

TL;DR

This work proposes symmetry-preserving interfaces as a robust diagnostic for symmetry-enriched criticality (SEC) in gapless phases, showing that any interface between distinct SECs with the same symmetry must be non-invertible when symmetry charges differ. It provides two complementary arguments—IR sweeping and two-point-function constraints—to establish this non-invertibility and derives concrete consequences such as vanishing cross-interface correlators and characteristic finite-size splittings. Through detailed Ising-CFT examples with Z2 × Z2^T symmetry, the authors classify possible interfaces, revealing a spectrum of non-invertible defects, degenerate ground states, and defect anomalies that generalize SPT edge phenomena to gapless settings. The work further connects these results to defect 't Hooft anomalies and SPT entanglers, showing how endpoints of defects encode projective G-representations, and discusses generalizations to higher dimensions, including the 2+1d Ising CFT. Overall, the paper establishes a bulk-defect (instead of bulk-edge) paradigm for diagnosing symmetry-enriched criticality with broad implications for topology and gapless phases.

Abstract

Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as $1/L^3$, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.

Figures (10)

• Figure 1: RG fate of $G$-symmetric interfaces $\mathcal{D}$ between symmetry-enriched criticalities. (a) "Symmetry-enriched criticalities" $\mathcal{T}_0$ and $\mathcal{T}_1$ share the same low-energy conformal field theory (CFT) but differ by how the global symmetry $G$ acts on (local or nonlocal) CFT operators. We classify the universal RG outcomes of any spatial interface $\mathcal{D}$ that preserves $G$. Bulk symmetry enrichment leaves concrete physically-observable interface signatures. In particular, symmetry forbids an invertible defect; instead $\mathcal{D}$ may flow to a non-invertible topological defect (with non-invertible fusion and quantum dimension $\langle \mathcal{D} \rangle>1$), to a partially or totally reflecting (factorizing) interface, or to a degenerate interface spontaneously breaking $G$. (b) In the IR, we say an interface $\mathcal{D}$ is $G$-symmetric when the UV network of $G$-defect lines and their junctions remains topological in its presence, i.e., it can slide across $\mathcal{D}$ without changing correlators or partition-function evaluations. Crucially, the UV $G$-defect lines map to IR defects and/or three-way junction phases in different ways in $\mathcal{T}_0$ vs. $\mathcal{T}_1$.
• Figure 2: Sweeping an invertible interface on a torus does not change its $G$-twisted partition function. If theories $\mathcal{T}_0$ (blue) and $\mathcal{T}_1$ (red) admit a $G$-symmetric invertible topological interface, then they have identical twisted partition functions and thus are not distinct symmetry enriched criticalities. Starting from $\mathcal{T}_0$ one can nucleate the invertible interface (left), apply two $F$-symbol moves to sweep it over the whole torus (middle), and contract it again (right), resulting in $\mathcal{T}_1$ and no changes to the partition function in the process. Due to $G$-symmetry, the same equalities hold in the presence of a background $G$-network (grey) defining the $(g,h)$ twisted partition functions.
• Figure 3: Selection rules from symmetry across an interface of symmetry-enriched criticalities For a $G$–symmetric interface $\mathcal{D}$, any operator whose $G$–charge differs on the two sides has a vanishing two-point function across the interface. In particular, if the local operator $\phi$ carries different $G$–charges in the two regions, then $\langle \phi^\dagger(-x)\,\phi(x)\rangle=0$ (top row). The same conclusion holds in $G$–twisted sectors: if the twisted primary operator $\mu$ has mismatched $G$-charges across the interface, then $\langle \mu^\dagger(-x)\,\mu(x)\rangle=0$ (bottom row). These vanishing correlators impose constraints on the IR defect.
• Figure 4: Interface phase diagrams for Ising symmetry--enriched criticalities. (a) Interface between $H_{\rm Ising}^\sigma$ and $H_{\rm Ising}$ (different $\sigma$ charge across the interface). Tuning $b$ moves along the Neumann family $N(\phi)$ with defect entropy $g = \sqrt{2}$, crossing the Kramers-Wannier defect at $b=1$; any $h \neq 0$ drives a flow to a factorizing defect with free boundaries on both sides ($g=1$). (b) Interface between $H_{\rm Ising}$ and $H_{\rm Ising}^\mu$ (different $\mu$ charge). The spins in the two stable phases are spontaneously aligned for $h>0$ and anti-aligned for $h<0$. (c) Interface between $H_{\rm Ising}^\sigma$ and $H_{\rm Ising}^\mu$ (different $\mu$ and $\sigma$ charge). The fine-tuned spontaneously fixed $g=2$ defect is unstable and any relevant perturbation flows to a factorizing $N(0)$ interface (free on one side, spontaneously fixed on the other). Here $h_X = h \cos \theta$ and $h_Z = h \sin \theta$.
• Figure 5: Algebraically localized stable interface modes. Interfaces between certain symmetry-enriched criticalities can host stable, degenerate modes with spontaneous symmetry breaking (SSB). Shown for Ising criticalities enriched with $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ symmetry, where twisted-sector operators carry different $\mathbb{Z}_2$-charges across the interface. Localization is quantified by the scaling of finite-size ground-state splitting $\Delta E(L)$ on a periodic chain of length $L$, computed by exact diagonalization using a free-fermion mapping. (a)$H_{\rm Ising}$ and $H_{\rm Ising}^{\mu}$ realize a stable, degenerate interface with $\Delta E \sim L^{-3}$ (computed for $h=b=1$ from Eq. \ref{['eq:H_Hu_detail']}). (b)$H_{\rm Ising}^\sigma$ and $H_{\rm Ising}^{\mu}$always realize a degenerate conformal interface; in the most stable case the gap also scales as $\Delta E \sim L^{-3}$ (computed here for $h = 1$, $\theta=0$ from Eq. \ref{['eq:Hs_Hmu_detail']} with additional irrelevant lattice perturbation $-Y_0 Y_1$).