Fate of entanglement in open quantum spin liquid: Time evolution of its genuine multipartite negativity upon sudden coupling to a dissipative bosonic environment

Abstract

Topological properties of many-body entanglement in quantum spin liquids (QSLs), persisting at arbitrarily long distances, have been intensely explored over the past two decades, but mostly for QSLs viewed as {\em closed} quantum systems. However, in experiments and potential quantum computing applications, candidate materials for this exotic phase of quantum matter will always interact with a dissipative environment, such as the one generated by bosonic quasiparticles in solids at finite temperature. Here we investigate the spatial structure and stability of entanglement in the Kitaev model of QSL made {\em open} by sudden coupling to an infinite bosonic bath of Caldeira-Leggett type and time-evolved using the Lindblad quantum master equation in the Markovian regime (i.e., for weak coupling) or tensor network methods for open quantum systems in the non-Markovian regime (i.e., for strong coupling). From the time-evolved density matrix of QSL and its subregions, we extract genuine multipartite negativity (GMN), quantum Fisher information, spin-spin correlators, and expectation value (EV) of the Wilson loop operator. In particular, time-dependence of GMN offers the most penetrating insights: (i) in the Markovian regime, it remains non-zero in larger loopy subregions of QSL (as also discovered very recently for closed QSLs) up to temperatures comparable to Kitaev exchange interaction at which other quantities, such as EV of the Wilson loop operator, vanish; (ii) in the non-Markovian regime with pronounced memory effects, GMN remains non-zero up to even higher temperatures, while also acquiring non-zero value in smaller non-loopy subregions. The non-Markovian dynamics can also generate emergent interactions between spins, thereby opening avenues for tailoring properties of QSL via environmental engineering.

Figures (3)

• Figure 1: Time dependence in the Markovian regime, as computed by the universal Lindblad QME Rudner2020, of: (a) GMN Lyu2025Jungnitsch2011Hofmann2014 for different subsystems enclosed within colored middle hexagon; (b) QFI defined in Eq. \ref{['eq:QFI']} for the wavevector $\mathbf{k}=(0,0)$; (c) equal-time spin-spin correlator $\langle \hat{\sigma}_i^z(t) \hat{\sigma}_j^z(t) \rangle$ for two NN sites $i$ and $j$; and (d) Wilson loop operator $\langle \hat{W}_p\rangle(t)$. The system considered is the gapless Kitaev QSL [Eq. \ref{['eq:Kitaev_Hamiltonian']}] composed of $N=14$ localized spins locally coupled to many baths [Eq. \ref{['eq:local']}]. The temperature of bosonic baths for (b)--(d) is indicated in panel (c), while in (a) it is set to $T=0.3J_z$.
• Figure 2: The same information as in Fig. \ref{['fig:Lindblad_results']}, but for the non-Markovian regime, as computed via PT-MPO+TEBD methodology Fux2023Fux2024Cygorek2024Cygorek2024a. The temperature of bosonic baths for (b)--(d) is indicated in panel (c), while in (a) is set to $T=0.5J_z$. Note that orange dots in panel (a) start from zero due to the unentangled initial state required for PT-MPO+TEBD calculations, unlike orange dots in Fig. \ref{['fig:Lindblad_results']}(a), where the initial state is highly entangled GS of QSL.
• Figure 3: Renormalized exchange interactions of Kitaev QSL Hamiltonian [Eq. \ref{['eq:Kitaev_Hamiltonian']}] for: (a) "local coupling" to many baths [Eq. \ref{['eq:local']}]; and (c) "global coupling" to a single bath [Eq. \ref{['eq:global']}]. (b) EV of Wilson loop operator for the case of "global coupling." (d) Static FM structure factor for the system corresponding to panel (c). These results were obtained by diagonalizing the effective Hamiltonian [Eq. \ref{['eq:Heff_global']}] produced by RC + polaron methodology, so they describe steady state in the long-time limit of non-Markovian dynamics generated by arbitrary strong coupling to a single global bath. The frequency of the RC was set to $\Omega=8J_z$. Note that the limit $\lambda/J_z \rightarrow 0$ corresponds to the Markovian dynamics in Fig. \ref{['fig:Lindblad_results']}.