Fermionic quantum spin liquids: Exact results for parametric pumping

The time dependence of the response of the quantum spin liquid with fermionic excitations to parametric pumping has been calculated exactly for the special case of periodic pumping, the steplike periodic field, without using the resonance approximation, and for any value of the pumping magnitude. In the closed regime each mode (related to each eigenstate of the system) oscillates with time. The oscillations persist about the steady-state values, which are determined by the parameters of the pumping. The magnitude of the oscillations is finite for any value of the magnitude of the pumping. Those features drastically differ from the well-known resonance parametric pumping of magnons in magnetically ordered system, where the number of resonance magnons grows exponentially in time. The interference of infinitely many modes generically yields only decaying-in-time modulated oscillations of the total number of quasiparticles per mode even in the closed regime. For the fermionic system only one mode is in resonance, while others are out of resonance. It is totally different from the bosonic behavior of magnons under parametric pumping, where all modes can exist in resonance. The inclusion of the linear relaxation in the open regime for that quantum spin liquid produces the decay of oscillations to the initial state.

Figure 1

Left: The time dependence of $\mathrm{\Delta }{m}_k$ for the honeycomb Kitaev model in the gapless case $J_x=J_z=1$, $J_y=0.5$ for $h=0.01$ in resonance ${\tau }_1=\pi /{\varepsilon }_{k,1}$ and ${\tau }_2=\pi /{\varepsilon }_k$ in the closed regime. The solid black line shows $k_x=0$, $k_y=\pi /2$; the dotted blue line shows $k_x=\pi /2$ and $k_y=0$, and the dashed red line shows $k_x=k_y=\pi /2$. Right: The same as in the left panel, but for the gapped case $J_x=1$, $J_y=0.5$, and $J_z=2$.

Figure 2

The time dependence of $n_{av}$ for the finite honeycomb Kitaev model in the gapless case out of resonance in the closed regime.

Figure 3

Left: The time dependence of the renormalization of the $z$ component of the averaged per site total spin of the Ising chain (with $J_x=2$) per site in the transverse field $h_t$ at $T=0.1$ in resonance (with ${\tau }_1={\tau }_2=\pi$) for $h=0.1$. Right: The same as in the left panel, but for $h=1.5$.

Figure 4

Top: The time dependence of the renormalization of the $z$ component of the averaged per site total spin of the Ising chain (with $J_x=2$) per site in the transverse field $h_t$ at $T=0.1$ in resonance (with ${\tau }_1={\tau }_2=\pi$) at $h=0.01$. Bottom: The same as in the top panel, but out of resonance (with ${\tau }_1=\pi /4$ and ${\tau }_2=\pi$) at $h=0.01$.

Figure 5

The time dependence of the renormalization of the $z$ component of the averaged per site total spin of the Ising chain (with $J_x=2$) per site in the transverse field $h_t$ at $T=0.1$ in resonance (black line, ${\tau }_1=\pi$ and ${\tau }_2=\pi$) and out of resonance (red line, with ${\tau }_1=\pi /2$ and ${\tau }_2=\pi$) at $h=0.01$.

Figure 6

Left: The quantum work associated with the considered time-dependent pumping for the Ising chain as a function of time out of resonance ${\tau }_1=\pi /2$ and ${\tau }_2=\pi$. Right: The same as in the left panel, but in resonance ${\tau }_1={\tau }_2=\pi$.

Figure 7

Left: The normalized power associated with the considered time-dependent pumping for the Ising chain as a function of time out of resonance ${\tau }_1=\pi /2$ and ${\tau }_2=\pi$. Right: The same as in the left panel, but in resonance ${\tau }_1={\tau }_2=\pi$.