Free Fermions Behind the Disguise

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. 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.

Integrable spin chains and the Clifford group

“…This includes, as a special case, the Clifford transformations that we have analysed and thus puts our results in context by relating them to classifications of integrable models. We also treat the case of spin chains solvable by transforming to free fermions, making connections to recent papers on families of exactly solvable spin-1 /2 chains [24][25][26][27]. Finally, we discuss further avenues of research.…”

A note on simplicial cliques

“…Each spin model is defined by a Hamiltonian operator, and to every such Hamiltonian one can associate a graph, called its frustration graph. In [4] a new method is given that allows us to "solve a model" (meaning in this case to find the spectrum and the eigenvectors of the Hamiltonian) whose frustration graph is even-hole-free, claw-free, and has a simplicial clique. This augments earlier results of [1] where it is shown that models whose frustration graphs are line-graphs are solvable using certain classical tools.…”

“…This augments earlier results of [1] where it is shown that models whose frustration graphs are line-graphs are solvable using certain classical tools. The solution method of [4] uses only the structure of the frustration graph, and it is an extension of both [5] and [6]. The authors of [4] raised a question:…”

“…The solution method of [4] uses only the structure of the frustration graph, and it is an extension of both [5] and [6]. The authors of [4] raised a question:…”

“…In [2] an algorithm is presented that finds a simplicial clique in a claw-free graph if one exists. The authors of [4] also asked if that algorithm can be simplified when the input is known to be even-hole-free. An easy corollary of our main result 1.2 is such a simpler, but slower, algorithm 4.1.…”