Gapless Liouvillian Spectrum in Open Systems

• The gapless Liouvillian spectrum is a phenomenon in open quantum systems where the steady-state eigenvalue and the rest of the spectrum become indistinguishable, leading to algebraic rather than exponential decay.
• It emerges in exactly solvable models like the dissipative Hubbard chains and quadratic fermionic chains, demonstrating universal relaxation and the impact of exceptional points.
• This property signals critical behavior and dissipation-induced correlations, with polynomial decay indicating complex spectral topologies and long-time dynamical signatures.

A gapless Liouvillian spectrum refers to the situation in open quantum systems where the Liouvillian superoperator, which governs the nonunitary time evolution of the system's density matrix under a Lindblad master equation, has vanishing real-part separation (“gap”) between its steady-state eigenvalue and the remainder of its spectrum. This characteristic signifies polynomial, rather than exponential, relaxation toward the steady state and is intricately connected to exceptional points, to the emergence of algebraic long-time dynamics, and to the interplay of system size and dissipation-induced correlations.

1. Liouvillian Eigenstructure and Gap: Generalities

In Lindblad-formulated open quantum systems, the master equation is

$$\frac{d\rho}{dt} = \mathcal{L}\rho = -i[H, \rho] + \sum_j\left(L_j \rho L_j^\dagger - \frac{1}{2} \{L_j^\dagger L_j, \rho\}\right)$$

where $H$ is the system Hamiltonian and $\{L_j\}$ are Lindblad operators representing dissipative jump processes. The spectrum of the Liouvillian $\mathcal{L}$ consists of complex eigenvalues $\{\lambda\}$, with the steady state associated to $\lambda=0$, while the real parts of nonzero eigenvalues set the relaxation rates toward stationarity.

The Liouvillian gap $\Delta$ is defined as the minimum of $-\Re \lambda$ over all $\lambda\ne 0$. A gapless Liouvillian, with $\Delta=0$, occurs either in the thermodynamic limit (e.g., scaling to zero with system size) or at particular parametric points/lines where eigenvalues with zero real part emerge, leading to non-exponential, often algebraic, approach to the steady state (Nakagawa et al., 2020, Xu et al., 31 Jan 2026, Zheng et al., 2022). This property governs universal late-time dynamics and is intimately linked to emergent criticality, exceptional points, and topological transitions in open quantum matter.

2. Gapless Spectra in Exactly Solvable Open Quantum Chains

Two paradigmatic families of models—dissipative Hubbard chains with two-body losses and quadratic open-fermion systems—exhibit analytically tractable gapless Liouvillian spectra.

(a) Dissipative 1D Hubbard Model

In the one-dimensional Hubbard model with local two-body loss, governed by

$$\mathcal{L}\rho = -i[H, \rho] + \sum_{j=1}^L (\sqrt{2\gamma}\, c_{j\downarrow}c_{j\uparrow}) \rho (\sqrt{2\gamma}\, c_{j\downarrow}c_{j\uparrow})^\dagger - \frac{1}{2}\{L_j^\dagger L_j, \rho\}$$

the non-Hermitian Bethe ansatz provides an exact construction of $\mathcal{L}$'s spectrum. The effective Hamiltonian $H_\text{eff} = H - (i/2)\sum_j L_j^\dagger L_j$ leads to complexified interaction $U_\text{eff} = U - i\gamma$. Bethe equations then specify eigenenergies, and the full Liouvillian eigenvalues take the form

$$\lambda_{ab}^{(N,n)} = -i [E_{N, a} - (E_{N+n, b})^*]$$

A key result is that, for spin-wave (M=1) excitations above the fully polarized steady manifold, the gap $\Delta$ scales as $L^{-2}$ and vanishes in the thermodynamic limit, signifying a gapless Liouvillian (Nakagawa et al., 2020). This engenders power-law ($\sim t^{-1/2}$) relaxation for arbitrary perturbations in one dimension.

(b) Quadratic Fermionic Chains and Higher-Order Exceptional Points

Quadratic open-fermion models with linear jump operators $L_j = \sqrt{\gamma}c_j$ realize exactly solvable, translationally invariant Liouvillian forms. Through third quantization, the Liouvillian is recast as a bilinear form

$$\hat{\mathcal{L}}_H = \sum_{j,k=1}^{4n} \hat{a}_j A_{jk} \hat{a}_k - \text{Tr} M^M$$

Diagonalization yields rapidities $\pm\beta_j$ and spectrum

$$\lambda_\nu = -2 \sum_j \nu_j \beta_j + \sum_j \beta_j - \text{Tr} M^M, \quad \nu_j \in \{0,1\}$$

In regimes where the Liouvillian is non-diagonalizable (i.e., matrix $A$ is not diagonalizable), large Jordan blocks appear, associated with higher-order exceptional points (EPs)—points where multiple eigenvalues and eigenvectors coalesce. For example, with $H = -t \sum_j c_j^\dagger c_{j+1} + \text{h.c.}$ and $L_j = \sqrt{\gamma} c_j$, the gap satisfies

$$\Delta = \min_{\lambda \neq \lambda_0} |\Re \lambda - \Re \lambda_0| = 0$$

and the largest Jordan block at $\lambda_0 = -n\gamma$ is of order $n+1$ (Xu et al., 31 Jan 2026). This establishes exact gaplessness for all system sizes and produces polynomial (algebraic) relaxation $\langle \tilde{N} \rangle_t \sim 1/t$ at long times.

3. Exceptional Points, Correlation Lengths, and Dynamical Signatures

Gapless Liouvillian spectra are generically associated with exceptional points (EPs), where the Liouvillian (or its effective Hamiltonian) becomes nondiagonalizable. At such points, eigenmodes merge non-analytically, and the system exhibits Jordan-block structure.

In the dissipative Hubbard model, the exceptional-point parameter regime is identified by the crossing of Bethe-equation poles with integration contours (specifically, where $\sin k = \lambda \pm iu$ hits the real-$k$ path), triggering non-analyticities and divergence of the boundary sensitivity length $\xi$. Explicitly,

$$\frac{1}{\xi} = \Re\left[\frac{1}{u} \int_1^\infty \frac{\ln(y + \sqrt{y^2-1})}{\cosh\left(\frac{\pi y}{2u}\right)} dy\right]$$

which vanishes at the EP, signifying $\xi \to \infty$ and indicating emergent gapless modes and critical dynamics (Nakagawa et al., 2020).

In quadratic chains, large EPs lead to polynomial temporal decay, rather than exponential. Experimental signatures include the characteristic crossover from algebraic decay (gapless, large EP) to exponential (gapped) behavior upon perturbing the system away from EP—e.g., by introducing many-body jump processes, which split the formerly degenerate eigenvalue and open a Liouvillian gap with universal fractional scaling $\Delta \sim z^{1/(d+1)}$ ($z$ the perturbation strength, $d$-body process) (Xu et al., 31 Jan 2026).

4. Exactly Solved Models: Phase Diagrams and Spectral Features

An explicit analytical solution is available for the Liouvillian spectrum of the boundary-dissipated transverse field Ising model, formulated via an imaginary Su-Schrieffer-Heeger (SSH) chain with boundary potentials and parity constraints (Zheng et al., 2022). The spectrum consists of “rapidities” $E_{j,P} = \pm\sqrt{1+h^2+2h\cos\theta_{j,P}}$ indexed by roots of special boundary characteristic equations.

The gapless condition occurs under two circumstances in the $(h, \gamma)$ parameter space:

• Along the Ising-critical line $h=1$.
• Along the dissipation-critical curve $h = \frac{1+\gamma^2}{2\gamma}$.

Across the phase diagram, the Liouvillian spectrum organizes into distinct “stripe” patterns in the complex plane, with the number and arrangement of stripes altered by $\gamma$ and $h$. The gapless transition lines separate phase regions with distinct spectral multiplicities (1, 3, 5, or 9 stripes), and the gap maxima occur at $\gamma=1$ for fixed $h$, with an exact duality $\Delta(h, \gamma) = \Delta(h, 1/\gamma)$.

5. Dissipation-Induced Correlations and Spin–Charge Separation

Dissipation can engender strong correlations via mechanisms such as Zeno blockade. In the dissipative Hubbard model, the quantum Zeno effect at large $\gamma$ localizes doublons, decoupling spin and charge degrees of freedom and producing spin-1/2 Heisenberg chain dynamics under continuous observation—even when $U=0$ (Nakagawa et al., 2020). In the $|u| \to \infty$ limit (either large $U$ or large $\gamma$), the Bethe equations separate, leading to a wavefunction factorized into charge (free-fermion-like) and spin (Heisenberg chain) components. The resulting spectrum is gapless in the spin sector, and the dissipation thus dynamically induces spin–charge separation entirely via loss.

6. Universality, Experimental Probes, and Power-Law Relaxation

The closure of the Liouvillian gap is directly responsible for the universal algebraic relaxation of observables, such as a $\tau^{-1/2}$ decay in the one-dimensional Hubbard chain. This behavior is robust and provides a clear dynamical fingerprint for gapless Liouvillian regimes, in contrast to the exponential decay characteristic of gapped spectra (Nakagawa et al., 2020, Xu et al., 31 Jan 2026). Experimental detection schemes include monitoring the dynamical structure factor, late-time decay of parity-even correlations, or particle number in cold atom or superconducting-circuit implementations.

The connection between gapless Liouvillian spectra and strong non-Hermitian degeneracies (EPs), as well as their resilience to weak perturbations and breaking of corresponding algebraic behavior upon the opening of a gap, has been confirmed in both analytical and numerical studies. This scenario provides a paradigmatic framework for understanding criticality, relaxation, and correlations in open quantum systems.

7. Phase Structure and Dualities in Open Quantum Chains

Gapless Liouvillian phases are most systematically understood via analytical continuation (through Bethe ansatz or third quantization) and through the characterization of spectral topologies. In the boundary-dissipated Ising chain, weak–strong dissipation duality imposes $\Delta(h, \gamma) = \Delta(h, 1/\gamma)$ in the thermodynamic limit, and spectral transitions are sharply demarcated by the gapless critical lines $h=1$ and $h = (1+\gamma^2)/(2\gamma)$. Between these lines, the phase structure reveals topological organization of decay modes, each region distinguished by different spectral segment counts and gap properties (Zheng et al., 2022).

A plausible implication is that similar dualities and exceptional-point-driven gapless transitions can be engineered or observed in broader classes of open quantum matter, with critical lines or manifolds manifesting as loci of gapless Liouvillian spectra in enlarged parameter spaces.