Entanglement Renormalization: an introduction

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)Comment: v2: new format. 24 pages, 10 figures, 2 tables, chapter of the book "Understanding Quantum Phase Transitions", edited by Lincoln D. Carr (Taylor & Francis, Boca Raton, 2010

A real space decoupling transformation for quantum many-body systems

We propose a real space renormalization group method to explicitly decouple into independent components a many-body system that, as in the phenomenon of spin-charge separation, exhibits separation of degrees of freedom at low energies. Our approach produces a branching holographic description of such systems that opens the path to the efficient simulation of the most entangled phases of quantum matter, such as those whose ground state violates a boundary law for entanglement entropy. As in the coarse-graining transformation of [Phys. Rev. Lett. 99, 220405 (2007)], the key ingredient of this decoupling transformation is the concept of entanglement renormalization, or removal of short-range entanglement. We demonstrate the feasibility of the approach, both analytically and numerically, by decoupling in real space the ground state of a critical quantum spin chain into two. Generalized notions of RG flow and of scale invariance are also put forward.Comment: 5 pages, 6 figures, includes appendi

Scaling of entanglement entropy in the (branching) multi-scale entanglement renormalization ansatz

We investigate the scaling of entanglement entropy in both the multi-scale entanglement renormalization ansatz (MERA) and in its generalization, the branching MERA. We provide analytical upper bounds for this scaling, which take the general form of a boundary law with various types of multiplicative corrections, including power-law corrections all the way to a bulk law. For several cases of interest, we also provide numerical results that indicate that these upper bounds are saturated to leading order. In particular we establish that, by a suitable choice of holographic tree, the branching MERA can reproduce the logarithmic multiplicative correction of the boundary law observed in Fermi liquids and spin-Bose metals in $D\geq 2$ dimensions.Comment: 17 pages, 14 figure

Algorithms for entanglement renormalization: boundaries, impurities and interfaces

We propose algorithms, based on the multi-scale entanglement renormalization ansatz, to obtain the ground state of quantum critical systems in the presence of boundaries, impurities, or interfaces. By exploiting the theory of minimal updates [G. Evenbly and G. Vidal, arXiv:1307.0831], the ground state is completely characterized in terms of a number of variational parameters that is independent of the system size, even though the presence of a boundary, an impurity, or an interface explicitly breaks the translation invariance of the host system. Similarly, computational costs do not scale with the system size, allowing the thermodynamic limit to be studied directly and thus avoiding finite size effects e.g. when extracting the universal properties of the critical system.Comment: 29 pages, 29 figure

Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension

Tensor networks offer a variational formalism to efficiently represent wave-functions of extended quantum many-body systems on a lattice. In a tensor network N, the dimension \chi of the bond indices that connect its tensors controls the number of variational parameters and associated computational costs. In the absence of any symmetry, the minimal bond dimension \chi^{min} required to represent a given many-body wave-function |\Psi> leads to the most compact, computationally efficient tensor network description of |\Psi>. In the presence of a global, on-site symmetry, one can use a tensor network N_{sym} made of symmetric tensors. Symmetric tensors allow to exactly preserve the symmetry and to target specific quantum numbers, while their sparse structure leads to a compact description and lowers computational costs. In this paper we explore the trade-off between using a tensor network N with minimal bond dimension \chi^{min} and a tensor network N_{sym} made of symmetric tensors, where the minimal bond dimension \chi^{min}_{sym} might be larger than \chi^{min}. We present two technical results. First, we show that in a tree tensor network, which is the most general tensor network without loops, the minimal bond dimension can always be achieved with symmetric tensors, so that \chi^{min}_{sym} = \chi^{min}. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that \chi_{sym}^{min} > \chi^{min}. We further argue, however, that in some situations there are important conceptual reasons to prefer a tensor network representation with symmetric tensors (and possibly larger bond dimension) over one with minimal bond dimension.Comment: 12 pages, 13 figure