Algorithms for entanglement renormalization

Abstract

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.

Entanglement Renormalization in Two Spatial Dimensions

“…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 the scheme by analysing the 2D quantum Ising model, for which we obtain accurate estimates of the ground state energy and magnetizations, as well as two-point correlators (shown to scale polynomially at criticality), the energy gap, and the critical magnetic field and beta exponent.…”

“…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.…”

“…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 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].…”

“…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].…”