An official website of the United States government
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
This content will become publicly available on September 1, 2026
Boundary conditions for the entanglement cut in two-dimensional conformal field theories
The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a two-dimensional (2D) conformal field theory (CFT), the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum, where one of the boundary conditions arise due to the “entanglement cut.” The latter has been argued to be conformal and has been numerically found to be the “free” boundary condition for Ising, Potts, and free boson theories. For these models, the free boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary minimal models of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the entanglement entropy from the CFT partition functions. These are compared with numerical results.
more »
« less
- Award ID(s):
- 2210187
- PAR ID:
- 10645828
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review B
- Volume:
- 112
- Issue:
- 12
- ISSN:
- 2469-9950
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We develop a nonequilibrium increment method in quantum Monte Carlo simulations to obtain the Rényi entanglement entropy of various quantum many-body systems with high efficiency and precision. To demonstrate its power, we show the results on a few important yet difficult (2 + 1) d quantum lattice models, ranging from the Heisenberg quantum antiferromagnet with spontaneous symmetry breaking, the quantum critical point with O(3) conformal field theory (CFT) to the toric code $${{\mathbb{Z}}}_{2}$$ Z 2 topological ordered state and the Kagome $${{\mathbb{Z}}}_{2}$$ Z 2 quantum spin liquid model with frustration and multi-spin interactions. In all these cases, our method either reveals the precise CFT data from the logarithmic correction or extracts the quantum dimension in topological order, from the dominant area law in finite-size scaling, with very large system sizes, controlled errorbars, and minimal computational costs. Our method, therefore, establishes a controlled and practical computation paradigm to obtain the difficult yet important universal properties in highly entangled quantum matter.more » « less
-
null (Ed.)A bstract Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.more » « less
-
Motivated by recent development of the concept of the disorder operator and its relation with entanglement entropy in bosonic systems, here we show the disorder operator successfully probes many aspects of quantum entanglement in fermionic many-body systems. From both analytical and numerical computations in free and interacting fermion systems in 1D and 2D, we find the disorder operator and the entanglement entropy exhibit similar universal scaling behavior, as a function of the boundary length of the subsystem, but with subtle yet important differences. In 1D they both follow the log(L) scaling behavior with the coefficient determined by the Luttinger parameter for disorder operator, and the conformal central charge for entanglement entropy. In 2D they both show the universal L\log(L) scaling behavior in free and interacting Fermi liquid states, with the coefficients depending on the geometry of the Fermi surfaces. However at a 2D quantum critical point with non-Fermi-liquid state, extra symmetry information is needed in the design of the disorder operator, so as to reveal the critical fluctuations as does the entanglement entropy. Our results demonstrate the fermion disorder operator can be used to probe quantum many-body entanglement related to global symmetry, and provide new tools to explore the still largely unknown territory of highly entangled fermion quantum matter in 2 or higher dimensions.more » « less
-
A bstract 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.more » « less
