Quantum Criticality and Dynamical Instability in the Kicked-Top Model

We investigate precursors of critical behavior in the quasienergy spectrum due to the dynamical instability in the kicked top. Using a semiclassical approach, we analytically obtain a logarithmic divergence in the density of states, which is analogous to a continuous excited state quantum phase transition in undriven systems. We propose a protocol to observe the cusp behavior of the magnetization close to the critical quasienergy.

Figure 1

(a) Exact quasienergy spectrum (black lines) and the eigenvalues of the effective Hamiltonian (red, open circles) as a function of $\kappa$ for $j=5$. (b) Zoom into the regime $\kappa \ll 1$ [dashed vertical line in Fig. 1] for $j=40$. The red, vertical line denotes the value of $\kappa$ used in Fig. 2. Features are marked by $C$ (clustering) and $D_M$ (discontinuity); see main text. The parameters are $p=0.1$ and $T=1$.

Figure 2

Comparison between the analytical (solid and dashed curves) and exact numerical results (symbols) for the DOQS (dashed, red) and the integrated DOQS (solid, black). The behavior is in direct correspondence with the spectrum along the vertical solid line in Fig. 1. Features are marked by $C$ (clustering) and $D_{M,m}$ [discontinuity due to maxima ($M$) and minimum ($m$)]. The parameters are $j=40$, $\kappa =0.2$, $p=0.1$, and $T=1$.

Figure 3

Numeric scaled magnetization in the quasienergy eigenstates $⟨J_x/j{⟩}_{\alpha }$ (filled circles), the time-averaged expectation value $\overline{⟨J_x/j⟩}$ (solid, dashed lines), and the semiclassical result $\overline{X}$ (open triangles, squares), see main text. Inset (I) depicts the paths on the Bloch sphere to perform the measurement protocol. The parameters are $j=40$, $\kappa =0.2$, $p=0.1$, and $T=1$.