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arXiv:2309.04112

Pseudo Rényi Entanglement Entropies For an Excited State and Its Time Evolution in a 2D CFT

Farzad Omidi

Published Sep 8, 2023• 0 citations
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Abstract

In this paper, we investigate the second and third pseudo Rényi entanglement entropies (PREE) for a locally excited state $| ψ\rangle $ and its time evolution $| ϕ\rangle = e^{- i H t} | ψ\rangle$ in a two-dimensional conformal field theory whose field content is a free massless scalar field. We consider excited states which are constructed by applying primary operators at time $t=0$, on the vacuum state. We study the time evolution of the PREE for an entangling region in the shape of finite and semi-infinite intervals at zero temperature. It is observed that the PREE is always a complex number for $t \neq 0$ and is a pure real number at $t=0$. Moreover, we discuss on its dependence on the location $x_m$ of the center of the entangling region.

DOI
10.48550/arxiv.2309.04112

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Abstract In this work, we study the real-time evolution of pseudo-(Rényi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(Rényi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-Rényi entropy in various limits and find some symmetries associated with the subsystem and the positions of the inserted operators. For excitation by linear combinations, the late time limit of the nth pseudo-Rényi entropy shows a simple form related to the coefficients of the combinations and Rényi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(Rényi) entropy remains real throughout the time evolution.
In this work we study the time evolutions of (Renyi) entanglement entropy of locally excited states in two dimensional conformal field theories (CFTs) at finite temperature. We consider excited states created by acting with local operators on thermal states and give both field theoretic and holographic calculations. In free field CFTs, we find that the growth of Renyi entanglement entropy at finite temperature is reduced compared to the zero temperature result by a small quantity proportional to the width of the localized excitations. On the other hand, in finite temperature CFTs with classical gravity duals, we find that the entanglement entropy approaches a characteristic value at late time. This behaviour does not occur at zero temperature. We also study the mutual information between the two CFTs in the thermofield double (TFD) formulation and give physical interpretations of our results.
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.
We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.
We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement. It is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in two, four and six dimensions. We find that these results are interpreted in terms of quantum entanglement of a finite number of states, including Einstein-Podolsky-Rosen states. They agree with a heuristic picture of propagations of entangled particles.
We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of log t term. Our analysis covers the entire parameter regions with respect to the replica number n and the conformal dimension h O of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov’s recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge c, drastically changes when the dimensions of external primary states reach the value c/32. In particular, when h O ≥ c/32 and n ≥ 2, we find a new universal formula ΔSA(n)≃nc24(n−1) \varDelta {S}_A^{(n)}\simeq \frac{nc}{24\left(n-1\right)} ΔSA(n)​≃24(n−1)nc​ log t. Our numerical results also confirm existing analytical results using the HHLL approximation.