# Differentiating between Tensor Networks

I am trying to study tensor networks and their application to quantum phase transitions. However, I had a question concerning the connection between the projected entangled-pair states (PEPS) and the multi-scale entanglement renormalization ansatz (MERA). In particular, I am looking at the following article by G. Vidal, which introduced the concept of MERA:

http://arxiv.org/abs/quant-ph/0610099

How is MERA different from PEPS, other than the fact that the former consists of tensors in a D+1 dimensional system, while the latter is for 2D systems? In particular, why would one use MERA as opposed to PEPS, and could MERA be used to describe long-range interactions?

• application domain of PEPS is wider ? According to G Vidal ( your link ) , MERA produces exact solutions and is particularly suited to describe states with quasi-long-range order, such as critical ground states. While PEPS - which has a much wider range of applications – local expectation values can only be obtained efficiently after a number of approximations ?
– user46925
Commented Jun 7, 2015 at 12:52

## 2 Answers

In one dimension, MERA naturally capture critical systems (i.e., systems with power-law decaying correlations and a log-divergence in the entanglement entropy). MPS (i.e., one-dimensional PEPS), one the other hand, have exponentially decaying correlations and a constant entanglement entropy. (Note: This is for a constant bond dimension and does not preclude using MPS to approximate critical systems.) One can indeed see that MPS are a subclass of MERA.

In two dimensions, on the other hand, PEPS are able to describe systems with algebraic correlations, and in fact, 2D MERA can be seen as a subclass of 2D PEPS. (One can however extent the entanglement renormalization idea behind MERA to branching MERA which cannot be written as PEPS).

One fundamental difference between MERA and PEPS is that there are two kinds of tensors in MERA - isometries and disentanglers. Isometries are maps from V $$\otimes V \to V$$, while disentanglers are V $$\otimes V \to V \otimes V$$, where $$V$$ is the vector space. In PEPS, there are tensors (say A & B) which are decomposed using isometries.

It turns out that disentanglers are important if you want to study critical systems on a classical computer because if you just use a simple tensor network then the local truncation space will be so large as to render classical simulations impossible. The growth of local truncation space is related to the accumulation of entanglement which is removed by disentanglers. This is known as "entanglement renormalization"(ER). If you do this ansatz on a multi-scale you get MERA (Multi-scale ER ansatz).