Dynamics of number entropy for free fermionic systems in presence of defects and stochastic processes

TL;DR

This paper studies how number entropy evolves in free fermionic chains with defects and stochastic processes, revealing logarithmic growth behaviors and connections to scattering properties, and compares it to entanglement entropy.

Contribution

It provides analytical and numerical analysis of number entropy dynamics in free fermions with defects and stochastic effects, highlighting new scaling behaviors and scattering relations.

Findings

Number entropy grows logarithmically in time for conformal defects.

Eigenvalue dynamics relate to scattering coefficients of defects.

Stochastic processes lead to distinct logarithmic scaling of number entropy.

Abstract

We investigate the dynamics of number entropy in a chain of free fermions subjected to both defects and stochastic processes. For a special class of defects, namely conformal defects, we present analytical and numerical results for the temporal growth of number entropy, the time evolution of the number distribution, and the eigenvalue profile of the associated correlation matrix within a subsystem. We show that the number entropy exhibits logarithmic growth in time, originating from the Gaussian structure of the number distribution. We find that the eigenvalue dynamics reveal a profound connection to the reflection and transmission coefficients of the associated scattering problem for a broad range of defects. When stochastic processes are introduced, specifically Stochastic Unitary Processes (SUP) and Quantum State Diffusion (QSD), the number entropy scales as $\ln(t)$ in the SUP case and shows strong hints of $\ln [\ln(t)]$ scaling in the QSD case. These findings establish compelling evidence that number entropy grows logarithmically slower than the corresponding von Neumann entanglement entropy across a wide class of systems.