Three-dimensional fracton topological orders with boundary Toeplitz braiding

In this paper, we theoretically study a class of three-dimensional (3D) nonliquid 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 the infinite-component Chern-Simons (iCS) field 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 nonzero 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 nontrivial $K$ 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. Symmetry fractionalization in iCS field theory is also briefly discussed. 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.