[Paper Review] Growth of R\'enyi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture

Bruno Bertini, Katja Klobas+3 more|arXiv (Cornell University)|Mar 31, 2022

Quantum many-body systems112 references70 citations

TL;DR

This paper establishes a spacetime duality that maps the linear growth slope of R\'enyi entropies in integrable quantum systems to the equilibrium entropy density of a dual model with exchanged space and time. The key result is an exact formula for the R\'enyi entropy growth slope in all integrable models solvable by thermodynamic Bethe ansatz, revealing that the quasiparticle picture fails for R\'enyi entropies beyond the von Neumann limit.

ABSTRACT

R\'enyi entropies are conceptually valuable and experimentally relevant generalisations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterisation is a key objective of current research. Here we show that the slope of R\'enyi entropies can be determined by means of a spacetime duality transformation. In essence, we argue that the slope coincides with the stationary density of entropy of the model obtained by exchanging the roles of space and time. Therefore, very surprisingly, the slope of the entanglement is expressed as an equilibrium quantity. We use this observation to find an explicit exact formula for the slope of R\'enyi entropies in all integrable models treatable by thermodynamic Bethe ansatz and evolving from integrable initial states. Interestingly, this formula can be understood in terms of a quasiparticle picture only in the von Neumann limit.

Motivation & Objective

• To resolve the long-standing puzzle of why R\'enyi entropies in integrable systems do not follow the quasiparticle picture, unlike von Neumann entropy.
• To establish a spacetime duality transformation that maps non-equilibrium entanglement growth to equilibrium entropy densities.
• To derive an exact, universal formula for the R\'enyi entropy growth slope in all integrable models treatable by thermodynamic Bethe ansatz (TBA).
• To demonstrate that the quasiparticle picture cannot consistently describe R\'enyi entropy dynamics, even in integrable systems.
• To provide a framework applicable to both quantum circuits and relativistic quantum field theories, with extension to general TBA-integrable models.

Proposed method

• Introduce a spacetime duality transformation that exchanges space and time, mapping the linear growth slope of R\'enyi entropy to the stationary entropy density in the dual model.
• Use dual-unitary quantum circuits to rigorously establish the duality correspondence in discrete spacetime, where dynamics are invariant under space-time exchange.
• Extend the duality to relativistic quantum field theories, showing the correspondence holds in the free limit and conjecturing its validity for interacting integrable QFTs.
• Derive an exact formula for the R\'enyi entropy growth slope using thermodynamic Bethe ansatz (TBA) for integrable models evolving from compatible initial states.
• Validate the formula against exact analytical results in free theories and numerical data in interacting models.
• Use the duality to show that the quasiparticle picture fails to describe R\'enyi entropy growth, as the slope cannot be expressed in terms of quasiparticle contributions.

Experimental results

Research questions

• RQ1 Can the slope of R\'enyi entropy growth in integrable quantum systems be expressed as an equilibrium quantity via spacetime duality?
• RQ2 Why does the quasiparticle picture fail to describe R\'enyi entropy dynamics despite working for von Neumann entropy?
• RQ3 Does the spacetime duality between entanglement growth and equilibrium entropy density hold beyond free theories and quantum circuits?
• RQ4 Can an exact formula for the R\'enyi entropy growth slope be derived for all TBA-integrable models?
• RQ5 What is the fundamental physical reason for the breakdown of the quasiparticle description in R\'enyi entropy dynamics?

Key findings

• The slope of R\'enyi entropy growth in integrable models is exactly equal to the equilibrium entropy density of the model obtained by exchanging space and time.
• The derived formula for the R\'enyi entropy growth slope is valid for all integrable models solvable by thermodynamic Bethe ansatz and evolving from compatible initial states.
• The quasiparticle picture can only describe the von Neumann entropy (α=1) in this context; it fails to describe R\'enyi entropies for α≥2.
• The spacetime duality is rigorously established in dual-unitary quantum circuits and shown to hold in the free limit of relativistic quantum field theories.
• The formula is validated against exact analytical results in free theories and numerical data in interacting models, confirming its universality.
• The breakdown of the quasiparticle picture for R\'enyi entropies reveals a fundamental distinction between von Neumann and higher R\'enyi entropies in non-equilibrium dynamics.