Liouvillian Kramers Degeneracy

The Kramers theorem guarantees the degeneracy of energy eigenstates in quantum systems with half-integer spin [1, 2, 3, 4]. Mathematically, it arises from antiunitary time-reversal symmetry with negative sign, which defines the symplectic class in the Wigner-Dyson threefold way of Hermitian random matrices [5, 6, 7] and plays a fundamental role in the characterization of quantum chaos [8, 9, 10, 11, 12]. The Kramers degeneracy also underlies various condensed matter phenomena, such as the Anderson localization transitions [13, 14, 15, 16, 17, 18] and Z$_2$ topological insulator phases [19, 20, 21, 22, 23, 24, 25, 26].

Meanwhile, the physics of open quantum systems has attracted considerable interest. Unlike isolated quantum systems, open quantum systems interact with their environment, and the reduced dynamics is no longer governed solely by Hermitian Hamiltonians [27, 28, 29]. Instead, the open quantum dynamics is generally described by Liouvillian superoperators acting on density matrices. Understanding such dynamics is important for protecting quantum coherence against dissipation. By contrast, engineered dissipation can also be useful as a resource for quantum computation and state preparation [30, 31]. In contrast to Hamiltonians, Liouvillians act on operator spaces with a double Hilbert-space structure, giving rise to a richer symmetry framework for open quantum systems [32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

The Kramers theorem was extended to generic non-Hermitian operators under generalized time-reversal symmetry†, with negative sign [42, 43]. Although many-body Liouvillians are also non-Hermitian superoperators, their double Hilbert-space structure makes the Kramers degeneracy elusive. Indeed, whereas the Kramers degeneracy was investigated in dissipative fermionic systems at thermal equilibrium within the Lindblad master equation [44], Ref. [45] subsequently argued that the double Hilbert-space structure of many-body Lindbladians excludes the Kramers degeneracy within individual symmetry-resolved subspaces. This issue also has important implications for dissipative quantum chaos [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. In particular, the complex spectral statistics of nonintegrable open quantum systems are expected to exhibit the universal behavior of non-Hermitian random matrices in the same symmetry classes. Among the 38 symmetry classes of non-Hermitian random matrices [43, 62], 11 support the Kramers degeneracy, whereas the remaining 27 do not. Consequently, only a restricted subset of universality classes can appear in open quantum systems, if the Kramers degeneracy is indeed forbidden in Liouvillians.

In this Letter, we show how the Kramers degeneracy arises within individual symmetry subspaces of many-body Liouvillians. We find that the argument of Ref. [45] relies on restrictive assumptions about the algebra of symmetry operators and demonstrate that the interplay between time-reversal symmetry† and parity symmetry enforces the Liouvillian Kramers degeneracy. In contrast to Ref. [44], our Liouvillian Kramers degeneracy theorem is independent of particle statistics and applicable to both fermionic and bosonic systems. To illustrate this, we explicitly construct Lindbladian models of interacting fermions and spins exhibiting the Kramers degeneracy. We further show that their complex spectral statistics follow those of non-Hermitian random matrices in the corresponding symmetry classes, providing characterization of nonintegrability for open quantum systems in the symplectic class.

**Kramers theorem.—**Suppose that a generic non-Hermitian operator $H$ respects time-reversal symmetry† [43]

where $T$ is an antiunitary operator (i.e., $T T^\dagger= T^\dagger T = 1$ and $T z T^{-1}=z^*$ for $z \in\mathbb{C}$). Although this symmetry is equivalent to time-reversal symmetry $T H T^{-1}=H$ for Hermitian $H$, the two symmetries are generally distinct for non-Hermitian $H$ [63]. Let $E \in\mathbb{C}$ be an eigenvalue of $H$, and $|\psi\rangle$ and $|\psi\rangle$ be the corresponding right and left eigenvectors, respectively (i.e., $H|\psi\rangle=E|\psi\rangle$, $\langle\psi|H=E\langle\psi|$). Equation (1) leads to

showing that $T|\psi\rangle$ is also a right eigenvector of $H$ with the same eigenvalue $E$.

Depending on $T^2=+1$ or $T^2=-1$, the relevant symmetry class becomes class AI${}^{\dagger}$ or AII${}^{\dagger}$, respectively. Specifically, $T^2=-1$ enforces the Kramers degeneracy for arbitrary complex eigenvalues $E\in\mathbb{C}$ [42, 43]. Indeed, from $T^{\dagger}=-T$, we have

leading to $\langle\!\langle\psi|T|\psi\rangle\!\rangle=0$. Accordingly, $T|\psi\rangle\!\rangle$ is biorthogonal to $|\psi\rangle\!\rangle$ [64] and constitutes an independent eigenvector with the same eigenvalue $E$, establishing the Kramers degeneracy [65]. The Kramers degeneracy underlies various physical phenomena, such as $\mathbb{Z}_2$ non-Hermitian topology and skin effect [66, 67, 68, 69]. Furthermore, time-reversal symmetry${}^{\dagger}$ governs the fundamental threefold universality classes of non-Hermitian random matrices [70], serving as a non-Hermitian counterpart of the Wigner-Dyson threefold way [5, 6].

In the presence of additional unitary symmetry that commutes with $H$, such as parity symmetry considered below, the symmetry classification must be performed separately within each subspace of the corresponding conserved charge. For example, analyzing the full spectrum without resolving these subspaces obscures the characteristic spectral correlations within individual subspaces, since the spectra of different subspaces are generally uncorrelated. For Lindbladians, Ref. [45] argued that the double Hilbert-space structure excludes the Kramers degeneracy within individual symmetry subspaces. By contrast, we below demonstrate the Liouvillian Kramers degeneracy by combining time-reversal symmetry${}^{\dagger}$ with parity symmetry.

Liouvillian Kramers theorem.—We consider a Liouvillian superoperator $\mathcal{L}$ acting on operators $X\in\mathcal{B}$, where $\mathcal{B}$ denotes operator space equipped with the Hilbert-Schmidt inner product. Similar to Eq. (1), we impose time-reversal symmetry${}^{\dagger}$ on $\mathcal{L}$:

where $\mathcal{T}$ is an antiunitary superoperator satisfying

with a parity superoperator $(-1)^{\mathcal{N}}$. If $\mathcal{L}$ is irreducible and possesses no conserved quantities, $\mathcal{T}^2$ must take a definite sign of either $+1$ or $-1$. However, since Eqs. (4) and (5) imply $[\mathcal{L},(-1)^{\mathcal{N}}]=0$, the entire operator space decomposes into invariant parity subspaces, for which $\mathcal{T}^2$ takes different signs. Specifically, in the subspace with $(-1)^{\mathcal{N}}=-1$, the Liouvillian $\mathcal{L}$ respects time-reversal symmetry${}^{\dagger}$ with $\mathcal{T}^2=-1$ and thus exhibits the Kramers degeneracy.

In general, physical Liouvillians $\mathcal{L}$ preserve Hermiticity of operators [i.e., $(\mathcal{L}\rho)^{\dagger}=\mathcal{L}\rho$ for $\rho^{\dagger}=\rho\in\mathcal{B}$]. Accordingly, $\mathcal{L}$ respects

where $\mathcal{J}$ is an antiunitary modular conjugation satisfying $\mathcal{J}X=X^{\dagger}$ for $X\in\mathcal{B}$ [41]. The symmetry classification of $\mathcal{L}$ is determined by the interplay between $\mathcal{T}$ and $\mathcal{J}$, as well as whether parity $(-1)^{\mathcal{N}}$ is strong or weak [32, 34]. Weak parity acts on an operator $X\in\mathcal{B}$ as $(-1)^{\mathcal{N}}X=(-1)^{\mathcal{N}}X(-1)^{\mathcal{N}}$, with a unitary and Hermitian operator $(-1)^{\mathcal{N}}\in\mathcal{B}$. In the following physical realizations, we take the representation $\mathcal{T}X=\mathcal{T}X\mathcal{T}^{-1}$ with $\mathcal{T}^2=\pm(-1)^{\mathcal{N}}$. Since $\mathcal{J}$ commutes with $(-1)^{\mathcal{N}}$, $\mathcal{J}$ is respected within each subspace of $(-1)^{\mathcal{N}}$. Consequently, the Liouvillian $\mathcal{L}$ belongs to

where $\mathcal{B}_{\pm}$ denotes the operator subspace with fixed weak parity $(-1)^{\mathcal{N}}=\pm1$, and the notation of symmetry classes follows the 38-fold classification of non-Hermitian operators [43].

On the other hand, when parity is strong symmetry, the left- and right-acting parity superoperators are separately conserved: $(-1)^{\mathcal{N}_{+}}X=(-1)^{\mathcal{N}_{+}}X$ and $(-1)^{\mathcal{N}_{-}}X=X(-1)^{\mathcal{N}_{-}}$. The operator space then decomposes into four invariant subspaces $\mathcal{B}_{pq}$ ($p,q=\pm1$), labeled by the corresponding strong parity $(-1)^{\mathcal{N}_{+}}$ and $(-1)^{\mathcal{N}_{-}}$. Within each subspace, the time-reversal superoperator $\mathcal{T}$ satisfies $\mathcal{T}^2=pq$. Moreover, since modular conjugation $\mathcal{J}$ exchanges $(-1)^{\mathcal{N}_{+}}$ and $(-1)^{\mathcal{N}_{-}}$, it preserves individual subspace only for $pq=1$. Consequently, the relevant symmetry classes are identified as

These constructions circumvent the argument of Ref. [45], which relies on specific assumptions about the representations of symmetry operators. In our constructions, $\mathcal{T}^2=\pm(-1)^{\mathcal{N}}$ yields $\mathcal{T}^2X=(-1)^{\mathcal{N}}X(-1)^{\mathcal{N}}$. Since $\mathcal{T}$ preserves each parity subspace of the Hilbert space, $\mathcal{T}$ acts within $\mathcal{B}_{+-}$ and $\mathcal{B}_{-+}$ separately while squaring to $-1$, enforcing the Kramers degeneracy even within each subspace. Notably, class AII${}^{\dagger}$ in Eq. (8) and class DIII${}^{\dagger}$ in Eq. (7) were previously argued to be absent from Lindbladians [45]. Below, we explicitly demonstrate their realizations in open quantum systems.

Lindblad master equation.—While the preceding discussion applies to generic Liouvillians, we now consider the Kramers degeneracy in the Lindblad master equation $d\rho/dt=\mathcal{L}\rho$, with [27, 28, 29, 72, 73]

where $H$ is a Hermitian Hamiltonian governing the unitary dynamics, and $L_n$’s are jump operators describing the dissipative coupling to the environment. To realize the Lindbladian Kramers degeneracy, we impose time-reversal symmetry on $H$, as in Eq. (1). We further require the set of jump operators to be invariant under both time reversal and Hermitian conjugation,

where the coefficient matrices $v$ and $w$ are unitary. A simple realization of these conditions is provided by the set of jump operators, $\{M,\mathcal{T}M\mathcal{T}^{-1},M^\dagger,\mathcal{T}M^\dagger\mathcal{T}^{-1}\}$, where $M$ denotes an arbitrary operator with definite parity, $(-1)^N M(-1)^N=\pm M$. Defining the time-reversal superoperator $\mathcal{T}$ by $\mathcal{T}X:=\mathcal{T}X\mathcal{T}^{-1}$ for $X\in\mathcal{B}$, we find that the Lindbladian $\mathcal{L}$ exhibits the Kramers degeneracy in the weak-parity subspace $\mathcal{T}^2=(-1)^N=-1$. In the following, we provide explicit fermionic and bosonic Lindbladians exhibiting the Kramers degeneracy. In the End Matter, we also demonstrate its realization in a random Lindbladian (Appendix A) and an exactly solvable dissipative spinful fermion (Appendix B).

Dissipative interacting fermions.— Let us consider the Hilbert space consisting of spinful complex fermions $c_{m\sigma}$ ($m=1,2,\cdots,M$; $\sigma=\uparrow,\downarrow$). Time reversal is implemented by the antiunitary transformation,

satisfying

We take the Hamiltonian to be

which respects time-reversal symmetry in Eq. (1) by construction. Here, $H_{\mathrm{SYK}}$ denotes the Sachdev-Ye-Kitaev model [74, 75, 76, 77, 78]

where $\psi_a$’s ($a=1,2,\cdots,4M$) are Majorana fermion operators ($\{\psi_a,\psi_b\}=\delta_{ab}$, $\psi_a^\dagger=\psi_a$) constructed from $c_{m\sigma}$, and the real-valued couplings $J_{abcd}\in\mathbb{R}$ are independently drawn from the Gaussian distribution $\mathcal{N}(0,3!J^2/(4M)^3)$ ($J\geq0$).

from the Gaussian distribution $\mathcal{N}(0,3!J^2/(4M)^3)$ ($J\geq0$). While $H_{\mathrm{SYK}}$ may possess additional symmetry [79, 80, 81, 82], the chemical potential term $-\mu N\ne0$ breaks such symmetry.

We introduce random dephasing through the jump operators

which transform under time reversal and Hermitian conjugation as

satisfying Eq. (10). Consequently, the Lindbladian exhibits the Kramers degeneracy in the subspace of weak fermion parity $(-1)^N=-1$. Furthermore, these jump operators preserve strong fermion parity [i.e., $(-1)^N L_{m\sigma}(-1)^N=L_{m\sigma}$], and the symmetry classification follows Eq. (8).

In Figure 1(a), we show the complex Lindbladian spectra for $\mathcal{B}_{++}$ and $\mathcal{B}_{+-}$. In our numerical calculations, we sample the dephasing rates $\gamma_m$ independently and uniformly from $[0.05,0.45]$ for each $m$. The overall spectra form a characteristic lemon-shaped structure, reminiscent of that observed in random Lindbladians [49, 50]. Whereas the spectrum in $\mathcal{B}_{++}$ is symmetric about the real axis, that in $\mathcal{B}_{+-}$ lacks this symmetry, consistent with the presence or absence of modular conjugation symmetry. To test the strongly correlated nature of the Lindbladian, we investigate the distributions of complex spacing ratios $\eta=re^{i\theta}$ [53, 71]. Given a reference eigenvalue $z\in\mathbb{C}$, let $z_{\mathrm{NN}}\in\mathbb{C}$ and $z_{\mathrm{NNN}}\in\mathbb{C}$ denote its nearest- and next-to-nearest-neighbor eigenvalues, respectively. Then, we define $\eta$ as

whose statistics characterize the local geometry and correlations of the complex spectrum. In $\mathcal{B}_{+-}$, each complex eigenvalue forms a Kramers pair, which is counted only once for calculations of $\eta$. In Figure 1 (b, Figure 1c), we show the probability densities of $r$ and $\theta$, which agree well with those for non-Hermitian random matrices with time-reversal symmetry¹. Specifically, the negative weak-fermion-parity subspace $(-1)^{N}=-1$ realizes the symplectic random-matrix statistics coexisting with the Kramers degeneracy, providing the spectral signatures of nonintegrability for open quantum systems in the symplectic class.

Dissipative spin chain.—Beyond fermionic systems, our Liouvillian Kramers theorem also applies to bosonic systems. As an explicit realization, we construct a dissipative quantum spin chain exhibiting the Kramers degeneracy. Let $X_n$, $Y_n$, and $Z_n$ denote Pauli matrices at site $n$ ($n=1,2,\cdots,L$) of a spin-1/2 chain with even length $L\ge4$. We choose the Hamiltonian as

where $J_n^{\perp}$ and $J_n^{z}$ are the bond-dependent transverse and longitudinal exchange interactions, respectively, and $D$ is the Dzyaloshinskii-Moriya interaction along the $z$ direction. We further introduce the jump operators

for $n=1,2,\cdots,L-1$, with $\kappa\in\mathbb{R}$. The two-spin terms in the jumps break conservation of magnetization $\sum_n Z_n$ while preserving parity $\prod_n Z_n$.

In contrast to the preceding fermionic model, the relevant antiunitary operation is not conventional spin time reversal but its combination with a uniform $\pi/2$ rotation about the $z$ axis. Specifically, we introduce an effective antiunitary time-reversal operator

satisfying

Under this effective time reversal, the Hamiltonian $H$ is invariant as in Eq. (1), and the jump operators satisfy $TL_nT^{-1}=-L_n$. Accordingly, the resulting Lindbladian exhibits the Kramers degeneracy in the weak-parity subspace $(-1)^N=-1$. Similarly to the fermionic model, the jump operators also preserve strong parity [i.e., $(-1)^N L_n(-1)^N=L_n$], leading to the symmetry classification in Eq. (8). Notably, in the discussion following Eq. (46) of Ref. [45], $\eta_{1,2}$ defined through $q_{1,2}=\eta_{1,2}q_{1,2}^{*}$ are assumed to be unimodular complex numbers (i.e., proportional to the identity), where

$q_{1,2}$ correspond to the unitary part $U_T$ of $T:=U_T\mathcal{K}$. By contrast, $U_T$ in our model satisfies $U_T=(-1)^{L/2}(-1)^N U_T^{*}$, violating the assumption made in Ref. [45].

In Figure 2, we investigate the complex-spacing-ratio distributions, focusing on the complex eigenvalues around the spectral center $\operatorname{Tr}\mathcal{L}_{\mathrm{pq}}/\operatorname{Tr}1=-\gamma(1+2\kappa^2)(L-1)$. These probability distributions show good agreement with the random-matrix distributions in the corresponding symmetry classes, despite deviations near $r=1$ within $\mathcal{B}_{+,-}$. Notably, the dissipative quantum spin chain considered here is spatially local, and the corresponding Lindbladians $\mathcal{L}_{\mathrm{pq}}$ are represented by sparse matrices, as opposed to dense random matrices. Nevertheless, the complex spectral statistics of $\mathcal{L}_{\mathrm{pq}}$ exhibit the random-matrix universality, providing a spectral diagnostic of nonintegrability for open quantum systems in the symplectic class.

Discussion.—We have established the Kramers degeneracy theorem for Liouvillians and demonstrated their random-matrix universality in the symplectic class. While we have identified classes AII$^{\dagger}$ and DIII$^{\dagger}$ as the simplest realizations, additional symmetry can yield a broader range of symplectic classes, which can be distinguished through the spectral statistics around the symmetry-invariant lines and point [83]. As in isolated quantum systems, the Liouvillian Kramers degeneracy can underlie unique phenomena of open quantum systems. Although a stationary density matrix can always be chosen in the $(-1)^N=+1$ subspace, the $(-1)^N=-1$ subspace contains no density matrices. Nevertheless, the Kramers pairs describe independent dynamical modes, which can be probed through two-time correlations of $(-1)^N=-1$ operators. Our results also motivate revisiting the fragility of time-reversalinvariant topological insulators in open quantum systems [37] and exploring possible connections to symmetry fractionalization [84].

K.K. is supported by JSPS KAKENHI Grants No. JP26H02015, No. JP26K06970, No. JP26K17046, and JST FOREST Program Grant No. JPMJFR256P.

End Matter

Symplectic random Lindbladian

In a similar manner to Refs. [49, 50], we construct a random-matrix model of a Lindbladian exhibiting the Kramers degeneracy. Let $4D$ ($D\in\mathbb{N}$) be the dimension of the Hilbert space. We take the Hamiltonian to be

where $H_{\mathrm{GOE}}$ and $H_{\mathrm{GSE}}$ are $2D\times2D$ Hermitian random matrices drawn from the Gaussian orthogonal and symplectic ensembles, respectively. Specifically, introducing Hermitian random matrices $G_{\pm}$ with the entries

we define

with $h\in\mathbb{R}$ and $\Sigma_{y}:=\sigma_{y}\otimes1_{D}$. Here, $\mathcal{N}(0,1)$ [$\mathcal{N}_{\mathbb{C}}(0,1)$] denotes a real (complex) Gaussian distribution with $\mathbb{E}z=0$ and $\mathbb{E}|z|^{2}=1$. We choose the jump operators as

where $A$ is a non-Hermitian random matrix drawn from the Ginibre unitary ensemble, with the independent entries distributed according to $A_{ij}\sim\mathcal{N}_{\mathbb{C}}(0,1/2D)$.

The Hamiltonian $H$ respects time-reversal symmetry in Eq. (1), implemented by the antiunitary operator

Here, $\mathcal{T}^{2}=\Sigma_{z}:=\sigma_{z}\otimes1_{2D}$ defines a parity operator. Moreover, the jump operators transform into each other under time reversal and Hermitian conjugation [Eq. (10)]:

Consequently, the Liouvillian exhibits the Kramers degeneracy in the subspace of weak parity $\mathcal{T}^{2}=-1$.

In Figure 3 (a), we show the complex spectrum for a single realization. In both weak-parity subspaces, the complex spectrum is symmetric about the real axis, owing to modular conjugation symmetry. We also note that we have $\mathbb{E}H_{\mathrm{GOE}}^{2}=\mathbb{E}H_{\mathrm{GSE}}^{2}=h^{2}1_{2D}$ and $\mathbb{E}(L_{1}^{\dagger}L_{1}+L_{2}^{\dagger}L_{2})=\gamma1_{4D}$. Accordingly, choosing $h$ and $\gamma$ to be constants of order unity ensures that the Hamiltonian and dissipation contributions have comparable characteristic scales. In Figure 3 (b, Figure 3c), we show the complex-spacing ratio distributions, which agree with those of non-Hermitian random matrices in class AI$^{\dagger}$ (AII$^{\dagger}$) for weak parity $+1$ ($-1$). Reference [45] argued that time-reversal symmetry$^{\dagger}$ in Eq. (4) with $\mathcal{T}^{2}=-1$ always leads to strong symmetry of Lindbladians. However, our random symplectic Lindbladian exhibits the Kramers degeneracy protected by time-reversal symmetry$^{\dagger}$ with $\mathcal{T}^{2}=-1$, despite the absence of strong symmetry.

Exactly solvable dissipative spinful fermion

We illustrate the Lindbladian Kramers degeneracy with a single spinful orbital, described by fermion operators $c_{\uparrow}$ and $c_{\downarrow}$. We take the Hamiltonian to be

with $\mu$, $U$, $\Delta\in\mathbb{R}$ and $n_{\sigma} := c_{\sigma}^{\dagger}c_{\sigma}$. This Hamiltonian respects time-reversal symmetry in Eq. (1) with

Although the superconducting pairing term breaks particle-number conservation, fermion parity remains preserved:

We introduce a set of four jump operators $\{L,\widetilde{L},L^{\dagger},\widetilde{L}^{\dagger}\}$

with $a,b\in\mathbb{R}$. The jump $L$ coherently combines single-fermion loss and gain for up spin, with the loss amplitude $1+ibn_{\downarrow}$ depending on the occupation of down spin. Since these jump operators change fermion parity, the resulting Lindbladian only preserves weak fermion parity $(-1)^{N}$. By construction, this set of jump operators is closed under both time reversal and Hermitian conjugation [Eq. (10)]. Consequently, the Lindbladian exhibits the Kramers degeneracy in the subspace of weak fermion parity $(-1)^{N}=-1$.

Specifically, the characteristic polynomials $\chi_{\pm}(\lambda):=\det(\lambda-\mathcal{L}_{\pm})$ in the respective weak-fermion-parity subspaces are given as

with $\Gamma:=2(1+a^{2})+b^{2}$. The perfect-square form of $\chi_{-}$ demonstrates the Kramers degeneracy.