Abstract
Entanglement renormalization refers to a sequence of real-space coarse-graining transformations in which short-range entanglement on successively longer length scales are distilled out. In this work, we introduce a state-based approach, “zipper entanglement renormalization” (ZER), for free-fermion systems. The name derives from a unitary we construct at every renormalization step, dubbed the zipper, which unzips the state into an approximate tensor product between a short-range entangled state and a renormalized one carrying the longer-range entanglement. By successively performing ZER on the renormalized states, we obtain a unitary transformation of the input state into a state that is approximately factorized over the emergent renormalization spacetime. Building upon the foundation of the 1-D zipper entanglement renormalization (ZER), we attempt to expand this state-based approach to two-dimensional free-fermion lattice systems to study more varieties of matter phases, notably the Chern insulators as chiral topological phases. The implementation of 2-D ZER follows the description of 1-D with more consideration of symmetry and self-similarity of the process. We will demonstrate the 2-D ZER on the honeycomb lattice that supports 3 matter phases, including trivial insulators, Dirac semi-metals and Chern insulators as a validation of the working principle.