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. DOI: 10.1103/PhysRevLett.115.180405 PACS numbers: 05.30.-d, 02.70.-c, 03.67.Mn, 75.10.Jm Understanding emergent phenomena in many-body systems remains one of the major challenges of modern physics. With sufficient knowledge of the microscopic degrees of freedom and their interactions, we can write the partition function of a classical system, namely, a weighted sum of all of its microscopic configurations, or the analogous Euclidean path integral of a quantum many-body system, where the weighted sum is now over all conceivable trajectories. These objects contain complete information on the collective properties of the many-body system. However, evaluating partition functions or Euclidean path integrals is generically very hard. Kadanoff's spin-blocking procedure [1] opened the path to nonperturbative approaches based on coarse graining a lattice [2,3]. More recently, Levin and Nave proposed the tensor renormalization group (TRG) [4], a versatile realspace coarse-graining transformation for 2D classical partition functions-or, equivalently, Euclidean path integrals of 1D quantum systems.The TRG is an extremely useful approach that has revolutionized how we coarse-grain lattice models [4][5][6][7][8][9]. However, this method fails to remove part of the shortrange correlations in the partition function and, as a result, the coarse-grained system still contains irrelevant microscopic information. Conceptually, this is in conflict with the very spirit of the renormalization group (RG) and results in a RG flow with the wrong structure of noncritical fixed points, as discussed in Ref. [7]. Computationally, the accumulation of short-range correlations over successive TRG coarse-graining transformation also has important consequences: as pointed out by Levin and Nave, it implies the breakdown of the TRG at criticality [4,10] (although universal information, such as critical exponents, can still be obtained from finite systems).An analogous problem, faced earlier in the context of ground state wave functions, was resolved with the introduction of entanglement renormalization techniques [18,19]. In this L...

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D = 1 dimensions, as well as projected entangled pair states (PEPS) in D > 1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D = 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary lawthat is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D > 1 dimensions.

The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and critical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit. The study of quantum critical phenomena through real-space renormalization group (RG) techniques [1,2] has traditionally been obstructed by the accumulation, over successive RG transformations, of short-range entanglement across block boundaries. Entanglement renormalization [3] was recently proposed as a technique to address this problem. By removing short-range entanglement at each iteration of the RG transformation, not only can arbitrarily large lattice systems be considered, but the scale invariance characteristic of critical phenomena is also seen to be restored [3,4].In this paper we explain how to use the multi-scale entanglement renormalization ansatz (MERA) [5] to investigate scale invariant systems [3,4,5,6,7]. It has been showed that the scale invariant MERA can represent the infra-red limit of topologically ordered phases [6]. Here we focus instead on its use at quantum criticality. We present the following results: (i) given a critical Hamiltonian, an adaptation of the algorithm of Ref.[8] to compute a scale invariant MERA for its ground state; then, starting from a scale invariant MERA, (ii) a procedure to identify the scaling operators/dimensions of the theory and (iii) a closed expression for two-point and threepoint correlators; (iv) a connection between the MERA and conformal field theory, which can be used to readily identify the continuum limit of a critical lattice model; finally (v) benchmark calculations for the Ising and Potts models.We note that result (ii) was already discussed by Giovannetti, Montangero and Fazio in Ref. [7] using the binary MERA of Ref. [5]. Our derivations are conducted instead with the ternary MERA of Ref [8] (see Fig. 1), in terms of which results (iii)-(iv) acquire a simple form.We start by considering a finite 1D lattice L made of N sites, each one described by a vector space V of dimension χ. The (ternary) MERA is a tensor network that serves as an ansatz for pure states |Ψ ∈ V ⊗N of the lattice, see Fig. 1. Its tensors, known as disentanglers and isometries, are organized in T ≈ log 3 N layers, each one implementing a RG transformation. Such transformations produce a sequence of lattices,where lattice L τ +1 is a coarse-graining of lattice L τ , and the top lattice L T is sufficiently small to allow exact numerical computations. Let o denote a local observable supported on two contiguous sites of L, and let ρ T be the density matrix that describes the state of the system on two contigu...

This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor network ansatz seems to produce quasiexact results in systems with sizes well beyond the reach of exact diagonalization techniques. We describe an algorithm to approximate the ground state of a local Hamiltonian on a L ϫ L lattice with the topology of a torus. Accurate results are obtained for L = ͕4,6,8͖, whereas approximate results are obtained for larger lattices. As an application of the approach, we analyze the scaling of the ground-state entanglement entropy at the quantum critical point of the model. We confirm the presence of a positive additive constant to the area law for half a torus. We also find a logarithmic additive correction to the entropic area law for a square block. The single copy entanglement for half a torus reveals similar corrections to the area law with a further term proportional to 1 / L.

Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-1/2 Heisenberg model on a kagome lattice. Lattices of N={36,144, infinity} sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal with a 36-site unit cell, compatible with a previous proposal. Its energy per site, E=-0.432 21, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state.

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We propose and test a scheme for entanglement renormalization capable of addressing large twodimensional quantum lattice systems. In a translationally invariant system, the cost of simulations grows only as the logarithm of the lattice size; at a quantum critical point, the simulation cost becomes independent of the lattice size and infinite systems can be analysed. We demonstrate the performance of the scheme by investigating the low energy properties of the 2D quantum Ising model on a square lattice of linear size L = {6, 9, 18, 54, ∞} with periodic boundary conditions. We compute the ground state and evaluate local observables and two-point correlators. We also produce accurate estimates of the critical magnetic field and critical exponent β. A calculation of the energy gap shows that it scales as 1/L at the critical point. The use of disentanglers leads to a real-space RG transformation that can in principle be iterated indefinitely, enabling the study of very large systems in a quasiexact way. This RG transformation also leads to the socalled multi-scale entanglement renormalization ansatz (MERA) [4] to describe the ground state of the system -or, more generally, a low energy sector of its Hilbert space. In a translation invariant lattice made of N sites, the cost of simulations grows only as log N [5]. In the presence of scale invariance, this additional symmetry is naturally incorporated into the MERA and a very concise description, independent of the size of the lattice, is obtained in the infrared limit of a topological phase [6] or at a quantum critical point [1,4,7,8,9,10,11].While the basic principles of entanglement renormalization are the same in any number of spatial dimensions, most available calculations refer to 1D models. Numerical work with 2D lattices incurs a much larger computational cost and has so far been limited to exploratory studies of free fermions [7] and free bosons [8] and of the Ising model in a square lattice of small linear size L ≤ 8 [12]. It must be emphasized, however, that the approach of Refs. [7,8] relies on the gaussian character of free particles and can not be generalised to the interacting case, whereas the results of Ref. [12] were obtained by exploiting a significant reduction in computational cost that occurs only for small 2D lattices.In this paper we present an implementation of the MERA that allows us to consider, with modest computational resources, 2D systems of arbitrary size, including infinite systems. In this way we demonstrate the scalability of entanglement renormalization in two spatial dimensions and decisively contribute to establishing the MERA as a competitive approach to systematically address 2D lattice models. The key of the present scheme is a carefully planned organization of the tensors in the MERA, leading to simulation costs that grow as O(χ 16 ), where χ is the dimension of the vector space of an effective site. This is drastically smaller than the cost O(χ 28 ) of the best previous scheme [7,8,12]. We also demonstrate the performance of t...