During this time of lockdown, the centre for quantum software and information (QSI) at the University of Technology Sydney has launched an online seminar series.  With talks once or twice a week from leading researchers in the field, meQuanics is supporting this series by mirroring the audio from each talk.  I would encourage if you listen to this episode, to visit and subscribe to the UTS:QSI YouTube page to see each of these talks with the associated slides to help it make more sense.

https://youtu.be/4KIXVQtR9Qw

Free-Fermion Solutions and Frustration Graphs  

TITLE: Characterization of free-fermion-solvable spin models via graph invariants 

SPEAKER: Dr Adrian Chapman 

AFFILIATION: ARC Centre of Excellence for Engineered Quantum Systems (EQUS), University of Sydney, Australia 

HOSTED BY: A/Prof Chris Ferrie, UTS Centre for Quantum Software and Information  

ABSTRACT:  Finding exact solutions to spin models is a fundamental problem of many-body physics. A workhorse technique for exact solution methods is mapping to an effective description by noninteracting fermions. The paradigmatic example of this is the Jordan-Wigner transformation for finding an exact solution to the one-dimensional XY model. Another important example is the exact free-fermion solution to the two-dimensional Kitaev honeycomb model.  I will describe a framework for recognizing general models which can be solved this way by utilizing the tools of graph theory. Our construction relies on a connection to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of this result is a complete set of constant-sized frustration structures which obstruct a free-fermion solution. We classify the kinds of Pauli symmetries which can be present in models for which a free-fermion solution exists, and we find that they correspond to either: (i) gauge qubits, (ii) cycles on the free-fermion hopping graph, or (iii) the fermion parity. Clifford symmetries, except in finitely-many cases, must be symmetries of the free-fermion Hamiltonian itself. We expect our characterization to motivate a renewed exploration of free-fermion-solvable models, and I will close with an elaborate discussion of how we expect to generalize our framework beyond generator-to-generator mappings.  

RELATED ARTICLES: Characterization of solvable spin models via graph invariants. Quantum 4, 278 (2020). Characterization of solvable spin models via graph invariants: quantum-journal.org/papers/q-2020-06-04-278/  

OTHER LINKS: Adrian Chapman Webpage: https://equs.org/users/adrian-chapman