Polaron-transformed dissipative Lipkin-Meshkov-Glick model

We investigate the Lipkin-Meshkov-Glick model coupled to a thermal bath. Since the isolated model itself exhibits a quantum phase transition, we explore the critical signatures of the open system. Starting from a system-reservoir interaction written in positive-definite form, we find that the position of the critical point remains unchanged, in contrast to the popular mean-field prediction. Technically, we employ the polaron transform to be able to study the full crossover regime from the normal to the symmetry-broken phase, which allows us to investigate the fate of quantum-critical points subject to dissipative environments. The signatures of the phase transition are reflected in observables such as magnetization, stationary mode occupation, or waiting-time distributions.

Figure 1

Lower part of the isolated LMG model spectrum for finite-size numerical diagonalization of Eq. (2) (thin curves) and using the bosonic representation (bold curves) based on Eq. (9) for the three lowest energies. For large $N$, the spectra are nearly indistinguishable. In the symmetry-broken phase (right), two numerical eigenvalues approach the same oscillator solution. These correspond to the two different parity sectors, formally represented by two possible displacement solutions $\pm \alpha (h,{\gamma }_x)$ in Eq. (8).

Figure 2

Contour plot of the magnetization density $〈J^z〉/N$ vs spin-spin interaction ${\gamma }_x$ and temperature $k_{B}T$ for $N=1000$. At the critical point ${\gamma }_x=h$, the magnetization density at low temperatures (bottom) suddenly starts to drop from a constant value in the normal phase (left) to a decaying curve in the symmetry-broken phase (right) as predicted by (27). At higher temperatures, the transition is smoother and the predictions from the bosonic representation [solid green contours, based on Eq. (26)] and the finite-size numerical calculation of the partition function [dashed contours, based on the Gibbs state with Eq. (2)] disagree for ${\gamma }_x\approx h$. For the finite-size calculation, weak coupling has been assumed, $k_{B}T\ll N{\omega }_c/\eta$, such that $U_p^{†}{J}_z{U}_p\approx J_z$ instead of (B4).

Figure 3

(a) LMG oscillator frequency $\omega (h,{\gamma }_x)$ or $\omega (h,{\tilde{\gamma }}_x)$; (b) diagonal frame steady-state mode occupations $\overline{〈d^{†}d〉}$ ($〈d^{†}d〉$); and (c) nondiagonal frame steady-state mode occupations $\overline{〈a^{†}a〉}\left(〈a^{†}a〉\right)$ for the polaron (solid) and nonpolaron (dashed) master equations. Divergent mode occupations indicate the position of the QPT where the excitation frequency vanishes. For the polaron treatment, the QPT position stays at ${\gamma }_x/h=1$ just as in the isolated LMG model, in contrast to the shift predicted by the BMS master equation. Parameters: $\eta =2\pi \left(0.1\right),{\omega }_c=0.5h,\beta =1.79/h$.

Figure 4

Waiting-time distributions (WTDs) between two emission (absorption and emission) events (a) ${\overline{\mathcal{w}}}_{ee\left(ae\right)}$ (solid, dot-dashed line) calculated in the polaron frame as a function of $\tau$ for a fixed ${\gamma }_x$ value and (b) distribution ${\overline{\mathcal{w}}}_{ee}$ as a function of ${\gamma }_x$ for two different fixed $\tau$ values. Additionally, the WTD in the nonpolaron frame is shown in (b) for $\tau =0$ case (dashed line), which wrongly diverges around the shifted critical point. At the true critical point, a nonanalytic dependence of the distribution on the intraspin-coupling strength ${\gamma }_x$ is clearly visible; within the polaron treatment, however, all WTDs remain finite. Parameters: $\eta =2\pi \left(0.1\right),{\omega }_c=0.5h,\beta =1.79/h$, (a) ${\gamma }_x=0.5h$.