Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and $\mathcal{PT}$ Symmetry Breaking

TL;DR

This work introduces an exactly solvable dissipative variant of the Yao–Lee spin–orbital model by mapping Lindbladian dynamics to a non-Hermitian bilayer Hamiltonian on a doubled Hilbert space, enabling analytic control over both steady-state structure and transient spectra. The model exhibits extensive strong and weak symmetries, yielding an exponentially large manifold of non-equilibrium steady states that realize a dissipative spin-liquid phase. In a translation-invariant flux sector, the single-particle Liouvillian spectrum features an exceptional ring in momentum space, with a dissipation-driven PT-symmetry breaking transition that drives the crossover from oscillatory to purely decaying relaxation of observables. The work provides a physically motivated, solvable setting to explore the coexistence of dissipative spin-liquid physics and Liouvillian spectral singularities in two dimensions, offering benchmarks and a platform for experimental realization and further theoretical development of Liouvillian topology and dynamics.

Abstract

Exactly solvable dissipative models provide an analytical tool for studying the relaxation dynamics in open quantum systems. In this work, we study an exactly solvable model based on an anisotropic variant of the Yao-Lee spin-orbital model, with dissipation acting in the spin sector. We map Liouvillian dynamics to fermions hopping in a doubled Hilbert space under a non-Hermitian Hamiltonian and demonstrate the model's exact solvability. We analyze the model's strong and weak symmetries, which protect an exponentially large manifold of non-equilibrium steady states, establishing the system as a physically feasible dissipative spin liquid. Furthermore, we analyze the transient dynamics in a translationally invariant sector and discover that the single-particle Liouvillian spectrum hosts an exceptional ring in momentum space. We map out a characteristic $\mathcal{PT}$ symmetry breaking transition driven by the dissipation strength, which governs the crossover from oscillatory to decaying relaxation of physical observables. Our work provides a physically motivated, solvable setting for exploring the coexistence of dissipative spin liquid physics and Liouvillian spectral singularities.

Figures (4)

• Figure 1: (a) Illustration of the mapping between an open system described by the Liouvillian $\mathcal{L}$ (left) and the non-Hermitian bilayer Hamiltonian $\mathcal{H}$ (right). In the doubled Hilbert space, the jump operators couple the bra and ket spaces. The two sublattices $A$ and $B$ are labeled by blue and red dots, respectively. (b) A hexagonal plaquette in the honeycomb lattice. There are three types of bonds: $x$, $y$, and $z$. (c) The Brillouin zone of the honeycomb lattice.
• Figure 2: Illustration of the strong and weak $\mathbb{Z}_2$ symmetries of the model in the bilayer framework. The hexagonal plaquettes $W_p$ and $\widetilde{W}_p$ (Eq. \ref{['eq:Wp']}) constitute strong symmetries of the model, while the vertical square plaquettes $V_p$'s (Eq. \ref{['eq:Vp']}) are the weak ($\mathbb{Z}_2$) symmetries.
• Figure 3: Real (a) and imaginary (b) parts of the energy gap $E_+-E_-$. The exceptional points are highlighted in yellow contours and form a ring in momentum space. Parameters: $\gamma=0.4$, $J = 1$.
• Figure 4: (a) Real-space Liouvillian spectrum for $\gamma=0.01$, $\gamma = 0.5$ and $\gamma = 0.8$. All eigenvalues are shifted by a constant $+\text{Tr}(\mathcal{L})/\text{Tr}(\mathbb I)$ for plotting purposes. The real-space lattice size is $10\times 10$. (b) Estimated $\gamma_\text{PT}$ and $\gamma^*$ for systems with different linear sizes $L$. The total number of sites is $N=L^2$. $\gamma^*$ approximately remains constant at $3J/4$ for all system sizes, while $\gamma_{\text{PT}}$ decays as $1/L$, as predicted in Ref. prosen_PT. Parameters: $L=4,8,16,32,64; J=1$. (c) Fraction of modes that are $\mathcal{P}\mathcal{T}$ broken. The ratio smoothly crosses over from $0$ for $\gamma < \gamma_{\text{PT}}$ to $1$ for $\gamma > \gamma^*$. Parameters: $L=20$, $J=1$.