Bosonic representation of a Lipkin-Meshkov-Glick model with Markovian dissipation

We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.

Figure 1

Results for the expectation value of the system Hamiltonian ${\mathcal{H}}_S$ and for the $m_x$ and $m_y$ components of the magnetization, plotted as functions of time for the initial state described in the text. Solid and dashed lines correspond to the predictions of Eqs. (57), (59a), and (59b). In the top figure the prediction of Eq. (57) is shown both with (solid line) and without (dashed line) the finite-size correction terms. The dots show exact numerical results obtained by solving the original spin-based Lindblad equation (8) for $S=150$. Parameters were set to $\mathrm{\Lambda }=0.1, \gamma =0.15, T=4$, and $\theta =1/\sqrt{S}$.

Figure 2

Spectrum of the Lindbladian (8) in a region of the left complex plane, computed for parameter values $\gamma =0.2$ and $T=4$. The left and right subfigures show results for the symmetric ($\mathrm{\Lambda }<1$) and symmetry-broken ($\mathrm{\Lambda }>1$) phases, respectively. Here ${\lambda }_{\pm }$ are numerical results for $S=3000$ for the eigenvalues corresponding to the ${\mathcal{K}}_{\pm }$ subspaces of $\mathcal{K}$, while ${\lambda }_{\mathrm{\Delta },n}$ are the predictions of equation (45), which is based on a Holstein-Primakoff approximation and assumes small coupling $\gamma$.

Figure 3

The eigenvalue ${\lambda }_{+,1}$ of $\mathcal{L}$ with the largest nonzero real part in the ${\mathcal{K}}_{+}$ subspace of $\mathcal{K}$, shown as a function of $\mathrm{\Lambda }$ and for parameter values $S=500, \gamma =0.005$, and $T=4$. Also shown are the real and imaginary parts of ${\lambda }_{-,0}$, the eigenstate from the ${\mathcal{K}}_{-}$ subspace with the largest real part. Note that ${\lambda }_{-,0}$ becomes nearly degenerate with ${\lambda }_{+,0}=0$ at large $\mathrm{\Lambda }$.

Figure 4

Diagonal entries of the stationary state ${\rho }_{+,0}$ (top) and nearly degenerate ${\rho }_{-,0}$ (center) eigenstate of $\mathcal{L}$ in the $S_x$ basis as functions of the eigenvalue of $S_x$ for $S=40$ and $\mathrm{\Lambda }=2.6$. The sum ${\rho }_{+,0}+{\rho }_{-,0}$ (bottom) corresponds to a state operator with support around the classical ground state with positive $m_x$ magnetization.

Figure 5

Exact numeric results for the dynamics of $\langle \mathit{S}\rangle$, obtained by solving the original Lindblad equation (8) with $S=60$. The choices of initial states are described in the text. Top: Parameter values $\gamma =0.05$ and $T=4$, with $\mathrm{\Lambda }=0.5$ in the symmetric phase. Center: As in the top panel, but for $\mathrm{\Lambda }=1.4$, which is slightly inside the symmetry-broken phase. Bottom: For parameter values $\gamma =0.64$ and $T=5$, with $\mathrm{\Lambda }=3.2$, a point deep inside the symmetry-broken phase. The dashed and solid lines correspond to different choices of initial state, as explained in the text.