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arXiv:2603.25512

Nonperturbative Resummation of Divergent Time-Local Generators

Dragomir Davidovic

quant-phhep-thmath-phSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations• Revised Jun 30, 2026 (v2)

v1 · Mar 26, 2026 v2 · Jun 30, 2026 v3 · Sep 27, 2026

Abstract

Perturbative van Kampen cumulant expansions of time-local generators of open quantum systems generically diverge at long times, even though the reduced dynamics remains regular. Here we show that these divergent cumulants nevertheless contain sufficient information to reconstruct the nonperturbative dynamical map. The resulting dynamical map reveals that the divergence does not signal a breakdown of the reduced dynamics, but the approach to isolated times at which the dynamical map becomes noninvertible. Rather than arising from special Lindblad-type constructions, the corresponding singular time-local generators emerge generically from microscopic open-system Hamiltonians. The onset of recurrent noninvertibility identifies the reduced-dynamical-map manifestation of the Khalfin effect, a transition from exponential to algebraic relaxation, establishing a direct connection between long-time quantum decay and noninvertibility of reduced open-system dynamics. Nevertheless, the distinguishability of quantum superposition states remains governed by an exponential decay law throughout the Khalfin regime, demonstrating that the Markovian loss of distinguishability survives even in the presence of long-lived non-Markovian memory.

Comments: 25 pages 5 figures

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Perturbative van Kampen cumulant expansions of time-local generators of open quantum systems generically diverge at long times, even though the reduced dynamics remains regular. Here we show that these divergent cumulants nevertheless contain sufficient information to reconstruct the nonperturbative dynamical map. The resulting dynamical map reveals that the divergence does not signal a breakdown of the reduced dynamics, but the approach to isolated times at which the dynamical map becomes noninvertible. Rather than arising from special Lindblad-type constructions, the corresponding singular time-local generators emerge generically from microscopic open-system Hamiltonians. The onset of recurrent noninvertibility identifies the reduced-dynamical-map manifestation of the Khalfin effect, a transition from exponential to algebraic relaxation, establishing a direct connection between long-time quantum decay and noninvertibility of reduced open-system dynamics. Nevertheless, the distinguishability of quantum superposition states remains governed by an exponential decay law throughout the Khalfin regime, demonstrating that the Markovian loss of distinguishability survives even in the presence of long-lived non-Markovian memory.

Master equations describing open quantum dynamics are typically first-order differential equations. When such dynamics brings the trajectories in state space of more than one initial state to the same point at finite instants in time, the generator of the corresponding master equation becomes singular while the dynamical map becomes noninvertible. The first-order, time-local, homogeneous master equations then fail to describe the dynamics beyond the singular point. Retaining time locality in the master equation necessitates a reformulation in terms of higher-order differential equations. We formulate a method to eliminate the divergent behavior of the generator by using a combination of higher-order derivatives of the generator with suitable weights and illustrate it with several examples. We also present a detailed study of the central spin model and we propose the average rate of information inflow in non-Markovian processes as a quantity that captures a different aspect of non-Markovian dynamics.

Perturbative master equations are essential for modeling open quantum systems but often exhibit late-time divergences when environmental correlations decay algebraically. In this work, we analyze the time-convolutionless (TCL) master equation expanded to order

{\mathrm{TCL}}_{2n}

and demonstrate that, while van Kampen's cumulants suppress early-time secular growth, they ultimately diverge at long times. To overcome this, we introduce a resummation technique based on the Hadamard trick, which incorporates time integrals directly into the bath's spectral density via elementwise multiplication. This approach establishes a maximum expansion order, ${n}_{\text{max}}=\ensuremath{\lceil}s+1\ensuremath{\rceil}$, and defines a precision limit of the asymptotic states of $O({\ensuremath{\lambda}}^{2\ensuremath{\lceil}s\ensuremath{\rceil}})$, where

s

is the power-law exponent and $\ensuremath{\lambda}$ is the weak-coupling constant. The resummed master equation features renormalized Bohr frequencies that capture decoherence and spectral overlap effects. In the unbiased spin-boson model, this results in an inflation of the generator at a temperature-independent rate equal to the decoherence rate ${\ensuremath{\nu}}_{2}$ and a finite validity time ${t}_{L}\ensuremath{\approx}(s+1)/{\ensuremath{\nu}}_{2}$. For exponentially decaying correlations, the method recovers a proper Markovian limit below a critical coupling threshold.

The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.

We introduce a systematic framework to derive the effective Hamiltonian governing the coherent dynamics of non-Markovian open quantum systems. By applying the minimal dissipation principle, we uniquely isolate the coherent contribution to the time-local generator of the reduced dynamics. We derive a general expression for the effective Hamiltonian and develop a recursive perturbative expansion that expresses it in terms of system-bath interaction terms and bath correlation functions. This expansion provides a systematic tool for analyzing energy renormalization effects across different coupling regimes. Applying our framework to paradigmatic spin systems, we reveal how environmental correlations influence energy shifts and eigenbasis rotations, offering insights into strong-coupling effects and non-Markovian quantum thermodynamics.

The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.

We demonstrate the existence of a universal, structurally unavoidable non-Markovian correction to effective exponential decay laws in open physical systems. The result is derived without assuming any fundamental dynamics, background time, spacetime structure, or microscopic model. The correction emerges solely from three generic ingredients: (i) the non-closure of correlational descriptions, (ii) projection to observable subsystems with unavoidable information loss, and (iii) spectral truncation of the induced structural operator. Under these minimal and representation-independent conditions, any effective exponential decay is necessarily accompanied by a long-time algebraic correction of the form

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, with a fixed exponent and sign that cannot be removed without violating structural consistency. This result is independent of gravity, cosmology, and specific microscopic dynamics, and applies generically to quantum-coherent, weakly dissipative regimes commonly described by Markovian semigroup models. The prediction provides a direct, experimentally testable signature distinguishing strictly Markovian descriptions from structurally non-closed correlational frameworks.