Multi-scale entanglement renormalization ansatz Tensor network

In summary, the conversation discusses a new approach for representing the ground state of a many-body Hamiltonian using tensor network renormalization, which involves applying the method to the Euclidean time evolution operator. This approach allows for a MERA representation of a thermal Gibbs state and connects various concepts such as entanglement, geometry, and renormalization. It is also mentioned that this approach has been proposed in the context of the AdS/CFT conjecture of string theory and has been used to understand the connection between condensed matter physics and holography.
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as a new proposal for QGhttp://arxiv.org/abs/1502.05385
Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
Glen Evenbly, Guifre Vidal
(Submitted on 18 Feb 2015)
We show how to build a multi-scale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed \textit{tensor network renormalization} (TNR) [G. Evenbly and G. Vidal, arXiv:1412.0732] to the Euclidean time evolution operator e−βH for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows TNR with a renormalization group flow in the space of wave-functions and Hamiltonians (and not just in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.
 
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It is not a new proposal. It is mainly about the AdS/CFT conjecture of string theory. If the conjecture is right, are there relatively simple ways we can understand how it works? The AdS/MERA picture was proposed by Brian Swingle (Physics Monkey) to help understand AdS/CFT using tools from condensed matter physics. Among the key papers preceding AdS/MERA are Juan Maldacena's thermofield double paper, the Ryu-Takayanagi proposal linking entanglement entropy and area, and Guifre Vidal's MERA. Also, there are many papers linking renormalization and holography, and the MERA is a form of renormalization. What the MERA seems to offer is a simple picture of how the various ideas (entanglement, geometry, renormalization) can be linked together.
 

1. What is the Multi-scale Entanglement Renormalization Ansatz (MERA) Tensor network?

The Multi-scale Entanglement Renormalization Ansatz (MERA) Tensor network is a mathematical framework used to represent and study quantum many-body systems. It is based on the idea of entanglement renormalization, where the entanglement between different parts of a system is used to build a more efficient representation of the system.

2. How does the MERA Tensor network work?

The MERA Tensor network is based on a hierarchical structure, where the system is divided into smaller subsystems and each subsystem is represented by a tensor. These tensors are then contracted in a specific way to form a network that represents the entire system. This network can then be used to calculate properties of the system, such as energy and correlation functions.

3. What are the advantages of using the MERA Tensor network?

The MERA Tensor network has several advantages over other methods used to study quantum many-body systems. It is a more efficient representation of the system, which allows for calculations to be done more quickly. It also has the ability to capture long-range correlations in the system, making it suitable for studying systems with critical behavior.

4. What are some applications of the MERA Tensor network?

The MERA Tensor network has been used to study a wide range of quantum many-body systems, including quantum spin chains, quantum field theories, and even black holes. It has also been applied to problems in condensed matter physics, such as the study of topological phases of matter.

5. How does the MERA Tensor network relate to other tensor network methods?

The MERA Tensor network is a specific type of tensor network that is closely related to other methods, such as the Tensor Network Renormalization (TNR) method. While all tensor network methods use the concept of entanglement renormalization, the MERA Tensor network has a unique hierarchical structure that sets it apart from other methods.

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