Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain

We deal with the problem of studying the effective theories and the symmetries of long-range models around critical points. We focus in particular on the Kitaev chain with long-range pairings decaying with distance as power-law with exponent $\alpha$. This is a quadratic solvable model, yet displaying non-trivial quantum phase transitions. By renormalization group approach we derive first the effective theory close to the critical line at positive chemical potential. This is the sum of two terms: a Dirac action $S_D$, found in the short-range Ising universality class, and a CS breaking term $S_{AN}$. While $S_D$ originates from low-energy excitations, $S_{AN}$ derives from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality $S_{AN}$ flows to zero if $\alpha > 2$, while if $\alpha < 2$ it dominates and determines the breakdown of the CS. Out of criticality $S_{AN}$ breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite $\alpha$. As $\alpha$ increases, such ELI breakdown becomes less and less pronounced and in the short-range limit $\alpha \to \infty$ the ELI is restored. To test the validity of the effective theory, we compared the two-fermion static correlations and the von Neumann entropy obtained from them with the ones calculated on the lattice, finding agreement. From the effective theory one can also see that new phases emerge for $\alpha < 1$. Finally we show that at every finite $\alpha$ the critical exponents are the same as for the short-range ($\alpha \to \infty$) model. At variance, for the critical line with negative chemical potential, only $S_{AN}$ ($S_D$) is present for $1 < \alpha < 2$ (for $\alpha > 2$). Close to this line, where the minimum of the spectrum coincides with the momentum where singularities develop, the critical exponents change exactly where CS is broken.