Long-range Ising and Kitaev models: phases, correlations and edge modes

Weanalyze the quantumphases, correlation functions and edgemodes for a class of spin-1/2 and fermionicmodels related to the one-dimensional Ising chain in the presence of a transverse field. Thesemodels are the Ising chainwith anti-ferromagnetic long-range interactions that decaywith distancer as a r 1 , as well as a related class of fermionicHamiltonians that generalize theKitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For thesemodels, we provide the phase diagram for all exponentsα, based on an analysis of the entanglement entropy, the decay of correlation functions, and the edgemodes in the case of open chains.We demonstrate that violations of the area law can occur for  a 1, while connected correlation functions can decaywith a hybrid exponential and power-law behavior, with a power that isα-dependent. Interestingly, for the fermionicmodels we provide an exact analytical derivation for the decay of the correlation functions at everyα. Along the critical lines, for allmodels breaking of conformal symmetry is argued at low enoughα. For the fermionicmodels we show that the edge modes,massless for  a 1, can acquire amass for a < 1. Themass of thesemodes can be tuned by varying the relative strength of the kinetic and pairing terms in theHamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edgemodes are not expected.We argue that, at least for the fermionic chains, thesemassive states correspond to the appearance of newphases, notably approached via quantumphase transitionswithoutmass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.