Fermionic Dissipation Numbers

• Fermionic Dissipation Numbers are quantitative parameters that define the impact of engineered dissipation on fermionic quantum systems.
• They are derived from Lindblad master equations and topological invariants, such as winding and Chern numbers, to signal phase transitions.
• These measures inform experimental setups by predicting steady-state purity and stabilizing Majorana zero modes in topological phases.

Fermionic Dissipation Numbers are quantitative measures, parameters, or indices that characterize dissipation processes, decoherence, and the interplay between coherent and dissipative dynamics in fermionic quantum systems. The terminology encompasses several key contexts, including engineered open-system quantum dynamics, symmetry-protected topological order induced by dissipation, thermodynamic entropy production in fermionic systems, and the quantification of quantum-to-classical transitions induced by dissipation. The concept gains technical significance in the paper and control of topological steady states, decoherence transitions, transport properties, and quantum information robustness in the presence of environmental coupling.

1. Definition and Theoretical Background

Fermionic dissipation numbers refer to specific parameters or invariants that quantify the impact of dissipation (such as particle loss, gain, dephasing, or engineered Lindblad dynamics) on the state and dynamics of fermionic many-body systems. The terminology emerges in the context of Markovian master equations of Lindblad form, e.g.,

$$\dot{\rho} = \sum_i \kappa_i\left( L_i\,\rho\,L_i^\dagger - \frac{1}{2}\{L_i^\dagger L_i,\rho\}\right),$$

where the $L_i$ are Lindblad (jump) operators. By careful design of these operators, dissipation becomes a central resource for driving the system towards desired steady states, such as topologically ordered phases or pure many-body dark states (Bardyn et al., 2013).

Fermionic dissipation numbers may materialize as:

• The ratio of dissipation rates (e.g., injection vs. removal, $\gamma = d\Gamma^{(i)}/d\Gamma^{(o)}$),
• Topological invariants determined under dissipative dynamics (e.g., winding numbers, Chern numbers of the covariance matrix),
• Gaps in the dissipative spectrum (damping gap) or spectrum of steady-state purity (purity gap),
• Parameters extracted as scaling exponents of quantum phase transitions or as normalization factors of entropy/response functions.

Unlike traditional Hamiltonian scenarios, where the energy gap dictates physical properties, in dissipative settings two independent gaps (dissipative and purity) appear. This dual-gapped structure requires new numerical and analytic measures to quantify fermionic dissipation.

2. Role of Dissipation in Inducing Topology and Quantum Phase Transitions

Dissipative dynamics can be engineered to "cool" a fermionic system into a topological phase from arbitrary initial states. Carefully chosen quasi-local Lindblad operators drive the system towards a steady state characterized by nonlocal correlations and topological order. For instance, in a 1D dissipative quantum wire, a typical jump operator is

$$L_n = \frac{1}{\sqrt{4+\kappa^2}} \left[\kappa\,a^\dagger_n + (a^\dagger_{n+1}+a^\dagger_{n-1}) + (a_{n+1} - a_{n-1})\right],$$

with a parameter $\kappa$ controlling the dissipative gap and topological invariant (winding number) (Bardyn et al., 2013). Dissipation numbers in this framework serve to label the "phase" of the system, where topological transitions are associated with the closure of damping or purity gaps.

In other scenarios, infinitesimal dissipation can drive quantum phase transitions with marked decoherence. In a dissipative tight-binding chain model,

• The ratio $\gamma = d\Gamma^{(i)}/d\Gamma^{(o)}$ controls local entropy in the bulk steady state,
• The transition from phase-coherent to decohered (fully local) states is signaled by jumps in response functions and entropy,
• Critical exponents (e.g., scaling of the lowest eigenvalue $\lambda_\mathrm{min} \sim (d\Gamma)^\beta$) can act as dissipation numbers for phase transitions (Medvedyeva et al., 2014).

3. Symmetry-Based Classification and Topological Invariants

Fermionic dissipation numbers also capture the symmetry-based classification of steady states:

• Mapping the steady-state covariance matrix to a fictitious quadratic Hamiltonian $H_\Gamma = i\sum_{ij} \Gamma_{ij}\,c_i c_j$, one can classify the dissipative steady states in Altland–Zirnbauer classes (e.g., BDI, D) (Bardyn et al., 2013).
• The winding number (in 1D with time-reversal symmetry) and the Chern number (in 2D with broken time-reversal symmetry) serve as fermionic dissipation numbers, encapsulating the topological character of the dissipative phase.
• For strictly quasi-local Lindblad operators, the Chern number may be forced to zero, requiring relaxation of locality to access nontrivial invariants.

Dissipation numbers in this sense are robust invariants emerging from the interplay of symmetry, nonlocal correlations, and Lindblad-engineered processes.

4. Steady-State Gaps: Dissipative and Purity Gaps

A fundamental distinction from Hamiltonian ground states is that in dissipative settings, the steady state can be either pure or mixed, and the steady state’s approach is characterized by two independent gaps:

• The dissipative (damping) gap governs the exponential relaxation to the steady state,
• The purity gap is associated with the flatness of the eigenvalue spectrum of $i\Gamma$, indicating the extent to which the state is pure vs. mixed.

These gaps can both serve as fermionic dissipation numbers. For example, a dissipative gap ensures stability against decoherence, while a large purity gap characterizes states close to a pure phase (dark state). The closure of either implies criticality (e.g., a transition to decoherence).

5. Bulk–Edge Correspondence and Majorana Zero–Damping Modes

In analogy to equilibrium systems, bulk–edge correspondence holds in dissipative fermionic settings, albeit modified. The number of localized, decoherence-free (zero-damping) Majorana modes at boundaries or vortex cores is bounded by the difference in bulk topological invariants: $$(m_\text{damping})_\text{edge} + (m_\text{purity})_\text{edge} \ge |\Delta\nu|,$$ where $|\Delta\nu|$ is the change in winding or Chern number across an edge (Bardyn et al., 2013). The dissipation number here is the count of protected zero-damping modes, with practical ramifications for the stabilization of topologically nontrivial excitations (such as Majorana zero modes), and for quantum information applications.

6. Experimental Implementations and Applications

Experimentally, cold atom systems provide a natural platform to realize and measure fermionic dissipation numbers. Quasi-local dissipative quantum wires and 2D lattices with engineered loss, gain, or phase noise have been implemented using auxiliary lossy sites, Raman transitions, and controlled coupling to reservoirs (Bardyn et al., 2013). Steady-state topological invariants, damping/purity gaps, and localized edge modes are accessible in such setups.

Moreover, dissipative metrics (e.g., local entropy per site $$S = \ln(1 + \gamma) - [\gamma/(1 + \gamma)]\ln\gamma$$ in the fully-dissipative phase (Medvedyeva et al., 2014)) and dynamical signatures (e.g., the persistent current, non-linear conductivity discontinuities) serve as experimentally observable dissipation numbers.

7. Broader Impact and Quantification Strategies

Fermionic dissipation numbers serve as unifying quantitative measures to:

• Distinguish between coherent and decoherent transport,
• Signal and classify dissipative quantum phase transitions,
• Quantify the protection and localization of edge states,
• Control the purity and robustness of many-body steady states under environmental coupling.

Depending on context, they are defined via ratios of reservoir couplings, topological invariants mapped from the steady-state correlation matrix, entropy jumps, scaling exponents of dissipative gaps, or explicit counts of protected steady-state zero modes. Their quantification is central for designing robust dissipative quantum simulators, understanding environment-induced decoherence, and steering dissipative preparation of topological phases.

In summary, fermionic dissipation numbers encapsulate the essential parameters and invariants that define, classify, and quantify the effects of dissipation on the physics of open fermionic quantum systems, providing a rigorous framework that underpins dissipative topological phenomena, decoherence transitions, and experimentally relevant observables.