[Paper Review] Entanglement spreading after local and extended excitations in a free-fermion chain

Viktor Eisler|arXiv (Cornell University)|Jun 30, 2021

Quantum many-body systems71 references20 citations

TL;DR

This paper investigates entanglement dynamics in a free-fermion chain following local and extended excitations, using Gaussian state techniques and quasiparticle picture analysis. It shows that single particle or hole excitations generate a constant excess entanglement of one bit, while double hole excitations exhibit non-additive entanglement due to coherence at finite separations—only becoming additive at large distances. For extended contiguous holes, excess entropy scales logarithmically with size, whereas finite separations yield linear scaling, indicating effective distinguishability of excitations.

ABSTRACT

We study the time evolution of entanglement created by local or extended excitations upon the ground state of a free-fermion chain. A single particle or hole excitation produces a single bit of excess entropy for large times and subsystem lengths. In case of a double hole, some of the coherence between the excitations is preserved and the excess entropy becomes additive only for large hole separations. In contrast, the coherence is always lost for particle-hole excitations. Multiple hole excitations on a completely filled chain are also investigated. We find that for an extended contiguous hole the excess entropy scales logarithmically with the size, whereas the increase is linear for finite separations between the holes.

Motivation & Objective

• To understand how entanglement spreads after local and extended excitations in a one-dimensional free-fermion chain.
• To determine the role of quantum coherence between quasiparticles in shaping the asymptotic entanglement entropy.
• To clarify whether excess entanglement is additive or non-additive for multiple excitations, depending on spatial separation and excitation type.
• To investigate the scaling of excess entanglement with excitation size, particularly for extended contiguous holes versus separated holes.
• To establish a rigorous connection between the quasiparticle picture and exact Gaussian state calculations in free-fermion systems.

Proposed method

• The study uses Gaussian state formalism, leveraging Wick’s theorem and time-evolved correlation matrices to compute entanglement entropy.
• The time evolution is computed via unitary transformation using Bessel functions, with U_{mn} = i^{n-m} J_{n-m}(t), to evolve the initial correlation matrix.
• Entanglement entropy is calculated from eigenvalues of the reduced correlation matrix via S(t) = ∑_{k=1}^L s(ζ_k(t)), where s(x) = −x ln x −(1−x) ln(1−x).
• Excess entanglement ∆S(t) = S(t) − S_0 is analyzed to isolate dynamics-induced entanglement from ground-state contributions.
• For multiple excitations, the method involves diagonalizing the perturbation ∆C(t) to extract eigenvalues governing the excess entropy.
• Asymptotic analysis and stationary phase approximations are used to derive analytical expressions for large times and large subsystems.

Experimental results

Research questions

• RQ1 How does the excess entanglement scale after a single local particle or hole excitation in a free-fermion chain?
• RQ2 What is the role of coherence between two hole excitations in determining the asymptotic entanglement entropy?
• RQ3 Does the entanglement from multiple hole excitations scale additively, and if not, under what conditions does it deviate?
• RQ4 How does the entanglement scaling differ between an extended contiguous hole and a set of separated holes?
• RQ5 To what extent does the quasiparticle picture accurately describe the entanglement dynamics in lattice systems with finite-size effects?

Key findings

• A single particle or hole excitation generates a constant excess entanglement of exactly one bit in the asymptotic limit, independent of the excitation location.
• For two hole excitations, coherence leads to non-additive entanglement at finite separations; excess entropy is not simply the sum of individual contributions.
• The excess entanglement for two holes becomes additive only in the limit of large separation, where coherence is lost.
• For particle-hole excitations, coherence is always lost at large times, resulting in additive excess entanglement regardless of separation.
• For multiple holes on a filled chain, an extended contiguous hole leads to logarithmic scaling of excess entanglement with size, while finite separations yield linear scaling.
• The logarithmic scaling for contiguous holes arises from preserved quantum coherence, whereas finite separations make excitations effectively distinguishable, leading to linear growth.