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arXiv:2508.02895

ETH-monotonicity and the black hole singularity

Nilakash Sorokhaibam

Published Aug 4, 2025• 0 citations
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Abstract

We study the enveloping function of the fluctuation term in eigenstate thermalization hypothesis (ETH) statement for holographic conformal field theories. We use this function to identify and examine black hole microstates. We set down a set of desirable criteria for this function called ETH-monotonicity. It reinforces the Kelvin statement of the second law of thermodynamics over and above the universal entropic contribution. We show that higher-dimensional holographic conformal field theories possess ETH-monotonicity. Stronger contribution from ETH-monotonicity to the second law of thermodynamics is observed in smaller black hole microstates. It dominates other quantum fluctuations. It also measures the curvature at the horizon of the small black holes. In the smallest size limit, the black hole curvature singularity is constructed of microstates for which ETH-monotonicity starts competing with the entropic contribution. We expect that ETH-monotonicity will persist even in the ultimate quantum theory of gravity. Because it is a property of many-body quantum chaotic systems which becomes more prominent with decreasing system size, unlike other physical properties which are usually more predictable with increasing system size. Two-dimensional holographic conformal field theory does not possess all features of ETH-monotonicity which is in agreement with the absence of curvature singularity in the BTZ black hole.

DOI
10.48550/arxiv.2508.02895

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We study the enveloping function of the fluctuation term in eigenstate thermalization hypothesis (ETH) statement for holographic conformal field theories. We use this function to identify and examine black hole microstates. We set down a set of desirable criteria for this function called ETH-monotonicity. It reinforces the Kelvin statement of the second law of thermodynamics over and above the universal entropic contribution. We show that higher-dimensional holographic conformal field theories possess ETH-monotonicity. Stronger contribution from ETH-monotonicity to the second law of thermodynamics is observed in smaller black hole microstates. It dominates other quantum fluctuations. It also measures the curvature at the horizon of the small black holes. In the smallest size limit, the black hole curvature singularity is constructed of microstates for which ETH-monotonicity starts competing with the entropic contribution. We expect that ETH-monotonicity will persist even in the ultimate quantum theory of gravity. Because it is a property of many-body quantum chaotic systems which becomes more prominent with decreasing system size, unlike other physical properties which are usually more predictable with increasing system size. Two-dimensional holographic conformal field theory does not possess all features of ETH-monotonicity which is in agreement with the absence of curvature singularity in the BTZ black hole.
We use Holevo information in the two-dimensional conformal field theory (CFT) with a large central charge c to distinguish microstates from the underlying thermal state. Holographically, the CFT microstates of a thermal state are dual to black hole microstate geometries in anti-de Sitter space. It was found recently that holographic Holevo information shows plateau behaviors at both short and long interval regions. This indicates that the black hole microstates are indistinguishable from the thermal state by measuring over a small region, and perfectly distinguishable over a region with its size comparable to the whole system. In this Letter, we demonstrate that the plateaus are lifted by including the 1/c corrections from both the vacuum and nonvacuum conformal families of CFT in either the canonical ensemble or microcanonical ensemble thermal state. Our results imply that the aforementioned indistinguishability and distinguishability of black hole microstate geometries from the underlying black hole are spoiled by higher order Newton constant G_{N} corrections of quantum gravity.
We consider the entanglement entropy in 2d conformal field theory in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval Renyi entropies and entanglement entropy, and conjecture that this dominates the answer in theories with a large central charge and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in three-dimensional gravity. High-energy eigenstates are dual to microstates of the BTZ black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that these individual microstates of holographic CFTs effectively thermalize at the level of the single-interval entanglement entropy. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.
Abstract We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theories with the same value of central charge. We argue that the ETH density matrix is close in trace distance to the reduced density matrix of the (micro)canonical ensemble. We support the argument in higher dimensions by comparing the Von Neumann entropy of the ETH density matrix with the entropy of a black hole in holographic systems in the low temperature limit. Finally, we generalize our analysis to the coherent states with energy density that varies slowly in space, and show that locally such states are well described by the ETH density matrix.
The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.
We study numerically the enveloping ffunction of the fluctuation term in eigenstate thermalization hypothesis statement. We concentrate on the energy (or entropy) dependence of this function in two-dimensional systems. Our numerical results show that it is, in general, a monotonically increasing function of the entropy. This is in agreement with the general expectation that fluctuations increase with increasing entropy. We observe that the ffunction locally flattens with increasing system size. The flattening rate is directly proportional to the system size. We also observe that the flattening rate is directly proportional to the particle number for systems of same spatial size. This variation of the f function is important for physics at subleading order of the system-size. So, it is relevant for intermediate-size systems (upto a few hundred qubits) which are experimentally accessible. One exception we find is that the ffunction of the order parameter of a thermal phase transition defies the monotonic behavior.
Abstract The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.