By analysing the Liouville gap in imaginary time, scientists reveal universal phase‑transition behaviour in both ground and finite‑temperature states
Attempts to understand quantum phase transitions in open systems usually rely on real‑time Lindbladian evolution, which tracks how a quantum state changes as it relaxes toward a steady state. This approach is powerful for studying decoherence, dissipation and long‑time behaviour, but it often fails to reveal the deeper structure of the system including the phase transitions, critical points and hidden quantum order that define its underlying physics.
In this work, the researchers introduce a new framework called imaginary‑time Lindbladian evolution, which allows them to define and classify quantum phases in open systems using the spectrum of an imaginary‑Liouville superoperator. This approach works not only for pure ground states but also for finite‑temperature Gibbs states of stabilizer Hamiltonians, showing its relevance for realistic, mixed‑state conditions.
A key diagnostic in their method is the imaginary‑Liouville gap, the spectral gap between the lowest and next‑lowest decay modes. When this gap closes, the system undergoes a phase transition, a change that is accompanied by diverging correlation lengths and nonanalytic shifts in physical observables. The closing of this gap also coincides with the divergence of the Markov length, a recently proposed indicator of criticality in open quantum systems.
To demonstrate the power of their framework, the researchers map out phase diagrams for systems with
symmetry, capturing both spontaneous symmetry breaking and average symmetry‑protected topological phases. Their method reveals universal critical behaviour that real‑time Lindbladian steady states fail to detect, highlighting why imaginary‑time evolution fills a missing piece in the theory of open‑system phases.
Importantly, the authors emphasise that real‑time Lindbladians remain essential for modelling dissipation in practical settings. Their new framework complements this conventional approach, offering a systematic way to study phase transitions in open systems. They also outline how phase diagrams can be constructed using both bottom‑up (state‑based) and top‑down (Hamiltonian‑based) strategies, illustrating the method with a dissipative transverse‑field Ising model.
Overall, this work provides a unified and versatile way to understand quantum phases in open systems, revealing critical behaviour and topological structure that were previously inaccessible. It opens new directions for studying mixed‑state quantum matter and advances the theoretical foundations needed for future quantum technologies.
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Focus on Quantum Entanglement: State of the Art and Open Questions guest edited by Anna Sanpera and Carlo Marconi (2025-2026)