Abstract

The Zipper Entanglement Renormalization(ZER) is a ‘state-based’  description for the free fermion ground states. Entanglement renormalization refers to a sequence of real-space coarse-graining transformations in which the short-range entanglement is discarded on a successively longer scale. In contrast to the ‘Hamiltonian-based’ descriptions such as the Matrix Products State(MPS) and Multiscale Entanglement Renormalization Ansatz(MERA), where the actual states are determined variationally, the free fermion states are exactly known based on the relation between the correlation matrix and the pure state. The name ‘Zipper’ follows from the ‘zipper unitary’ we constructed at every renormalization step, in which it unzips the states into an approximation of tensor product over short-range degrees of freedom and a renormalized states containing the long-range information. By successively applying ZER on the renormalized states, we obtain a unitary transformation that factorizes the input states over the emergent renormalization spacetime. As a proof of principle, we perform the ZER in several one-dimensional free Fermion models, including the Su-Schriffer-Heeger model, a scale-invariant critical state, and a more general gapless state with two sets of Fermi points, in order to demonstrate the flexibility of the ZER description.