Entanglement Renormalization: an introduction

TL;DR

The paper introduces entanglement renormalization and the Multi-scale Entanglement Renormalization Ansatz (MERA) as a real-space RG framework for quantum lattice systems at zero temperature. It combines disentanglers that remove short-range entanglement with isometries that coarse-grain blocks, yielding a scalable RG flow on two-site interactions and a scale-invariant fixed-point description via a MERA network. By analyzing the quantum Ising chain, it demonstrates how scaling operators, critical exponents, correlation functions, and boundary critical phenomena can be extracted from the scaling superoperator, with finite-$\chi$ MERA reproducing conformal data to high accuracy. The work highlights practical computational advantages, a clear circuit interpretation of ground-state preparation, and potential connections to holography and universal quantum critical behavior in both 1D and 2D systems.

Abstract

We present an elementary introduction to entanglement renormalization, a real space renormalization group for quantum lattice systems. This manuscript corresponds to a chapter of the book "Understanding Quantum Phase Transitions", edited by Lincoln D. Carr (Taylor & Francis, Boca Raton, 2010)

Figures (10)

• Figure 1. 1: (i) Coarse-graining transformations characterized by an isometry $w$ that maps blocks of three sites of lattice $\mathcal{L}$ into single sites of a coarse-grained lattice $\mathcal{L}'$. (ii) Graphical representation of $w^{\dagger} w=I_{\mathbb{V}'}$. (iii) An operator $o$ supported on e.g. two blocks of sites of $\mathcal{L}$ becomes a two-site operator $o'$, cf. Eq. \ref{['V:eq:oo']}.
• Figure 1. 2: (i) Coarse-graining transformation with disentanglers $u$, acting on two contiguous sites of $\mathcal{L}$ across the boundary of two blocks, followed by isometries $w$, which map a block of three sites into a single site of the coarse-grained lattice $\mathcal{L}'$. (ii) Graphical representation of $uu^{\dagger} = I_{\mathbb{V}^{\otimes 2}}$. (iii) A local operator $o$ supported on two contiguous sites of $\mathcal{L}$ is mapped into an effective operator $o'$ on two contiguous sites of $\mathcal{L}'$. The linear transformation $o\rightarrow o' = \mathcal{A}(o)$ is described by a superoperator (i.e., a linear map in the space of operators) referred to as the ascending superoperator $\mathcal{A}$.
• Figure 1. 3: To illustrate the role of disentanglers $u$, a simple example is considered where the ground state $\hbox{$| \Psi_{\hbox{\tiny GS}} \rangle$}$ of a one-dimensional lattice $\mathcal{L}$ factorizes into the product of entangled states $(\hbox{$| 1_r 1_s \rangle$} + \hbox{$| 2_r 2_s \rangle$})/\sqrt{2}$ involving only two nearest neighbor sites $r,s \in \mathcal{L}$ (on both sides of the boundaries between blocks) and states of single sites (in the interior of each block). (i) When only isometries are used, entanglement across the boundary of the blocks is preserved in $\mathcal{L}'$. (ii) By using disentanglers $u$ such that transform the state $(\hbox{$| 1_r 1_s \rangle$} + \hbox{$| 2_r 2_s \rangle$})/\sqrt{2}$ of the two boundary sites $r,s\in\mathcal{L}$ into an unentangled state, e.g. $\hbox{$| 1_r1_s \rangle$}$, entanglement across the boundary of the blocks can be removed before the isometries are applied, and the ground state of $\mathcal{L}'$ has no entanglement.
• Figure 1. 4: Diagramatic representation of two iterations of the coarse-graining transformation, producing a sequence of increasingly coarse-grained lattices $\mathcal{L}^{(0)}$, $\mathcal{L}^{(1)}$, and $\mathcal{L}^{(2)}$. At each iteration $u^{(\tau)}$ is chosen as to remove short-range entanglement and $w^{(\tau)}$ follows White's rule. The ascending superoperator $\mathcal{A}$ maps $o^{(\tau)}$ into $o^{(\tau+1)}$ while the descending superoperator $\mathcal{D}$ maps $\rho^{(\tau+1)}$ into $\rho^{(\tau)}$.
• Figure 1. 5: Multi-scale entanglement renormalization ansatz made of disentanglers and isometries corresponding to two iterations of the coarse-graining transformation in Fig. \ref{['V:fig:disentangler']}. Notice the periodic boundary conditions.