Restoring phase coherence in a one-dimensional superconductor using power-law electron hopping

In a one-dimensional (1D) superconductor, zero-temperature quantum fluctuations destroy phase coherence. Here we put forward a mechanism which can restore phase coherence: power-law hopping. We study a 1D attractive-U Hubbard model with power-law hopping using Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, noninteger spatial dimensionality d(eff). For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppresses fluctuations and restores superconducting long-range order for any d(eff) > 1, at zero temperature. A detailed DMRG analysis fully supports these findings. These results are also of direct relevance to quantum magnetism as our model can be mapped onto an S = 1/2 XXZ spin chain with power-law decaying couplings, which can be studied experimentally with cold-ion-trap techniques.