Analogs of deconfined quantum criticality for non-invertible symmetry breaking in 1d

The spontaneous breaking of non-invertible symmetries can lead to exotic phenomena such as coexistence of order and disorder. Here we explore second-order phase transitions in 1d spin chains between two phases that correspond to distinct patterns of non-invertible symmetry breaking. The critical point shares several features with well-understood examples of deconfined quantum critical points, such as enlarged symmetry and identical exponents for the two order parameters participating in the transition. Interestingly, such deconfined transitions involving non-invertible symmetries allow one to construct a whole family of similar critical points by gauging spin-flip symmetries. By employing gauging and bosonization, we characterize the phase diagram of our model in the vicinity of the critical point. We also explore proximate phases and phase transitions in related models, including a deconfined quantum critical point between invertible order parameters that is enforced by a non-invertible symmetry.

💡 Research Summary

This paper investigates novel types of second-order phase transitions in one-dimensional quantum spin chains, focusing on the spontaneous breaking of “non-invertible symmetries.” Unlike conventional group symmetries, non-invertible symmetries lack an inverse operation and can lead to exotic phenomena, such as the coexistence of order and disorder, which fall outside the standard Landau-Ginzburg-Wilson paradigm.

The core model consists of a spin-1/2 chain with a distinction between odd and even sites. The system respects two conventional Z2 symmetries (η_o, η_e) acting on odd and even sites separately, supplemented by additional symmetry generators: the site-swap operator (S) and a composite non-invertible operator (S D_o D_e), which combines site-swap with Kramers-Wannier transformations (D_o, D_e) on the two sublattices. A key technical tool employed is the “Baxter transformation,” a specific gauging procedure that maps the original model with non-invertible symmetries to a dual model possessing only invertible (group-like) symmetries, including lattice translation (T). This mapping renders the problem more tractable, connecting it to well-studied models like the XY chain.

Using this approach, the authors first analyze a single chain. They identify four distinct gapped phases near a decoupled Ising critical point: a partially ordered phase (breaking η_o and η_e), a symmetric phase, and two non-invertible symmetry-breaking phases—the (D_o D_e, S D_o D_e)-breaking phase and the (S, D_o D_e)-breaking phase. These latter phases are characterized by local order parameters (O_j and Q_j, respectively) that are charged under the non-invertible symmetries, representing new phases of matter. In the single-chain setup, the two non-invertible broken phases are not directly connected; instead, they are separated by a stable gapless region.

To realize a direct transition between two non-invertible symmetry-breaking phases—an analog of a deconfined quantum critical point (DQCP) in 1D—the authors couple two identical copies of such chains. By introducing symmetry-allowed inter-chain interactions, they demonstrate that a single tuning parameter (g) can drive a continuous phase transition between the (D_o D_e, S D_o D_e)-breaking phase and the (S, D_o D_e)-breaking phase. This critical point is described by a two-component bosonized field theory with central charge c=2 and features characteristic DQCP traits: (i) the two order parameters break unrelated symmetries, (ii) there is an enhanced self-duality at the critical point that exchanges the two order parameters (up to a sign), ensuring they have identical scaling dimensions, and (iii) an emergent continuous non-invertible “cosine” symmetry appears near criticality.

A profound finding is the robustness of this DQCP structure under the process of “gauging” the spin-flip symmetries (η_o, η_e). This means that starting from one such DQCP, one can generate an infinite family of related DQCPs by gauging different subgroups or in different ways. For instance, the non-invertible symmetry-breaking phases map to translational symmetry-breaking phases (like a valence bond solid) in the gauged model. This establishes a powerful mechanism for constructing new critical theories and highlights a deep connection between non-invertible symmetries in (1+1)D and anomalies or topological order in higher dimensions.

In summary, this work extends the concept of deconfined criticality to systems with non-invertible symmetries, provides a concrete and analyzable 1D platform for studying such phenomena, and unveils a generative link between gauging procedures and families of quantum critical points. It bridges condensed matter physics, quantum field theory, and the theory of topological phases.