Toeplitz non-liquids and Toeplitz braiding

We study a class of $3$D non-liquid states called ``Toeplitz non-liquids''. These states consist of a stack of $2$D twisted ZN topologically ordered layers along the z-direction; nearby layers are coupled while keeping translational symmetry along z. The effective field theory is described by infinite Chern-Simons (iCS) theory, with a coefficient matrix called ``K-matrix'' that is of block-tridiagonal Toeplitz matrix-type. With open boundary conditions (OBC) along the z-direction, certain K-matrices exhibit an exotic phenomenon called ``Toeplitz braiding'', where the mutual braiding statistical phase between two anyons at opposite boundaries oscillates and remains non-zero in the thermodynamic limit. As a necessary condition, this requires boundary zero modes in the K-matrix spectrum under OBC. A key example is the K-matrix resembling the Hamiltonian of the $1$D Su-Schrieffer-Heeger insulator. Since the gauge invariance of Chern-Simons theory guarantees integer quantized entries for K-matrices, no usual global symmetries are needed to protect these zero modes or Toeplitz braiding. In order to obtain the general theory, we categorize K-matrices that support Toeplitz braiding into three types and analyze the conditions for each. We further numerically study the analytical results for all types of K-matrices. For comparison, a trivial case is numerically shown, where the mutual statistical phase angle decays exponentially to zero in the thermodynamic limit.