Tensor network method for real-space topology in quasicrystal Chern mosaics

Abstract

Computing topological invariants in two-dimensional quasicrystals and super-moire matter is a remarkable open challenge, due to the absence of translational symmetry and the colossal number of sites inherent to these systems. Here, we establish a method to compute local topological invariants of exceptionally large systems using tensor networks, enabling the computation of invariants for Hamiltonians with hundreds of millions of sites, several orders of magnitude above the capabilities of conventional methodologies. Our approach leverages a tensor-network representation of the density matrix using a Chebyshev tensor network algorithm, enabling large-scale calculations of topological markers in quasicrystalline and moire systems. We demonstrate our methodology with two-dimensional quasicrystals featuring $C_8$ and $C_{10}$ rotational symmetries and mosaics of Chern phases. Our work establishes a powerful method to compute topological phases in exceptionally large-scale topological systems, providing the required tool to rationalize generic supe-moire and quasicrystalline topological matter.