Toeplitz Braiding in Topological Phases

• Toeplitz braiding is a nonlocal phenomenon where the mutual braiding phase between boundary excitations remains finite or oscillatory even as the number of layers increases.
• It arises from the spectral properties of block-Toeplitz matrices and the presence of boundary zero modes that sustain robust long-range topological correlations.
• Analytical and numerical studies confirm that this effect underpins new fracton topological orders and connects with non-Hermitian amplification in driven quantum systems.

Toeplitz braiding is a form of nonlocal anyonic or particle–loop braiding that emerges in higher-dimensional topological phases constructed via stacking and coupling lower-dimensional layers, with the defining feature that, in the thermodynamic limit, the mutual braiding phase between excitations on opposing boundaries remains finite or oscillatory rather than decaying. This phenomenon is governed by the spectral and localization properties of boundary zero (singular) modes of block-Toeplitz coupling matrices in the effective low-energy field theory. Toeplitz braiding both signals and generates new types of fracton topological order with nontrivially entangled boundary excitations, and exhibits deep connections to non-Hermitian boundary amplification in driven open quantum systems (Li et al., 2024, Li et al., 12 Nov 2025).

1. Definition and Theoretical Background

In the context of 3D and 4D fracton topological phases, Toeplitz braiding is defined via an effective infinite-component Chern-Simons (iCS) or $BF$ (iBF) field theory, whose coupling constants are encoded in large integer-valued Toeplitz matrices $K_{IJ}$. Under open boundary conditions (OBC) in the stacking direction, the key diagnostic of Toeplitz braiding is the limiting value of the off-diagonal elements of $K^{-1}$: for boundary excitations (such as anyons, particles, or loops) localized at opposite surfaces, the gauge-invariant braiding phase is

$\Theta_{a,b}(N) = 2\pi (K^{-1})_{1,N} \bmod 2\pi,$

where $N$ is the number of layers. In the Toeplitz braiding regime, $(K^{-1})_{1,N}$ does not decay but remains finite or oscillatory as $N \to \infty$ (Li et al., 2024, Li et al., 12 Nov 2025).

This distinguishes Toeplitz braiding from "trivial" boundary-to-boundary braiding, where such phases vanish exponentially with increasing separation, as occurs in most conventional topological orders where boundary excitations become decoupled in the thermodynamic limit.

2. Infinite-component Topological Field Theories and Toeplitz $K$ Matrices

The emergence of Toeplitz braiding is rooted in the construction of topological field theories built by stacking lower-dimensional layers with translationally invariant inter-layer couplings. In 3D, this is exemplified by stacking $N$ twisted $\mathbb{Z}_m$-ordered layers with nearest-neighbor Chern-Simons couplings, yielding a $2N \times 2N$ block-tridiagonal Toeplitz matrix $K^{(2N)}$ of the form:

$$K^{(2N)} = \begin{pmatrix} A & B^\top & 0 & \cdots & 0 \ B & A & B^\top & & \vdots \ 0 & B & A & \ddots & 0 \ \vdots & & \ddots & \ddots & B^\top \ 0 & \cdots & 0 & B & A \end{pmatrix},$$

with $A$ and $B$ integer-valued blocks encoding intra- and inter-layer couplings (Li et al., 2024). In the 4D iBF case, higher-dimensional analogues with block Toeplitz matrices built from several matrix-valued blocks arise, such as the Hatano–Nelson or non-Hermitian SSH chain (Li et al., 12 Nov 2025).

The translation invariance in the stacking direction is critical, forcing the Toeplitz property $K_{IJ}=K_{I-J}$ for $|I-J|$ within the interaction range, and ensuring the existence of characteristic zero (singular) modes at certain parameter choices.

3. Anatomy of the Braiding Phase and Its Persistence

The physical significance of Toeplitz braiding is that it endows boundary excitations with robust long-range entanglement. The mutual braiding of two anyons (in 3D) or of a loop and a particle (in 4D) at distant boundaries is quantified by

$\theta_{ab}(N) = 2\pi (K^{-1})_{1,N} \bmod 2\pi,$

computed for excitations localized at the first and $N$-th layers. In trivial stacking, $(K^{-1})_{1,N}$ decays as $c e^{-N/\xi}$ for some correlation length $\xi$; in the Toeplitz regime, it behaves as

$$(K^{-1})_{1,N} = \frac{1}{m} \left(-\frac{l}{m}\right)^{N-1},$$

with $|l/m|>1$ yielding persistent, oscillatory nonlocal phases (Li et al., 2024). This is confirmed analytically and numerically for paradigmatic cases such as the SSH and Hatano–Nelson matrices (Li et al., 2024, Li et al., 12 Nov 2025). For the iBF construction, analogous boundary-to-boundary phases survive and can be directional or bi-directional depending on the structure of $K$.

The origin is traced to boundary zero modes (termed "zero singular modes" (ZSMs) in the non-Hermitian language), whose exponentially small eigenvalues or singular values result in large weights in $K^{-1}$, sharply localized at the system corners, and controlling the long-range braiding phase.

4. Spectral Characterization and Boundary Zero Modes

A necessary and sufficient condition for Toeplitz braiding is the existence of (one or more) boundary zero modes in the OBC spectrum of $K$. For block Toeplitz $K$-matrices, these zero modes correspond to solutions of the characteristic equation (symbol) $D(\beta) = \det(B + A\beta + B^\top \beta^2)$ featuring roots $|\beta_j|<1$ (localized at the lower boundary) and $|\beta_j|>1$ (upper boundary) (Li et al., 2024). The zero modes are explicitly constructed as edge-localized eigenvectors $\mathbf{u}$, exponentially decaying from their respective boundaries.

In the SVD framework, for non-Hermitian or asymmetric Toeplitz matrices, the singular vectors $u_j$ and $v_j$ localize at opposite boundaries. Their contribution to the inverse is

$K^{-1} = \sum_j \sigma_j^{-1} v_j u_j^T,$

with $\sigma_j$ small and $u_j$, $v_j$ supported near boundary 1 and $N$, yielding non-vanishing $(K^{-1})_{1,N}$ (Li et al., 12 Nov 2025). The number and localization of ZSMs directly controls the nontriviality and directionality of Toeplitz braiding, e.g., one or two ZSMs corresponding respectively to Type I or II phase diagrams (Li et al., 2024).

5. Representative Examples: SSH and Hatano–Nelson Toeplitz Matrices

Explicit examples demonstrate the mechanism and classification:

Model	$K$ Structure	Braiding Regime
SSH (Hermitian)	Block tridiagonal, $2\times 2$ blocks	Type I: two boundary ZMs $\longrightarrow$ nonzero $(K^{-1})_{1,N}$ and $(K^{-1})_{N,1}$
Hatano–Nelson (Non-Hermitian)	Scalar Toeplitz with asymmetric hoppings	Directional Toeplitz braiding: nonzero $(K^{-1})_{1,N}$, zero $(K^{-1})_{N,1}$
Non-Hermitian SSH	$2\times2$ non-symmetric blocks	Cases I–III: uni- or bi-directional Toeplitz braiding; IV: trivial

For the SSH model, $K$ is identical to the Su-Schrieffer-Heeger Hamiltonian; its inverse yields two exact boundary zero modes with eigenvalues vanishing as $(m/l)^N$ and an explicit, oscillatory $(K^{-1})_{1,2N}$. For the Hatano–Nelson case, non-Hermitian amplification is directly mapped to Toeplitz braiding, with one ZSM pair dominating the relevant off-diagonal inverse (Li et al., 2024, Li et al., 12 Nov 2025).

6. Physical Implications and Relation to Fracton and Non-Hermitian Physics

Toeplitz braiding signals the breakdown of purely local bulk screening in the stacked system, instead manifesting robust, boundary-to-boundary topological correlations. In fracton topological orders, this leads to point or loop excitations with restricted mobility anchored at distinct boundaries, whose mutual braiding phase cannot be screened by increasing bulk thickness (Li et al., 2024, Li et al., 12 Nov 2025). The phenomenon is rooted in the spectral "near-singularity" of the coupling matrices and persists despite the lack of global symmetries, solely due to large-gauge-invariance enforced integer quantization.

The mathematical structure underlying Toeplitz braiding is isomorphic to mechanisms of non-Hermitian boundary amplification, wherein open systems with non-Hermitian couplings exhibit directional signal transmission. Here, the long-range (corner-to-corner) contributions to $K^{-1}$ map precisely onto the amplification response in non-Hermitian cavity arrays (Li et al., 12 Nov 2025).

Extensions include theories with three-loop or Borromean ring statistics (by stacking twisted $\mathbb{Z}_m$ or $BF$ layers with additional topological terms), symmetry-enriched fracton phases, entanglement renormalization by layer addition/removal, and a foliation or "parallel-universe" analogy in which stacked boundaries communicate via nonlocal topological "wormholes" (Li et al., 12 Nov 2025).

7. Diagnostic Criteria, Numerical Evidence, and Classification

The occurrence of Toeplitz braiding is established by (i) existence of boundary zero modes under OBC, and (ii) non-integral, persistent values of $(K^{-1})_{1,N}$ as $N\to\infty$. Numerical studies confirm this diagnostic by plotting $\Theta_{IJ} = 2\pi (K^{-1})_{IJ}$, revealing non-vanishing matrix elements in corners associated with boundary-to-boundary braiding, corroborated by finite-size scaling (the only non-decaying contributions arise from the zero-mode sector) (Li et al., 2024, Li et al., 12 Nov 2025).

The classification of Toeplitz $K$-matrices supporting Toeplitz braiding is cataloged by the number and localization of their boundary zero modes and the block structure of $K$. This underlies the rich directional and multi-excitation braiding possibilities in higher-dimensional fracton and non-Hermitian systems.

---

Toeplitz braiding delineates a nontrivial class of boundary-dependent topological phenomena, exemplifying the fusion of infinite-component field theory, advanced spectral analysis, and novel nonlocal order in both fracton and non-Hermitian physics (Li et al., 2024, Li et al., 12 Nov 2025).