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arXiv:2512.03167

Energy Reflection and Transmission of Interfaces in $T\bar{T}$-deformed CFT

Avik Banerjee, Giuseppe Policastro

Published Dec 2, 2025• 0 citations
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Abstract

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 $T\bar T$-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 $T\bar T$ 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 $T\bar T$-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.

DOI
10.48550/arxiv.2512.03167

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Energy Reflection and Transmission of Interfaces in TTˉT\bar{T}TTˉ-deformed CFT
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 TTˉT\bar TTTˉ-deformed 2D CFTs across an interface, which requires the condition cLμL=cRμRc_L μ_L = c_R μ_R cL​μL​=cR​μR​ to be obeyed. We show that, at least when the interface admits a holographic description, the TTˉT\bar TTTˉ 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 TTˉT\bar TTTˉ-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.
A bstract We investigate the reflected entropy for various mixed state configurations in the two dimensional holographic conformal field theories sharing a common interface (ICFTs). In the AdS 3 /ICFT 2 framework, we compute the holographic reflected entropy for the required configurations in the vacuum state of the ICFT 2 which is given by twice the entanglement wedge cross section (EWCS) in a spacetime involving two AdS 3 geometries glued along a thin interface brane. Subsequently, we evaluate the EWCS in the bulk geometry involving eternal BTZ black strings with an AdS 2 interface brane, which is dual to an ICFT 2 in the thermofield double (TFD) state. We explore the system from a doubly holographic perspective and determine the island contributions to the reflected entropy in the two dimensional semi-classical description involving two CFT 2 s coupled to an AdS 2 brane. We demonstrate that the results from the island formula match precisely with the bulk AdS 3 results in the large tension limit of the interface brane. We illustrate that the phase structure of the reflected entropy is quite rich involving many novel induced island phases and demonstrate that it obeys the expected Page curve for the reflected entropy in a radiation bath coupled to the AdS 2 black hole.
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.
Motivated by recent results in spin chains we study dissipation and relaxation in a two-dimensional \ak{holographic} interface conformal field theory (ICFT) in which degrees of freedom on one side of the interface are coupled to an external bath, while the other side remains isolated. In the bulk description this setup is realized by gluing a supersymmetric Janus geometry to a BTZ black hole region, with the coupling implemented through a double-trace deformation. We determine the quasinormal modes in the bulk by solving the double-trace matching conditions of the system and bath. Focusing on the fixed AdS2_22​ descendant channel at large frequency, we introduce the dimensionless ratio cUVc_{UV}cUV​ as a projected measure of interface-induced suppression of dissipation rate. Analytically we find that in this limit, cUVc_{UV}cUV​ is independent of coupling details to the bath. It is a strong candidate for a universal interface observable characterizing dissipation and relaxation across the interface, in additional to the crelaxc_{\rm relax}crelax​ raised in earlier work for lattice models.
We construct directly on the lattice a class of non-unitary interfaces that are both exactly conformal and exactly solvable, and establish their corresponding boundary and interface conformal field theory (CFT) descriptions. The construction is obtained by analytically continuing the scattering data of known exact unitary conformal interfaces on the lattice, yielding an SL(2,C)SL(2,\mathbb C)SL(2,C)-parametrized family, which is non-compact and breaks probability-current conservation. Exploiting the exact lattice-continuum correspondence, we derive the conformal boundary states in the folded picture. We show that a proper definition of the Hilbert space in the closed-string channel requires the incoming and outgoing boundary states to be specified independently by boundary data associated with a pair of dual biorthogonal bases, in close analogy with the right and left eigenvectors of a non-Hermitian Hamiltonian. This requirement determines a consistent CFT construction of non-unitary boundaries and interfaces, and leads to a non-unitary generalization of the conventional Cardy's condition for unitary boundary CFT. Beyond their formal construction, these non-unitary interfaces are shown to exhibit logarithmic entanglement scaling governed by an effective central charge that is generally complex. For the SU(1,1)SU(1,1)SU(1,1) subclass, the effective central charge remains real but grows without bound as the transmission coefficient increases. This result is demonstrated through analytical and numerical lattice calculations, as well as an interface CFT analysis in the unfolded picture. Finally, we present a general CFT analysis of a class of global quantum quenches whose initial states are prepared with non-unitary boundaries. We relate their effective temperature to the conformal dimension of the boundary-condition-changing operators associated with non-unitary boundary conditions.
Boundary conformal field theory (BCFT) and interface conformal field theory (ICFT) attract attention in the context of the information paradox problem. On this background, we develop the idea of the reflected entropy in BCFT/ICFT. We first introduce the left-right reflected entropy in BCFT and show that its holographic dual is given by the area of the entanglement wedge cross section through AdS/BCFT. We also present how to evaluate the reflected entropy in ICFT. By using this technique, we can show the universal behavior of the reflected entropy in some special classes.
This thesis is dedicated to the analysis of correlation structure in two quite different ex-amples. In the first one we investigate the entanglement entropy (EE) through conformal inter-faces in two-dimensional conformal field theories. The EE is a measure for the strength of entanglement between two subsystems of a system in a pure state. Interfaces are one-dimensional objects on a two-dimensional surface along which two conformal field theories are glued together. They are called conformal if their gluing conditions is invariant under conformal transformations that preserve the shape of the interface. Special interfaces are topological ones. They are called topological because the partition function does not change when the interface is continuously deformed. We in particular show that in a vast class of conformal field theories the presence of a topological interface at the boundary between two subregions makes the EE get dressed with an additional sub-leading but universal contribution that solely depends on the interface data. This sub-leading contribution can be interpreted as a relative entropy measuring the relative loss of entanglement compared to no interface insertion. We also compute the EE through general conformal interfaces in the critical Ising model and find that its leading order is affected by the interfaces if and only if it is not topological. Any conformal defect can be characterized by its transmissivity. As physically expected, the EE though conformal interfaces in the Ising model decreases for lower transmissivities. The second example is what we call classical holographic codes. They are classical probabilistic codes defined by a network on a uniform tiling of a constant time slice of AdS 3 -spacetime. They share some remarkable properties with certain quantum error correcting codes that are designed to mimic particular properties of the AdS/CFT correspondence. Under these features are the Ryu-Takayanagi formula and bulk reconstruction properties that both reflect deep connections between the correlation structure of a theory and the ge- ometry of its dual description. Our classical codes can be seen as toy models for holography that show that the latter features do not necessarily originate from a quantum description.