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We analyze the quantum phases, correlation functions and edge modes for aclass of spin-1/2 and fermionic models related to the 1D Ising chain in thepresence of a transverse field. These models are the Ising chain withanti-ferromagnetic long-range interactions that decay with distance $r$ as$1/r^\alpha$, as well as a related class of fermionic Hamiltonians thatgeneralise the Kitaev chain, where both the hopping and pairing terms arelong-range and their relative strength can be varied. For these models, weprovide the phase diagram for all exponents $\alpha$, based on an analysis ofthe entanglement entropy, the decay of correlation functions, and the edgemodes in the case of open chains. We demonstrate that violations of the arealaw can occur for $\alpha \lesssim1$, while connected correlation functions candecay with a hybrid exponential and power-law behaviour, with a power that is$\alpha$-dependent. Interestingly, for the fermionic models we provide an exactanalytical derivation for the decay of the correlation functions at every$\alpha$. Along the critical lines, for all models breaking of conformalsymmetry is argued at low enough $\alpha$. For the fermionic models we showthat the edge modes, massless for $\alpha \gtrsim 1$, can acquire a mass for$\alpha < 1$. The mass of these modes can be tuned by varying the relativestrength of the kinetic and pairing terms in the Hamiltonian. Interestingly,for the Ising chain a similar edge localization appears for the first andsecond 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 new phases, notably approachedvia quantum phase transitions without mass gap closure. Finally, we discuss thepossibility to detect some of these effects in experiments with cold trappedions.

Long-range Ising and Kitaev Models: Phases, Correlations and Edge Modes