Exact steady states of interacting driven dissipative fermionic systems with hidden time-reversal symmetry

Abstract: We present exact solutions for the non-equilibrium steady states of a class of dissipative spinless fermionic systems with arbitrary Hamiltonian pairing terms, global charging energy interactions, and uniform single particle loss on every site. Our exact solution is found by generalizing the coherent quantum absorber technique to fermionic systems, and our result establishes the existence of hidden time-reversal symmetry in driven-dissipative fermionic models. The steady state exhibits a first order phase transition in the particle density, with the resulting jump discontinuity in density persisting even for finite dissipation rates. A mean-field description of the model exhibits a bistable regime that encompasses the first-order transition line yet which fails to accurately predict its precise location via a Maxwell construction. We also show that the model's hidden time-reversal symmetry results in an Onsager symmetry of certain two-time correlation functions.

• The paper generalizes the coherent quantum absorber method to fermionic systems with linear loss, producing closed-form nonequilibrium steady states for arbitrary antisymmetric pairing matrices and global charging interactions.
• The exact solution reveals a first-order dissipative phase transition with a persistent density jump at finite loss, including a critical pairing of approximately 0.02138 for μ = 0.2 and κ = 10⁻³.
• The paper verifies fermionic Onsager time-reversal relations, identifies a limited mapping to dissipative transverse-field Ising dynamics, and shows that naive Maxwell constructions can misplace the phase boundary.

The paper presents an exact solution for the non-equilibrium steady state of a class of interacting driven-dissipative spinless fermion models, and in doing so establishes that hidden time-reversal symmetry (hTRS) — previously known only for bosonic and spin systems — extends to open fermionic systems. The construction generalizes the coherent quantum absorber (CQA) technique to fermions with odd (linear) jump operators, a step that requires care because composite fermionic Hilbert spaces are built from exterior products rather than tensor products. The resulting steady state exhibits a first-order dissipative phase transition whose density discontinuity persists at finite dissipation rates, and the associated Onsager symmetries of two-time correlation functions are verified.

Model

The system consists of $L$ spinless fermions $\hat c_j$ on a lattice with chemical potential $\mu$, an all-to-all charging energy $E_C$ via $(E_C/2L)(\sum_j \hat n_j)^2$, and an arbitrary antisymmetric pairing matrix $\Delta_{ij}$, subject to uniform single-particle loss at rate $\kappa$ on every site:

$$\hat H = -\mu \sum_j \hat n_j + \frac{E_c}{2L}\Big(\sum_j \hat n_j\Big)^2 + \sum_{ij} (\Delta_{ij}\hat c_i^\dagger \hat c_{j}^\dagger + {\rm H.c.}),$$

with Lindbladian $$\mathcal{L}\hat\rho = -i[\hat H,\hat\rho] + \kappa\sum_j \mathcal{D}[\hat c_j]\hat\rho$$. The model deliberately omits tunneling, since hopping breaks solvability when $E_C\neq 0$. The concrete case analyzed is the 1D Kitaev wire with nearest-neighbor $\hat c_j$0-wave pairing $\hat c_j$1 under periodic boundary conditions. Steady-state uniqueness is assumed rather than proven: the uniqueness theorem of Ref. (Granek, 2019) does not formally apply to fermionic Lindbladians, though exact diagonalization of small systems finds no evidence of multiple steady states.

Coherent quantum absorber solution

The CQA method introduces a mirrored absorber $\hat c_j$2 with $\hat c_j$3 and jump operators $\hat c_j$4, cascaded downstream through unidirectional waveguides. For fermions with linear dissipators, the standard cascade assumption $\hat c_j$5 fails because odd operators anticommute across subsystems. The authors show nevertheless that nonreciprocity of all even-parity observables holds under parity superselection, with the cascade tuning $\hat c_j$6, $\hat c_j$7; remarkably, the cascaded interaction Hamiltonian takes the same sign as in the bosonic/spin case despite the opposite gauge phase.

Working in dark/bright mode combinations $\hat c_j$8, the pure steady state must be annihilated by all bright modes and satisfy a zero-energy eigenstate condition of the effective non-Hermitian Hamiltonian. Using hard-core-boson-like pair creation operators $\hat c_j$9 and "defect" operators $\mu$0, term-by-term destructive interference yields a one-term recurrence whose solution is

$\mu$1

where $\mu$2 creates nearest-neighbor pairs delocalized over the chain and $\mu$3. This result holds for any antisymmetric pairing matrix $\mu$4 and any dimensionality. Each $\mu$5-pair component sees an effective chemical potential shifted by its charging-energy cost $\mu$6; at $\mu$7 the state reduces to a Gaussian BCS form. Normalization reduces to counting nearest-neighbor dimer coverings, giving $\mu$8 for periodic chains, with the series terminating at $\mu$9 because the maximally filled state cancels exactly by a sign argument.

First-order phase transition

In the thermodynamic limit, rewriting the wavefunction in orthonormal pair-number states and applying Stirling's approximation to the coefficients defines an effective free energy $E_C$0 over the density coordinate $E_C$1. For $E_C$2 and small enough $E_C$3, $E_C$4 develops two local minima straddling $E_C$5; since their energy difference is monotonic in both $E_C$6 and $E_C$7, the global minimum switches discontinuously at a critical pairing $E_C$8, producing a jump in particle density. Numerically, at $E_C$9 the transition occurs at $(E_C/2L)(\sum_j \hat n_j)^2$0 in the weak-dissipation limit, and shifts only slightly to $(E_C/2L)(\sum_j \hat n_j)^2$1 at finite $(E_C/2L)(\sum_j \hat n_j)^2$2 for $(E_C/2L)(\sum_j \hat n_j)^2$3. The paper emphasizes that this discontinuity survives at finite dissipation, not merely as a $(E_C/2L)(\sum_j \hat n_j)^2$4 artifact; finite $(E_C/2L)(\sum_j \hat n_j)^2$5 renormalizes the boundary and moves its terminal point below $(E_C/2L)(\sum_j \hat n_j)^2$6. The same mechanism explains the transition analytically through a resonance factor in $(E_C/2L)(\sum_j \hat n_j)^2$7 near $(E_C/2L)(\sum_j \hat n_j)^2$8 competing against the exponential suppression $(E_C/2L)(\sum_j \hat n_j)^2$9.

Mean-field theory and bistability

Mean-field decoupling of the global interaction retains only the Hartree term $\Delta_{ij}$0 in the thermodynamic limit, yielding the self-consistency equation

$\Delta_{ij}$1

This equation possesses three solutions in a region of the $\Delta_{ij}$2–$\Delta_{ij}$3 plane confined to $\Delta_{ij}$4 that encompasses the exact first-order line. The two stable branches match the exact density well away from the transition, but a Maxwell equal-area construction fails to locate the actual transition point correctly. This discrepancy is notable: it indicates that even in this infinite-range-interaction setting, where mean-field intuition might be expected to be quantitatively reliable, the naive thermodynamic construction misplaces the phase boundary.

Mapping to a dissipative transverse-field Ising model

Under the Anderson pseudospin mapping $\Delta_{ij}$5, $\Delta_{ij}$6, the Hamiltonian maps exactly to a transverse-field Ising model with inhomogeneous Rabi drives $\Delta_{ij}$7 and longitudinal field renormalized by magnetization. Although single-particle fermion loss has no direct spin counterpart, the authors prove that within the pseudospin subspace, and for the mean-field (or $\Delta_{ij}$8) theory, fermion loss maps to the combination of spin loss and dephasing,

$\Delta_{ij}$9

so that expectation values of corresponding operators evolve identically, with higher moments matching by Wick's theorem. Physically, single-particle loss halves the density reduction relative to pair loss, acting as effective dephasing of Cooper pairs. The correspondence breaks down for the full interacting Hamiltonian: dissipation removes population from the pseudospin subspace, and the charging energy then distinguishes in-subspace from out-of-subspace configurations ($\kappa$0 versus $\kappa$1). A further consequence of the drive inhomogeneity is that the self-consistency equation is quartic rather than cubic in $\kappa$2, unlike standard dissipative TFIM mean-field results.

Hidden time-reversal symmetry

Because the model admits a perfect absorber, it possesses hTRS, implying Onsager symmetry of two-time correlation functions. One modification arises from anticommutation: correlations between two linear jump operators are antisymmetric under time reversal, $\kappa$3, while correlations involving the bilinear $\kappa$4 take the standard symmetric form. Numerically, adding a weak pumping impurity $\kappa$5 with $\kappa$6 breaks both the perfect absorber condition and the correlation-function time-(anti)symmetry, confirming the diagnostic value of the Onsager relations.

Limitations and open questions

Several caveats are stated explicitly. Solvability requires the absence of tunneling once $\kappa$7, so the solvable class excludes models with single-particle hopping. Steady-state uniqueness is assumed on numerical evidence rather than established rigorously for fermionic Lindbladians. The tension between Kramers-type degeneracy of open fermionic systems (associated with $\kappa$8) and the known hTRS structure, where each steady state has its own $\kappa$9 with $$\hat H = -\mu \sum_j \hat n_j + \frac{E_c}{2L}\Big(\sum_j \hat n_j\Big)^2 + \sum_{ij} (\Delta_{ij}\hat c_i^\dagger \hat c_{j}^\dagger + {\rm H.c.}),$$0, is left unresolved. Normalization of the exact state for a general pairing matrix $$\hat H = -\mu \sum_j \hat n_j + \frac{E_c}{2L}\Big(\sum_j \hat n_j\Big)^2 + \sum_{ij} (\Delta_{ij}\hat c_i^\dagger \hat c_{j}^\dagger + {\rm H.c.}),$$1 remains a nontrivial combinatorial problem solved here only for nearest-neighbor dimers in 1D. Finally, the fermion-to-spin dynamical mapping is exact only at the mean-field level and fails for the fully interacting model, so its use as an approximation tool for interacting spin problems is proposed but not demonstrated.

Conclusion

This work establishes that the coherent quantum absorber technique and hidden time-reversal symmetry extend to interacting fermionic Lindblad systems with linear jump operators, providing closed-form steady states for arbitrary pairing matrices combined with global charging energy and uniform loss. The exact solution reveals a first-order dissipative transition with a density jump robust to finite dissipation, exposes the failure of Maxwell-construction mean-field predictions despite a surrounding bistable region, and uncovers an unexpected dynamical correspondence with a dissipative transverse-field Ising model. The main open questions are whether uniqueness can be proven for such fermionic Liouvillians, how hTRS reconciles with Kramers degeneracy in open fermionic systems, and whether the fermion–spin mapping can be developed into a controlled approximation scheme beyond mean field.