Colliders and conformal interfaces

Abstract

We set up a scattering experiment of matter against an impurity which separates two generic one-dimensional critical quantum systems. We compute the flux of reflected and transmitted energy, thus defining a precise measure of the transparency of the interface between the related two-dimensional conformal field theories. If the largest symmetry algebra is Virasoro, we find that the reflection and transmission coefficients are independent of the details of the initial state, and are fixed in terms of the central charges and of the two-point function of the displacement operator. The situation is more elaborate when extended symmetries are present. Positivity of the total energy flux at infinity imposes bounds on the coefficient of the two-point function of the displacement operator, which controls the free-energy cost of a small deformation of the interface. Finally, we study out-of-equilibrium steady states of a critical system connecting two reservoirs at different temperatures. In the absence of extended symmetries, our result implies that the energy flux across an impurity is proportional to the difference of the squared temperatures and controlled by the reflection coefficient.

Holo-ween

“…It may be useful to note that the interface brane tension in holographic models has also been related [30] to the transmission coefficient [31,32], a quantitative measure the extent to which excitations can pass from one CFT to another across a particular interface.…”

Steady states of holographic interfaces

“…Let us pause here to take stock of the situation. We found that (i) the dual of an isolated interface must correspond to a brane that enters the ergoregion, and (ii) that the brane equations determine in this case the flow of heat in accordance with the CFT result of [24,25] and the transmission coefficients found in [10]. 14 To complete the story, we must make sure that once inside the ergoregion the brane does not come out again.…”

“…As noticed in [25], reflection positivity of the Euclidean theory gives a weaker bound [40] than this Lorentzian bound. Note also that in the asymmetric case (c 2 strictly bigger than c 1 ) part of the energy incident from side 2 is necessarily reflected.…”

“…In section 3 we explain, using the results of refs. [24,25], why the flow of heat across a 2d conformal interface is proportional to the energy-transmission coefficient(s) of the interface. It is a special feature of two dimensions that these transport coefficients are universal, that is independent of the nature of incident excitations [25].…”

“…We have used here the key fact that the energy-transport coefficients across a conformal interface in 2d are universal, i.e independent of the nature of the incident excitations. The proof [25] assumes that the Virasoro symmetry is not extended by extra spin-2 generators, which is true in our holographic model. We have also used that the incoming and outgoing excitations do not interact away from the interface.…”