Conformal Properties of Hyperinvariant Tensor Networks

TL;DR

This paper explores the mathematical properties and optimization challenges of hyperinvariant tensor networks (hyMERA), demonstrating their potential for simulating conformal field theories and introducing new tensor decompositions that expand their analytical framework.

Contribution

It analyzes tensor optimization issues in hyMERA, introduces alternative tensor decompositions, and shows compatibility with minimal model CFT spectra through randomized trials.

Findings

Tensor decompositions differ from original hyMERA construction.

Constraints on spectra are compatible with several minimal model CFTs.

Optimization of tensors in hyMERA faces specific challenges.

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.