A unified graph-theoretic framework for free-fermion solubility

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• Sam Elman (UTS Sydney)
• Friday 21 July 2023, 14:00-15:00
• MR14.

If you have a question about this talk, please contact Sergii Strelchuk.

An invaluable method for probing the physics of a quantum many-body spin system is a mapping to a system of non-interacting fermions. The canonical transformation from spin operators to fermionic operators is, of course, the Jordan-Wigner transform, which is both generic, in that it is not dependent on fine tuning of coupling coefficients, and generator-to-generator. In recent work, it was shown that a Jordan-Wigner-like mapping to free fermions is possible if and only if the frustration graph of the Hamiltonian is a line graph. More recently, a free-fermion model outside of the Jordan-Wigner framework, called the four-fermion model, was presented by Fendley in his paper Free fermions in disguise. The solution holistically maps the Hamiltonian to free fermions and is generic despite transcending the Jordan-Wigner structure. Surprisingly, the existence of a solution of this form is also revealed by the structure of the Hamiltonians frustration graph. In this work, we show that a quantum spin model has an exact description by free fermions if its frustration graph is claw-free and contains a structure called simplicial clique, which is a class of graphs containing the line graph class. In this way, we unify models which can be solved by Fendley’s method and those soluble by Jordan-Wigner. Extending the solution method to arbitrary spatial dimensions is made possible by the identification of a new class of Hamiltonian symmetries corresponding to graphical structures called generalised even holes.

This talk is part of the CQIF Seminar series.

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