Free Fermions Behind the Disguise
Abstract
An invaluable method for probing the physics of a quantum manybody 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 (evenhole, claw)free, we construct an explicit freefermion solution for H using only this structure of G, even when no JordanWigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic LiebSchultzMattis 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 freefermion 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 singleparticle 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 (evenhole, 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 clawfree only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no JordanWigner solution exists, and we give a detailed analysis of such a spin chain having 4body couplings using this method.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 2021
 DOI:
 10.1007/s0022002104220w
 arXiv:
 arXiv:2012.07857
 Bibcode:
 2021CMaPh.388..969E
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 29 pages, 9 figures