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Entanglement Entropy, the Multi-scale Entanglement Renormalisation Ansatz and Lifshitz Field Theories

K.J. Kavanagh

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

In this thesis we will summarise recent developments in tensor network methods and extend results for entanglement entropy using cMERA in Lifshitz field theories. After a brief recounting of the important aspects of entanglement entropy we will present a description of pertinent tensor networks. This will include both matrix product states (MPS) and the multi-scale entanglement renormalisation ansatz (MERA). We will recount previous area law results for the entanglement entropy in these ansatze. Following this we will continue to discuss MERA and in particular its continuous realisation (cMERA). Finally we apply this cMERA framework to free field theories and extend known results to the case of field theories with anisotropic scaling, that is, Lifshitz field theories. In particular we find a new result for the entanglement entropy of fermionic theories with anisotropic scaling.

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Entanglement and correlations in the continuous multi-scale entanglement renormalization ansatz

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We investigate the entanglement structure of the continuous multi-scale entanglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett. 110 (2013) 100402] for ground states of quantum field theories (QFTs). The cMERA, proposed as an extension to QFTs of the lattice MERA, is defined directly in the continuum but is nevertheless naturally equipped with a short-distance scale 1/Λ that acts as a UV regulator for quantum fluctuations. We consider the simplified setting of Gaussian cMERA for free QFTs, where explicit calculations can be performed. For relativistic free massless bosonic and fermionic QFTs in both 1+1 and 2+1 spacetime dimensions, we show that the cMERA state indeed displays no UV divergences in two-point correlation functions or entanglement entropy, in sharp contrast with the exact ground states.

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In this thesis we will summarise recent developments in tensor network methods and extend results for entanglement entropy using cMERA in Lifshitz field theories. After a brief recounting of the important aspects of entanglement entropy we will present a description of pertinent tensor networks. This will include both matrix product states (MPS) and the multi-scale entanglement renormalisation ansatz (MERA). We will recount previous area law results for the entanglement entropy in these ansatze. Following this we will continue to discuss MERA and in particular its continuous realisation (cMERA). Finally we apply this cMERA framework to free field theories and extend known results to the case of field theories with anisotropic scaling, that is, Lifshitz field theories. In particular we find a new result for the entanglement entropy of fermionic theories with anisotropic scaling.
Abstract We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in [1] obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent z and we exhibit a transition from an area law to a volume law analytically in the limit of large z. We move on to explore the effects of a massive deformation, obtaining results for any z in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.
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.
This chapter introduces the tensor network ansatz for a quantum state whose entanglement entropy obeys the so-called area law with or without logarithmic corrections. This ansatz represents a quantum many-body wave function by a network product of local tensors defined on the lattice sites and treats all tensor elements as variational parameters. It includes, for example, one-dimensional matrix product states (MPS) and two-dimensional projected entangled pair states (PEPS) or projected entangled simplex states (PESS). A typical example is the spin-1 AKLT chain, whose ground state can be exactly represented as an MPS. If a logarithmic correction to the entanglement area law emerges, a tensor network state termed the multi-scale entanglement renormalization ansatz (MERA) describes the entanglement structure of the ground state more accurately in one dimension.
Abstract The continuous Multi Scale Entanglement Renormalization Anstaz (cMERA) consists of a variational method which carries out a real space renormalization scheme on the wavefunctionals of quantum field theories. In this work we calculate the entanglement entropy of the half space for a free scalar theory through a Gaussian cMERA circuit. We obtain the correct entropy written in terms of the optimized cMERA variational parameter, the local density of disentanglers. Accordingly, using the entanglement entropy production per unit scale, we study local areas in the bulk of the tensor network in terms of the differential entanglement generated along the cMERA flow. This result spurs us to establish an explicit relation between the cMERA variational parameter and the radial component of a dual AdS geometry through the Ryu‐Takayanagi formula. Finally, we argue that the entanglement entropy for the half space can be written as an integral along the renormalization scale whose measure is given by the Fisher information metric of the cMERA circuit. Consequently, a straightforward relation between AdS geometry and the Fisher information metric is also established.
We investigate the entanglement structure of the continuous multi-scale entanglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett., 110, 100402 (2013)] for ground states of quantum field theories (QFTs). The cMERA, proposed as an extension to QFTs of the lattice MERA, is defined directly in the continuum but is nevertheless naturally equipped with a short-distance scale 1/Λ1/\Lambda1/Λ that acts as a UV regulator for quantum fluctuations. We consider the simplified setting of Gaussian cMERA for free QFTs, where explicit calculations can be performed. For relativistic free massless bosonic and fermionic QFTs in both 1+1 and 2+1 spacetime dimensions, we show that the cMERA state indeed displays no UV divergences in two-point correlation functions or entanglement entropy, in sharp contrast with the exact ground states.
We investigate the entanglement structure of the continuous multi-scale entanglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett. 110 (2013) 100402] for ground states of quantum field theories (QFTs). The cMERA, proposed as an extension to QFTs of the lattice MERA, is defined directly in the continuum but is nevertheless naturally equipped with a short-distance scale 1/Λ that acts as a UV regulator for quantum fluctuations. We consider the simplified setting of Gaussian cMERA for free QFTs, where explicit calculations can be performed. For relativistic free massless bosonic and fermionic QFTs in both 1+1 and 2+1 spacetime dimensions, we show that the cMERA state indeed displays no UV divergences in two-point correlation functions or entanglement entropy, in sharp contrast with the exact ground states.