Universal logarithmic correction to R\'enyi (Shannon) entropy in generic systems of critical quadratic fermions

Abstract

The R\'enyi (Shannon) entropy, i.e. $Re_{\alpha}(Sh)$, of the ground state of quantum systems in local bases normally show a volume-law behavior. For a subsystem of quantum chains at critical point there is an extra logarithmic subleading term with a coefficient which is universal. In this paper we study this coefficient for generic time-reversal translational invariant quadratic critical free fermions. These models can be parameterized by a complex function which has zeros on the unit circle. When the zeros on the unit circle do not have degeneracy and there is no zero outside of the unit circle we are able to classify the coefficient of the logarithm. In particular, we numerically calculate the R\'enyi (Shannon) entropy in configuration basis for wide variety of these models and show that there are two distinct classes. For systems with $U(1)$ symmetry the coefficient is proportional to the central charge, i.e. one half of the number of points that one can linearize the dispersion relation of the system; for all the values of $\alpha$ with transition point at $\alpha=4$. For systems without this symmetry, when $\alpha>1$ this coefficient is again proportional to the central charge. However, the coefficient for $\alpha\leq 1$ is a new universal number. Finally, by using the discrete version of Bisognano-Wichmann modular Hamiltonian of the Ising chain we show that these coefficients are universal and dependent on the underlying CFT.