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I want to understand Multi-scale Entanglement Renormalization Ansatz (MERA) with very elementary examples. So far I could find references which are mostly based on numerics. It would be a great help if someone could give a very simple example explaining the basic concepts of MERA or give some references (papers/theses/reviews) that implement the operations (i.e, isometry and disentangler) in analytic fashion (even for a simplest possible system).

Actually, I do not have any 'practical' knowledge about tensor networks in general as I am from theoretical HEP background. So references on (analytic) MPS or PEPS with simple examples will be really useful.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

  • $\begingroup$ @Qmechanic What is the motivation to convert this to a resource question? After all, the OP asks whether someone could give a very simple example explaining the basic concepts of MERA or give some references. As far as I can see, making it a resource question invalidates the first part of the question? $\endgroup$ – Norbert Schuch Nov 7 '16 at 9:13
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    $\begingroup$ @NorbertSchuch: It is a borderline judgement. The CW could perhaps be converted e.g. if the thread receive another type of answers. $\endgroup$ – Qmechanic Nov 7 '16 at 9:57
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The canonical example for MPS (in fact, the first MPS ever) is the AKLT model (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.799, https://projecteuclid.org/euclid.cmp/1104161001). The 2nd reference also discusses the 2D (=PEPS) version of the state.

Another example of an exact MPS/PEPS model are (nearest-neighbor) RVB states (https://arxiv.org/abs/quant-ph/0601075, http://arxiv.org/abs/1203.4816); in one dimension, this is known as the Majumdar-Ghosh model.

Also, all string-net models (non-chiral topological fixed point models) can be exactly expressed as PEPS: http://arxiv.org/abs/0809.2393, https://arxiv.org/abs/0809.2821.

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You may want to start from the paper Entanglement Renormalization: an introduction. In this work author describes basic consepts of MERA, things like isometries and disentanglers and why one needs to use them in order to produce MERA.
Article Algorithms for entanglement renormalization is more advanced and is focused mainly on description of algorithm itself. It will be a good exercise to build MERA for simple Hamiltonian based on information provided in aforementioned article.

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