We propose and analyze a generalization of the Kitaev chain for fermions with long-range p-wave pairing, which decays with distance as a power law with exponent α. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range (α > 1) or purely algebraically (α < 1). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small α. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap, and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions. [4,5] has sparked renewed interest in novel properties of topological models as well as in experimental realizations. For example, Kitaev chains with long-range hopping and pairing have been recently proposed as models for helical Shiba chains, made of magnetic impurities on an s-wave superconductor [6].Intimately related to the Kitaev chain, Ising-type spin chains with tunable long-range interactions can now be realized using trapped ions coupled to motional degrees of freedom or, alternatively, using neutral atoms coupled to photonic modes [7][8][9][10][11]. Very recently, theory and experiments have provided evidence for novel static and dynamic phenomena in these systems, such as, e.g., the non-local propagation of correlations [8,9,[12][13][14] or the possible violation of the area law in one dimension [15]. While some of these phenomena can be explained theoretically using approximate analytical and numerical methods [15,16], it remains a fundamental challenge to determine basic properties of long-range interacting systems, where methods based on short-range models may fail.In this work, we introduce and analyze an exactly solvable model for one-dimensional fermions with long-range pairing, decaying with distance r as a power-law ∼ 1/r α . We analyze the phase diagram as a function of the power α of the pairing, finding several novel features. These include: (i) gapped phases for α > 1 where the decay of correlation functions evolves from exponential to algebraic from short to long distances and (ii) a gapped phase with a purely algebraic decay of correlations for α < 1. For the open chain, we find that (iii) the localization of the edge modes, similar to the case of correlations, varies from hybrid (exponential followed by algebraic) for α > 1 to purely algebraic for α < 1, where these modes become gapped. Throughout the phase diagram, (iv) the entanglement entropy fails to capture some of the main features of the energy spectrum and correlation functions. However, it correctly predicts (v) an exotic transition along one of the two critical lines induced by long-range pairing from an Ising-type theory for α > 3/2 to a Luttinger-liquid-type theory for α < 3/2. This corresponds to (vi) a breaki...

We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distancer as a r 1 , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α, based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for  a 1, while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α-dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α. Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α. For the fermionic models we show that the edge modes, massless for  a 1, can acquire a mass for a < 1. The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. 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 edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.

Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finitedimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.