Tensor Networks for Quantum Field Theories II
I will propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class shall be demonstrated to retain key features of the multi-scale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible.
We will describe how the reconstruction of a bulk operator can be organised systematically. With a suitable parametrisation, an analogue of the HKLL formula emerges, involving a smearing function satisfying a Klein Gordon equation in the graph. The parametrisation also allows us to read off interaction vertices, and build up loop diagrams systematically. When we interpret the Bruhat-Tits tree as a tensor network, we recover (partially) features of the p-adic AdS/CFT dictionary discussed recently in the literature.
In this talk I discuss the problem of introducing dynamics for holographic codes. To do this it is necessary to take a continuum limit of the holographic code. As I argue, a convenient kinematical continuum limit space is given by Jones’ semicontinuous limit. Dynamics are then furnished by a unitary representation of a discrete analogue of the conformal group known as Thompson’s group T. I will describe these representations in detail in the simplest case of a discrete AdS geometry modelled by trees.
Tensor network is a constructive description of many-body quantum entangled states starting from few-body building blocks. Random tensor networks provide useful models that naturally incorporate various important features of holographic duality, such as the Ryu-Takayanagi formula for entropy-area relation, and operator correspondence between bulk and boundary. In this talk I will overview the setup and key properties of random tensor networks, and then discuss how to describe quantum superposition of geometries in this formalism.
According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the Einstein-Hilbert action of a certain region in the dual space-time. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries.
I will describe some recent work studying proposals for computational complexity in holographic theories and in quantum field theories. In particular, I will discuss some interesting properties of the new gravitational observables and of complexity in the boundary theory.
In this talk, I would like to discuss how we can realize the correspondence between AdS/CFT and tensor network in quantum field theories (i.e. the continous limit). As the first approach I will discuss a possible connection between continuous MERA and AdS/CFT. Next I will introduce the second approach based on the optimization of Euclidean path-integral, where the strcutures of hyperbolic spaces and entanglement wedges emerge naturally. This second appraoch is closely related to the idea of tensor network renormalization.
We will review the topic of tensor network renormalization, relate it to real space Hamiltonian flows, and discuss the emergence of matrix product operator algebras as symmetries of the renormalization fixed points.
joint work with Matthias Bal, Michael Marien and Jutho Haegeman
I will discuss analytic approaches to construct tensor network representations of quantum field theories, more specifically conformal field theories in 1+1 dimensions. A key insight is that we should understand how well the tensor network can reproduce the correlation functions of the quantum field theory. Based on this measure of closeness, I will present rigorous results allowing for explicit error bounds which show that both Matrix product states (MPS) as well as the multiscale renormalization Ansatz (MERA) do approximate conformal field theories.