In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
Collapse as a Dissipative Phase Transition: Liouvillian Gap Opening, Pointer Stability, and τ-Flow Dynamics
Abstract We present a rigorous operator-theoretic formulation of quantum collapse as a dissipative phase transition in the spectrum of a completely positive dynamical semigroup. Within a finite-dimensional Hilbert space, we define collapse in terms of exponential contraction toward a pointer algebra induced by a Liouvillian spectral gap in the coherence sector. We prove that coherence decays at a rate determined by the minimal nonzero real part of the Liouvillian spectrum and establish a sharp stability inequality quantifying competition between dissipative gap opening and Hamiltonian coherence regeneration. We further show that in macroscopic amplification regimes, the gap scales superlinearly in redundancy, producing thermodynamic sharpening analogous to conventional phase transitions. Finally, we demonstrate that this structure emerges naturally from a $ \tau $-recursive dynamical generator of the form $ \mathcal{A} = -iH - \Gamma $, where $ \Gamma $ is constructed from commutator tension $ [H,M]^\dagger[H,M] $. Conclusion: Collapse thus appears not as a primitive stochastic postulate, nor as purely epistemic update, but as a dynamical transition in the spectral structure of the evolution generator. Key Highlights & Physical Implications Novel Resolution to the Measurement Problem: Replaces axiomatic stochastic jumps with a deterministic, spectral phase transition. Commutator Tension Mechanism: Identifies $ [H,M]^\dagger[H,M] $ as the dynamical origin of quantum friction and decoherence. Macroscopic Scaling Laws: Proves superlinear gap growth in redundant apparatuses, explaining why large systems behave classically. Pointer Algebra Stability: Establishes a strict inequality ($ \Delta > \kappa $) that dictates when a system crosses from the coherent phase to the classical pointer phase. Keywords & Index Terms Quantum Measurement Problem, Decoherence Theory, Dissipative Phase Transitions, Open Quantum Systems, Liouvillian Spectral Gap, Pointer States, Lindblad Master Equation, Onto-Physical Entanglement, CPT-Coherence Theory, Commutator Tension, Quantum Trajectories, Thermodynamic Limit.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by M M random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, m=M/(2N_F) m = M / ( 2 N F ) , we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs for m=1/2 m = 1 / 2 and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below m=1/2 m = 1 / 2 , the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.
We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We perform a thorough numerical study, focusing on global spectral features, the spectral gap, and the steady-state purity and statistics. We establish that all properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular observable. Within each regime, we determine the scaling exponents with the dissipation strength and system size. On the analytical side, we compute the average spectral gap at arbitrary dissipation and provide simple closed-form expressions for the asymptotic values at strong and weak dissipation strength. We also consider spectral correlations in generic complex spectra, such as that of the Liouvillian. We introduce and study, both analytically and numerically, the complex ratio of the nearest-neighbor spacing by next-to-nearest-neighbor spacing. Besides the usual level repulsion, we find that complex spacing ratios have a nontrivial angular dependence which offers a distinct signature of quantum chaos. It distinguishes integrable systems, which support the flat ratio distribution characteristic of Poisson statistics, and chaotic dissipative quantum systems, which conform to Random Matrix Theory statistics. We apply this new signature of quantum chaos to random Liouvillians and boundary-driven dissipative spin-chains. The results of this thesis can help understand the long-time dynamics, steady-state properties, and spectral correlations of generic dissipative systems.
The nonequilibrium dissipative phase transition, arising from the competition of cooperative behavior and coherent field driving, discovered in the 1970s by Narducci et al. [Phys. Rev. A18, 1571 (1978)] and Walls et al. [Prog. Theor. Phys. Suppl.64, 307 (1978)], has been found to exhibit time-crystal behavior when the driving field exceeds the cooperative decay rate. This was seen through the study of the eigenvalues of the Liouvillian superoperator that describes the joint effect of drive and cooperativity. The cooperative decay depends on the nature of the reservoir correlations. If the reservoir correlations have phase-sensitive behavior, then the eigenvalues of the Liouvillian will be different. We investigate the time-crystal behavior of the nonequilibrium dissipative phase transitions under the influence of a squeezed vacuum reservoir. We analyze the steady-state phase diagram as a function of the control parameter and demonstrate that increasing the squeezing strength sharpens the dissipative phase transition. Spectral analysis of the Liouvillian reveals gap closings and the emergence of purely imaginary eigenvalues in the thermodynamic limit, indicating the time-crystal phase. We find that the real parts of subleading eigenvalues exhibit nonmonotonic behavior with increasing squeezing, reflecting the sensitivity of relaxation dynamics to the reservoir properties. Time-domain simulations confirm that the oscillation frequencies correspond to the imaginary parts of the Liouvillian eigenvalues. We also present results on quantum fluctuations in the time-crystal phase. Our results call attention to the study of time crystals in models of cooperativity based on engineered environments.
Nonequilibrium dissipative phase transition, arising from the competition of cooperative behavior and coherent field driving, discovered in the 1970s by Narducci et al. and Walls et al., has been found to exhibit time-crystal behavior when the driving field exceeds the cooperative decay rate. This was seen through the study of the eigenvalues of the Liouvillian superoperator that describes the joint effect of drive and cooperativity. The cooperative decay depends on the nature of the reservoir correlations. If the reservoir correlations have phase-sensitive behavior, then the eigenvalues of the Liouvillian will be different. We investigate the time-crystal behavior of the nonequilibrium dissipative phase transitions under the influence of a squeezed vacuum reservoir. We analyze the steady-state phase diagram as a function of the control parameter and demonstrate that increasing the squeezing strength sharpens the dissipative phase transition. Spectral analysis of the Liouvillian reveals gap closings and the emergence of purely imaginary eigenvalues in the thermodynamic limit, indicating the time-crystal phase. We find that the real parts of subleading eigenvalues exhibit nonmonotonic behavior with increasing squeezing, reflecting the sensitivity of relaxation dynamics to the reservoir properties. Time-domain simulations confirm that the oscillation frequencies correspond to the imaginary parts of the Liouvillian eigenvalues. We also present results on quantum fluctuations in the time-crystal phase. Our results call attention to the study of time crystals in models of cooperativity based on engineered environments.