Entanglement Renormalization in Two Spatial Dimensions

Abstract

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...

Exotic topological order in fractal spin liquids

“…It may be interesting to analyze how classical and quantum fractal liquids behave under RG transformations in the language of matrix and tensor product state representations. [41][42][43][44] An underlying difficulty in physically realizing the Sierpinski triangle model lies in the fact that the model has three-body interaction terms. Yet, one may simulate three-body interactions easily by using hopping particles.…”

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

“…To complete the example, let us assume that the index a is described by the vector space where we notice that tensorT is a matrix as in Eq. (38). Similarly, we can split incoming index d of tensorT back into outgoing index b and incoming index c of tensorT according to…”

“…By using an eigenbasis {|nt n } of the particle number operatorn, these constraints imply the presence of the zeros in Eqs. (36)- (38). Thus a reduced number of complex coefficients is required in order to describe U(1)-symmetric vectors and operators.…”

“…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. As variational Ansätze, PEPS and 2D MERA are particularly interesting since they can be used to address large two-dimensional lattices, including systems of frustrated spins 31,40 and interacting fermions, [41][42][43][44][45][46][47][48][49][50] where Monte Carlo techniques fail due to the sign problem.…”

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

“…For instance, the 2D MERA presented in Ref. 5 has a numerical cost of O(χ 16 ln L), which on current computers restricts χ < 8. For many systems, this does not allow for enough entanglement to accurately describe the ground state, limiting the accuracy of the approach.…”