---
type: education
title: Ph.D in Condensed Matter Theory
period: 2018 - 2021
location: Imperial College London
subtitle: "The one-dimensional Long-Range Falikov-Kimball Model"

image: /assets/images/koala_logo.svg
alt: "A colourful scientific figure from my work."

layout: cv_entry
read_more: true

funding: https://gtr.ukri.org/project/145404DD-ABAD-4CFB-A2D8-152A6AFCCEB7#/tabOverview
---

When you have lots of 

Disorder Free Localisation in the 1D Falikov-Kimball Model. EPSRC Project N.<a href="{{page.funding}}">2120140</a> 

My research focuses on the behaviour of electrons in crystaline solids.<br>

Supervisor: Dr Johannes Knolle<br>


### Plain English

Building upon the work of others, I study a very simplified model of electrons moving about in a crystal. A crystal is a structure that repeats through space, like the pattern on a chess board. We work in one dimension so in our case it would be more like patterns of coloured beads on a necklace. In addition the crystal is slightly disordered, as if the person making the necklace sometimes gets it a little wrong. This disorder tends to disrupt the movement of the electrons, often meaning that they cannot move at all.

### Technical English

Both disorder or interactions can turn metals into insulators. One of the simplest settings to study this physics is given by the Falikov-Kimbal (FK) model describing itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, it has been shown that in-homogenous configurations of the background field give rise to effective disorder physics leading to a rich phase diagram in two (or more) dimensions with finite temperature charge density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalised FK model in one dimension with long-range interactions which shows a similarly rich phase diagram. Using an exact Markov Chain Monte Carlo method we map out the phase diagram and compute the energy resolved localisation properties of the fermions. We compare the behaviour of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field and discuss its experimental implications.

![A figure from the paper]({{"assets/images/phase_diagram.png" | absolute_url}})

<!-- <span class="link-block">
                  <a href="https://drive.google.com/file/d/19HmB8Ls_H_3higF_QOHrIWu9hISXVIwZ/view?usp=sharing" class="paper-link-button">
                    <span class="icon">
                      <svg class="svg-inline--fa fa-file-pdf fa-w-12" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="file-pdf" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 384 512" data-fa-i2svg=""><path fill="currentColor" d="M181.9 256.1c-5-16-4.9-46.9-2-46.9 8.4 0 7.6 36.9 2 46.9zm-1.7 47.2c-7.7 20.2-17.3 43.3-28.4 62.7 18.3-7 39-17.2 62.9-21.9-12.7-9.6-24.9-23.4-34.5-40.8zM86.1 428.1c0 .8 13.2-5.4 34.9-40.2-6.7 6.3-29.1 24.5-34.9 40.2zM248 160h136v328c0 13.3-10.7 24-24 24H24c-13.3 0-24-10.7-24-24V24C0 10.7 10.7 0 24 0h200v136c0 13.2 10.8 24 24 24zm-8 171.8c-20-12.2-33.3-29-42.7-53.8 4.5-18.5 11.6-46.6 6.2-64.2-4.7-29.4-42.4-26.5-47.8-6.8-5 18.3-.4 44.1 8.1 77-11.6 27.6-28.7 64.6-40.8 85.8-.1 0-.1.1-.2.1-27.1 13.9-73.6 44.5-54.5 68 5.6 6.9 16 10 21.5 10 17.9 0 35.7-18 61.1-61.8 25.8-8.5 54.1-19.1 79-23.2 21.7 11.8 47.1 19.5 64 19.5 29.2 0 31.2-32 19.7-43.4-13.9-13.6-54.3-9.7-73.6-7.2zM377 105L279 7c-4.5-4.5-10.6-7-17-7h-6v128h128v-6.1c0-6.3-2.5-12.4-7-16.9zm-74.1 255.3c4.1-2.7-2.5-11.9-42.8-9 37.1 15.8 42.8 9 42.8 9z"></path></svg><!-- <i class="fas fa-file-pdf"></i> Font Awesome fontawesome.com -->
                    </span>Paper
                  </a>
                </span> -->