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  • Open Access

Tensor-Network Framework for Lindbladian Spectra and Steady States

Philipp Westhoff1,*, Mattia Moroder2,3,†, Ulrich Schollwöck1,‡, and Sebastian Paeckel1,§

  • 1Department of Physics, Arnold Sommerfeld Center for Theoretical Physics (ASC), Munich Center for Quantum Science and Technology (MCQST), Ludwig-Maximilians-Universität München, 80333 München, Germany
  • 2School of Physics, Trinity College Dublin, College Green, Dublin 2, D02K8N4, Ireland
  • 3Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, D02YN67, Ireland

  • *Contact author: p.westhoff@physik.uni-muenchen.de
  • †Contact author: moroderm@tcd.ie
  • ‡Contact author: schollwoeck@lmu.de
  • §Contact author: sebastian.paeckel@physik.uni-muenchen.de
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Phys. Rev. X 16, 041020 – Published 21 October, 2026

DOI: https://doi.org/10.1103/7kd1-kxxz

Abstract

Quantum systems coupled to (non-) Markovian environments attract increasing attention due to their peculiar physical properties. Exciting prospects such as unconventional nonequilibrium phases beyond the Mermin-Wagner limit or dissipative state preparation demand a systematic analysis of quantum many-body phases out of equilibrium. Akin to the equilibrium case, this requires the computation of the low-lying eigenstates of Lindbladians, a problem challenging conventional approaches for simulating quantum many-body systems. Here, we undertake a first step to overcome this limitation and introduce a tensor-network-based framework to systematically compute not only steady states, but also low-lying excited states for large, driven quantum many-body systems. Our framework is based on recent advances utilizing complex-time Krylov spaces, and we leverage these ideas to create a toolbox tailored to solve the challenging non-Hermitian eigenvalue problem ubiquitous in open quantum systems. With an example of the interacting Bose-Hubbard model driven by dissipation-assisted hopping, we demonstrate the high efficiency and accuracy. From a reliable finite-size-scaling analysis of the spectral gap, we find strong evidence for nonlinear hydrodynamic behavior consistent with Kardar-Parisi-Zhang-type superdiffusive relaxation and establish the existence of exponentially accelerated, anomalous relaxation. This method unlocks the capability of spectral analysis of generic open quantum many-body systems suitable also for non-Markovian environments.

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Physics Subject Headings (PhySH)

Popular Summary

Many of the most intriguing quantum phenomena do not emerge in isolation, but are driven by constantly exchanging energy and information with the environment. Treating these open quantum systems theoretically is notoriously difficult because the dynamics of their slowly decaying modes is crucial, but current methods that reach large systems are essentially restricted to the steady state and, in some cases, the dissipative gap. We introduce a framework that makes these hidden modes accessible by evolving along carefully chosen paths in complex time, extracting long-time information from short-time simulations. Applied to a driven Bose-Hubbard model, we reach system sizes significantly beyond previous approaches, uncover signatures of Kardar-Parisi-Zhang superdiffusive relaxation, and reveal experimentally realistic initial states that relax exponentially faster toward a condensate. Our approach opens the door to the systematic exploration of relaxation, thermalization, and emergent collective behavior in quantum systems coupled to complex environments.

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Phys. Rev. X 16, 041020– Published 21 October, 2026
  • Received 16 September 2025
  • Revised 5 June 2026
  • Accepted 7 August 2026
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