Conformal properties of hyperinvariant tensor networks

Matthew Steinberg,Javier Prior

doi:10.1038/s41598-021-04375-5

Abstract

Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. Although this new class of tensor network shows much potential for simulating conformal field theories arising from hyperbolic bulk manifolds with quasiperiodic boundaries, many issues are unresolved. In this manuscript we analyze the challenges related to optimizing tensors in a hyMERA with respect to some quasiperiodic critical spin chain, and compare with standard approaches in MERA. Additionally, we show two new sets of tensor decompositions which exhibit different properties from the original construction, implying that the multitensor constraints are neither unique, nor difficult to find, and that a generalization of the analytical tensor forms used up until now may exist. Lastly, we perform randomized trials using a descending superoperator with several of the investigated tensor decompositions, and find that the constraints imposed on the spectra of local descending superoperators in hyMERA are compatible with the operator spectra of several minimial model CFTs.

Highlights

Hyperinvariant tensor networks were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence
Many different models have emerged over the years proving to be very useful for the parameterization of local, gapped H amiltonians5,6), tensor networks such as the multiscale entanglement renormalization ansatz have proven to be useful in the simulation of quantum critical lattice models[7]; such work concerns the study of conformal field theory[7,8,9,10,11,12,13] and the AdS/CFT c orrespondence[14,15,16]
We have accomplished this analysis as follows: firstly, we have examined the standard Hamiltonianbased variational algorithms from MERA10–12, and have shown that modifications are required in order to account for the boundary quasiperiodicity inherent to regular tessellations of hyperbolic bulk manifolds

Summary

Introduction

Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. The purpose of these layers is twofold: first, we wish to decouple the bond dimension χ of the network from the local Hilbert-space dimension d of sites related to the original lattice on which our Hamiltonian is defined; we wish to minimize the effect of irrelevant operators in the RG flow once the original lattice has been renormalized into the scale-invariant version of the MERA network.

Results

Conclusion