Tensor Networks

One of the first steps towards designing a quantum computer is simulating quantum algorithms. A natural framework in which to achieve this is tensor networks, which have a direct correspondence to the unitary circuits executed on quantum processors, such as the IBM Eagle. The key feature of such algorithms is that they can truncate the amount of entanglement in the system in a controlled way, which enables simulation on a classical computer. Moreover, applications of tensor networks go beyond quantum simulators, as they can also be leveraged to efficiently analyze a large class of quantum many-body systems. In my current research, I am developing 2D generalizations of popular 1D tensor network algorithms to improve system-size scaling for the simulation of real materials.

• “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).

Topological Matter

Topological phases of matter cannot be characterized using a local order parameter and are robust to external perturbations. This topological protection can lead to revolutionary physical phenomena, such as the macroscopic quantization seen in quantum Hall systems, which is now used to define fundamental physical constants. Moreover, applications of topological matter go beyond metrology and device physics, as theory suggests that they can be exploited to design better quantum computers, immune to many of the obstacles associated with decoherence and error correction. In my current research, I am analyzing the stability of fractional quantum Hall states in lattice models to guide experiments in constructing anyon braid gates, which are the building blocks of a topological quantum computer.

• “Self-similarity of spectral response functions for fractional quantum Hall states with long-range interactions'', B. Andrews and G. Möller, arXiv:2201.04704 [cond-mat.str-el], Proceedings of the Royal Society A 479, 20230021 (2023).
• “Fractional Chern insulators with a non-Landau level continuum limit'', D. Bauer, S. Talkington, F. Harper, B. Andrews, and R. Roy, arXiv:2110.09565 [cond-mat.str-el], Physical Review B 105, 045144 (2022).
• “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).

Electronic Structure

Not all interesting problems in quantum many-body systems involve strongly-correlated electrons. In fact, some of the most pressing and challenging questions reduce to precisely analyzing the energy spectrum of conventional materials. Electronic structure methods leverage advanced algorithms and physical insight to achieve this goal. In my research, I am interested in the stability of molecules and crystals, in order to determine why some configurations are more energetically favorable than others. To this end, I have implemented matrix of force constants calculations directly in quantum Monte Carlo, to compute eigenmodes more efficiently than competing methods, and I have investigated the theory of non-harmonic crystal structures, to unify paradigmatic models in the literature.

• “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).
• “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

Exotic Phases

Aside from topological matter, there are several other exotic phases in quantum many-body systems that are not fully understood. In my research, I study these new phases of matter to understand and leverage their unusual properties. To this end, I have analyzed the stability of periodically-driven quantum spin systems with a non-thermal subset of eigenstates, I have investigated the extent to which skyrmions can be used to model the baryons of quantum chromodynamics, and I have modeled the anomalous expansion of turbulent Bose-Einstein condensates released from an ellipsoidal trap. In my current research, I am using a combination of electronic structure and tensor network methods to simulate the non-Hermitian skin effect in interacting graphene nanoribbons.

• “erg_loc”, B. Andrews, Python Package, UC Los Angeles, 2022.
• “Skyrmions”, B. Andrews, MASt Thesis, University of Cambridge, 2015.
• “Simulating the Expansion of Turbulent Bose-Einstein Condensates”, B. Andrews, MSci Thesis, Imperial College London, 2013.

HofstadterTools

HofstadterTools is an open-source Python package used for analyzing the Hofstadter model, which describes the behavior of non-interacting quantum particles hopping on a lattice coupled to a gauge field. Motivated by the model’s generalizability, interdisciplinary appeal, and recent experimental realization, I created this software package independently for use by the community. For more information, please see the git repo or documentation.

TeNPy

TeNPy is an open-source Python repository used for tensor network calculations. I have contributed to TeNPy by adding several examples for characterizing topological phases, as well as tutorials on simulating fractional Chern insulators and symmetry-protected topology. Currently, I am developing a closed-source branch based on algorithms for isometric projected entangled pair states. For more information, please see the git repo or release article.

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 have contributed to CASINO 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 research paper or the forum.

DiagHam

DiagHam is an open-source C++ repository used for exact diagonalization calculations. I have contributed to DiagHam by adding the functionality to compute density-density correlation functions and dynamical structure factors for fractional quantum Hall states, together with Gunnar Möller. Currently, I am implementing tools to analyze the quantum geometry of fractional Chern insulators. For more information, please see the svn repo or wiki.