Gaussian Dissipative Quantum Phase Transitions

• Gaussian dissipative quantum phase transitions are critical phenomena in open quantum systems described by Gaussian states under quadratic Hamiltonians and linear Lindblad dissipation.
• They reveal both thermodynamic singularities and topologically driven changes, with observables like entropy density and Loschmidt echo indicating discontinuities.
• Experimental realizations in cold atom chains, photonic lattices, and superconducting qubits validate GDQPT predictions and non-equilibrium scaling laws.

A Gaussian dissipative quantum phase transition (GDQPT) refers to critical phenomena occurring in quantum systems whose dynamics and stationary states are described exclusively by Gaussian density matrices and correlation functions, owing to quadratic Hamiltonians and linear Lindblad dissipation. These transitions encapsulate both conventional thermodynamic singularities and topologically driven changes, manifesting under open-system Lindblad evolution—even in the absence of strong interactions—across bosonic and fermionic platforms. GDQPTs unify dynamical (time-dependent) and steady-state (NESS) critical scenarios, bridging nonanalytic phenomena, scaling laws, information geometry, and the emergence or destruction of order parameters under dissipation.

1. Fundamental Lindblad Gaussian Dynamics

The dynamics are governed by master equations of the Lindblad type: $$\frac{d\rho}{dt} = -i[H,\rho] + \sum_\alpha \left(2L_\alpha\,\rho\,L_\alpha^\dagger - \{L_\alpha^\dagger L_\alpha, \rho\}\right),$$ with $H$ quadratic in bosonic/fermionic modes and $L_\alpha$ linear (usually creation/annihilation operators or Majorana combinations) (Carollo et al., 2017, Banchi et al., 2013, Verstraelen et al., 2019, Deng et al., 22 Aug 2025). The Gaussianity is strictly preserved throughout both transient and steady-state evolution, ensuring that full information is encoded in covariance matrices:

• Fermionic: $$\Gamma_{jk}(t) = \frac{1}{2} \mathrm{Tr}[\rho(t) [\omega_j, \omega_k]]$$ (Majorana basis)
• Bosonic: $$\Theta_{ij}(t) = \frac{1}{2}\langle R_i R_j + R_j R_i\rangle - \langle R_i\rangle\langle R_j\rangle$$ (quadrature basis)

The Lyapunov/Sylvester equations,

$X\Gamma + \Gamma X^T = Y,$

capture stationary solutions, with $X$, $Y$ derived from $H$ and Lindblad parameters.

2. Thermodynamic and Decoherence Phase Transitions

GDQPTs can exhibit first-order behavior, exemplified by the dissipative tight-binding chain (Medvedyeva et al., 2014):

• The bulk entropy density,

$s = \lim_{L\to\infty} \frac{1}{L} S(\rho_{ss}),$

jumps discontinuously at vanishing dissipation, marking a first-order transition between coherent and decoherent phases.

• Bulk transport properties such as nonlinear conductivity $\sigma$ exhibit discontinuous behavior across the transition, with diverging values in the coherent regime and finite ones under dissipation.

In boundary regions, critical phenomena persist with power-law divergences in correlation length $\xi$ and response functions, even as bulk observables become strictly local.

3. Dynamical (Non-equilibrium) Phase Transitions: Loschmidt and RLE Nonanalyticities

Time-dependent GDQPTs arise from quantum quenches between distinct Gaussian phases:

• The Loschmidt echo (unitary evolution) and generalized echos for mixed states are used to define rate functions $l(t)$ with possible nonanalytic "cusps" at critical times

$l(t) = -\frac{1}{2\pi}\int_0^\pi dk\,\ln|L^k(t)|.$

• For open systems, the reduced Loschmidt echo (RLE) is computed via correlation matrices of subsystems:

$$f_A(t) = \det\left[\frac{1}{2}(I_\ell + J_0 J_A(t))\right],\quad \zeta_A(t) = -\frac{1}{\ell}\log f_A(t).$$

• DQPT nonanalyticities (cusps) in $\zeta_A$ correspond to eigenvalue crossings of $-1$. Pure gain or loss channels may preserve these singularities, but simultaneous gain+loss strictly smears all DQPTs regardless of channel strengths (Parez et al., 25 Sep 2025, Sedlmayr et al., 2017).

Nested light-cone spreading patterns in the RLE, typically subleading in unitary evolution, become prominent under dissipation due to mixing effects among correlation matrix components.

4. Universality, Scaling Laws, and Information Geometry

GDQPTs often exhibit universal scaling and robust critical exponents, closely paralleling classical and quantum equilibrium transitions:

• In driven-dissipative photonic or Kerr lattices, second-order transitions interpolate between quantum and thermal Ising universality, depending on the loss-to-nonlinearity ratio (Verstraelen et al., 2019).
• Slow driving (ramp protocols) across GDQPTs demonstrates thermodynamic Kibble–Zurek scaling for nonadiabatic entropy production $\Sigma_\mathrm{na}$, with leading behavior:

$\Sigma_\mathrm{na} \sim \tau_q^{(α-2)/(γ+1)}$

but for bosonic Gaussian models, $\Sigma_\mathrm{na}$ is speed-independent ($\tau^0$), indicating breakdown of adiabaticity and unavoidable irreversibility (Bettmann et al., 4 Dec 2025).

Information-geometry methods (Bures metric, quantum Fisher tensor, Uhlmann curvature) provide sharp, basis-independent identifiers for criticality:

• The metric $g_{\mu\nu}$ in covariance space grows super-extensively at critical points,
• Uhlmann curvature $\Ucal_{\mu\nu}$ singularities directly track diverging correlation lengths, even where Liouvillian gaps close without forming collective order (Banchi et al., 2013, Carollo et al., 2017).

Observable	Critical Scaling	Context
Entropy density	Discontinuous	First-order decoherence QPT
Uhlmann curvature	$N^\alpha$, $\alpha>1$	Diverging $\xi$ at criticality
Nonadiabatic entropy	$\tau^0$	Gaussian bosonic QPTs

5. Topology and Dissipation-Driven Transitions

Topological GDQPTs occur when the modular Hamiltonian's symmetry class and associated invariants are preserved under Lindbladian evolution:

• For a quadratic density matrix $\rho = e^{-K_M}$, topological invariants (winding number $\nu$, Fu–Kane $\mathbb{Z}_2$ indices) are defined from the single-particle spectrum of $K_M$ or covariance matrices.
• Dissipation-induced transitions, such as in 1D class D, can drive gap closings and abrupt changes of the $\mathbb{Z}_2$ invariant at finite times $t_c$, determined purely by dissipation matrix parameters—entirely independent of the system Hamiltonian (Mao et al., 2023, Deng et al., 22 Aug 2025).
• Entanglement spectral analysis reveals bulk–edge correspondence via single-particle spectral gap closures and emergent zero modes, confirming the transition's topological nature.
• In symmetry-preserving Lindbladians, transitions at $t_c$ may be observed via purity-gap closings, $1/n$-scaling of Rényi entropies, or the disappearance of edge modes.

6. Experimental Realization and Physical Signatures

GDQPTs are experimentally accessible in platforms where dissipation and Lindblad engineering are finely controlled:

• Cold fermionic atom chains with engineered loss/gain,
• Nonlinear photonic lattices under quadratic driving and loss,
• Superconducting qubit arrays enabling explicit measurement of covariance matrices and entanglement spectra.

Physical signatures include:

• Jumps or singularities in entropy density, bulk transport, or purity-gap,
• Observation of DQPT cusps in time-resolved Loschmidt return rates,
• Detection of entanglement zero modes at edges signifying bulk–boundary correspondence for topological indices.

7. Open Problems and Context within Quantum Criticality

GDQPTs challenge conventional paradigms of closed-system quantum phase transitions by:

• Demonstrating criticality and universality in open, Markovian settings,
• Revealing dissipation as a sole or dominant driver of topology and collective phenomena,
• Uncovering regions of quantum-classical crossover and breakdowns of adiabatic control.

Outstanding issues include:

• Extension of symmetry-protected topological order to non-Gaussian open-system settings under more general Lindbladians,
• Classification of critical exponents and scaling functions in higher dimensions or with interacting, non-Gaussian baths.

Gaussian dissipative quantum phase transitions thus unite thermodynamic and topological critical phenomena under a common, analytically tractable framework, enabling insight into non-equilibrium quantum matter far beyond closed-system dynamics.