ac-driven quantum phase transition in the Lipkin-Meshkov-Glick model

We establish a set of nonequilibrium quantum phase transitions in the Lipkin-Meshkov-Glick model under monochromatic modulation of the interparticle interaction. We show that the external driving induces a rich phase diagram that characterizes the multistability in the system. Interestingly, the number of stable configurations can be tuned by increasing the amplitude of the driving field. Furthermore, by studying the quantum evolution, we demonstrate that the system exhibits a set of quantum phases that correspond to dynamically stabilized states.

Figure 1

Comparison between the quasienergies obtained numerically (black curves) and the eigenvalues of the effective Hamiltonian ${\widehat{h}}_0^{\left(0\right)}$ (red curves) as a function of the driving frequency $\Omega$. We consider a finite-size system of $N=10$ particles and the parameters $\frac{1}{\left|h\right|}(h,{\gamma }_0^x,{\gamma }_1^x,{\gamma }^y)=(-1,-1,210,2)$. The straight lines correspond to the boundaries ${\varepsilon }_{\alpha }=\pm \Omega /2$ of the first Brillouin zone.

Figure 2

Phase diagram of the nonequilibrium QPT in the Lipkin-Meshkov-Glick model. The number of local minima of the quasienergy surface $E_G^{\left(0\right)}(Q,P)$ in the colored zones are indicated by the labels. (a) Depicts the phase diagram as a function of ${\gamma }^y$ and the driving amplitude ${\gamma }_1^x$ for ${\gamma }_0^x/\left|h\right|=0.5$, and (b) as a function of ${\gamma }_0^x$ and ${\gamma }_1^x$ for ${\gamma }^y/\left|h\right|=2$. The dashed red (light gray) line in (a) resembles the second-order QPT in the undriven LMG model that occurs at ${\gamma }^y/\left|h\right|=1$; in the driven case, however, this line separates the regions with even and odd number of minima. The dashed black lines in (a) and (b) depict the transition of the symmetric phase $(Q,P)=(0,0)$ from a saddle point to a local maximum and correspond to the contour ${\lambda }_2=0$. We consider the parameters $\Omega /\left|h\right|=40$ and $h/\left|h\right|=-1$.

Figure 3

Quasienergy landscape $E_G^{\left(0\right)}(Q,P)$ for the parameters $\frac{1}{\left|h\right|}(h,{\gamma }_0^x,{\gamma }_1^x,{\gamma }^y)=(-1,-1,210,2)$. In the undriven system, these parameters correspond to the symmetry-broken phase. The quantum evolution for $N=100$ particles within one period is calculated when the system is initialized in a spin coherent state. (a) Depicts the quantum evolution of the operators in the laboratory frame and (b) the effective evolution in the rotating frame. The green (light gray) line on the surface depicts the evolution of an initial wave packet centered at minimum B and the red line shows the corresponding evolution for a wave packet initially centered at A. We consider the parameters $m=0$ and $\Omega /\left|h\right|=40$.

Figure 4

Quantum evolution of the observables for finite size $N=100$ and parameters $\frac{1}{\left|h\right|}(h,{\gamma }_0^x,{\gamma }_1^x,{\gamma }^y)=(-1,-1,210,2)$. The expectation values $\langle J_x\rangle /j,\langle J_y\rangle /j$ and $\langle J_z\rangle /j$ are depicted by the orange (medium gray), cyan (light gray), and magenta (dark gray) curves, respectively. For an initial wave packet centered at the minimum A in Fig. 3, (a) depicts the quantum evolution within one period, (b) the effective evolution in the rotating frame, and (c) depicts the stroboscopic dynamics. Correspondingly, when the system is initialized in a wave packet centered at the minimum B in Fig. 3, (d) depicts the quantum evolution within one period, (e) the effective evolution, and (e) the stroboscopic evolution. We consider the parameters $m=0$ and $\Omega /\left|h\right|=40$.