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Statistical and Computational Physics

Physics professor Valéry Rousseau is currently working on two topics, an electronic analog quantum simulator that he designed, and quantum Monte Carlo simulations of various forms of the Hubbard model by using the Stochastic Green Function method that he developed.

Analog Quantum Simulator

With the help of one of his former students at the College of Wooster, Matthew King-Smith, prof. Rousseau is working on an alternative approach to quantum problem solving. The idea is based on an electronic implementation of a recurrent neural network, where the matrix elements of the Hamiltonian to be simulated are encoded in the neurons. The outputs of this linear neural network are fed back to its inputs, which forces the system to fall into an eigenstate. This method has been experimentally applied by Matthew King-Smith during his senior Independent Study thesis, by programming analog micro-controllers to act as neurons. The experimental device was able to correctly determine the eigenvectors and their associated eigenvalues for a two-dimensional Hilbert space, which constituted a proof of concept. The focus is now to implement this method in a microchip in order to be able to simulate Hilbert spaces with much higher dimensions. This method could be used to solve systems of fermionic particles on lattices, a problem that is still unsolvable (except for very small systems) by quantum Monte Carlo methods due to the so-called ``sign problem".

The Stochastic Green Function Method

Almost all quantum Monte Carlo methods for the simulation of lattice Hamiltonians require the Hamiltonian to be expandable as a sum of two-site coupling terms. However, many models of interest currently under investigation have four-site, and even six-site, coupling terms. Adapting these methods to these models is challenging and model-dependent. With the aim of solving this problem, prof. Rousseau developed the Stochastic Green Function method which can be directly applied to any Hamiltonian that does not suffer from the nemesis sign problem. With this method, prof. Rousseau is currently investigating various challenging models, such as bosons in a three-dimensional pyrochlore lattice with six-site coupling terms, of bosonic mixtures of atom-molecules undergoing a Feshbach resonance.