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Free Fermions Behind the Disguise

Elman, Samuel J. and Chapman, Adrian and Flammia, Steven T. (2021) Free Fermions Behind the Disguise. Communications in Mathematical Physics, 388 (2). pp. 969-1003. ISSN 0010-3616. doi:10.1007/s00220-021-04220-w.

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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. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable 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.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Elman, Samuel J.0000-0003-2140-4787
Flammia, Steven T.0000-0002-3975-0226
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 20 December 2020; Accepted 07 September 2021; Published 29 October 2021. We thank Paul Fendley and Sergey Bravyi for initially pointing us to Ref. [5]. We also thank Ryan Mann for enlightening discussions on graph theory and independence polynomials early in this project. Finally, we thank Maria Chudnovsky and Paul Seymour for providing additional details to us about the algorithm in Ref. [69] for finding a simplicial clique in a claw-free graph in polynomial time, and for sharing the result from Ref. [3] that every ECF graph has a simplicial clique. This research was supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009).
Group:AWS Center for Quantum Computing
Funding AgencyGrant Number
Australian Research CouncilCE170100009
Issue or Number:2
Record Number:CaltechAUTHORS:20211122-165424843
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Official Citation:Elman, S.J., Chapman, A. & Flammia, S.T. Free Fermions Behind the Disguise. Commun. Math. Phys. 388, 969–1003 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111965
Deposited By: Tony Diaz
Deposited On:22 Nov 2021 17:02
Last Modified:22 Nov 2021 17:02

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