Exactly solvable higher-order Liouvillian exceptional points in dissipative fermionic systems

We propose a general class of open fermionic models where quadratic Liouvillians governing the dissipative dynamics feature exactly solvable higher-order exceptional points (EPs). Invoking the formalism of third quantization, we show that, among the multiple EPs of Liouvillian, an EP with its order approaching the system size arises as the quasisteady state of the system, leading to a gapless Liouvillian spectrum. By introducing perturbations, in the form of many-body quantum-jump processes, these higher-order EPs break down, leading to finite Liouvillian gaps with fractional power-law scalings. While the power-law scaling is a signature of the higher-order EP, its explicit form is sensitively dependent on the many-body perturbation. Finally, we discuss the steady-state approaching dynamic which can serve as detectable signals for the higher-order Liouvillian EPs.

💡 Research Summary

The paper introduces a novel class of open fermionic systems whose dissipative dynamics are governed by quadratic Liouvillians that can be solved exactly and exhibit higher‑order exceptional points (EPs) whose order scales with the system size. Starting from the standard Lindblad master equation, the authors impose a partial post‑selection that discards quantum‑jump trajectories associated with a complementary set of jump operators. This leads to a “hybrid” Liouvillian consisting of a coherent part, a set of retained jump terms, and a uniform shift term –c ρ (with c = Tr MM). The shift moves the entire spectrum without changing eigenvectors, allowing the quasisteady state to have a negative eigenvalue rather than the usual zero.

To analyze the quadratic Liouvillian, the authors employ the third‑quantization formalism, which maps the density‑matrix dynamics onto a quadratic form in a 4n‑dimensional Majorana fermion space. The resulting shape matrix A can be written as A = T₁⊗I₂ + T₂⊗iσ_y, where T₁ and T₂ are built from the Hamiltonian matrix h and the jump‑coefficient matrix M. By a unitary transformation the matrix becomes block‑diagonal with two triangular blocks T₊ = T₁ + iT₂ and T₋ = T₁ − iT₂. The eigenvalues of A (the rapidities) are ±β_j, where β_j are the eigenvalues of i h. Crucially, T₊ is generically non‑diagonalizable: each β_j is associated with a Jordan block of size n_j ∈ {1,2}. When n_j = 2 the block is of order two, and the total number of such blocks, K₂, determines the maximal EP order through 1 + K₂.

A concrete example is provided: a one‑dimensional chain of n sites with nearest‑neighbour hopping H = −t∑j(c_j†c{j+1}+h.c.) and local loss L_j = √γ c_j. In momentum space the block T₊(q) = −i t cos q σ_z − iγ σ_y is never diagonalizable; each momentum mode yields a Jordan block of size two. Consequently the Liouvillian eigenvalues are λ_ν = 2i∑j ν{j,1} cos(2πj/n) − nγ with ν_{j,1}∈{0,1,2}. The configuration ν_{j,1}=1 for all j produces a single eigenvalue λ₀ = −nγ whose associated Jordan block has order n + 1. This is the quasisteady state of the system and represents the highest‑order EP, whose order grows linearly with system size. The spectrum is gapless (Re λ = −Tr MM) and the dynamics toward the quasisteady state is algebraic rather than exponential: observables decay as t^{−(n+1)} due to the large Jordan block.

To probe the robustness of these EPs, the authors introduce many‑body quantum‑jump perturbations, e.g. two‑particle loss operators L_{jk}=√γ’ c_j c_k. These perturbations add off‑diagonal elements to T₊, lifting the degeneracy of the high‑order EP. Perturbation theory shows that the Liouvillian gap opens with a fractional power law |γ’|^{α}, where α depends on both the EP order and the order of the perturbation (e.g., α = 1/(n+1), 2/(n+1), …). Thus the appearance of a gap scaling with a non‑integer exponent is a clear fingerprint of an underlying higher‑order EP.

The dynamical consequences are illustrated using the particle‑number operator N = ∑_j c_j†c_j. The exact solution for ρ(t) involves sums over Jordan chains, leading to terms proportional to t^{k}e^{λ₀t} with k up to n. Consequently, quantities such as exp(nγt)⟨N⟩_t grow linearly in time for the unperturbed system, while 1/⟨\tilde N⟩_t decays linearly, as shown in numerical plots. When many‑body jumps are added, the algebraic decay crosses over to exponential relaxation with a rate set by the fractional‑power gap, providing an experimentally accessible signature.

In summary, the paper delivers four major contributions: (i) formulation of a hybrid Liouvillian with partial post‑selection, (ii) exact third‑quantization analysis revealing non‑diagonalizable shape matrices and high‑order EPs, (iii) demonstration that the EP order can scale with system size, and (iv) identification of fractional‑power gap scaling and characteristic dynamical observables as detection tools. These results open a pathway toward engineering and exploiting higher‑order non‑Hermitian phenomena in many‑body quantum systems, with potential applications in sensing, state preparation, and the study of non‑equilibrium quantum phases.