Dissipative Phase Transitions

• Dissipative phase transitions are non-equilibrium critical phenomena in open quantum or classical systems where steady states change abruptly due to external driving and dissipation.
• The topic distinguishes first-order transitions with discontinuous jumps from second-order ones with continuous changes and characteristic Liouvillian gap closures.
• Experimental and theoretical methods, including dynamic hysteresis and spectral analysis, are used to probe these transitions in platforms like superconducting circuits and photonic resonators.

Dissipative phase transitions are non-equilibrium critical phenomena exhibited by open quantum or classical systems subject to external driving and loss. They fundamentally differ from their equilibrium counterparts in their microscopic mechanisms, order parameter structure, and the role of system-environment interactions. Dissipative phase transitions manifest as abrupt or non-analytic changes in the steady-state of the system’s density matrix upon tuning external parameters such as drive strength, dissipation rates, or disorder, and are characterized by dramatic alterations in the spectrum of the dynamical generator—typically the Liouvillian superoperator. These phenomena have been investigated analytically, numerically, and experimentally in a diverse range of platforms including spin ensembles, photonic and optomechanical resonators, superconducting circuits, polaritonic microcavities, cold atoms, and mesoscopic quantum devices.

1. Theoretical Framework and Classification

The theoretical description of dissipative phase transitions relies on the master equation formalism, where the evolution of the system’s density matrix ρ is governed by a Liouvillian superoperator ℒ,

$$\frac{d\rho}{dt} = \mathcal{L}\rho = -i[H, \rho] + \sum_j \mathcal{D}[L_j]\rho,$$

with dissipators $$\mathcal{D}[L_j]\rho = L_j \rho L_j^\dagger - \frac{1}{2}\{L_j^\dagger L_j, \rho\}$$. The steady state(s) $\rho_{ss}$ satisfy $\mathcal{L}\rho_{ss} = 0$. Phase transitions are signaled by non-analytic changes in the dependencies $\rho_{ss}(g)$ as external control parameter $g$ is tuned.

Two principal universality classes are typically distinguished:

• First-order dissipative phase transitions: The steady-state changes discontinuously at a critical parameter value. The Liouvillian gap (real part of the smallest nonzero eigenvalue) closes exactly at criticality, and the steady state at the critical point is an equal mixture of two macroscopically distinct branches, e.g. $\rho_{ss}(g_c) = (\rho_+ + \rho_-)/2$ (Minganti et al., 2018), with the corresponding eigenmatrix being proportional to their difference.
• Second-order (continuous) dissipative phase transitions: The steady-state changes continuously, with local observables often remaining continuous but showing discontinuities in their first derivative (akin to symmetry-breaking in equilibrium systems), and the Liouvillian gap closes over a finite range or at a point, often associated with spontaneous symmetry breaking (Minganti et al., 2018, Hannukainen et al., 2017).

Extending to non-Markovian systems, where the environment exhibits memory effects, the spectrum of the generalized HEOM (Hierarchical Equations of Motion) Liouvillian encodes the critical properties and can exhibit gap closing at transition points, even when conventional Lindblad approaches fail (Debecker et al., 10 Oct 2024).

2. Spectral Properties and Liouvillian Gap

The Liouvillian spectral gap $\lambda=\mathrm{Re}(\lambda_1)$, where $\lambda_1$ is the leading nonzero eigenvalue of $\mathcal{L}$, plays a central role in diagnosing and characterizing dissipative phase transitions (Minganti et al., 2018). Its closure signals critical slowing down and the dynamical emergence of multiple (effectively degenerate) steady states in the thermodynamic limit. Spectral decomposition reveals how, at criticality, the steady state is constructed as a mixture of macroscopic branches: $$\rho_{ss}(g_c) = (\rho_+ + \rho_-)/2, \qquad \mathcal{L}(\rho_+ - \rho_-)=0,$$ for first-order transitions, and as an equal mixture of symmetry-broken components in second-order transitions with, for example, $Z_n$ symmetry.

In non-Markovian systems, the HEOM Liouvillian spectrum generalizes all these results, allowing description of memory-induced phase boundary reshaping, and the exact reproduction of, e.g., the superradiant phase transition in U(1) symmetric two-mode Dicke models (Debecker et al., 10 Oct 2024).

3. Mechanisms and Model Systems

Dissipative phase transitions arise from nontrivial interplay between coherent Hamiltonian dynamics and incoherent dissipative processes.

Examples of microscopic mechanisms:

• Central-spin models: Coherent driving and dissipation of an electron spin, hyperfine-coupled to a many-spin environment, exhibits both first- and second-order transitions, with regions of bistability and spin squeezing. Analytical methods include self-consistent Holstein-Primakoff expansion and adiabatic elimination of fast degrees of freedom (Kessler et al., 2012).
• Ising and p-spin models: Fully-connected driven-dissipative Ising models with p-spin interactions, solved semiclassically and with Dicke basis numerics, display multistability, first- and continuous-order transitions, with the nature and symmetry of transitions depending on p and the interaction term (Wang et al., 2020, Haack et al., 2023).
• Driven-dissipative resonators: Kerr resonators with single- or multi-photon drive/dissipation display a spectrum of critical phenomena, including second-order transitions (n=2,4) and first-order transitions (odd n, or multistability for higher order) depending on the order of nonlinearity and symmetry constraints (Minganti et al., 2023, Minganti et al., 2018).
• Nonlinear photonic/optomechanical systems: Bistability and abrupt discontinuities in photon/phonon occupation, with critical lines derived from stability (Routh–Hurwitz) criteria, are observed in driven-dissipative optomechanical settings subject to strong drive and weak nonlinearity (Bibak et al., 2022).
• Low-dimensional electronic systems: Amorphous thin films (NbSi) exhibit multiple metallic—i.e., dissipative—phases between superconducting and insulating phases, with transitions driven by disorder, film thickness, and temperature; fluctuation-induced metallicity violates standard localization expectations (Couëdo et al., 2016).

4. Symmetry, Spontaneous Symmetry Breaking, and Non-Uniqueness

The role of symmetry is subtle in dissipative phase transitions. While second-order transitions are often associated with spontaneous symmetry breaking (SSB) and the multiplicity of steady states, this is not universally required. Certain classes of models demonstrate criticality in the absence of SSB, with transitions marked by nonanalyticity of the steady state, yet remaining unique (Minganti et al., 2021, Hannukainen et al., 2017). In such cases, SSB can be artificially "removed" via engineered dissipators, with criticality remaining robust.

For models with weak or strong symmetries (e.g., $Z_n$, $U(1)$), the structure of the Liouvillian determines whether multiple distinct steady states form in the symmetry-broken phase, or whether the solution remains unique. Strong symmetries enforce conservation laws and the existence of multiple steady states, while weak symmetries only block-diagonalize the Liouvillian, often resulting in unique but critical steady states.

5. Experimental Probes and Observations

Experimental detection of dissipative phase transitions relies on both static and dynamical observables:

• Output current fluctuations: The power spectrum and autocorrelation function of quantum jump events (e.g., photon emission or atomic loss) display definitive crossover—overdamped to underdamped decay, emergence of finite-frequency peaks, and maximal critical slowing down—across the DPT (Matsumoto et al., 3 Feb 2025). The timescale $\tau_s = (1/C(0)) \int_0^\infty d\tau\, C(\tau)$ peaks at criticality and aligns with spectral gap closures.
• Dynamic hysteresis: By sweeping dissipation strength or external drive, experiments record hysteresis loops in occupation or transmitted intensity. These loops scale with sweep time as a power law (e.g., $A \sim \tau_s^\alpha$ with $\alpha\approx-1$ for dissipation sweeps), and enhanced atomic/photonic number fluctuations are observed at the steepest part of the transition (Röhrle et al., 2023, Benary et al., 2022).
• Entanglement and discord: In driven-dissipative spin models, quantum entanglement peaks at the critical point, while quantum discord increases monotonically with drive into the low-purity phase, indicating robust nonclassical correlations even when entanglement is suppressed by dissipation (Wang et al., 2023).
• Geometry and dimensionality: Experiments with polariton microcavities have demonstrated the necessity of two-dimensional geometry and diffusive boundary conditions for the emergence of first-order DPT, while one-dimensional configurations fail to display non-analytic behavior (Li et al., 2021).
• Time crystals and spectral features: Dissipative phase transitions in nonlinear photonic systems can correspond, in a different reference frame, to the emergence of dissipative time crystals. The closure of the Liouvillian gap in the rotating frame implies persistent oscillations in the laboratory frame (Minganti et al., 2020).

6. Geometric and Information-Theoretic Approaches

Geometric properties of the steady-state density matrix, notably the mean Uhlmann curvature (MUC), have been identified as sensitive probes of dissipative criticality. The MUC,

$$\mathcal{U}_{\mu\nu} = \frac{i}{4} \mathrm{Tr} [\rho [L_\mu, L_\nu]],$$

where $L_\mu$ are the symmetric logarithmic derivatives with respect to control parameters, becomes singular precisely when the correlation length diverges (Carollo et al., 2017). The scaling of the MUC differentiates quantum (super-extensive, e.g., $n^2$ scaling) and classical (constant in system size) criticality, and provides a geometric perspective on parameter incompatibility in quantum estimation theory.

7. Implications, Control, and Perspectives

Dissipative phase transitions provide a framework for stabilizing nontrivial non-equilibrium phases—including bistable, excited-state, and time-crystalline regimes—that are unattainable in equilibrium scenarios (Soriente et al., 2021, Minganti et al., 2020). By tuning system-environment coupling or using engineered reservoirs (potentially with non-Markovian memory), phase boundaries can be reshaped and novel steady states can be stabilized (Debecker et al., 10 Oct 2024). This underscores the importance of dissipation as a resource in quantum engineering, with applications in quantum sensing (critical enhancement of sensitivity near second-order DPTs), reservoir engineering, photonic devices, and quantum computation (e.g., cat-state stabilization).

Advanced numerical techniques—such as the spectral analysis of the HEOM Liouvillian, cumulant expansions, and semidefinite programming for entanglement verification—support both the characterization and universal classification of dissipative phase transitions. Experimental validation is increasingly available in platforms ranging from ultracold atoms, optomechanically coupled cavities, and semiconductor microcavities to solid-state spin ensembles (NV centers, quantum dots), superconducting circuits, and trapped-ion quantum simulators.

Dissipative phase transitions thus unify concepts from non-equilibrium statistical mechanics, quantum information geometry, and condensed-matter physics, and offer a robust platform for the paper and control of emergent collective behavior far from equilibrium.