Dissipation-Driven Topological Phase Transitions

• Dissipation-driven topological phase transitions are transformations in open systems where topology is governed by engineered dissipation rather than coherent Hamiltonian dynamics.
• The framework leverages Lindblad jump operators and reservoir-induced couplings to modulate steady-state invariants and observable edge phenomena.
• Key insights include analytical predictions of critical time dynamics, phase boundaries, and Floquet transitions, highlighting non-Hermitian effects in topological classifications.

Searching arXiv for recent and foundational papers on dissipation-driven topological phase transitions. Dissipation-driven topological phase transitions are changes between topologically distinct phases of an open system in which the control parameter is the dissipative structure itself—Lindblad jump operators, loss and gain profiles, reservoir-induced couplings, or their Floquet generalizations—rather than, or in addition to, a closed-system Hamiltonian. In this setting, topology is assigned not only to Hamiltonian bands but also to steady-state density matrices, covariance matrices, Liouvillian generators, damping matrices, or dissipation-rate bands. A recent one-dimensional class D analysis makes the point sharply: for Gaussian Lindblad dynamics with linear dissipation operators, the steady-state $\mathbb{Z}_2$ invariant and the critical times of dynamical topological transitions depend exclusively on the dissipation operators and are independent of the system Hamiltonian (Deng et al., 22 Aug 2025).

1. Formal setting and topological objects

The common starting point is the Lindblad equation,

$$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$

with $\hat{H}$ quadratic and $\hat{L}_\mu$ linear in the Gaussian fermionic constructions that admit closed correlation-matrix dynamics (Deng et al., 22 Aug 2025). In those cases, the density matrix remains Gaussian, and topology can be formulated through the modular Hamiltonian, the correlation matrix, or an equivalent flattened covariance-matrix projector. This density-matrix-centered viewpoint was systematized in the general framework of dissipative fermionic topology, where the relevant notions are the steady state, the covariance matrix, the dissipative gap, and the purity gap rather than a Hamiltonian ground state alone (Bardyn et al., 2013).

Several inequivalent but related topological carriers appear across the literature. In Gaussian Lindblad systems, the modular Hamiltonian of the mixed state can fall into an Altland–Zirnbauer class and inherit the associated invariants; in one dimension, class D yields a Pfaffian $\mathbb{Z}_2$ index evaluated at high-symmetry momenta (Deng et al., 22 Aug 2025). In engineered non-Hermitian lattices, the effective Bloch object is instead a complex coupling matrix or damping matrix, and the relevant invariant is often a winding of a complex phase or of a ratio of eigenvector components around a dark mode or exceptional structure (Rosenthal et al., 2018). In Floquet open systems, the natural object can be a Floquet Liouvillian or a Floquet damping matrix, whose complex quasienergies support $0$- and $\pi$-gap topology (Gogoi et al., 14 Jul 2025).

This plurality of topological objects is not a contradiction. It indicates that in open systems the spectrum, the state, and the relaxation operator are not rigidly locked together. That separation is one of the principal differences from conventional Hamiltonian band topology (Bardyn et al., 2013).

2. Steady-state topology controlled by dissipation

The most direct realization of a dissipation-driven topological phase transition is a change of the steady-state invariant upon tuning dissipative parameters while keeping the coherent Hamiltonian fixed. In the one-dimensional fermionic Lindblad problem with linear dissipation operators, the steady-state correlation matrix $\Delta_s(k)$ defines a class D Pfaffian invariant through the signs of $\mathrm{Pf}\,\Delta_s(0)$ and $\mathrm{Pf}\,\Delta_s(\pi)$. The central result is that the corresponding $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$0 invariant reduces to a sign structure determined only by dissipative coefficients, so the steady-state topology is independent of the Hamiltonian parameters (Deng et al., 22 Aug 2025). For the explicit nearest-neighbor dissipator

$$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$1

the phase diagram is controlled by the dissipative amplitudes $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$2 and is independent of the Hamiltonian parameters $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$3 (Deng et al., 22 Aug 2025).

The broader dissipative-topology program pushed this logic further by asking for topological states generated even when the Hamiltonian is absent or secondary. In the density-matrix framework of quadratic fermionic Liouvillians, dissipation can be engineered so that the unique steady state is the analog of a topological superfluid or insulator, with classification inherited from the flattened covariance matrix rather than a filled-band projector of a closed Hamiltonian (Bardyn et al., 2013). In two dimensions, engineered reservoirs can stabilize a Chern-insulator-like steady state, but transport need not follow equilibrium expectations: the steady-state Chern number can remain nontrivial while the Hall conductance ceases to be quantized unless the coherent Hamiltonian, the dissipative preparation protocol, and the occupancy structure are mutually compatible (Bribian et al., 2019).

A recurring consequence is that “topology by dissipation” is not merely topology surviving environmental coupling. It is topology whose defining invariant is set by the reservoir engineering itself.

3. Dynamical and Floquet transitions in time

Dissipation-driven transitions are not restricted to asymptotic steady states. In the class D Lindblad quench problem, the system is prepared in the steady state of one Lindbladian and then evolved under another. The time-dependent Pfaffian at $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$4 acquires a closed analytic form in terms of the initial and final dissipative parameters alone, and critical times are obtained from its zeros. A notable consequence is that the topology can change at analytically predictable times even when the initial and final steady states have the same overall $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$5 index, so the non-equilibrium trajectory traverses intermediate topological sectors that are absent from the endpoints (Deng et al., 22 Aug 2025).

A distinct dynamical mechanism appears in two-dimensional superconductors with self-consistent order-parameter relaxation under a Lindblad equation. There the dissipative evolution drives a transition from a topologically trivial planar $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$6 phase to a topologically nontrivial $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$7 phase of class C. The transition is detected by a kink in the fidelity-based rate function and by the time-dependent spin-Hall conductance approaching the topological value of the post-quench phase. The critical time $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$8 depends strongly on the system–bath coupling and decreases approximately as a power of the dissipation strength (Nava et al., 2023).

Floquet settings add another layer. In dissipative periodically driven XY and extended XY chains, the effective Floquet non-Hermitian Hamiltonian separates into pure real-gap, pure imaginary-gap, and complex-gap regions. Dissipative Floquet dynamical phase transitions occur in the real-gap region, and the relevant frequency window shrinks as the dissipation coupling increases until it collapses to a single point. Importantly, the work shows that a non-Hermitian topological phase is not an essential condition for these dissipative Floquet dynamical phase transitions (Naji et al., 2021). In a periodically driven Rashba nanowire with Markovian loss, the periodic Liouvillian can be reduced by third quantization to a Floquet damping matrix supporting edge-localized Majorana $$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}] + \sum_\mu \left(2\hat{L}_\mu \hat{\rho}\hat{L}_\mu^\dagger - \{\hat{L}_\mu^\dagger \hat{L}_\mu,\hat{\rho}\}\right),$$9-modes and $\hat{H}$0-modes, while dissipation can also generate topologically trivial $\hat{H}$1- and $\hat{H}$2-modes tied to exceptional points rather than bulk invariants (Gogoi et al., 14 Jul 2025).

Taken together, these results suggest that in open systems the “phase transition” can occur in steady-state topology, in the topology of the time-evolving density matrix, or in the Floquet damping spectrum, depending on which operator governs the long-time physics.

4. Bulk–edge correspondence, edge protection, and exceptional structures

Bulk–edge correspondence survives in several dissipative settings, but in modified form. In the modular-Hamiltonian formulation of Gaussian open systems, the non-equilibrium density matrix exhibits a bulk gap closing in the single-particle entanglement spectrum under periodic boundary conditions and topologically protected zero modes under open boundary conditions, yielding a direct bulk–edge correspondence for the density matrix rather than the physical Hamiltonian (Deng et al., 22 Aug 2025). In the earlier dissipative classification program, bulk topology of the covariance matrix implied boundary Majorana modes, but the correspondence was refined: edge degrees of freedom can appear either as zero-damping modes or as zero-purity modes, because in open systems the state and the Liouvillian spectrum are not identical objects (Bardyn et al., 2013).

Strong system–environment coupling introduces another variant. In a topological emitter array coupled to a one-dimensional electromagnetic environment, the environment induces long-range coherent interactions and correlated dissipation while preserving chiral symmetry for appropriate spacings. A topological phase transition occurs at a critical single-emitter decay rate $\hat{H}$3, the spectrum width of the emitter array. On one side of this point, edge states are dissipative; on the other, the edge-state dissipation rates vanish in the thermodynamic limit, producing robust dissipationless edge states protected by the symmetry structure of the Lindblad operator (Nie et al., 2021).

Exceptional points complicate the picture further. In the driven dissipative Rashba nanowire, topological Majorana $\hat{H}$4- and $\hat{H}$5-modes are distinguished from trivial edge-localized $\hat{H}$6- and $\hat{H}$7-modes whose emergence is tied to exceptional points and does not carry a bulk invariant (Gogoi et al., 14 Jul 2025). In non-Hermitian SSH-type chains with a gain–loss domain wall, bulk–boundary correspondence can break down continuously as the system is “gradually torn” by tuning the domain-wall strength. The resulting global phase diagrams in the thermodynamic limit become hybrids of periodic-boundary and open-boundary phase diagrams, and several phase transitions occur during that crossover (Ren et al., 22 May 2025).

A common misconception is that edge localization alone diagnoses dissipative topology. The literature shows that it does not: edge localization may arise from bulk topology, from exceptional points, or from non-Hermitian boundary sensitivity, and these mechanisms are not equivalent.

5. Representative realizations and observables

The phenomenon spans quantum, photonic, circuit, and metamaterial platforms. In a topological Josephson junction, Majorana bound states generate a double sine-Gordon action with a $\hat{H}$8 term, so the relevant tunneling event is a correlated $\hat{H}$9 phase slip. Ohmic dissipation then drives a zero-temperature transition between resistive and superconducting phases, but the critical dissipation is shifted: the critical resistance is $\hat{L}_\mu$0, four times the conventional $\hat{L}_\mu$1, equivalently the critical damping is four times smaller than in a conventional junction (Matthews et al., 2013). This is a dissipation-driven transition whose location is altered by topology.

In a dissipative radio-frequency metamaterial, topology is encoded by the winding of the complex inter-sublattice coupling phase $\hat{L}_\mu$2 around a dark mode. The measured observable is the mean displacement of dissipated energy,

$\hat{L}_\mu$3

which is exactly quantized as $\hat{L}_\mu$4 for $\hat{L}_\mu$5 and $\hat{L}_\mu$6 for $\hat{L}_\mu$7, with a transition at the dark-state point $\hat{L}_\mu$8 (Rosenthal et al., 2018). The transition is non-Hermitian in the precise sense that it is characterized by the vanishing of the imaginary part of an eigenvalue rather than by a conventional real bandgap closing (Rosenthal et al., 2018).

A time-multiplexed photonic resonator network realizes dissipatively coupled SSH and Harper–Hofstadter models with $\hat{L}_\mu$9, so topology resides in bands of dissipation rates rather than energies. The SSH transition is driven purely by the ratio of dissipative couplings $\mathbb{Z}_2$0, with the phase boundary at $\mathbb{Z}_2$1 and edge states appearing as isolated loss rates in the gap between bulk dissipation bands (Leefmans et al., 2021).

A conceptually minimal quantum model is the single-particle dissipative walker on a one-dimensional bipartite lattice, where the steady-state velocity is

$\mathbb{Z}_2$2

with $\mathbb{Z}_2$3 the winding number of the complex hopping amplitude $\mathbb{Z}_2$4 and $\mathbb{Z}_2$5 the average time between quantum jumps. Here the average velocity itself is a non-equilibrium order parameter, and boundaries between regions with different winding numbers host topological bound states whose number equals the difference of the winding numbers. When that difference exceeds one, the bound-state sector becomes a dark decoherence-free subspace (Kastoryano et al., 2018).

These realizations also define the main diagnostics: Pfaffian sign changes and entanglement-spectrum gap closures (Deng et al., 22 Aug 2025), spin-Hall conductance and fidelity rate functions (Nava et al., 2023), dissipation-band spectroscopy and edge localization (Leefmans et al., 2021), quantized mean dissipative displacement (Rosenthal et al., 2018), transport thresholds (Matthews et al., 2013), and direct reconstruction of correlation matrices or damping matrices (Gogoi et al., 14 Jul 2025).

6. Conceptual tensions, limitations, and current directions

Several results delimit the scope of the subject. First, topology and transport can decouple in open systems. In dissipatively prepared Chern-insulator steady states, the usual relation between the Chern number and the Hall conductance is broken; quantized Hall response requires not only a nontrivial steady-state Chern number but also a compatible coherent Hamiltonian and appropriate occupation structure (Bribian et al., 2019). Second, non-Hermitian bulk–boundary correspondence can fail or interpolate between boundary conditions rather than holding in its Hermitian form, as shown explicitly by gain–loss-domain-wall tearing (Ren et al., 22 May 2025). Third, a dissipative dynamical topological transition need not coincide with a non-Hermitian bulk topological phase, as demonstrated in the Floquet XY analysis (Naji et al., 2021).

The robustness of dissipation-driven topology is also pattern-dependent. In the bosonic Kitaev chain, uniform dissipation destroys topological amplification at a finite critical loss, whereas dissipation on every other site leaves the system topological even for arbitrarily large loss, while odd-length unit-cell patterns with loss on the first site do induce a topological phase transition at an analytically determined critical dissipation (Fortin et al., 2024). In a non-Hermitian Ising chain, an imaginary transverse field can induce topological degeneracy in a phase that is topologically trivial in the Hermitian limit, and that degeneracy remains robust against random imaginary fields because of an emergent nonlocal symmetry in the thermodynamic limit (Zhang et al., 2020).

Most exact treatments currently rely on restrictive structures: Gaussian states, linear or quadratic Lindblad operators, Markovian baths, one-dimensional class D or BDI settings, or specially engineered dissipative couplings (Deng et al., 22 Aug 2025). This suggests that the field’s present precision comes from solvable corners. A plausible implication is that extending these results to genuinely interacting, non-Gaussian, or strongly non-Markovian environments will require topological notions that are neither purely Hamiltonian nor purely single-particle Liouvillian.

Dissipation-driven topological phase transitions therefore occupy a distinct conceptual domain. They are not merely Hamiltonian transitions perturbed by loss, and they are not reducible to generic non-Hermitian band phenomena. They are transitions in which the reservoir architecture, the decay channels, and the state-space geometry become topological control parameters in their own right.