Nonequilibrium Quantum Criticality and Non-Markovian Environment: Critical Exponent of a Quantum Phase Transition

We show that the critical exponent of a quantum phase transition in a damped-driven open system is determined by the spectral density function of the reservoir. We consider the open-system variant of the Dicke model, where the driven boson mode and also the large $N$ spin couple to independent reservoirs at zero temperature. The critical exponent, which is 1 if there is no spin-bath coupling, decreases below 1 when the spin couples to a sub-Ohmic reservoir.

Figure 1

Thick lines show the real part (a) and imaginary part (b) of the soft mode frequency ${\omega }_s$ as a function of the coupling $y$ for sub-Ohmic dissipation strengths $\gamma =0$ (solid red line) and $0.5{\omega }_b$ (dashed blue line). Thin lines represent the pair of the soft mode pole. The bath low-frequency exponent is $s=4/5$. Other parameters are ${\omega }_a=\kappa =2{\omega }_b$.

Figure 2

Power spectrum of the “photonic” mode $a$ (a) and of mode $b$ (b) for various coupling strengths approaching the critical point: $y=0$ (solid red line), 0.5 (dashed green line), $0.9y_c$ (dotted blue line), and $0.95y_c$ (dash-dotted brown line). Parameters are ${\omega }_a=\kappa =2{\omega }_b$, $\gamma =0.5{\omega }_b$, $s=4/5$. In the photonic spectrum (a), the peaks for $y=0.9y_c$ and $y=0.95y_c$ are divided by a factor of 10 to fit into the plotted range.

Figure 3

Critical exponent as a function of the exponent $s$ of the sub-Ohmic bath, which is extracted from the scaling of the population in mode $a$ (“photon number,” solid red line) and in mode $b$ (green circles). The power law scaling of the photon number is shown in the inset for $s=0.8$ (red crosses), 0.7 ($\mathrm{green}\times \mathrm{symbols}$), 0.6 (blue stars), 0.5 (brown squares). Other parameters are the same as in Fig. 2. The solid black line shows the $\gamma =0$ case for reference.