Topological Matter

Topological phases of matter is an active and rapidly-growing field of research. So far, my focus has been broadly on the fractional quantum Hall effect and the Hofstadter model. During my PhD, I studied a series of lattice-generalised fractional quantum Hall states called fractional Chern insulators. Specifically, we numerically verified a predicted filling fraction series for higher Chern bands in the Hofstadter model using exact diagonalization. Currently, I am investigating fractional quantum Hall states in the flat bands of Moiré superstructures using infinite cylinder density matrix renormalization group. Motivated by the progress in the field of twisted bilayer graphene, we are finding ways to characterize and engineer exotic fractional states in realistic bilayer materials.

• “Scaling of spectral response functions for fractional quantum Hall states with long-range interactions'', B. Andrews and G. Möller (in preparation).
• “Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model'', B. Andrews, T. Neupert, and G. Möller, arXiv:2107.06047 [cond-mat.str-el], Physical Review B 104, 125107 (2021).
• “Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models'', B. Andrews, M. Mohan, and T. Neupert, arXiv:2007.08870 [cond-mat.str-el], Physical Review B 103, 075132 (2021).
• “Fractional quantum Hall states for moiré superstructures in the Hofstadter regime'', B. Andrews and A. Soluyanov, arXiv:2004.06602 [cond-mat.str-el], Physical Review B 101, 235312 (2020).
• “Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1'', B. Andrews and G. Möller, arXiv:1710.09350 [cond-mat.str-el], Physical Review B 97, 035159 (2018).

Quantum Monte Carlo

Quantum Monte Carlo is a leading numerical technique for electronic structure calculations, and unlike competing techniques, such as density functional theory, can potentially compute energies to arbitrary accuracy. Efficiently computing forces with quantum Monte Carlo however, was a great challenge for the community, mainly due to the heavy tails of the force distribution. After several decades work, forces can now be efficiently computed and are routinely used to relax atoms in molecules and crystals. Using the lessons learned from calculating forces, we implemented a method to compute the matrix of force constants in quantum Monte Carlo. We showed that, for simple molecules, this method can relax atoms more quickly and compute the vibrational eigenfrequencies more accurately, compared to existing algorithms.

• “Direct evaluation of the force constant matrix in quantum Monte Carlo'', Y. Y. Liu, B. Andrews, and G. Conduit, arxiv:1901.04396 [physics.comp-ph], The Journal of Chemical Physics 150, 034104 (2019). Editor’s Pick

Crystal Stability

Much of what we currently understand about crystals today dates back to the Drude theory of solids from 1904. We intuitively think of atoms in a crystal as balls connected by springs, when discussing concepts such as harmonic modes or phonons. However, in this work we show that certain crystals lattices have a zero diagonal matrix of force constants and so cannot be thought of with springs. Instead these crystal lattices are (de)stabilized at fourth order. We construct simplified models for crystals in the tight-binding and free-electron limits, and derive a phase diagram for the most energetically favorable crystal lattices in these regimes. Finally, we use this to gain insight into the distribution of crystal structures for elemental solids in the periodic table.

• “Absence of diagonal force constants in cubic Coulomb crystals'', B. Andrews and G. Conduit, arXiv:2007.00476 [cond-mat.mtrl-sci], Proceedings of the Royal Society A 476, 20200518 (2020).

Bose-Einstein Condensation

Motivated by recent experiments, I studied numerically the anomalous aspect ratio inversion of ellipsoidal Bose-Einstein condensates once they are released from a harmonic trap. Typically, one would expect that the aspect ratio of a released ellipsoidal condensate would invert due to the larger kinetic energy along the minor axis. The experiments showed, however, that if the condensate contains vortices, or is turbulent, then the aspect ratio stays approximately constant as the cloud expands. For my MSci project, I developed a simulation of this expansion based on solutions of the Gross-Pitaevskii equation, and studied the effect of vortices on aspect ratio inversion.

• “Simulating the Expansion of Turbulent Bose-Einstein Condensates”, B. Andrews, MSci Thesis, Heidelberg University, 2013.

TeNPy

Tensor Network Python (TeNPy) is an open-source Python repository used for density network normalization group calculations. I am developing this code for my current research, together with Johannes Hauschild and Leon Schoonderwoerd. My main contribution has been the addition of examples for characterizing topological phases from the ground states on an infinite cylinder. For more information, please see the github repo or forum.

CASINO

CASINO is a commercial Fortran repository used for quantum Monte Carlo calculations. Originally designed as an in-house code for the University of Cambridge, CASINO is now used by researchers all over the world. I contributed by adding the capability to directly compute the matrix of force constants in quantum Monte Carlo, together with Yu Yang Liu and Gareth Conduit. For more information, please see our recent paper or the forum.

DiagHam

DiagHam is an open-source C++ repository used for exact diagonalization calculations. I contributed to this code at the start of my PhD and am now developing it further for my current research, together with Gunnar Möller. My main contribution has been the capability to compute density-density correlation functions for the Hofstadter model. For more information, please see the svn repo or wiki.