Add some more explanation and images.

README.md

@@ -1,10 +1,45 @@
-# Long-Range Falikov Kimball Model in One Dimension: Phase Transition, Disorder-Free Localisation and Delocalisation
-
+## Markov Chain Montecarlo for Quantum Tight Binding Models with Classical Sectors.
 [![DOI](https://zenodo.org/badge/174118363.svg)](https://zenodo.org/badge/latestdoi/174118363)
 
-This repository contains code to perform Markov Chain Monte Carlo (MCMC) simulations of a 1D tight binding model with a mixture of classical and quantum degrees of freedom. The strategy used is to perform an MCMC walk over the classical configurations of the system, diagonalising the quantum hamiltionian at each step. Includes code for submission to cluster systems and analysis of the results, such as computing an energy-resolved Inverse Participation Ratio.
+This repo contains code to perform [Markov Chain Monte Carlo][mcmc] over [the Falikov-Kimball model][fk]. It was written for the publication "Long-Range Falikov Kimball Model in One Dimension: Phase Transition, Disorder-Free Localisation and Delocalisation" coming soon to the arXiv! I provide the code here as a record of what was done, it's likely not reproducable without extra help but if you wish to know more please get in touch.
+
+---
+
+### Applicability and Optimisations 
+
+The code exploits two special properties of this model and would work well on other models with either property:
+
+<p align="center">
+  <img width="460" src="https://user-images.githubusercontent.com/2063944/111747025-17724e00-888f-11eb-928a-a98f2f65d70f.png">
+</p>
+
+1. The model contains a classical degree of freedom (in this case S_i) and a quantum degree of freedom (c_i) such that when the classical degree of freedom is fixed the model is quadratic in c_i or otherwise easy to solve.
+2. The model contains entirely classical terms like J_ij S_i S_j
+
+Property 1 allows the MCMC to define a walk over the classical states of the system and solve the quantum system at each point, effectively factoring the classical subspace out of the full Hilbert space. 
+
+---
 
-## Installation
+Property 2 allows for an optimisation where the standard Metropolis-Hastings alogorithm is modified to compute the classical terms first, perform a probabalistic accept/reject step, and only on acceptance compute the quantum terms, saving upto 90% computation time in our model.
+
+<p align="left">
+  <img height="300" src="https://user-images.githubusercontent.com/2063944/111749795-8f8e4300-8892-11eb-9d0a-afab6a83d964.png">
+</p>
+
+Also includes code for submission to cluster systems (PBS and SLURM) and analysis of the results, such as computing energy-resolved Inverse Participation Ratios.
+
+[mcmc]: https://arxiv.org/abs/cond-mat/9612186
+[fk]: https://arxiv.org/abs/math-ph/0502041
+
+---
+A phase diagram for the above model showing the Charge Denisity Wave phase where the spins order antiferromagnetically, the gapped Mott Insulator phase and the gapless Anderson Insulator phase.
+<p align="left">
+  <img height="300" src="https://user-images.githubusercontent.com/2063944/111750304-5bffe880-8893-11eb-8e4d-73276c1240fb.png">
+</p>
+
+---
+
+### Installation and Use
 1. Clone the repo
     ```sh
     git clone https://github.com/TomHodson/FKMC
@@ -40,4 +75,3 @@ A typical use would be to:
 ```sh
 ./batchscripts/submit.py 
 ```
-1.