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.
Conformal interfaces gluing a pair of two-dimensional conformal field theories enjoy a large degree of universality in terms of the coefficients of reflection and transmission of energy, that describe the scattering of conformal matter at the interface. In this article, we study these coefficients beyond conformality, by gluing a pair of
TTˉ-deformed 2D CFTs across an interface, which requires the condition $c_L μ_L = c_R μ_R $ to be obeyed. We show that, at least when the interface admits a holographic description, the
TTˉ deformation of the CFTs can be extended to the interface. We propose a generalization of the linear matching condition in the universal sector of the undeformed ICFT to a non-linear one, which is captured by a universal antisymmetric \emph{transmission function} of the incoming fluxes. We employ the flow equations of the
TTˉ-deformed CFTs to compute this function in two special classes of states, namely the non-equilibrium steady state (NESS) and scattering state. We show that the results can also be reproduced using holographic techniques in the bulk dual of these states.
Conformal interfaces play an important role in quantum critical systems. In closed systems, the transmission properties of conformal interfaces are typically characterized by two quantities: One is the effective central charge
ceff, which measures the amount of quantum entanglement through the interface, and the other is the transmission coefficient
cLR, which measures the energy transmission through the interface. In the present work, to characterize the transmission property of conformal interfaces in open quantum systems, we propose a third quantity
crelax, which is defined through the ratio of Liouvillian gaps with and without an interface. Physically,
crelax measures the suppression of the relaxation rate towards a steady state when the system is subject to a local dissipation. We perform both analytical perturbation calculations and exact numerical calculations based on a free fermion chain at the critical point. It is found that
crelax decreases monotonically with the strength of the interface. In particular,
0≤crelax≤cLR≤ceff, where the equalities hold if and only if the interface is totally reflective or totally transmissive. Our result for
crelax is universal in the sense that
crelax is independent of (i) the dissipation strength in the weak dissipation regime and (ii) the location where the local dissipation is introduced. Comparing to the previously known
cLR and
ceff in a closed system, our
crelax shows a distinct behavior as a function of the interface strength, suggesting its novelty to characterize conformal interfaces in open systems and offering insights into critical systems under dissipation.
Abstract We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.
We present a construction of non-equilibrium steady states within conformal field theory. These states sustain energy flows between two quantum systems, initially prepared at different temperatures, whose dynamical properties are represented by two, possibly different, conformal field theories connected through an impurity. This construction relies on a real time formulation of conformal defect dynamics based on a field scattering picture parallelizing—but yet different from—the Euclidean formulation. We present the basic characteristics of this formulation and give an algebraic construction of the real time scattering maps that we illustrate in the case of -based conformal field theories.
Conformal interfaces provide an important setting for studying universal transmission phenomena in one-dimensional quantum critical systems. While energy and information transmission across such interfaces are characterized by universal quantities in closed systems, the corresponding role of conformal interfaces in dissipative dynamics is less understood. In this work, we study relaxation in locally dissipative quantum critical chains with a conformal interface and show that a universal characterization emerges at the level of individual relaxation modes. We first show that the relaxation coefficient defined in the recent work [1] from the Liouvillian gap can become non-universal for certain boundary conditions because the mode determining the smallest decay rate can change as the interface transmission is varied. To resolve this ambiguity, we introduce a mode-resolved relaxation coefficient
crelax by continuously tracking the same Liouvillian rapidity mode as a function of the interface transmission. Using analytical and numerical calculations for a critical harmonic chain and a critical free-fermion chain, we find that, in the weak-dissipation regime,
crelax follows the same universal dependence on the interface transmission in all cases considered, independent of microscopic details such as the boundary conditions, dissipation strength, and location of the local dissipation. For boundary dissipation, this universal behavior persists even at finite dissipation strength. Our results establish a universal mode-resolved characterization of relaxation across conformal interfaces in open quantum critical systems.
The effective central charge (denoted by c_{eff}) is a measure of entanglement through a conformal interface, while the transmission coefficient (encoded in the coefficient c_{LR} of the two-point function of the energy-momentum tensor across the interface) is a measure of energy transmission through the interface. It has been pointed out that these two are generally different. In this Letter, we propose the inequalities, 0≤c_{LR}≤c_{eff}≤min(c_{L},c_{R}). They have the simple but important implication that the amount of energy transmission can never exceed the amount of information transmission. We verify them using the AdS/CFT correspondence, using the perturbation method, and in examples beyond holography. We also show that these inequalities are sharp by constructing a class of interfaces that saturate them.