Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
Building on the infinite-component Chern--Simons theory of three-dimensional fracton phases by Ma et al. [Phys. Rev. B 105, 195124 (2022)] and the Toeplitz braiding of anyons by Li et al.~[Phys. Rev B 110, 205108 (2024)], we show that stacking $(3+1)$D $BF$ topological field theories, which serve as low-energy effective descriptions of a class of three-dimensional topological orders, along a fourth spatial direction gives rise to an exotic class of four-dimensional fracton phases. Their low-energy physics is governed by a new field-theoretic framework, namely \textit{infinite-component $BF$} (i$BF$) \textit{theories}, characterized by asymmetric integer Toeplitz $K$ matrices. Under open boundary conditions along the stacking direction, i$BF$ theories with properly chosen $K$ matrices exhibit a striking phenomenon termed \textit{Toeplitz particle--loop braiding}, where a particle and a loop placed on opposite three-dimensional boundaries acquire a strongly oscillating yet robustly nonvanishing braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: adiabatically transporting the particle induces a winding boundary trajectory on the opposite boundary that encircles the loop. We show that this robustness originates from boundary zero singular modes (ZSMs) of Toeplitz $K$ matrices revealed by singular value decomposition, rather than from boundary zero eigenmodes responsible for previously known Toeplitz braiding of anyons, and that the same ZSM mechanism also underlies directional amplification in the rapidly developing field of non-Hermitian physics. We analytically and numerically study representative i$BF$ theories with Hatano--Nelson--type and non-Hermitian Su--Schrieffer--Heeger--type $K$ matrices, establishing a universal correspondence between ZSMs and Toeplitz particle--loop braiding.