Infinite-component $BF$ field theory: Nexus of fracton order, Toeplitz braiding, and non-Hermitian amplification

Abstract: Building on the recent study of Toeplitz braiding by Li et al. [Phys. Rev. B 110, 205108 (2024)], we introduce \textit{infinite-component} $BF$ (i$BF$) theories by stacking topological $BF$ theories along a fourth ($w$) spatial direction and coupling them in a translationally invariant manner. The i$BF$ framework captures the low-energy physics of 4D fracton topological orders in which both particle and loop excitations exhibit restricted mobility along the stacking direction, and their particle-loop braiding statistics are encoded in asymmetric, integer-valued Toeplitz $K$ matrices. We identify a novel form of particle-loop braiding, termed \textit{Toeplitz braiding}, originating from boundary zero singular modes (ZSMs) of the $K$ matrix. In the thermodynamic limit, nontrivial braiding phases persist even when the particle and loop reside on opposite 3D boundaries, as the boundary ZSMs dominate the nonvanishing off-diagonal elements of $K^{-1}$ and govern boundary-driven braiding behavior. Analytical and numerical studies of i$BF$ theories with Hatano-Nelson-type and non-Hermitian Su-Schrieffer-Heeger-type Toeplitz $K$ matrices confirm the correspondence between ZSMs and Toeplitz braiding. The i$BF$ construction thus forges a bridge between strongly correlated topological field theory and noninteracting non-Hermitian physics, where ZSMs underlie the non-Hermitian amplification effect. Possible extensions include 3-loop and Borromean-rings Toeplitz braiding induced by twisted topological terms, generalized entanglement renormalization, and foliation structures within i$BF$ theories. An intriguing analogy to the scenario of parallel universes is also briefly discussed.