Critical quasienergy states in driven many-body systems

We discuss singularities in the spectrum of driven many-body spin systems. In contrast to undriven models, the driving allows us to control the geometry of the quasienergy landscape. As a consequence, one can engineer singularities in the density of quasienergy states by tuning an external control. We show that the density of levels exhibits logarithmic divergences at the saddle points, while jumps are due to local minima of the quasienergy landscape. We discuss the characteristic signatures of these divergences in observables such as the magnetization, which should be measurable with current technology.

Figure 1

Applying an external control $g\left(t\right)$ to the undriven system (a) allows us to engineer geometrical features of the quasienergy landscape, which leaves measurable signatures in observables. Depending on the shape of the external control, e.g., $\delta$-kick-type (solid red) or monochromatic (dashed blue), the emergent saddle points are connected via (b) homoclinic or (c) heteroclinic orbits, giving rise to characteristic density of quasienergy states.

Figure 2

QEL for (a) the kicked top with $K=0.3$ and (b) the ac-driven model with $GT=20$. The yellow (light gray) and blue (dark gray) regions have energy $E>h$ and $E<h$ for the kicked top, and $E>0$ and $E<0$, for the ac-driven model, respectively. Solid lines show levels of constant energy. Features are denoted by S (saddle point), M (maxima), and m (minima). Other parameters are $\Omega T=2\pi$ and $hT=0.1$.

Figure 3

DOQS $\rho \left({\varepsilon }_{\alpha }\right)$ and integrated DOQS $N\left({\varepsilon }_{\alpha }\right)$ of (a) the kicked top with $K=0.3$ and $j=100$ and (b) the ac-driven model with $GT=20$ and $j=100$ calculated analytically (red, solid lines) and numerically exact (black symbols). Other parameters are $\Omega T=2\pi$ and $hT=0.1$.

Figure 4

Magnetization ${\langle J_x/j\rangle }_{\mu }$ for (a) the kicked top with $K=0.3$, and (b) the ac-driven LMG model with $GT=20$. Calculations in quasienergy states (filled, black squares) are compared to time-averaged expectation values (blue circles and red triangles) with initial conditions shown in insets (I), respectively. Insets (II) show details of the cusp arising due to the CQS for the different drivings. Features of the corresponding QEL are denoted by S (saddle point), M (maxima), and m (minima). Other parameters are $j=50$, $\Omega T=2\pi$, and $hT=0.1$.