Entanglement entropy with localized and extended interface defects

Abstract

The quantum Ising chain of length, L, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is non-universal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time t > 0. For the localized defect the increase of the entropy with log L or with log t involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated, but the non-equilibrium entropy has a logarithmic time-dependence the prefactor of which depends on the strength of the defect.

Fano resonances and entanglement entropy

“…An example is given by critical lattice models where single defective links separate the two subsystems. This can lead to a modified prefactor for the logarithm varying continuously with the defect strength in models with free fermions [6][7][8][9] , while for interacting electrons the defect either renormalizes to a cut or to the homogeneous value in the L → ∞ scaling limit 10,11 . In the more general context of a conformal interface separating two CFTs the effective central charge has been calculated recently and was shown to depend on a single parameter 12 .…”

Non-equilibrium dynamics of one-dimensional isolated quantum systems

“…The ground state [122], [163][164][165] and the dynamical [122,166] entanglement entropy across a defect also shows a similar behavior: the so-called effective central charge is a prefactor of the entropy, and a continuous function of the defect parameters. It is expected, that the non-equilibrium relaxation of the magnetization is also non-universal.…”

“…Numerical investigations about the dynamics of the entanglement entropy were done in [56] and [122]. In this subsection the main numerical results about the after quench dynamics are recapitulated.…”

“…Later this type of description was modified to describe zero temperature non-equilibrium dynamics [37] of the local magnetization and the correlation functions and also has been used to interpret the dynamical entanglement entropy [122] [56]. In this subsection we follow articles [37] and [122] and describe the dynamics of the magnetization and the entanglement entropy with the use of quasi-particles.…”

“…These conformal methods usually work for appropriate boundary conditions: κ = 0 (uncoupled half chains) or κ = ∞ (fixed local spin) and κ = 1, i.e., uniformly coupled chain. Among other quantities the after quench entanglement entropy has been studied by CFT ( [122,[144][145][146][147]), after joining two initially independent systems (changing from κ 1 = 0 to κ 2 = 1), the entropy grows logarithmically [142], and the prefactor of the logarithm is universal S(t) = (c/3) ln t + const, it is one third of the central charge of the CFT. In finite systems the entropy cannot grow without limit, in a system of length L the entropy starts to grow for short times, but for long times it shows a periodic variation in terms of t/L [143].…”