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

TL;DR

The paper develops infinite-component BF (i$BF$) field theories by stacking 3+1D BF layers along a stacking direction to realize 4+1D fracton topological orders. It identifies boundary zero singular modes (ZSMs) of asymmetric Toeplitz $K$ matrices as the microscopic engine of Toeplitz braiding, producing nonlocal particle–loop braiding along the stacking direction when ZSMs localize on opposite boundaries; these braiding phases are read off from $K^{-1}$ corners and connected to non-Hermitian directional amplification. Two concrete $K$-matrix families are analyzed: Hatano–Nelson type and non-Hermitian SSH type, with SVD (and, for symmetric cases, eigen-decomposition) clarifying how ZSMs encode the nonlocal braiding, including regimes with two ZSM sets enabling braiding at both boundaries. The study builds a bridge between fracton/topological field theories and non-Hermitian physics, offering extensions to multi-loop braiding, entanglement renormalization, foliation, and possible lattice realizations, and hints at a speculative parallel-universe interpretation via wormhole condensates. Overall, the work provides a concrete mechanism (ZSMs) for Toeplitz braiding in higher-dimensional topological orders and reveals rich connections to non-Hermitian amplification phenomena.

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.

Figures (6)

• Figure 1: Illustration of stacking and braiding in iCS theory. (a) Stacking and braiding process: each layer is a Chern–Simons theory on a torus, with wavy lines denoting interlayer couplings. Two particles on opposite $z$ boundaries are shown in red and their projections in gray. (b) Matrix plot of the braiding phase $\Theta_{I,J}=2\pi(K^{-1})_{IJ}$ encoded in the iCS theory with $A=\bigl(0220\bigr)$ and $B=\bigl(0300\bigr)$, where we set the size of $K$ to $N=40$ for illustrative purpose. (c1) and (c2) show two boundary zero modes of $K$. All data are generated using the theory of Ref. li2024.
• Figure 2: Illustration of stacking and particle–loop braiding in i$BF$ theories. Each layer, defined on a three-torus, is represented by blue cubes in (a) and (b), while wavy lines denote interlayer $BF$ couplings along the stacking (or $w$) direction. The two 3+1D boundaries along this direction are labeled as the $w_1$ and $w_2$ boundaries. Panel (a) shows the particle–loop braiding between a loop at the $w_1$ boundary and a particle at the $w_2$ boundary, quantitatively encoded in the lower-left elements of $K^{-1}$, highlighted by the red disk in (c). Panel (b) depicts the opposite process, where a loop at the $w_2$ boundary braids with a particle at the $w_1$ boundary, corresponding to the upper right elements of $K^{-1}$ highlighted in (d). Panel (e) summarizes the relation between the locations of the LZSMs and RZSMs, the patterns of $K^{-1}$, and the resulting braiding statistics.
• Figure 3: Braiding statistics encoded in i$BF$ theories with Hatano-Nelson-type (HN-type) $K$ matrices. (a1)–-(e1) and (a2)--(e2) demonstrate the Toeplitz braiding encoded in HN-type i$BF$ theories, with the $K_{\mathrm{HN}}'$ matrix specified by the parameters $n=5$, $b=1$, $c=5$ and $n=5$, $b=5$, $c=1$, respectively. (a3)--(e3) demonstrates a trivial i$BF$ theory with $K_{\mathrm{HN}}'$ matrix specified by $n=5$, $b=3$ and $c=1$. (a1)--(a3) are the plots of the left singular modes of these matrices, while (b1)--(b3) are the plots of the right singular modes of these matrices. (c1)--(c3) are the matrix plots of the braiding phases $\Theta_{I,J}=2\pi ({K_{\mathrm{HN}}'}^{-1})_{IJ}$. For illustrative purpose, we take the system size $N$ to be 40. (d1)--(d3) and (e1)--(e3) demonstrate how the braiding phase $\Theta_{1,N}$ and $\Theta_{N,1}$ vary as the system size $N$ approaches infinity.
• Figure 4: Braiding phase $\Theta_{I,J}=2\pi (K_{\mathrm{HN}}^{-1})_{IJ}$ encoded in i$BF$ theories with local braiding statistics along the stacking direction, where we select system size $N=80$ for demonstrative purpose. (a) is the matrix plot of $\Theta_{I,J}=2\pi ({K_{\mathrm{HN},\mathrm{PBC}}'}^{-1})_{IJ}$ with $K_{\mathrm{HN},\mathrm{PBC}}'$ specified by $n=5,b=5,c=1$ and periodic boundary condition applied in the stacking direction; (b) is the matrix plot of $\Theta_{I,J}$ with $K_{\mathrm{HN}}'$ specified by $n=2,b=4,c=0$.
• Figure 5: (a1) and (a2) display skin modes and eigenvalues of $K_{\mathrm{HN}}'$ matrix specified by $n=5,b=5,c=1$. (b1) and (b2) display skin modes and eigenvalues of $K_{\mathrm{HN}}'$ matrix specified by $n=5,b=3,c=1$. For illustrative purpose, we depict the skin modes and eigenvalues of the $K_{\mathrm{HN}}'$ matrices with system size $N=40$.