Local operator quench induced by two-dimensional inhomogeneous and homogeneous CFT Hamiltonians

TL;DR

This paper analyzes non equilibrium dynamics of 2d CFTs under local operator quenches generated by inhomogeneous Möbius and homogeneous SSD Hamiltonians. It contrasts Lorentzian and Euclidean evolutions to determine how time ordering affects partition functions, energy density and entanglement entropy, illustrating a universal distinction between holographic and integrable theories. In free boson and RCFTs the entanglement dynamics follow a quasiparticle picture and are insensitive to the time order, while holographic CFTs exhibit ordering dependent EE and non unitary evolution can preserve initial operator information longer. The work employs twist operator methods and cross ratio analysis for EE, analyzes energy flow and regulator behavior, and provides a comprehensive view of information scrambling and operator growth across different CFT regimes with clear implications for holography and quantum quenches.

Abstract

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.

Figures (6)

• Figure 1: A sketch depicting the smearing of a local operator. After acting $e^{-\epsilon H_2}$ on a state with an insertion of a local operator, the local operator is smeared within a region of size $\mathcal{O}(\epsilon)$. The circle and red curve illustrate the local and smeared local operators respectively.
• Figure 2: Unitary and non-unitary processes. In [a], we smear the local operator, then evolve the system in real time. In [b], we evolve the system in real time before smearing the local operator. The orange region illustrates the space-time region where the local operator is delocalized by the dynamics. In the case of $\left| \Phi_a \right>$, the effect of smearing is delocalized during the real time evolution. In the case of $\left| \Phi_b \right>$, the effect of smearing is spatially localized.
• Figure 3: The subsystems considered in this paper. In (a), the subsystem includes only the origin. In (b), the subsystem includes only $x=\frac{L}{2}$. In (c), the subsystem does not contain both the origin and $\frac{L}{2}$. The origin and $\frac{L}{2}$ are the fixed points during the SSD time evolution, as explained in Section \ref{['Sec:trajectory-of-LO']}.
• Figure 4: The picture illustrating the spatial trajectory of the operator during the time evolution. Panel [a] illustrates the time evolution of the local operator during the Möbius time evolution, while [b] does it during the SSD time evolution. The initial insertion points of the operator are marked by blue and green, and their trajectories are denoted by blue and green curves. The two fixed points, $X_1^f=0$ and $X_2^f=L / 2$, are marked in red.
• Figure 5: Four typical cases. The details of the subsystems illustrated here are in (\ref{['eq:generalsubsystems']}).