# Entanglement renormalization for chiral topological phases

## TL;DR

This paper investigates the application of MERA to chiral topological phases, establishing fundamental limits on bond dimension growth and the impossibility of reaching a fixed point with constant bond dimensions.

## Contribution

It provides a rigorous proof of the monotonicity of a correlation length functional in MERA layers and introduces a no-go theorem for fixed bond dimension in chiral topological phases.

## Key findings

- Bond dimension must grow with the number of layers for accurate representation.
- Constant bond dimension limits the height, preventing convergence to a fixed point.
- A lower bound exists for the correlation length in chiral states.

## Abstract

We considered the question of applying the multiscale entanglement renormalization ansatz (MERA) to describe chiral topological phases. We defined a functional for each layer in the MERA, which captures the correlation length. With some algebraic geometry tools, we rigorously proved its monotonicity with respect to adjacent layers, and the existence of a lower bound for chiral states, which shows a trade-off between the bond dimension and the correlation length. Using this theorem, we showed the number of orbitals per cell (which roughly corresponds to the bond dimension) should grow with the height. Conversely, if we restrict the bond dimensions to be constant, then there is an upper bound of the height. Specifically, we established a no-go theorem stating that we will not approach a renormalization fixed point in this case.