Entanglement renormalization, scale invariance, and quantum criticality

Abstract

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...

Tensor Network States and Geometry

“…3 for the scale invariant MERA. Instead of reproducing the original derivation of this result for MPS 1-3 and MERA 16,18,19 , here we will focus on the geometrical interpretation of Eq. 13 in terms of the structure of geodesics.…”

“…The asymptotic decay of correlations has long been known to be exponential in an MPS 1-3 and polynomial in the scale invariant MERA 16,18,19 . In this section we point out that such behaviour is dictated by the structure of geodesics in the geometry attached to each of these tensor network states.…”

“…The MPS is also used to simulate dynamics with the time-evolving block decimation (TEBD) [9][10][11][12] algorithm and variations thereof, often referred to as time-dependent DMRG. Other tensor network states for one-dimensional systems include the tree tensor network (TTN) 13,14 and the multi-scale entanglement renormalization ansatz (MERA) [15][16][17][18][19][20] , with the later being particularly successful at describing ground states at quantum critical points.…”

Tensor network states and algorithms in the presence of a global U(1) symmetry

“…Examples of tensor network states for one-dimensional systems include the matrix product state [1][2][3] (MPS), which results naturally from both Wilson's numerical renormalization group 4 and White's density-matrix renormalization group [5][6][7][8] (DMRG) and is also used as a basis for simulation of time evolution, e.g., with the time evolving block decimation (TEBD) [9][10][11] algorithm and variations thereof, often collectively referred to as time-dependent DMRG; 9-14 the tree tensor network 15 (TTN), which follows from coarsegraining schemes where the spins are blocked hierarchically; and the multiscale entanglement renormalization Ansatz [16][17][18][19][20][21] (MERA), which results from a renormalization-group procedure known as entanglement renormalization. 16,21 For twodimensional (2D) lattices there are generalizations of these three tensor network states, namely projected entangled pair states [22][23][24][25][26][27][28][29][30][31] (PEPS), 2D TTN, 32,33 and 2D MERA, [34][35][36][37][38][39][40] respectively.…”

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

“…Optimization algorithms to approximate ground states, and to evaluate local expectation values and correlators, are present in the literature. 11,15 The numerical cost of finding an expectation value or performing a single optimization iteration using the binary MERA scales as O(χ 9 ln L) for a translation-invariant system. For more complex MERA structures, such as those representing two-dimensional (2D) lattices, the power of χ for the cost increases dramatically.…”