--- title: Dark-State Phase Transition url: https://www.emergentmind.com/topics/dark-state-phase-transition type: topic --- # Dark-State Phase Transition A dark-state phase transition is a nonequilibrium phase transition in an open quantum system whose stationary structure is governed by dark states, i.e. stationary states of a dissipative, Lindblad-type time evolution with zero von Neumann entropy and therefore pure steady quantum states [1803.08514]. In the most specific formulation, it denotes a transition within a dark manifold itself: irreversible decay, interactions, and quantum interference compete so that the system switches between distinct non-fluctuating stationary states rather than between a fluctuating active phase and a classical absorbing state [2105.06729]. Related usages occur in driven-dissipative light-matter systems, monitored fermionic chains, cavity-QED dark subspaces, and dissipative superfluids, where the transition can be encoded in the Liouvillian spectrum, in the dynamical response, or in the dimension of the dark sector [1909.09550]. ## 1. Conceptual definition In open many-body quantum systems, a dark state is a pure steady state annihilated by the dissipative dynamics. The notion becomes nontrivial when the set of dark states is not a single trivial absorbing configuration but a structured dark space with competing stationary states. The paper introducing the term “dark space phase transition” identifies precisely this situation: two competing dark states, a trivial one corresponding to a classical absorbing state and an emergent one which is quantum coherent, together with a nonequilibrium phase transition within this dark space [2105.06729]. This formulation differs from the classical absorbing-state paradigm. In classical models such as epidemic spreading, the transition is from a fluctuating active phase to an absorbing configuration state. In the quantum setting of the dark space phase transition, both phases are dark and non-fluctuating, but the nature of the dark state changes. The emergent phase is collective and quantum coherent, and its existence relies on interference between different dynamical paths, a phenomenology that cannot be encountered in classical systems [2105.06729]. The literature also uses the expression in a broader sense for transitions involving a dark state on one side and a mixed state on the other. In a driven open Rydberg spin ensemble, for example, the transition connects a zero entropy dark state with a finite entropy mixed state and thus goes beyond the realm of equilibrium statistical mechanics and becomes of genuine nonequilibrium character [1803.08514]. A plausible implication is that “dark-state phase transition” now names a family of nonequilibrium critical phenomena unified by the central role of dissipation-protected stationary states rather than a single universality class. ## 2. Microscopic framework of the dark space transition A concrete many-body realization is an array of \(N\) three-level units, in one dimension or higher dimension, with local basis states \(\ket{\bullet}\) (contagious infected), \(\ket{*}\) (non-contagious infected), and \(\ket{\circ}\) (healthy) [2105.06729]. The deterministic evolution of the density matrix is governed by the Lindblad quantum master equation \[ \frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t], \] with dissipator \[ \mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*}, \] and Hamiltonian \[ H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ]. \] Here \(\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}\), \(\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}\), and \(\Pi_\bullet^k\) is a facilitation operator indicating the presence of contagious neighbors [2105.06729]. The three ingredients have sharply differentiated roles. Irreversible decay \((\gamma)\) drives the system toward the healthy absorbing configuration. Interactions enter through facilitation: \(\ket{\circ} \leftrightarrow \ket{*}\) is only allowed if at least one neighbor is in \(\ket{\bullet}\). Quantum coherence and interference arise from the coherent transitions \(\ket{\bullet}\leftrightarrow\ket{*}\) and from the facilitated channel. It is this interplay of irreversible decay, interactions, and quantum interference that produces the dark space phase transition [2105.06729]. In the thermodynamic limit, the model yields two fundamentally different dark states. The trivial, classical absorbing state is \[ \ket{\rm D}=\bigotimes_{k=1}^N \ket{\circ}^{(k)}. \] The emergent quantum-coherent dark state is \[ \ket{\rm D_e} = \bigotimes_{k=1}^N \left( \alpha_+ \ket{\bullet} - \alpha_- \ket{\circ} \right)^{(k)}, \] a collective state with finite coherence between contagious and healthy states, measurable through off-diagonal observables such as \(\langle \lambda_4\rangle\), where \(\lambda_4=\ket{\bullet}\bra{\circ}+\text{h.c.}\) [2105.06729]. The emergent state has no classical analog and only appears collectively in the thermodynamic limit. ## 3. Order parameters, criticality, and first-order phenomenology For the nonequilibrium dark space transition, the control parameter is the coherent infection rate \(\Omega_2\). At fixed \(\Omega_1\) and \(\gamma\), the threshold is \[ \Omega_2^{\rm c}=2\Omega_1. \] For \(\Omega_2<\Omega_2^{\rm c}\), the system ends in the trivial dark state \(\ket{\rm D}\); for \(\Omega_2>\Omega_2^{\rm c}\), it ends in the emergent dark state \(\ket{\rm D_e}\) [2105.06729]. The order parameter is the density of contagious sites, \[ \varrho_\bullet = \frac{1}{N} \sum_{k=1}^N \langle n_\bullet^{(k)} \rangle. \] It vanishes in \(\ket{\rm D}\) and is nonzero in \(\ket{\rm D_e}\). For fully connected or infinite-dimensional lattices, the stationary solutions are \[ \varrho_\bullet = \begin{cases} 0 & \text{for } \Omega_2 < 2\Omega_1,\\[4pt] \frac{1}{2}[1 \pm \sqrt{1-x^2}] & \text{for } \Omega_2 \ge 2\Omega_1, \end{cases} \qquad x=\frac{2\Omega_1}{\Omega_2}. \] At \(\Omega_2=\Omega_2^{\rm c}\), the gap in the imaginary part of the effective non-Hermitian Hamiltonian \(H_{\rm eff}\) closes as \(N\to\infty\), indicating a phase transition [2105.06729]. A distinct but related first-order dark-state phenomenology appears in a driven open Rydberg spin ensemble on a \(d\)-dimensional square lattice [1803.08514]. There the order parameter is the local Rydberg density \(\nu=\langle n_l\rangle\), with \(\nu=0\) in the dark state and \(\nu>0\) in the mixed phase. The coarse-grained dynamics is described by \[ \partial_t \nu_X = D \nabla^2 \nu_X - V'(\nu_X) + \xi_X, \] with Landau potential \[ V(\nu_X)=\tfrac{\Delta}{2}\nu_X^2+\tfrac{\mu}{3}\nu_X^3+\tfrac{\lambda}{4}\nu_X^4, \] and multiplicative non-thermal noise \[ \langle \xi_X \xi_Y \rangle = \delta(X-Y)\kappa \nu_X. \] The multiplicative character is crucial: the noise vanishes when \(\nu_X=0\), so fluctuations vanish strictly in the dark state [1803.08514]. The functional RG analysis of that first-order dark-state transition shows strong dimensional dependence. In \(d=1\), fluctuations smooth out the potential barrier during the RG flow and the first-order transition is replaced by a second-order transition. For \(d\ge 2\), the barrier survives the RG flow and the first-order transition persists [1803.08514]. At coexistence, the long-wavelength theory develops a flat deterministic potential together with a vanishing noise kernel over an extended domain \(0<\nu<\nu_f\), supporting an extensive set of noiseless steady configurations. This suggests a sharp distinction between equilibrium first-order transitions and absorbing or dark-state coexistence. ## 4. Driven-dissipative realizations in light-matter and superfluid systems In the interpolating Dicke-Tavis-Cummings model, infinitesimal dissipation stabilizes new steady-state solutions and extends the region in which the normal phase exists into domains that would be occupied by the superradiant phase in the closed system [1909.09550]. The resulting dissipation-stabilized regime, termed the excited Normal Phase (e-NP), is statically identical to the usual normal phase: in both cases the cavity is empty and all spins are unexcited. The transition between them is therefore invisible in static or equal-time observables and appears only in the dynamical response. The cavity response function \[ \mathcal{A}(\omega)=-2\,\mathrm{Im}[G^R(\omega)] \] shows a peak inversion across the transition, and the dominant fluctuations flip from being particlelike to holelike. The Liouvillian eigenvalues exhibit behavior akin to exceptional points [1909.09550]. A different driven open Dicke model exhibits a transition from a predominantly superradiant to a predominantly subradiant state [1705.02889]. The transition is controlled by the interplay of resonant driving, cavity decay, spontaneous emission, and dephasing. Reported signatures include a kink in the mean TLS excitation as a function of driving, a local minimum in the mean cavity photon number, a peak in the second-order photon correlation function \(g^{(2)}(0)\), and a switch in a collectivity measure \(R(l)\) from enhancement of the maximal-spin Dicke sector to enhancement of low-\(l\) subradiant sectors [1705.02889]. After the drive is switched off, relaxation generates a cascade of dark Dicke states, with dark state populations up to unity. In a Bose-Hubbard chain with a single lossy site, the identified transition is first-order and occurs between a discrete dark soliton and a uniform superfluid [2509.25707]. The Lindblad dynamics is \[ \frac{d\hat{\rho}}{dt} = -i[\hat{H}, \hat{\rho}] + \gamma \left(\hat{a}_0 \hat{\rho} \hat{a}_0^\dagger - \frac{1}{2} \{\hat{a}_0^\dagger \hat{a}_0, \hat{\rho}\}\right), \] with jump operator \(\hat{\Gamma}=\sqrt{\gamma}\hat{a}_0\). The dark soliton satisfies \(n_0=\langle \hat{a}_0^\dagger \hat{a}_0\rangle=0\), or equivalently \(\psi_0=0\) in the discrete Gross-Pitaevskii description, so the lossy site is empty and the state is dark. Classical-field simulations and Bogoliubov stability analysis identify a bistable region and an upper bistability boundary \(J=\gamma/4\) in the ideal theory [2509.25707]. ## 5. Dark subspaces, order parameters from nullity, and stability of degenerate manifolds In the Tavis-Cummings model, dark states are many-body quantum states with nonzero excitation in the collection of qubits but from which no photon emission is possible [2509.14313]. With no photons in the cavity and emission operator \[ \hat{O}_{\rm dark}=\sum_j g_j S_{j,-}, \] a dark state satisfies \(a^\dagger a|d\rangle=0\) and \(\hat{O}_{\rm dark}|d\rangle=0\). The number of linearly independent dark states in the \(s\)-excitation sector is \[ N_{{\rm dark},N,s}= \begin{cases} 0, & s>N/2,\\[4pt] \binom{N}{s}-\binom{N}{s-1}, & s\le N/2. \end{cases} \] This formula holds even when the couplings \(g_j\) are completely non-uniform and disordered, provided they are all nonzero [2509.14313]. The same work identifies a dark-bright phase transition in which the number of independent dark states plays the role of an order parameter. With tuning parameter \(\alpha_{s,N}=s/N\) and order parameter \[ o_{s,N}=\frac{N_{{\rm dark},N,s}}{\binom{N}{s}}, \] the thermodynamic-limit behavior is \[ o_{N\to\infty}= \begin{cases} 0, & \alpha>1/2,\\[4pt] \dfrac{1-2\alpha}{1-\alpha}, & \alpha\le 1/2. \end{cases} \] The critical point is \(\alpha_c=1/2\), and the transition is second-order [2509.14313]. Experimentally, the dark-state number can be accessed by a no-photon protocol: if no photons are detected from a lossy cavity, the qubits collapse onto a dark state, and summing over product-state preparations yields the dimension of the dark subspace [2509.14313]. The dark-space transition of the facilitated three-level model has a more compact dark manifold: the two dark states \(\{\ket{D},\ket{D_e}\}\) together span a two-dimensional steady-state dark subspace, \[ \ket{\Psi}=\alpha\ket{D}+\beta\ket{D_e}, \] which can encode a qubit [2105.06729]. The proposed significance is collective encoding of quantum information based on non-fluctuating, collectively robust steady states. A contrasting many-body setting is the \(\Lambda\)-Dicke model with two bosonic modes [1109.2456]. There a family of dark states exists only when the two ground states are degenerate, \(\delta=E_2-E_1=0\), and is characterized by \(\varphi_{1,2}=0\) and \(\Psi_3=0\). In that limit, \(\Psi_2\) is arbitrary within normalization, so the dark manifold reflects a degeneracy rather than a robust ordered phase. The spectrum contains a zero mode, \[ \varepsilon_0=0, \] and the dark state is stable only for \(\delta=0\) and below the critical couplings, with stability criterion \[ \left( \frac{g_1}{g_{1,c}} \right)^2 (1-\Psi_2^2) \leq 1 - \left( \frac{g_2}{g_{2,c}} \right)^2\Psi_2^2. \] The authors emphasize that the many-body dark state is meta-stable or even unstable and is not robust against perturbations [1109.2456]. ## 6. Measurement-induced dark-state transitions Measurement-induced phase transitions can also be framed as quantum phase transitions in the dark state of an effective non-Hermitian Hamiltonian [2105.08076]. In a one-dimensional ring of free fermions with algebraically decaying hopping \(t_{s,m}=|s-m|^{-p}\), continuous local density measurements at rate \(\gamma\), and \(p>1\), the competition between long-range coherent hopping and disentangling local measurements produces three phases: an area law phase, a critical phase with logarithmic entanglement growth, and a new algebraic scaling phase [2105.08076]. For \(13/2\), the transition between the area law and critical phases is Berezinskii-Kosterlitz-Thouless-like and depends on \(\gamma\). The effective field theory is a non-Hermitian sine-Gordon type theory with an additional long-range purely imaginary hopping term. The first-order RG equations for the long-range coupling \(\Delta\) and stiffness \(\eta\) are \[ \partial_s \Delta=(3-2p-\eta)\Delta,\qquad \partial_s \eta=-\eta^2\Delta. \] For \(p<3/2\), the long-range term is always relevant, which explains why the algebraic phase is independent of the measurement strength \(\gamma>0\) [2105.08076]. Exact numerical simulations of the monitored wave functions and analytical predictions from replica field theory show excellent quantitative agreement, reinforcing the interpretation of monitored criticality as a dark-state transition of an effective non-Hermitian generator. ## 7. Terminological scope and adjacent usages The phrase “dark-state phase transition” should be distinguished from phase transitions in a dark sector. Several recent arXiv works use “dark” to denote dark matter or dark-energy sectors rather than dissipation-protected pure stationary states. One study rescues overabundant fermionic dark matter through a strongly first order phase transition after freeze-out, where the relic abundance is diluted by entropy injection, \[ \Delta=(1+\alpha)^{3/4}, \] and the resulting stochastic gravitational-wave signal can lie within the sensitivity of future experiments [2602.16822]. Another considers a nearly-conformal dark sector in which a confining first-order phase transition can generate the NANOGrav signal of a stochastic gravitational wave background [2306.17086]. Other examples include the population of approximately Weyl invariant secluded dark sectors through a phase transition controlled by the Hubble parameter, with the condition \(H(t_{\rm PT})\sim M\) and heavy dark matter generically predicted [2210.03108]; a conformal freeze-in scenario where a dark CFT confines and the cosmological phase transition completes promptly without significant supercooling [2502.06965]; and a galactic model positing a phase transition from radially emitted dark matter to dark energy, with Bose-Einstein condensation proposed as one mechanism [2110.00366]. These usages are conceptually separate from dark-state phase transitions in open quantum dynamics. Within quantum many-body physics proper, the term therefore refers most precisely to transitions in which dark states determine the stationary critical structure: transitions inside dark space, transitions from dark to mixed steady states, or transitions whose order parameter is the dimension or dynamical response of a dark sector. This suggests a unifying viewpoint in which the relevant object is neither an equilibrium free energy nor a classical absorbing configuration, but the dark manifold of a Lindbladian or effective non-Hermitian evolution. Source: https://www.emergentmind.com/topics/dark-state-phase-transition