<p>We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math> dimensions, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>th Rényi entropy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>S</mi><mi>A</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup></math> behaves as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi>f</mi><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>n</mi><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mtext>Area</mtext><mo stretchy="false">(</mo><mo stretchy="false">∂</mo><mi>A</mi><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">−</mo><mn>2</mn><mo stretchy="false">)</mo><msup><mrow><mi>ε</mi></mrow><mrow><mi>d</mi><mo stretchy="false">−</mo><mn>2</mn></mrow></msup><mo stretchy="false">]</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mrow></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mo stretchy="false">→</mo><mn>0</mn></math> limit when the boundary of the entanglement domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math> is spherical with the UV cutoff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ε</mi></math>. The theory dependence is encapsulated in the cosmological constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math> in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.</p>

Japan

Yuya Kusuki

United States

Hirosi Ooguri

Sridip Pal

Universality of Rényi Entropy in Conformal Field Theory

A state in a quantum system with a given global symmetry, , can be sensitive to the presence of boundaries, which may either preserve or break this symmetry. In this work, we investigate how conformal invariant boundary conditions influence the -symmetry breaking through the lens of the entanglement asymmetry, a quantifier of the “distance” between a symmetry-broken state and its symmetrized counterpart. By leveraging 2D boundary conformal field theory (BCFT), we investigate the symmetry breaking for both finite and compact Lie groups. Beyond the leading order term, we also compute the subleading corrections in the subsystem size, highlighting their dependence on the symmetry group and the BCFT operator content. We further explore the entanglement asymmetry following a global quantum quench, where a symmetry-broken state evolves under a symmetry-restoring Hamiltonian. In this dynamical setting, we compute the entanglement asymmetry by extending the method of images to a BCFT with non-local objects such as invertible symmetry defects.

Sara Murciano

Entanglement asymmetry and symmetry defects in boundary conformal field theory

The effective central charge (denoted by <math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>eff</mi></mrow></msub></mrow></math>) is a measure of entanglement through a conformal interface, while the transmission coefficient (encoded in the coefficient <math><msub><mi>c</mi><mrow><mi>L</mi><mi>R</mi></mrow></msub></math> 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, <math><mrow><mn>0</mn><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>L</mi><mi>R</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>eff</mi></mrow></msub><mo>≤</mo><mi>min</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></mrow></math>. 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.

Andreas Karch

Hao-Yu Sun

Mianqi Wang

Universal Bound on Effective Central Charge and Its Saturation

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Symmetry-resolved entanglement entropy, spectra &amp; boundary conformal field theory

When an interface connects two CFTs, the entanglement entropy between the two CFTs is determined by a quantity called the effective central charge. The effective central charge does not have a simple form in terms of the central charges of the two CFTs, but intricately depends on the transmissive properties of the interface.

Universality of effective central charge in interface CFTs

While recent progress in the black hole information problem has shown that the entropy of Hawking radiation follows a unitary Page curve, the quantum state of Hawking radiation prior the Page time is still treated as purely thermal, containing no information about the microstructure of the black hole. We demonstrate that there is significant quantum information regarding the quantum state of the black hole in the Hawking radiation prior to the Page time. By computing of the quantum fidelity in a 2D boundary conformal field theory (BCFT) model of black hole evaporation, we demonstrate that an observer outside of an evaporating black hole may distinguish different black holes via measurements of the Hawking radiation at any time during the evaporation process, albeit with an exponentially large number of measurements. Furthermore, our results are universal, applicable to general BCFTs including those with large central charge and rational BCFTs. The techniques we develop for computing the fidelity are more generally applicable to excited states in CFT. As such, we are able to characterize more general aspects of thermalization in 2D conformal field theory.

Jonah Kudler-Flam

On quantum information before the Page time

We initiate a conformal bootstrap program to study AdS3/BCFT2 with heavy excitations. We start by solving the bootstrap equations associated with two-point functions of scalar/non-scalar primaries under the assumption that one-point functions vanish. These correspond to gravity with a brane and a non-spinning/spinning particle where the brane and the particle do not intersect with each other. From the bootstrap equations, we obtain the energy spectrum and the modified black hole threshold. We then carefully analyze the gravity duals and find the results perfectly match the BCFT analysis. In particular, brane self-intersections, which are usually considered to be problematic, are nicely avoided by the black hole formation. Despite the assumption to solve the bootstrap equations, one-point functions of scalar primaries can be non-zero in general. We construct the holographic dual for a non-vanishing one-point function, in which the heavy particle can end on the brane, by holographically computing the Rényi entropy in AdS/BCFT. As a bonus, we find a refined formula for the holographic Rényi entropy, which appears to be crucial to correctly reproduce the boundary entropy term. On the other hand, we explain why one-point functions of non-scalar primaries always vanish from the gravity dual. The non-sensitivity of the solution for the bootstrap equation to the boundary entropy helps us to construct gravity duals with negative tension branes. We also find a holographic dual of boundary primaries.

Zixia Wei

AdS/BCFT from conformal bootstrap: construction of gravity with branes and particles

The interpretation of semiclassical gravity as an ensemble-average of CFTs has provided much progress in understanding the factorization problem and the information paradox. In this article, we consider an averaging over boundary conditions in two-dimensional boundary conformal field theories (BCFTs). We show that the boundary averaging plays an important role not only in producing wormhole contributions but also in resolving the following problem. In general setups in AdS/BCFT, one can find unphysical solutions where branes have (self-)intersections. We show that the averaging nicely resolves this intersection problem. This provides another piece of evidence that averaging plays an important role in reproducing Einstein gravity. We also propose that the averaging provides a BCFT dual of the island model.

Semiclassical gravity from averaged boundaries in two-dimensional boundary conformal field theories

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.

Reflected entropy in boundary and interface conformal field theory

Recently, boundary conformal field theories (BCFTs) have attracted much attention in the context of quantum gravity. This is because a BCFT can be dual to gravity coupled to a heat bath CFT, known as the island model. On this background, it would be interesting to explore the duality between the boundary and the braneworld. However, this seems to be a challenging problem. The reason is because although there has been much study of rational BCFTs, there has been comparatively little study of irrational BCFTs, and irrational BCFTs are expected to be the boundary duals of the braneworlds. For this reason, we explore properties of boundary ingredients: the boundary primary spectrum, the boundary-boundary-boundary OPE coefficients and the bulk-boundary OPE coefficients. For this purpose, the conformal bootstrap is extremely useful. This is the first step in providing an understanding of BCFTs in the context of braneworld holography by using the conformal bootstrap. The techniques developed in this paper may be useful for further investigation of irrational BCFTs.