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arXiv:2512.18781 · Physical review. D/Physical review. D.

Entanglement Dynamics by (Non-)Unitary Local Operator Quenches in a 2D Holographic CFT

Weibo Mao, Akihiro Miyata, Masahiro Nozaki, Farzad Omidi

hep-thcond-mat.str-elquant-phSubmitted Dec 21, 2025• Announced Dec 23, 2025• 0 citations• Revised Jul 16, 2026 (v2)
v1 · Dec 21, 2025v2 · Jul 16, 2026
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Abstract

In this paper, we investigate the time evolution of entanglement entropy and mutual information for the spatially-infinite systems where we act with a primary operator on the vacuum state and then time-evolve it with the sequence of the Euclidean and Lorentzian time evolutions. Two-dimensional holographic conformal field theories describe the systems under consideration in this paper. The Euclidean time evolution is induced by the Rindler Hamiltonian and behaves as the regulator that tames the divergence induced by the local operator, while the Lorentzian one is induced by the uniform Hamiltonian. Under these time evolutions, we investigate the time ordering effect of the Rindler Euclidean and uniform Lorentzian time evolution operators. Consequently, we find the remarkable differences between those time evolutions are induced by whether those are unitary or non-unitary. Especially, we find that the unitary time evolution induces the late-time logarithmic growth of the entanglement entropy, while the non-unitary time evolution induces the late-time constant behavior. Furthermore, we investigate the dual gravity of the systems under consideration. Especially, we investigate the gravity duals of the systems with the insertion of the heavy primary operator and show that it is a black brane with a spacetime-dependent horizon.

Comments
60 pages, 11 figures, 4 appendices

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In this paper, we investigate the time evolution of entanglement entropy and mutual information for the spatially-infinite systems where we act with a primary operator on the vacuum state and then time-evolve it with the sequence of the Euclidean and Lorentzian time evolutions. Two-dimensional holographic conformal field theories describe the systems under consideration in this paper. The Euclidean time evolution is induced by the Rindler Hamiltonian and behaves as the regulator that tames the divergence induced by the local operator, while the Lorentzian one is induced by the uniform Hamiltonian. Under these time evolutions, we investigate the time ordering effect of the Rindler Euclidean and uniform Lorentzian time evolution operators. Consequently, we find the remarkable differences between those time evolutions are induced by whether those are unitary or non-unitary. Especially, we find that the unitary time evolution induces the late-time logarithmic growth of the entanglement entropy, while the non-unitary time evolution induces the late-time constant behavior. Furthermore, we investigate the dual gravity of the systems under consideration. Especially, we investigate the gravity duals of the systems with the insertion of the heavy primary operator and show that it is a black brane with a spacetime-dependent horizon.
A bstract In this paper, we investigate the entanglement dynamics induced by a composite operator, defined as the local operator evolved with the time evolution operator constructed out of the Euclidean and Lorentzian ones. The systems under consideration are described by two-dimensional holographic conformal field theories (2d holographic CFTs), the theories described by gravity. Then, we find that the time ordering of Euclidean and Lorentzian time evolutions determines whether the time evolution, inducing the operator growth, is unitary. We also establish the relation between entanglement entropy and energy densities. This relation provides the entanglement entropy as a function of those densities. By exploiting this relation, we find that the time ordering, i.e., whether the time evolution is unitary, determines the late-time behavior of bipartite entanglement and non-local correlation. We also investigate how the decay of the quasiparticle influences the bipartite entanglement and non-local correlation. Furthermore, we investigate the gravity dual of the systems considered in this paper.
We investigate the complex time evolution of a vacuum state with the insertion of a local primary operator in two-dimensional conformal field theories (2d CFTs). This complex time evolution can be considered as a composite process constructed from Lorentzian time evolution and a Euclidean evolution induced by a post-selected measurement. Our main finding is that in the spatially-compact system, this complex time evolution drives the state of the subsystems to those of the primary state with the same conformal dimensions of the inserted operator. Contrary to the compact system, the subsystems of the spatially non-compact system evolve to states that depend on the non-unitary process during a certain time regime. In holographic systems with a compact spatial direction, this process induced by a heavy local operator can correspond to the relaxation from a black hole with an inhomogeneous horizon to that with a uniform one, while in the ones with a non-compact spatial direction, it can correspond to the relaxation to that with a horizon depending on the non-unitary process.
A bstract We explore non-equilibrium processes in two-dimensional conformal field theories (2d CFTs) due to the growth of operators induced by inhomogeneous and homogeneous Hamiltonians by investigating the time dependence of the partition function, energy density, and entanglement entropy. The non-equilibrium processes considered in this paper are constructed out of the Lorentzian and Euclidean time evolution governed by different Hamiltonians. We explore the effect of the time ordering on entanglement dynamics so that we find that in a free boson CFT and RCFTs, this time ordering does not affect the entanglement entropy, while in the holographic CFTs, it does. Our main finding is that in the holographic CFTs, the non-unitary time evolution induced by the inhomogeneous Hamiltonian can retain the initial state information longer than in the unitary time evolution.
A bstract We will explore the dynamical property of non-equilibrium phenomena induced by two-dimensional holographic conformal field theory (2d holographic CFT) Hamiltonian on the curved spacetime by studying the time dependence of the entanglement entropy and mutual information. Here, holographic CFT is the CFT having the gravity dual. We will start from the boundary and thermofield double states, evolve the systems in Euclidean time with the Hamiltonian on the curved background, and then evolve them in real-time with the same Hamiltonian. We found that the early- and late-time entanglement structure depends on the curved background, while the entanglement growth does not, and is linear. Furthermore, in the gravity dual for the thermofield double state, this entanglement growth is due to the linear growth of the wormhole, while in the one for the boundary state, it is due to the in-falling of the end of the world brane to the black hole. We discussed the low temperature system can be regarded as the dynamical system induced by the multi-joining quenches. We also discussed the effective description of the high temperature system, called line tension picture.
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.
We calculate the evolution of the geometric entanglement entropy following a local quench in the D1-D5 conformal field theory, a two-dimensional theory that describes a particular bound state of D1 and D5 branes. The quench corresponds to a localized insertion of the exactly marginal operator that deforms the field theory off of the orbifold (free) point in its moduli space. This deformation ultimately leads to thermalization of the system. We find an exact analytic expression for the entanglement entropy of any spatial interval as a function of time after the quench and analyze its properties. This process is holographically dual to one stage in the formation of a stringy black hole.