# Colliders and conformal interfaces

## TL;DR

This paper investigates energy transport across conformal interfaces in one-dimensional critical quantum systems, deriving universal reflection and transmission coefficients and exploring their bounds and behavior in equilibrium and out-of-equilibrium states.

## Contribution

It provides a universal framework for computing energy fluxes at conformal interfaces, including bounds and state-independence results, extending understanding of quantum impurity problems.

## Key findings

- Reflection and transmission coefficients depend only on central charges for Virasoro symmetry.
- Positivity constraints impose bounds on interface deformation parameters.
- Energy flux in out-of-equilibrium states is proportional to temperature differences and reflection coefficients.

## 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.