  In this paper, we theoretically study a class of 3D non-liquid states that
show exotic boundary phenomena in the thermodynamical limit. More concretely,
we focus on a class of 3D fracton topological orders formed via stacking 2D
twisted \(\mathbb{Z}_N\) topologically ordered layers along \(z\)-direction.
Nearby layers are coupled while maintaining translation symmetry along \(z\)
direction. The effective field theory is given by infinite-component
Chern-Simons theory, with an integer-valued symmetric block-tridiagonal
Toeplitz \(K\)-matrix whose size is thermodynamically large. With open boundary
conditions (OBC) along \(z\), certain choice of \(K\)-matrices exhibits exotic
boundary ``Toeplitz braiding'', where the mutual braiding phase angle between
two anyons at opposite boundaries oscillates and remains non-zero in the
thermodynamic limit. In contrast, in trivial case, the mutual braiding phase
angle decays exponentially to zero in the thermodynamical limit. As a necessary
condition, this phenomenon requires the existence of boundary zero modes in the
\(K\)-matrix spectrum under OBC. We categorize nontrivial \(K\)-matrices into
two distinct types. Each type-I possesses two boundary zero modes, whereas each
type-II possesses only one boundary zero mode. Interestingly, the
integer-valued Hamiltonian matrix of the familiar 1D ``Su-Schrieffer-Heeger
model'' can be used as a non-trivial $K$-matrix. Importantly, since
large-gauge-invariance ensures integer quantized \(K\)-matrix entries, global
symmetries are not needed to protect these zero modes. We also present
numerical simulation as well as finite size scaling, further confirming the
above analytical results. Motivated by the present field-theoretical work, it
will be interesting to construct 3D lattice models for demonstrating Toeplitz
braiding, which is left to future investigation.
