Infinite-component BF field theory: Nexus 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 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 - infinite-component BF (iBF) theories - characterized by asymmetric integer Toeplitz K matrices. Under open boundary conditions, iBF theories exhibit a striking phenomenon: Toeplitz particle-loop braiding, where a particle and a loop placed on opposite three-dimensional boundaries acquire a finite braiding phase even at infinite separation. This nonlocal braiding admits a geometric interpretation: 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 (SVD), rather than from the eigenvalue zero modes responsible for previously known Toeplitz braiding of anyons. We analytically and numerically study representative iBF 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. Our results identify boundary zero singular modes as the mechanism behind Toeplitz particle-loop braiding and establish infinite-component BF theory as a predictive framework for higher-dimensional fracton topological orders.