Integrable Quantum Dynamics of Open Collective Spin Models

We consider a collective quantum spin $s$ in contact with Markovian spin-polarized baths. Using a conserved superoperator charge, a differential representation of the Liouvillian is constructed to find its exact spectrum and eigenmodes. We study the spectral properties of the model in the large-$s$ limit using a semiclassical quantization condition and show that the spectral density may diverge along certain curves in the complex plane. We exploit our exact solution to characterize steady-state properties, in particular at the discontinuous phase transition that arises for unpolarized environments, and to determine the decay rates of coherences and populations. Our approach provides a systematic way of finding integrable Liouvillian operators with nontrivial steady states as well as a way to study their spectral properties and eigenmodes.

Figure 1

Spectrum of the Liouvillian computed for $h=1$, $\mathrm{\Gamma }=1.2$, $p=0.9$, ${\mathrm{\Gamma }}_0=0.2$, and $s=17$. The eigenvalues corresponding to $q=6$ are shown in red. The gray curves are analytical predictions $\{{\lambda }_k[x=\mathrm{Im}\mathrm{\Lambda }/(2hs)]=\mathrm{Re}\mathrm{\Lambda }/s\}$ for the edges of the spectrum for $s\to \infty$. Left insets: Root structure of the eigenmodes. The (inverse) roots corresponding to some of the eigenmodes are plotted in the complex plane—black points. The branch points $r_{k=1,\dots ,4}$ of $G_0$ are depicted as orange crosses.

Figure 2

Upper panel: contours of integration taken to obtain the quantization condition in region I and II of the spectrum. Lower panel: Comparison between the numerics (red dots for $s=17$, $q=8$, and blue dots for $s=50$, $q=23$) and the leading order analytic prediction (crossings between the horizontal red or blue lines with the black curve) computed for $h=1$, $\mathrm{\Gamma }=1.2$, $p=0.9$, ${\mathrm{\Gamma }}_0=0.2$, and $x=q/2s\simeq 0.23$.

Figure 3

Density of eigenvalues of the Liouvillian in the thermodynamic limit, $\mathcal{D}$, plotted for $h=1$, $\mathrm{\Gamma }=1.2$, $p=0.9$, ${\mathrm{\Gamma }}_0=0.2$, and $s=17$. Inset: cut of the 3D plot for $\mathrm{\Lambda }/s=-0.3$ (upper) and $\mathrm{\Lambda }/s=-1.5$ (lower).

Figure 4

Steady state magnetization along the $z$ direction (left) and entropy (right) of the nonequilibrium steady state as a function of $p$ for $h=1$, $\mathrm{\Gamma }=1.2$, $p=0.9$, ${\mathrm{\Gamma }}_0=0.2$, and several values of $s$. Dots are numerical data obtained by exact diagonalization and lines are analytic results.