Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering

Abstract: We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.

• The paper characterizes gapped phases' bulk properties using smeared boundary conformal field theories (SBCFTs), emphasizing the role of Higgs/Nambu-Goldstone duality.
• A central finding is that the Hilbert space in gapped phases cannot be constructed using conventional boundary Cardy states but requires smeared Ishibashi states.
• Results suggest a reevaluation of boundary phenomena approaches, noting a categorical algebraic method in understanding ordering in gapped phases.

Characterizing Gapped Phases via Smeared Boundary Conformal Field Theories and Higgs/Nambu-Goldstone Duality

Introduction and Context

The paper "Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering" (2605.07734) addresses the quantum field theoretic classification of gapped phases and their bulk properties, especially under renormalization group (RG) flows that are dual via a sign reversal of relevant perturbations—termed Higgs/Nambu-Goldstone (NG) duality. The work situates itself within the modern framework where symmetry is generalized from group- to ring-like and even fusion categories, capturing symmetries beyond those handled by traditional group-theoretic approaches, including noninvertible and generalized symmetries.

Central to the paper is a systematic analysis of how bulk ordering in gapped (massive) phases can be characterized using smeared boundary conformal field theories (SBCFTs), a C-linear extension of the boundary CFT formalism, with attention to the nontrivial algebraic and categorical aspects underpinning the correspondence between UV and IR theories.

Theoretical Framework

The study leverages several key formalisms:

• Fusion Ring Symmetry: The algebraic structure of the set of generalized symmetry operators, viewed as a complex-linear (C-linear) ring, is the backbone for analyzing symmetry in both UV and IR fixed-point theories. The RG flow is modeled as a ring homomorphism between such fusion rings.
• Smeared Boundary CFT (SBCFT): Building on Cardy's proposal that bulk gapped phases can be related to boundary states "smeared" into the bulk, SBCFT extends the conventional boundary CFT framework by emphasizing its C-linear character, which is crucial when dealing with non-group-like symmetries.
• Ishibashi and Cardy States: The distinction between these is central; Cardy states characterize physical boundary conditions in BCFT (with positivity and integer coefficients ensuring locality and unitarity), whereas Ishibashi states, although unphysical in the BCFT context, serve as a natural spanning set for Hilbert spaces arising in some gapped phases under the dual RG flow.
• Higgs/NG Duality: For an RG flow triggered by a relevant perturbation $X_{\mathrm{pert}}$, the sign reversal ($-X_{\mathrm{pert}}$) induces a dual flow, interchanging high- and low-energy sectors but preserving the unbroken symmetry algebra. This duality often leads to a mismatch between the structures of gapped phases and those expected from boundary critical phenomena.

Formal Results and Numerical Insights

A principal observation is that, for a wide class of dual massive RG flows (i.e., those related by Higgs/NG duality), the Hilbert space of the resultant gapped phase cannot be constructed from smeared Cardy boundary states but must instead be built from smeared Ishibashi states. This phenomenon persists even in the simplest unitary minimal model transitions, such as from tricritical Ising ($M(4,5)$) to Ising ($M(3,4)$).

Applying the general strategy to the $M(4,5) \rightarrow M(3,4)$ flow, the authors rigorously track idempotents (associated with symmetry operators) through the RG flow via conservation of algebraic generalized quantum dimensions (AGQDs). For this case, three unpreserved idempotents give rise to a three-dimensional degenerate module in the IR, with basis vectors forming nontrivial linear combinations of Cardy states characterized by non-integer coefficients—a direct signal that these ground states are not physical boundary states in the BCFT sense. Thus, the characterization of ordering is governed by quantum dimensions and fusion rules, not solely by the AGQD structure.

The paper argues that these modules correspond to "order-disorder coexistence" phases, as observed, e.g., in [170], and that their ground state degeneracies and operator content follow from these categorical and algebraic mechanisms. The Hilbert space structure admits an explicit description, and the action of generalized symmetry (Verlinde) operators is nontrivial and, in the dual RG context, transfers between the Ishibashi states consistent with the fusion ring, not with the naive group-like symmetry action.

Implications, Contradictions, and Extensions

The results show that ordering in dual massive RGs cannot, in general, be approximated by boundary critical phenomena. This invalidates the previously presumed universality relating bulk gapped phases and boundary states in CFT, at least in the presence of Higgs/NG duality and non-group-like symmetry. The phenomenon is tightly linked to the C-linearity of the underlying algebraic structures: the vector spaces associated with these CFTs require the complex linear combinations allowed by Ishibashi state bases.

This is both a categorical generalization and a contradiction to the expectation that any gapped phase emerging from RG flow can be understood via boundary CFT. Instead, the module structure of gapped phases can reside strictly outside the standard module category generated by Cardy states, reflecting fundamentally new types of ordering tied to the spontaneous breaking of noninvertible symmetries.

Practically, this observation motivates new approaches to identifying and classifying gapped phases in critical quantum lattice models, especially in settings where non-group-like or categorical symmetries (fusion categories or their C-linear versions) arise. The understanding of polarization amplitudes, entanglement structure, and ground state degeneracies in these models must be revisited in light of the distinction between Cardy and Ishibashi basis constructions.

Theoretically, this work demands further exploration of how categorical algebra and module theory interplay with RG phenomena, quantum quench dynamics, entanglement spectrum analysis, and the implementation of such structures in tensor network states and matrix product operator frameworks. The extension to systems with symmetry-enforced gaplessness, higher-dimensional analogues, and connections to TQFT/BCFT correspondence is an open and fertile area.

Conclusion

This paper provides a rigorous algebraic and categorical analysis of gapped phases in two-dimensional quantum systems with non-group-like symmetry, showing that dual RG flows (via Higgs/NG duality) naturally give rise to module structures outside the space of boundary critical phenomena. The construction via smeared Ishibashi states signals a new class of ordering and spontaneous symmetry breaking, not visible in previous group-centric approaches. This development prompts a reevaluation of the mapping between bulk and boundary in topologically ordered and symmetry-enriched phases, and places categorical algebra and C-linearity at the forefront of the future classification and understanding of quantum phases of matter.