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arXiv:2610.02309

Entanglement Requires Fluctuations at Conformal Interfaces

Yuya Kusuki, Kosei Suzuyama, Yin Tang

hep-thcond-mat.stat-mechquant-phSubmitted Oct 1, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

We conjecture that the universal logarithmic coefficient for capacity of entanglement $c_{\mathrm{cap}}$ is bounded below by the corresponding entropy coefficient $c_{\mathrm{eff}}$ at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.

Comments
18 pages, 8 figures

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We conjecture that the universal logarithmic coefficient for capacity of entanglement ccapc_{\mathrm{cap}}ccap​ is bounded below by the corresponding entropy coefficient ceffc_{\mathrm{eff}}ceff​ at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.
The entanglement entropy of an arbitrary spacetime region A in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, F(A). For general theories, the value of F(A) is minimized when A is a round disk, F_{0}, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity F(A)/F_{0} is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies C_{T}/F_{0}≤3/(4π^{2}log2-6ζ[3])≃0.14887 for general CFTs. We verify the validity of this bound for free scalars and fermions, general O(N) and Gross-Neveu models, holographic theories, N=2 Wess-Zumino models and general ABJM theories.
The entanglement entropy in many gapless quantum systems receives a contribution from the corners in the entangling surface in 2+1d, which is characterized by a universal function $a(\ensuremath{\theta})$ depending on the opening angle $\ensuremath{\theta}$, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\pi}$ yields the stress tensor two-point function coefficient CT{C}_{T}CT​. Little is known about $a(\ensuremath{\theta})$ beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the two- and three-point correlators of the stress tensor. We conjecture that it encodes four-point data, making it much richer. Further, we establish strong constraints on this and higher-order smooth-limit coefficients. We also show that $a(\ensuremath{\theta})$ is lower-bounded by a nontrivial function multiplied by the central charge CT{C}_{T}CT​, e.g., $a(\ensuremath{\pi}/2)\ensuremath{\ge}({\ensuremath{\pi}}^{2}ln2){C}_{T}/6$. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the R\'enyi entropies. We illustrate our findings using O(N)\text{O}(N)O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d.
We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R1,d−1 {\mathrm{\mathbb{R}}}^{1,d-1} R1,d−1 . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient σCT=π224 \frac{\sigma }{C_T}=\frac{\pi^2}{24} CT​σ​=24π2​ in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.
We explore interfaces and junctions joining multiple two-dimensional conformal field theories, with the goal of calculating entanglement entropies in their presence and exploring their holographic duals. In chapter 1 we start with an overview of the three subjects, collecting various well-known results and reviewing some foundational works.In chapter 2 we calculate the holographic entanglement entropy in the presence of a conformal interface for a geometric configuration in which the entangling region lies on one side of the interface. For the supersymmetric Janus solution we find exact agreement between the holographic and conformal field theory calculation of the entanglement entropy.In chapter 3 we calculate the entanglement entropy for topological interfaces in rational conformal field theories for the case where the interface lies at the boundary of the entangling interval and for the case where it is located in the center of the entangling interval. We compare the results to each other and also to the left/right entropy of a related boundary conformal field theory. We also comment on the entanglement entropies for topological interfaces in Liouville theory.In chapter 4 we consider entanglement through permeable junctions of N free boson and free fermion conformal field theories. We constrain the form of the general boundary state and calculate the replicated partition functions with interface operators inserted, from which the entanglement entropy is calculated. We find the functional form of the universal and constant terms to be similar to the N=2 case, depending only of the total transmission of the junction and the unit volume of the zero mode lattice. For N>2 we see a subleading divergent term which does not depend on the parameters of the junction. For N=3 we consider some specific geometries and discuss various limits.In chapter 5 we investigate topological interfaces between three-dimensional Abelian Chern-Simons theories in the context of the AdS_3/CFT_2 correspondence. We show that it is possible to connect the topological interfaces in the bulk Chern-Simons theory to topological interfaces in the dual conformal field theory on the boundary. In addition to the [U(1)]^{2N} Chern-Simons theory on AdS_3, we show that it is possible to find boundary counter terms which lead to the N conserved currents in the dual two-dimensional conformal field theory.
We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.