Quantum Information Journal Club: Continuous and field tensor network states

Abstract: Tensor network methods have provided us with a powerful set of tools with which to study strongly interacting many-body systems on the lattice. Understanding the limitations of this ansatz is paramount for the success of both numerical algorithms and exact analytical representations of the many-body wavefunction. In recent years, several generalizations making use of different notions of the continuum have appeared in the literature to tackle such different limitations. Firstly, continuous tensor network states (cTNS) is an ansatz that allows us to work directly in the continuum to answer questions related to quantum field theories, removing the need for sending the lattice spacing to zero in standard tensor network approaches. Secondly, field tensor network states (fTNS) aim to provide an exact tensor network approach to the analytical description of chiral gapped topological states in two spatial dimensions, such as the Laughlin wavefunction. While these states had been previously considered to be out of reach for the most prominent two dimensional tensor network, projected entangled pair states, by upgrading the virtual space of the network to an infinite dimensional one, we circumvent the previous no-go theorems to provide the first exact representations. In this talk, I will present an overview of both cTNS and fTNS. I will present the main ideas behind both generalizations and relation to quantum field theories. I will also showcase some of the applications that we are currently working on, as well as some of the problems that one encounters along the way. This talk is based on both published and ongoing work with Arkadiusz Bochniak, Germán Sierra and Ignacio Cirac.