# Projects past and present

Below are semi-detailed descriptions of the research projects I have worked on over the course of my time studying physics. I will try to include hyperlinks for concepts that maybe deserve a bit more exposition. Some projects are still active, while others have run their course. They can be roughly lumped into the following categories

Note: This section is still  under construction.

Non-equilibrium quantum many-body physics
The Quantum Hamiltonian Mean Field model
Caustics in coupled condensates

Effective Field Theory
Point particle effective field theory in the lab

###### The Quantum Hamiltonian Mean Field model

The HMF model is the toy model of long-range interacting systems (in the way the Ising model is for critical phenomena). It describes $N$ particles of mass $m$ interacting on a ring of radius $R$ by the following Hamiltonian

$\displaystyle H=\sum\limits_i \frac{L_i^2}{2mR^2} + \frac{\epsilon}{N} \sum\limits_{i

where $L_i$  is the (angular) momentum conjugate to $\theta_i$, and $\epsilon/N$ is the inter-particle coupling. The coupling is re-scaled to ensure a non-trivial limit as $N\rightarrow \infty$, this is known as the Kac prescription.

We have been studying the quantum version of the model. Our first result was to determine the fate of the so-called bi-cluster two of our main results are shown below. We managed to make  non-equilibrium phase diagram that demonstrates how quantum zero-point energy modifies the dynamics, resulting in the interference pattern shown below

Non-equilibrium phase diagram that shows how quantum effects change the model’s non-eq. behaviour.

Bi-cluster seeded by  small spatial variations in the phase of $\Psi$

Papers:

• Violent relaxation in a long-range interacting Bose condensed gas: quantum dynamics of the HMF model. (submitted for publication)
• Emergent solitons in a long-range interacting Bose gas (in preparation)
• Many-body ground state of the bosonic HMF model (research ongoing)
• Realizing the HMF model in a lab (future work)

###### Point particle effective field theory in the lab

While, PPEFT is interesting in its own right, the emphasis of my interest in this project is the latter half of the title; namely  “in the lab”.

The seeds of the project started with Cliff Burgess’ proposal to develop a systematic treatment of field theories interacting with a compact body (PPEFT). A powerful, and insightful setting to understand features of PPEFT is the quantum mechanics of the inverse square potential. The inverse square potential is interesting because of a phenomena known as “fall to the centre” and has connections to

• Near horizon physics of black holes
• Efimov physics
• Conformal symmetry breaking

This project has three basic goals

1.  Identify how to parameterize the inverse square potential as a PPEFT, and identify a universal set of RG invariants.
2.  Propose an explicit proposal to realize, and test this system in a lab.
3. Connect the results of any future experiment to other physical systems such as black hole and Efimov physics by using the universal parameterization mentioned above.

Papers:

• Paper 1
• Paper 2