## Summary

Use the Julia language^{1} to implement the dTWA and cTWA algorithms and apply them to various scenarios to learn something about the physics.

Note: Previous experience with Julia is not required if you have a strong programming background.

## Background

The discrete truncated Wigner approximation (dTWA)^{2} is a semi-classical method for simulating the time evolution of quantum spin systems. The core idea is to sample classical initial states for each spin from the quantum initial state and propagate them with classical mean-field equations. It can be shown, that this is exact at short times but surprisingly it turns out to work remarkable well even at late times. The downside of dTWA is its uncontrolled nature. There is no good way of knowing whether your results are accurate.

There is a generalization of the dTWA procedure, called cluster truncated Wigner approximation (cTWA)^{3}, where spins are clustered together before sampling and propagation. This means that quantum mechanical interactions within the clusters are treated exactly and only the inter-cluster interaction is approximated semi-classically. In turn, this gives a tuning parameter to check the results for convergence, as in the limit of large cluster size (when there is only 1 cluster containing ALL spins) the procedure equivalent to the full quantum dynamics.

The system of interest is a Heisenberg-type spin system where the couplings arise from power-law interaction between randomly positioned sites. Such systems arise naturally in cold atomic gases, e.g. the Rydberg experiment of the Weidemüller group who we are collaborating with. Crucially, we can control the strength of the randomness via the system’s density. At strong disorder, we find that the system becomes many-body localized^{4}, which essentially means the dynamics are slowed down to the point where the system decomposes into smaller parts that can’t really talk to each other anymore. In this setting, dTWA does not work well (because it always predicts a diffusive behavior at late times) but cTWA should become (almost?) exact for smallish clusters - if they are chosen right.

## Project outline

First milestone: Implement dTWA and cTWA in Julia

Apply (some of) the following questions/problems:

- Study convergence of dTWA/cTWA in thermal and MBL systems and influence of choide of clustering.
- No experiment is perfectly 1D/2D. What is the effect of “small dimensions” on the dynamics?
- Thermalising quantum systems become hydrodynamic at late times
^{5}, but dTWA gets the diffusion constant wrong. Can cTWA do better and can we study the change in diffusion towards the MBL regime?