A note on simplicial cliques

Chudnovsky, Maria; Scott, Alex; Seymour, Paul; Spirkl, Sophie

doi:10.48550/arxiv.2012.05287

Cited by 1 publication

(4 citation statements)

References 6 publications

(19 reference statements)

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“…A graph G is said to be (even-hole, claw)-free, or ECF, if it contains no even holes and no claws among its induced subgraphs (see It can be shown that all (even-hole, claw)-free graphs are simplicial, meaning they contain a simplicial clique [3]. If a Hamiltonian H is ECF then its frustration graph is necessarily simplicial, so we say that H is simplicial as well.…”

Section: Definition 3 (Ecf)mentioning

confidence: 99%

“…It is therefore computationally efficient to recognize a simplicial clique in an ECF graph. Moreover, every (nonempty) ECF graph has at least one simplicial clique [3].…”

Section: Relation To Prior Workmentioning

confidence: 99%

“…A graph is claw-free or even-hole-free if none of its vertex-induced subgraphs contain (b) the "claw" K 1,3 or (c) an even hole C 2k . If a graph is (even-hole, claw)-free, it necessarily contains a simplicial clique [3].…”

Section: Introductionmentioning

confidence: 99%

“…[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).…”

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confidence: 97%

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Free fermions behind the disguise

Elman,

Chapman,

Flammia

2020

Preprint

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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. 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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Section: Definition 3 (Ecf)mentioning

confidence: 99%

“…It is therefore computationally efficient to recognize a simplicial clique in an ECF graph. Moreover, every (nonempty) ECF graph has at least one simplicial clique [3].…”

Section: Relation To Prior Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 97%

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Free fermions behind the disguise

Elman,

Chapman,

Flammia

2020

Preprint

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A note on simplicial cliques

Cited by 1 publication

References 6 publications

Free fermions behind the disguise

Free fermions behind the disguise

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