Entanglement entropy and negativity in the Ising model with defects

TL;DR

This paper investigates how energy and duality defects in the Ising conformal field theory affect entanglement entropy and negativity, revealing unique defect signatures and finite-size effects through numerical DMRG methods.

Contribution

It provides the first detailed numerical analysis of entanglement measures in the Ising model with defects, confirming some recent free fermion results and challenging earlier field theory predictions.

Findings

Duality defect shows different EE behavior than energy defect due to zero modes.

Finite-size corrections cause deviations from standard logarithmic scaling.

Logarithmic negativity scales with separation, revealing defect fingerprints.

Abstract

Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we compute the entanglement entropy (EE) and the entanglement negativity (EN) of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial `finite-size correction' in the EE obtained recently using free fermion computations. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar `finite size corrections' like the EE which leads to deviations from its characteristic logarithmic scaling.