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Journal of High Energy Physics

Entanglement Renormalization for Quantum Field Theories with Discrete Wavelet Transforms

Daniele S. M. Alves

Published Jul 10, 2024• 1 citation
Publisher

Abstract

A bstract We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the Multiscale Entanglement Renormalization Ansatz (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.

DOI
10.1007/jhep07(2024)081

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A bstract We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the Multiscale Entanglement Renormalization Ansatz (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.
We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the \emph{Multiscale Entanglement Renormalization Ansatz} (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.
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 establish a precise connection between discrete wavelet transforms and entanglement renormalization, a real-space renormalization group transformation for quantum systems on the lattice, in the context of free particle systems. Specifically, we employ Daubechies wavelets to build approximations to the ground state of the critical Ising model, then demonstrate that these states correspond to instances of the multiscale entanglement renormalization ansatz (MERA), producing the first known analytic MERA for critical systems.
Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multiscale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. Here we propose the use of symmetric tensors as a mechanism to build a symmetry-protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry-protected entanglement renormalization produces the proper structure of RG fixed points, namely, a fixed-point for each symmetry-protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.
We adapt the techniques of entanglement renormalization tensor networks to weakly interacting quantum field theories in the continuum. A key tool is ``quantum circuit perturbation theory,'' which enables us to systematically construct unitaries that map between wave functionals which are Gaussian with arbitrary perturbative corrections. As an application, we construct a local continuous multiscale entanglement renormalization ansatz (cMERA) circuit that maps an unentangled scale-invariant state to the ground state of ${\ensuremath{\varphi}}^{4}$ theory to one loop. Our local cMERA circuit corresponds exactly to one-loop Wilsonian renormalization group (RG) flow on the spatial momentum modes. In other words, we establish that perturbative Wilsonian RG on spatial momentum modes can be equivalently recast as a local cMERA circuit in ${\ensuremath{\varphi}}^{4}$ theory and argue that this correspondence holds more generally. Our analysis also suggests useful numerical ans\"atze for cMERA in the nonperturbative regime.
This thesis reviews the multiscale entanglement renormalisation ansatz or MERA, a numerical tool for the study of quantum many-body systems and a discrete realisation of the AdS/CFT duality. The thesis covers an introduction to the necessary background concepts of entanglement, entanglement entropy and tensor network states, the structure and main features of MERA and its applications in condensed matter theory and holography. Also covered are details on the algorithmic implementation of MERA and some of its generalisations and extensions. MERA belongs to a class of variational ansatze for quantum many-body states known as tensor network states. It is especially well-suited for the study of scale invariant critical points. MERA is based on a real-space renormalisation group procedure called entanglement renormalisation, designed to systematically handle entanglement at different length scales along the coarse-graining flow. Entanglement renormalisation has be used for example to efficiently describe Kitaev states of the toric code, the prime example of topological order, and numerically study the ground state of the highly frustrated spin-1/2 Heisenberg model on a kagome lattice and various other one- and two-dimensional lattice models. The geometric and causal structure of MERA, which underlies its effectiveness as a numerical tool, also makes it a discrete version of the AdS/CFT duality. This duality describes a conformal field theory by a gravity theory in a higher dimensional space, and vice versa. The duality is manifest in the scaling of entanglement entropy in MERA, which is governed by a law highly analogous to the Ryu-Takayanagi formula for holographic entanglement entropy, in the connection between thermal states and a black-hole-like MERA and in the connection between correlation functions and holographic geodesics in a scale invariant MERA. The aim of this thesis is to lead the reader to an understanding of what MERA is, how it works and how it can be used. MERA's core features and uses are presented in a comprehensive and explicit way, and a broad view of possible applications and further directions is given. Plenty of references are also offered to direct the reader to further research on how MERA may relate to his/her interests.