Compressing many-body quantum states in continuous space-time with tensor networks

Introduction

Many-body quantum systems with strong correlations are particularly difficult to understand in the continuum, where non-perturbative techniques are in scarce supply. Direct diagonalization methods are not available, since the Hilbert space is simply too large to be manageable. This inhibits progress in high energy physics, nuclear physics, and in the study of exotic topological phases of matter.

Advances in Lattice Techniques

On the lattice, tensor network states, a variational class of wavefunctions coming from quantum information theory, have allowed for the compression of exponentially large Hilbert spaces down to a smaller numerically manageable corner. This has allowed substantial theoretical and numerical advances on the many-body problem on the lattice.

Project Overview

This project will develop continuous tensor network states, a new framework to extend the recent lattice progress to the continuum and quantum field theory (QFT). The originality of the approach is that it will not rely on any discretization of space-time.

Methodology

I will work directly in the continuum, without any cutoff. Low energy states of quantum field theories, which a priori live in a continuously infinite-dimensional Hilbert space, will be compressed down to a finite and small number of parameters. This will then allow for the numerical solution of very generic (non-integrable) strongly coupled theories in a fully non-perturbative manner. Such a compression was long thought to be impossible, particularly in the relativistic case, but I overcame crucial theoretical hurdles in the past year, making the proposal particularly timely.

Applications

I will construct this framework with three main applications in mind:

1. Non-relativistic problems in 2 space dimensions and more, including e.g. fractional quantum Hall states.
2. Relativistic QFT, starting with a 1+1 dimensional toy model and gradually increasing complexity to get closer to nonabelian gauge theories.
3. Critical quantum systems (and classical statistical mechanics).