Quantum Critical Dynamics of Tensor Networks

One of the strangest aspects of quantum mechanics is the idea of action at a distance, whereby if two particles are prepared in a certain state relative to one another and then separated to opposite sides of the galaxy, a measurement on one of them will instantaneously tell one about the state of the other. This so worried Einstein that it gave him a deep suspicion about the theory of quantum mechanics. Nevertheless, experiment after experiment has confirmed that that is just how the universe is. If an intuitive understanding of this escapes us, we at least now know how to quantify the amount of this "quantumness" in a measure called entanglement.

The situation is even trickier when we think about systems with many particles. In this case we can imagine that every pair of particles might be entangled with one another. An important insight about the real world, however, is that for a variety of reasons usually pairs of particles are not entangled with one another if they are more than a certain distance apart. A certain type of wavefunction can be written down that encodes this idea of a finite range of entanglement mathematically. These states are known as tensor network states.

The use of these states is relatively new, but already their impact is profound and far-reaching. They have enabled the behaviour of many-particle quantum systems to be simulated on a computer with unprecedented accuracy. At the same time, insights flowing from the way in which these states behave has enabled us to make some hitherto unsuspected connections between quantum systems - including ultimately a link between the theory of black holes and the quantum mechanics of electrons in certain types of crystals known a quantum critical.

In this project we will use tensor network states to try to understand the dynamics of quantum critical systems - systems that in a certain sense are balanced between being classical and quantum mechanical. Our hope is that in doing so, we will find new ways of modelling quantum systems efficiently on computers, new ways of thinking about the quantum world and new ways to harness it to technological ends.

Entanglement is rapidly proving to be a notion that has universal applicability in the study of quantum mechanics. The idea that it might be a resource that persists in real systems only up to a given scale is embodied in tensor network wavefunctions. Such states are proving powerful in generating both accurate and efficient numerical simulations and analytical insights. They provide a common language to discuss entanglement across a very broad range of situations and disciplines from quantum chemistry to string theory. By its very formulation, it provides a fixed point for discourse between analytical and numerical physicists and the potential to facilitate detailed calculations that can engage experiment.

As technology embraces the quantum limit, it will increasingly require this facility. The very subject of this project, then provides the main pathway to its impact. We will work to develop a fruitful interaction between analytical and numerical theorists - the PI and PDRA respectively - communicating the benefits of this to the UK condensed matter community (for example, through the Thomas Young Centre) and geographically and intellectually further afield though our network of collaborations.

This project is a pilot towards long-term goals that address some of the most fundamental problems in quantum mechanics. By generating new knowledge in the interplay between numerical and analytical theory, it aims to give an early impetus to such research and, by focusing on a few clearly defined questions, some concrete results to build upon. It is hoped that the generation of these new ideas and the international collaborations that we foster in the course of working on them will enhance the international profile of UK science.

A major impact of this project is the personnel that it will train. By guiding the PDRA towards work in tensor networks tied closely to analytical understanding, I will train him/her to engage productively with diverse research and researchers in quantum mechanics. The experience of the PhD student working alongside this PDRA will be salutary. The particular area of scientific expertise will fulfill a specific need in the UK physics community, however, the experience of this challenging and stimulating environment will allow them to develop a skill set needed to achieve across a range of walks of life.