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arXiv:2206.11761

Zipper Entanglement Renormalization for Free Fermions

Sing Lam Wong, Ka Chun Pang, Hoi Chun Po

Published Jun 23, 2022• 0 citations

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. As a demonstration, we apply ZER to one-dimensional models and show that it efficiently disentangles the ground states of the Su-Schrieffer-Heeger model, a scale-invariant critical state, as well as a more general gapless state with two sets of Fermi points.

DOI: 10.48550/arxiv.2206.11761

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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. As a demonstration, we apply ZER to one-dimensional models and show that it efficiently disentangles the ground states of the Su-Schrieffer-Heeger model, a scale-invariant critical state, as well as a more general gapless state with two sets of Fermi points.

In a recent contribution [P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, arXiv:0904.4151 (unpublished)] entanglement renormalization was generalized to fermionic lattice systems in two spatial dimensions. Entanglement renormalization is a real-space coarse-graining transformation for lattice systems that produces a variational ansatz, the multiscale entanglement renormalization ansatz (MERA), for the ground states of local Hamiltonians. In this paper we describe in detail the fermionic version of the MERA formalism and algorithm. Starting from the bosonic MERA, which can be regarded both as a quantum circuit or in relation to a coarse-graining transformation, we indicate how the scheme needs to be modified to simulate fermions. To confirm the validity of the approach, we present benchmark results for free and interacting fermions on a square lattice with sizes between $6\ifmmode\times\else\texttimes\fi{}6$ and $162\ifmmode\times\else\texttimes\fi{}162$ and with periodic boundary conditions. The present formulation of the approach applies to generic tensor network algorithms.

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.