An official website of the United States government
We study subsystem entropies in 2d CFTs for subsystems constituting a finite fraction of the full system. We focus on the extensive contribution, which scales linearly with the subsystem size in the thermodynamic limit. We employ the so-called diagonal approximation to evaluate subsystem entropy for chaotic CFTs in the thermal state (canonical ensemble), the microcanonical ensemble, and in a primary state, matching previously known results. We then proceed to find analytic expressions for the subsystem entropy at leading order in , when the global CFT state is the KdV-generalized Gibbs ensemble or the KdV-microcanonical ensemble. Previous studies of primary eigenstates have shown that, akin to the fixed-area states in AdS/CFT, the corresponding subsystem entanglement spectrum is flat. This behavior is seemingly in sharp contradiction with that of the thermal (microcanonical) state, and thus in apparent contradiction with the subsystem eigenstate thermalization hypothesis (ETH). In this paper, we resolve this issue by comparing the primary state with the KdV-(micro)canonical ensemble. We show that the results are consistent with the KdV-generalized version of the subsystem ETH, in which local properties of quantum eigenstates are governed by their values of conserved KdV charges. Our paper solidifies evidence for the KdV-generalized ETH in 2d CFTs and emphasizes Rényi entropy as a sensitive probe of the reduced-density matrix.
more »
« less
This content will become publicly available on January 1, 2026
Holographic Renyi entropy of 2d CFT in KdV generalized ensemble
The eigenstate thermalization hypothesis (ETH) in chaotic two-dimensional CFTs is subtle due to the presence of infinitely many conserved KdV charges. Previous works have shown that primary CFT eigenstates exhibit a flat entanglement spectrum, which is very different from that of the microcanonical ensemble. This appears to contradict conventional ETH, which does not account for KdV charges. In a companion paper \cite{1}, we resolve this discrepancy by analyzing the subsystem entropy of a chaotic CFT in KdV-generalized Gibbs and microcanonical ensembles. In this paper, we perform parallel computations within the framework of AdS/CFT. We focus on the high-density limit, which corresponds to the thermodynamic limit in conformal theories. In this regime, holographic Rényi entropy can be calculated using the so-called *gluing construction*. We specifically study the KdV-generalized microcanonical ensemble where the densities of the first two KdV charges are fixed: $$ \langle Q_1 \rangle = q_1, \quad \langle Q_3 \rangle = q_3 $$ with the condition $$q_3 - q_1^2 \ll q_1^2$$. In this regime, we find that the refined Rényi entropy $$\tilde{S}_n$$ becomes independent of $$n$$ for $$n > n_{\text{cut}}$$, where $$n_{\text{cut}}$$ depends on $$q_1$$ and $$q_3$$. By taking the primary state limit $$q_3 \to q_1^2$$, we recover the flat entanglement spectrum characteristic of fixed-area states, consistent with the behavior of primary states. This result supports the consistency of KdV-generalized ETH in 2d CFTs.
more »
« less
- Award ID(s):
- 2310426
- PAR ID:
- 10608126
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2025
- Issue:
- 1
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called “efficient basis” over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.more » « less
-
A<sc>bstract</sc> We construct a Type II∞von Neumann algebra that describes the largeNphysics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative 1/Ncorrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in theG →0 limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a “free product” von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to “bulge” quantum extremal surfaces contribute with a negative sign.more » « less
-
A<sc>bstract</sc> We employ semiclassical quantization to calculate spectrum of quantum KdV charges in the limit of large central chargec. Classically, KdV chargesQ2n−1generate completely integrable dynamics on the co-adjoint orbit of the Virasoro algebra. They can be expressed in terms of action variablesIk, e.g. as a power series expansion. Quantum-mechanically this series becomes the expansion in 1/c, while action variables become integer-valued quantum numbersni. Crucially, classical expression, which is homogeneous inIk, acquires quantum corrections that include terms of subleading powers innk. At first two non-trivial orders in 1/cexpansion these “quantum” terms can be fixed from the analytic form ofQ2n−1acting on the primary states. In this way we find explicit expression for the spectrum ofQ2n−1up to first three orders in 1/cexpansion. We apply this result to study thermal expectation values ofQ2n−1and free energy of the KdV Generalized Gibbs Ensemble.more » « less
-
We study the entanglement contour and partial entanglement entropy (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the entanglement contour, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the ALC proposal should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the ALC proposal. We derive the entanglement contour function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.more » « less
