QSI Seminar: Dr Adrian Chapman, USyd, Characterization of free-fermion-solvable spin models 07/07/20

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).

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