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

TL;DR

The study addresses continuous transitions between non-invertible symmetry-breaking phases in 1D spin chains by combining Baxter gauging and bosonization to reveal deconfined-quantum-critical-point–like behavior. It identifies a DQCP between non-invertible SSB phases in a single $Z_2^o imesZ_2^e$ chain and develops a two-copy construction with an emergent non-invertible cosine symmetry and self-duality, described by a Luttinger parameter $K<1$. Crucially, the DQCP persists under various gauging operations, enabling a large family of related DQCPs and connections to invertible anomalous theories via twisted or diagonal gauging. The results provide analytical control over non-invertible critical points in 1D, offering a pathway to generalize DQCP concepts beyond group-like symmetries and toward richer fusion-category frameworks.

Abstract

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.

Figures (4)

• Figure 1: Main results of the paper. (a) Imposing $S D^o D^e$ as an additional symmetry generator on top of a $\mathbb{Z}^o_2 \times \mathbb{Z}^e_2$ symmetric spin chain induces a second-order transition tuned by a single parameter $\lambda_{\phi'}$ [see Eq.\ref{['Eq:singe_copy_Sint']}] between two conventional symmetry-breaking phases. (b) Imposing $S D^o D^e$ and $S$ as additional symmetry generators leads to two distinct non-invertible SSB phases separated by a stable gapless regime. (c) By considering two coupled $\mathbb{Z}^o_2 \times \mathbb{Z}^e_2$ symmetric spin chains with similar non-invertible symmetries imposed in (b), a second-order transition tuned by a single parameter $g$ exists [see Eq.\ref{['Eq:rewrtie_Lint']} and Fig.\ref{['fig:DQCP']}] between two non-invertible SSB phases.
• Figure 2: Schematic representation of the Baxter transformation in Eq.\ref{['Eq:baxter_trans']} (time flows upward).
• Figure 3: Phase diagram of Eq.\ref{['Eq:action_two_copy']} in the regime $|\Lambda_\theta| + |\Lambda_\phi| \gg ||\Lambda_\theta| - |\Lambda_\phi||, |g-1|$ . There exists a critical surface between the $(\prod_{r = 1}^2 D^{o,r} D^{e,r}, \prod_{r = 1}^2 S^{(r)} D^{o,r} D^{e,r})$-breaking and $(\prod_{r = 1}^2 S^{(r)}, \prod_{r = 1}^2 D^{o,r} D^{e,r})$-breaking phases, and thus a second-order transition can be tuned by a single parameter. The theory at the critical surface is described by a single Luttinger parameter $K = 1- \frac{2 \pi}{v} (|\Lambda_\phi| + |\Lambda_\theta|)<1$ and the scaling dimensions of the order parameters in both phases are equivalent to the Luttinger parameter $K$.
• Figure 4: New DQCPs from gauging. Starting from the two-coupled $\mathbb{Z}^o_2 \times \mathbb{Z}^e_2$ symmetric spin chain that exhibits a non-invertible DQCP in (a), one can generate another non-invertible DQCP by gauging the $\mathbb{Z}_2^o$ symmetry on both copies, as shown in (b). From (b), one can obtain a new invertible DQCP with a type-III anomaly in (d) by applying twisted gauging (i.e., the Kennedy--Tasaki transformation). Alternatively, one can arrive at the previously studied invertible DQCP with an LSM anomaly in (c) using spin-chain bosonization by further gauging the diagonal $\mathbb{Z}_2^o \times \mathbb{Z}_2^e$ symmetry. Each subfigure lists the symmetry generators (excluding spin-flip and exchange symmetries), the ground-state degeneracies (GSDs) of each phase, the fixed-point wavefunctions, and the symmetries preserved in each phase of the corresponding DQCP.