A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
Yuchen Guo, Ke Ding, Shuo Yang
TL;DR
This work addresses the challenge of defining quantum phases in open systems, where real-time Lindbladian steady states often fail to capture phase transitions or converge to pure-state limits. It develops an imaginary-time Lindbladian framework based on the imaginary-Liouville operator $\mathcal{L}^I$, with a gap $\Delta^I$ that governs the open-system phase structure. The authors construct explicit realizations, show that finite-temperature Gibbs states of stabilizer Hamiltonians arise as steady states of $\mathcal{L}^I$, and analyze a concrete open-system phase diagram with $\mathbb{Z}_2^{\sigma}\times \mathbb{Z}_2^{\tau}$ symmetry, revealing ASPT, symmetry-breaking, and trivial sectors as well as universal critical properties like nonanalytic observables, diverging correlation lengths, and gap closing. A key result is the connection in 1D between $\Delta^I \to 0$ and divergent Markov length, linking the spectrum of $\mathcal{L}^I$ to long-range correlations, which supports the proposed phase classification. The paper also contrasts imaginary-time with real-time Lindbladian formalisms, showing that the latter can obscure phase distinctions, thereby highlighting the practical relevance of the imaginary-time approach for open-system quantum phase transitions.
Abstract
This study delves into the concept of quantum phases in open quantum systems, examining the shortcomings of existing approaches that focus on steady states of Lindbladians and highlighting their limitations in capturing key phase transitions. In contrast to these methods, we introduce the concept of imaginary-time Lindbladian evolution as an alternative framework. This new approach defines gapped quantum phases in open systems through the spectrum properties of the imaginary-Liouville superoperator. We find that, in addition to all pure gapped ground states, the Gibbs state of a stabilizer Hamiltonian at any finite temperature can also be characterized by our scheme, demonstrated through explicit construction. Moreover, the closing of the imaginary Liouville gap is associated with the divergence of the Markov length, which has recently been proposed as an indicator of phase transitions in open quantum systems. To illustrate the effectiveness of this framework, we apply it to investigate the phase diagram for open systems with $\mathbb{Z}_2^σ\times \mathbb{Z}_2^τ$ symmetry, including cases with nontrivial average symmetry protected topological order or spontaneous symmetry breaking order. Our findings demonstrate universal properties at quantum criticality, such as nonanalytic behaviors of steady-state observables, divergence of correlation lengths, and closing of the imaginary-Liouville gap. These results advance our understanding of quantum phase transitions in open quantum systems. In contrast, we find that the steady states of real-time Lindbladians do not provide an effective framework for characterizing phase transitions in open systems.