Schwinger-Keldysh field theory for operator R´enyi entropy and entanglement growth in non-interacting systems with sub-ballistic transports - Research summary

Sep 15, 2026

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Operator growth in quantum systems links transport of conserved quantities to entanglement generation during unitary dynamics. The study defines a state-independent diagnostic using a subsystem operator R´enyi entropy, showing it captures both spatial and temporal structure and connects to transport for local operators tied to conservation laws. A unified Schwinger-Keldysh framework is built to compute operator and state R´enyi/Von Neumann entropies, expressed via Keldysh Green’s functions at infinite temperature and vacuum. Applications to non-interacting quasiperiodic and disordered models reveal that subsystem operator R´enyi and state entanglement entropies distinguish ballistic, sub-ballistic, diffusive, anomalous diffusive, and localization regimes.

LanguageEnglish

CategoryResearch & Report

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Pages18

Reading Time45 min

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Introduction

Operator growth and entanglement/transport diagnostics

R´enyi entropies and expected growth behaviors

Sub-ballistic transport and motivation for the present work

Proposed diagnostic and framework

Subsystem operator second R´enyi entropy

State-independent operator growth measure and spatial content

arXiv :2602 .22331v1 [ quant-ph] 25 Feb 2026 Schwinger-Keldysh field theory for operator R´enyi entropy and entanglement growth in non-interacting systems with sub-ballistic transports Priesh Roy, and Sumilan Banerjee Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India ∗ The notion of operator growth in quantum systems furnishes a bridge between transport and the generation of entanglement between different parts of the system under quantum dynamics. We define a measure of operator growth in terms of subsystem operator R´enyi entropy, which provides a state-independent measure of operator growth, unlike entanglement entropies, and the usual measures of operator growth like out-of-time-order correlators. We show that the subsystem operator R´enyi entropy encodes both spatial and temporal information, and thus can directly connect to transport for a local operator related to a conserved quantity. We construct a unified Schwinger-Keldysh (SK) field theory formalism for the time evolution of operator R´enyi entropy and entanglement entropies of initial pure states. We use the SK field theory to obtain the operator R´enyi and state entanglement entropies in terms of infinite-temperature and vacuum Keldysh Green’s functions, respectively, for non-interacting systems. We apply the method to explore the connection between operator and entanglement growth, and transport in non-interacting systems with quasiperiodic and random disorder, like the one- and two-dimensional Aubry-Andr´e models and the two-dimensional Anderson model. In particular, we show that the growth of subsystem operator R´enyi entropy and state von Neumann and R´enyi entanglement entropies can capture both ballistic and sub-ballistic transport behaviors, like diffusive and anomalous diffusive transport, as well as localization in these systems. I. INTRODUCTION The investigation of quantum information propagation, for example, characterized by growth of operators and entanglement of quantum states under quantum manybody time evolution, has emerged as a pivotal research area in condensed matter and high-energy physics [1– 5] . In particular, it is an exciting avenue to study the connections between diagnostics of quantum information propagation and conventional characterizations of quantum many-body systems, such as in terms of transport of conserved quantities. Quantum information propagation is commonly quantified in terms of out-of-time-order correlators (OTOC) [6], and operator and state entanglement entropies [2, 3 , 7 , 8] . For example, entanglement of a pure state with reduced density matrix ρA of a subsystem is quantified in terms of n-th R´enyi entanglement entropies, SA(n) ≡ [1/(1 − n)]ln Tr(ρnA), like the second R´enyi entropy S(2)A and (n → 1) von Neumann entropy SA = SA(n→ 1) . Starting with unentangled initial states, in generic interacting thermalizing systems with diffusive transport of conserved quantities, the R´enyi entanglement entropies SA(n>1)(t) are found to grow diffusivelyas ∼ √t with time t, while the SA (t) follows ballistic growth ∼ t [9–11] . In contrast, all R´enyi entropies and the von Neumann entropy are expected to grow ballistically in clean integrable systems [12–19] with ballistic transport, as well as in systems without any conservation laws [20–23] . The above correspondence between transport and state entanglement growth has been rationalized based on the time dependence of entanglement eigenvalues or Schmidt values of the reduced density matrix in numerical stud- ies of integrable and non-integrable many-body models [10, 24] . Analytical bounds [11] and calculations in unitary quantum circuits with or without conservation laws [3, 20 , 21] also support the numerical results. In an interacting diffusive system, the maximum entanglement eigenvalue Λmax , which dominates SA(n>1)(t) in the longtime limit, is found to decay diffusively as lnΛmax (t) ∼− √t. Thi

FAQs

What diagnostic does the paper propose to measure operator growth?

It proposes a subsystem operator second R´enyi entropy, defined via a partial trace of a time-evolved local operator. This yields a state-independent measure of operator growth.

How does subsystem operator R´enyi entropy relate to transport?

The paper shows that subsystem operator R´enyi entropy encodes both spatial and temporal information, enabling a direct connection to transport for a local operator related to a conserved quantity.

What theoretical framework is used to compute the entropies?

A unified Schwinger-Keldysh (SK) field theory formalism is constructed for the time evolution of operator R´enyi entropy and entanglement entropies from initial pure states.

Sep 15, 2026

Uploaded by Mabel