Stability of ground state degeneracy to long-range interactions

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g. power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (a) the stability of the gap, and (b) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.

Automated summary

We show that certain gapped quantum many-body systems have a stable ground state degeneracy to long-range perturbations, with exponentially small ground state energy splitting induced by such perturbations. Specifically, we study an Ising symmetry-breaking Hamiltonian with degenerate ground states and a energy gap, and perturb the system with Ising symmetric long-range interactions. Our proof relies on a convergent polymer expansion adapted to handle the long-range interactions. We also discuss applications to physical models, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.