Dissipation-Induced Topology in Open Systems

• Dissipation-induced topology is a framework where engineered dissipation and system–environment coupling generate and stabilize topological properties in quantum systems.
• It utilizes non-Hermitian band theory, Liouvillian spectra, and density matrix invariants to characterize phenomena such as Majorana edge modes and symmetry-protected order.
• Recent approaches use tensor-network simulations and Floquet dynamics to derive design recipes and understand limitations for realizing pure or mixed topological steady states in open systems.

Dissipation-induced topology denotes the emergence, stabilization, modification, or diagnosis of topological structure through coupling to an environment. In the open-system literature, this encompasses purely dissipative preparation of topological steady states, symmetry-protected topology in mixed nonequilibrium steady states, Liouvillian and damping-matrix topological invariants, non-Hermitian point-gap topology, and transport phenomena in which loss itself closes, reopens, or reshapes effective gaps (Bardyn et al., 2013). The central shift relative to Hamiltonian band theory is that topology may reside not only in a ground-state wavefunction, but also in a density matrix, covariance matrix, modular Hamiltonian, Liouvillian, Floquet damping matrix, or even in bands of dissipation rates (Goldstein, 2018).

1. Open-system formulations and topological data

The basic dynamical setting is the Gorini–Kossakowski–Sudarshan–Lindblad equation

$$\partial_t \rho = -i[H,\rho] + \sum_\mu \left(L_\mu \rho L_\mu^\dagger - \tfrac12\{L_\mu^\dagger L_\mu,\rho\}\right),$$

with a Hamiltonian contribution $H$ and jump operators $L_\mu$ encoding Markovian dissipation. In fermionic Gaussian settings, quadratic $H$ and linear or quadratic jump operators preserve Gaussianity, so the full state is encoded by single-particle correlators or covariance matrices; in bosonic quadratic Markovian dynamics, the analogous role is played by the second-moment matrix $Q$ (Goldstein, 2018).

Several technically distinct but closely related representations recur. In mixed-state tensor-network treatments, the density matrix is vectorized by the Choi isomorphism, yielding a superoperator evolution in Liouville space that can be simulated directly, including in the thermodynamic limit by infinite-TEBD applied to matrix-product operators (Veríssimo et al., 2022). In third quantization, quadratic Liouvillians are mapped to quadratic forms in adjoint Fock or Liouville-Fock space, and spectral as well as topological information is encoded in a non-Hermitian damping matrix such as $X$ or a Floquet generalization $X_F$ (Gogoi et al., 14 Jul 2025). In bosonic problems, a non-Hermitian dynamical matrix $\mathbf G$ governs the Heisenberg evolution of Nambu operators and determines rapidity bands, pseudospectra, and metastable edge operators (Flynn et al., 2021).

A defining structural difference from equilibrium is that spectral and state-based notions of topology separate. Bardyn et al. distinguished a dissipative gap, set by the smallest nonzero damping rate, from a purity gap, which controls whether a mixed Gaussian steady state can still be flattened into a topological projector (Bardyn et al., 2013). In non-Hermitian band formulations, the relevant gap may instead be a point gap in the complex plane, characterized by a winding of $\det[H(k)-E_0]$ around a base point $E_0$ (Lu et al., 2023). In purely dissipative photonic and atomic lattices, the topological object can even be a band of decay rates rather than an energy band (Leefmans et al., 2021).

2. Density-matrix topology, modular Hamiltonians, and classification

A central idea of dissipative topology is that the density matrix itself can carry topological information. For Gaussian steady states, Bardyn et al. introduced the covariance matrix

$H$0

and its spectrally flattened form $H$1, from which one builds

$H$2

This permits a symmetry-based classification parallel to Altland–Zirnbauer band theory. In 2D class D one obtains a Chern number from $H$3, while in 1D class BDI one obtains an integer winding number. Crucially, these invariants remain meaningful for mixed Gaussian steady states provided the purity gap stays open (Bardyn et al., 2013).

Diehl’s no-go theorem and construction sharpened this perspective for 2D Chern insulators. For any quadratic Gaussian steady state, even mixed, one may define a flattened single-particle operator

$H$4

with a reference “purity chemical potential” $H$5 inside the occupancy gap. This places dissipative steady states into the same classifying spaces as equilibrium flattened Hamiltonians and extends the classification to other symmetry classes and to superconductors in a Majorana basis (Goldstein, 2018).

A more recent formulation uses the modular Hamiltonian. In one-dimensional class D systems with linear dissipation operators, the density matrix remains Gaussian and can be written as $H$6, with $H$7 quadratic in Majoranas and related to the correlation matrix by $H$8. In this framework, the steady-state $H$9 invariant is

$L_\mu$0

and depends only on the antisymmetric part $L_\mu$1 of the dissipator, not on the system Hamiltonian $L_\mu$2. Under a quench from one Lindbladian to another, topological transitions occur at analytically predictable critical times $L_\mu$3, and the single-particle entanglement spectrum exhibits bulk–edge correspondence through gap closures under periodic boundary conditions and exact zero modes under open boundary conditions (Deng et al., 22 Aug 2025).

These constructions imply that dissipative topology is not confined to steady-state purification. Mixed-state invariants, purity-gap protection, and modular-Hamiltonian topology make it possible for nonequilibrium density matrices to exhibit sharp topological transitions even when the initial and final steady states share the same topological index (Deng et al., 22 Aug 2025).

3. Dissipative Majorana physics and topological superconductors

The canonical one-dimensional example is the dissipative Kitaev wire. In “Topology by Dissipation in Atomic Quantum Wires,” the dynamics is purely dissipative, with $L_\mu$4, and jump operators

$L_\mu$5

These quasi-local operators cool the bulk into a pure p-wave paired dark state identical to the BCS ground state of Kitaev’s Hamiltonian at $L_\mu$6 and $L_\mu$7. The bulk has a dissipative gap $L_\mu$8, while a finite chain supports two Majorana edge modes $L_\mu$9 and $H$0 that span a decoherence-free subspace. Adiabatic deformations of the jump operators then implement non-Abelian braiding within that subspace (Diehl et al., 2011).

The broader density-matrix framework of Bardyn et al. showed how Majorana zero-damping modes and zero-purity modes arise from bulk–edge correspondence in dissipative topological superconductors. In pure-state regimes, the number of edge zero-damping modes saturates the bulk invariant jump; in mixed-state regimes, zero-purity modes can share the burden of bulk–edge matching. The formal distinction has no direct equilibrium analogue and reflects the independent roles of damping and purity gaps (Bardyn et al., 2013).

A complementary algebraic viewpoint was developed for dissipative topological superconductors with linear Lindblad operators. In third quantization, equilibrium Majorana zero modes of the isolated Hamiltonian become kinetic zero modes of the open-system matrix $H$1. If $H$2 denotes the number of equilibrium Majorana zero modes and $H$3 the overlap matrix between the Majorana wavefunctions and the dissipator fields, then the number of kinetic zero modes is

$H$4

This relation yields explicit design principles: dissipators acting only in the gapped bulk, self-adjoint dissipators, and dissipation localized away from edges all enlarge the dark-space degeneracy (Shustin et al., 15 Aug 2025).

Periodic drive enriches the same theme. In a driven Rashba nanowire proximitized by an $H$5-wave superconductor, the periodic Liouvillian defines a Floquet damping matrix $H$6 with pseudo-anti-Hermitian symmetry $H$7. Two winding numbers extracted from half-period decompositions combine into $H$8 and $H$9, which count Majorana $Q$0-modes and $Q$1-modes per edge. Dissipation not only shifts phase boundaries but can induce topological phases absent at $Q$2. The same system also supports trivial $Q$3- and $Q$4-modes tied to exceptional points rather than to bulk invariants, making the distinction between topological and exceptional-point-induced boundary modes operationally important (Gogoi et al., 14 Jul 2025).

An explicitly bosonic analogue exists in metastable rather than asymptotically stable dynamics. In quadratic bosonic chains undergoing Markovian dissipation, nontrivial winding of rapidity bands can produce edge-localized “Majorana bosons,” paired with symmetry generators on the opposite edge. Their signatures are exponentially long mixing times and a zero-frequency peak in the steady-state power spectrum (Flynn et al., 2021).

4. Symmetry-protected topological order in mixed states

Dissipation-induced topology is not restricted to fermionic superconductors. Ver et al. studied the spin-1 Affleck–Kennedy–Lieb–Tasaki chain coupled to a Markovian environment and showed that local dissipation can preserve or destroy symmetry-protected topological order depending on the symmetry of the jump operators. The closed-system Hamiltonian is

$Q$5

Two local dissipations were compared: the time-reversal-breaking choice $Q$6 and the time-reversal-preserving choice $Q$7 (Veríssimo et al., 2022).

The mixed-state steady state was computed directly in the thermodynamic limit by an MPO-based tensor-network method. For time-reversal-preserving dissipation, the nonequilibrium steady state retains two hallmark signatures of the Haldane phase: a finite long-distance string correlator

$Q$8

and exact doublets in the “entanglement energies” $Q$9 obtained from the Schmidt values of the vectorized MPO. For $X$0, by contrast, $X$1 already for $X$2 and no systematic degeneracies remain (Veríssimo et al., 2022).

This leads to a generalized definition of dissipative SPT order in mixed states based on two jointly necessary criteria: nonzero long-distance string correlators and stable entanglement-spectrum degeneracies corresponding to irreducible projective representations of the protecting symmetry group. The same work identifies the key mechanism succinctly: the jump operators must commute with the projective representation of the protecting symmetry, here time reversal, in order not to spoil the SPT order (Veríssimo et al., 2022).

A related but distinct non-Hermitian route to topological degeneracy appears in the Ising chain with a real and an imaginary transverse field. There the condition

$X$3

defines a nontrivial $X$4 phase with twofold ground-state degeneracy and a nonlocal strong zero mode, while $X$5 gives the critical lines. The degeneracy remains robust against random imaginary fields, which the paper interprets as a form of disordered dissipation (Zhang et al., 2020). This suggests that dissipation-induced topology spans both Lindbladian steady-state constructions and non-Hermitian effective descriptions with edge-protected degeneracies.

5. Non-Hermitian topology, dissipation bands, and transport

In many systems the relevant topology is attached not to a density matrix but to a non-Hermitian band structure generated by loss. A paradigmatic example is the radio-frequency metamaterial of harmonic oscillators studied by Helbig et al. A one-dimensional lattice of coupled LC resonators with resistive loss only on the $X$6 sublattice has a complex function $X$7 whose winding

$X$8

changes from $X$9 for $X_F$0 to $X_F$1 for $X_F$2, with a sharp transition at the “dark” mode $X_F$3. The mean dissipation position $X_F$4 equals this winding exactly, $X_F$5, so the topological invariant is measured directly as quantized bulk energy transport (Rosenthal et al., 2018).

A closely related purely dissipative SSH structure was realized in atomic spinwave lattices. There the Bloch coupling matrix

$X_F$6

has chiral symmetry $X_F$7, winding number

$X_F$8

and a dissipative gap $X_F$9. For $\mathbf G$0, open chains host exponentially localized zero-dissipation edge modes. EIT spectroscopy directly reconstructs the dissipation spectrum and confirms the edge-mode pair inside the dissipative gap (Hao et al., 2022).

The same logic extends to synthetic photonic dimensions. In a time-multiplexed resonator network, purely dissipative versions of the SSH and Harper–Hofstadter models were implemented with vanishing conservative Hamiltonian. The topological phases are carried by bands of decay rates, with SSH winding numbers and Harper–Hofstadter Chern numbers computed from the effective coupling matrix $\mathbf G$1. Edge states appear as isolated dissipation rates in the gaps between bulk dissipation bands rather than as midgap energies in a Hermitian spectrum (Leefmans et al., 2021).

Loss can also reshape transport in systems that are already topological in the Hermitian limit. In a narrow quantum anomalous Hall ribbon described by the Qi–Wu–Zhang model, dissipation applied only on one edge produces non-Hermitian point-gap topology characterized by

$\mathbf G$2

The effective two-edge theory shows that the real hybridization gap closes when $\mathbf G$3, and between the two exceptional points the decaying coefficient is reduced. As a result, transport is suppressed on the dissipative edge but significantly enhanced on the other edge, producing dissipation-enhanced unidirectional transport (Lu et al., 2023).

Transport in dissipatively prepared 2D Chern states reveals a further departure from equilibrium intuition. Shavit and Goldstein considered a Chern-insulator steady state sustained by “out” and “in” Lindblad processes and showed that the usual relation between Chern number and Hall conductance is broken. In the “Hamiltonian regime” $\mathbf G$4 with a gauge-compatible Hamiltonian, the Hall conductance approaches a plateau at $\mathbf G$5 and $\mathbf G$6. In the “dissipative regime” $\mathbf G$7, both conductances are small, with $\mathbf G$8 and $\mathbf G$9 (Shavit et al., 2019).

Dissipation-driven topological transitions also occur at strong system–environment coupling. In an emitter array with SSH dimerization and long-range photon-mediated couplings, a dissipative topological phase transition occurs at $\det[H(k)-E_0]$0, where the bands touch at $\det[H(k)-E_0]$1. Below that coupling the edge state has finite decay, whereas above it the edge state becomes effectively dissipationless while bulk modes remain radiative. A particularly robust protection window appears at emitter spacing $\det[H(k)-E_0]$2 (Nie et al., 2021).

In three-dimensional continuous Weyl materials, the symmetry class controls whether loss can annihilate topological charge. For the nonreciprocal model $\det[H(k)-E_0]$3 perturbed by $\det[H(k)-E_0]$4, loss produces two Weyl exceptional rings on the planes $\det[H(k)-E_0]$5 with radius $\det[H(k)-E_0]$6, each carrying Chern number $\det[H(k)-E_0]$7. Because opposite charges remain on distinct $\det[H(k)-E_0]$8 slices, no loss-induced annihilation occurs for any finite $\det[H(k)-E_0]$9. In reciprocal inversion-broken realizations, by contrast, opposite-charge rings can expand and annihilate at a critical loss, making the Weyl topology far more fragile (Shastri et al., 2020).

6. No-go theorems, design recipes, and conceptual issues

The strongest general limitation presently established concerns pure topological steady states above one dimension. Diehl proved that in $E_0$0 a strictly finite-range quadratic Lindbladian cannot simultaneously have a unique pure steady state of nonzero Chern number and a finite Liouvillian gap. The obstruction is ultimately tied to the impossibility of realizing a gapped, exactly flat Chern band with a finite-range Hermitian operator. Thus a finite-range Lindbladian may prepare either a mixed state or a state with asymptotically slow relaxation, but not a unique pure topological state with finite-rate exponential decay (Goldstein, 2018).

The same work gave two constructive workarounds. If exponentially decaying couplings are allowed, one can define jump operators from the projector $E_0$1 of a gapped Chern Hamiltonian and obtain a unique pure steady state with finite decay rates. If strict finite range is retained, an “evaporation” Lindbladian together with uniform refilling produces a mixed state exponentially close to the desired pure Chern insulator, with deviations in few-body observables of order $E_0$2 and a Liouvillian gap bounded below by $E_0$3 (Goldstein, 2018).

Several conceptual cautions follow from the literature. First, dissipation does not merely degrade topology: it can create topological phases absent at $E_0$4, lower critical fields for Majorana modes, or protect edge states at strong coupling (Gogoi et al., 14 Jul 2025). Second, bulk–edge correspondence in open systems is more intricate than in equilibrium because zero-damping and zero-purity modes can both participate (Bardyn et al., 2013). Third, not every boundary mode pinned near $E_0$5 or $E_0$6 is topological; exceptional-point-induced trivial modes can mimic topological Majorana signatures (Gogoi et al., 14 Jul 2025). Fourth, a nonzero topological invariant of a steady-state projector need not imply quantized Hall transport unless the coherent probe Hamiltonian is compatible with the dissipative gauge and dominates the dissipative scale (Shavit et al., 2019).

Experimentally, dissipation-induced topology has already been realized across disparate platforms: radio-frequency metamaterials with quantized bulk energy transport, vapor-cell atomic spinwave SSH lattices, and time-multiplexed photonic resonator networks with dissipative SSH and Harper–Hofstadter bands (Rosenthal et al., 2018). Additional proposals or analyzed implementations include ultracold $E_0$7Yb in optical lattices, semiconductor nanowires with superconducting proximity, quantum anomalous Hall devices with edge-selective loss, emitter arrays coupled to the electromagnetic vacuum, photonic and acoustic metamaterials, circuit-QED architectures, and magnetized plasma or chiral media (Goldstein, 2018).

Taken together, these results establish dissipation-induced topology as a framework in which topology can be encoded in nonequilibrium states, Liouvillian spectra, damping matrices, or dissipation bands, with symmetry, locality, and the distinction between purity and damping playing the decisive roles. A plausible implication is that future classifications of topological matter will increasingly treat engineered reservoirs not as perturbations to be minimized, but as independent structural elements of the phase itself.