Stability of gapped quantum chains under small perturbations

Abstract: We consider a family of quantum chains that has attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of on-site terms and with a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant.

In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain, that can be also applied to complex Hamiltonians obtained by considering complex values of the coupling constant.

We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved, in contrast to most approaches. Furthermore the method can be extended to higher spatial dimensions.

(Joint work with S. Del Vecchio, J. Fröhlich, and S. Rossi.)

mathematical physics

Audience: researchers in the discipline

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