Free fermions behind the disguise

• URL: http://arxiv.org/abs/2012.07857v2
• Date: Mon, 8 Nov 2021 02:07:21 GMT
• Title: Free fermions behind the disguise
• Authors: Samuel J. Elman, Adrian Chapman and Steven T. Flammia
• Abstract summary: We find mappings to noninteracting fermions in a quantum many-body spin system. We generalize both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain dubbed "free fermions in disguise"
• Score: 0.3222802562733786
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph $G$ of a Hamiltonian $H$, i.e., the network of anticommutation relations between the Pauli terms in $H$ in a given basis. Specifically, when $G$ is (even-hole, claw)-free, we construct an explicit free-fermion solution for $H$ using only this structure of $G$, even when no Jordan-Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed "free fermions in disguise." Like Fendley's original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of $G$. The associated single-particle energies are calculated using the roots of the independence polynomial of $G$, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. We give several example families of solvable models for which no Jordan-Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

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