Tensor Networks and the AdS/MERA correspondence
Lecture 1 – An introduction to tensor network methods for many-body quantum systems.
Abstract: Fairly recently it has been realised that locality of interactions in quantum many-body systems has profound consequences for the entanglement and correlation structure of certain relevant states within these systems. A tensor network representation of a many-body quantum state is a representation in which this correlation structure is made explicit, and over the last twenty years such representations have proven to be powerful tools for both the theoretical and numerical study of many-body quantum systems. In this talk I will provide a motivation for tensor networks as well as introduction to the formalism of tensor network diagrams through a discussion of the Matrix product State ansatz for one-dimensional systems.
Lecture 2 – Entanglement renormalisation, MERA and holography.
Abstract: In this talk I will begin by describing the procedure of entanglement renormalisation, a coarse graining procedure for many-body quantum states inspired by real space renormalisation group techniques, but generalised to ensure that the entanglement structure of the original state is preserved at each length scale. I will then proceed to discuss the Multi-Scale Entanglement Renormalisation (MERA) ansatz, which is a tensor network description of a class of many-body quantum states inspired by entanglement renormalisation and capable of representing efficiently the ground state of certain many-body models at their critical points. Finally, I will discuss how one can develop a holographic-esque bulk/boundary correspondence in MERA tensor networks, in which the bulk geometry is determined from the entanglement structure of the many body state represented by the MERA. In particular I will show how the bulk geometry of such networks naturally realises a discrete anti de Sitter space when the MERA represents the ground state of a critical system, a property now known as the AdS/MERA correspondence.