Entanglement through conformal interfaces

Abstract

We consider entanglement through permeable interfaces in the c = 1 (1+1)dimensional conformal field theory. We compute the partition functions with the interfaces inserted. By the replica trick, the entanglement entropy is obtained analytically.The entropy scales logarithmically with respect to the size of the system, similarly to the universal scaling of the ordinary entanglement entropy in (1+1)-dimensional conformal field theory. Its coefficient, however, is not constant but controlled by the permeability, the dependence on which is expressed through the dilogarithm function.The sub-leading term of the entropy counts the winding numbers, showing an analogy to the topological entanglement entropy which characterizes the topological order in (2+1)-dimensional systems.

Fano resonances and entanglement entropy

“…We find that the entropy scaling of a block neighboring the impurity is of the form (1) with a prefactor c eff which decreases monotonously as the parameters are tuned towards the Fano resonance and depends only on a well-defined scattering amplitude. The numerical analysis leads to the same functional form of c eff that has recently been derived for simpler fermionic models with interface defects 9 and also seems to be closely related to the one found for conformal interfaces 12 . If the subsystem contains an open boundary we find a rapidly oscillating subleading term in the entropy that persists even in the large L limit.…”

“…(12). The remaining integral is taken along the contour C 2 which is parametrized as z = e ±iqF e −q ′ and using the analytic continuation of the scattering phase it reads…”

“…where 1/2 comes from the unmodified boundary while the integral is, up to a sign, the same as in 12 and describes the contribution of the impurity. The integral (19) depends on the model parameters only through the variable…”

“…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-geometric backgrounds based on topological interfaces

“…The torus partition functions with multiple insertions of the general u(1)-preserving conformal interfaces have been evaluated in [33]. The evaluation of (3.9) is much simpler and, once it is obtained explicitly, we can uniquely determine the fiber partition function with general winding Z fiber (w,m) (τ ) ≡ Z fiber λ=wτ +m (τ ) so that the total partition function (3.8) becomes modular invariant.…”