Topological Instabilities in ac-Driven Bosonic Systems

Under nonequilibrium conditions, bosonic modes can become dynamically unstable with an exponentially growing occupation. On the other hand, topological band structures give rise to symmetry protected midgap states. In this Letter, we investigate the interplay of instability and topology. Thereby, we establish a general relation between topology and instability under ac driving. We apply our findings to create dynamical instabilities which are strongly localized at the boundaries of a finite-size system. As these localized instabilities are protected by symmetry, they can be considered as topological instabilities.

Figure 1

(a) Sketch of the system. (b) and (c) depict quasienergy spectra with colored lines for ${\nu }_0=1.5$, ${\nu }_0^{\prime }=0$, ${\nu }_1=3$, $\mu =-5$ and $\mathrm{\Omega }=5.2$. We choose ${\nu }_1^{\prime }=11$ and ${\nu }_1^{\prime }=6$ in (b) and (c), respectively. Black lines depict the spectrum of an effective Hamiltonian [36]. All quantities are expressed in units of $g$.

Figure 2

(a) Topological phase diagram. The parameters are as in Fig. 1. In the green (dark gray) areas, the system is not globally stable, so the topological invariant $W^S$ in Eq. (9) is not defined there. In the white and yellow (light gray) areas, we find that $W^S=0$ and $W^S=2$, respectively. The topological phases are separated by instability areas (green). Dashed lines are obtained by using an effective Hamiltonian [36]. (b) Stability diagram as a function of $h_{x,1}$ and $h_{y,1}$ corresponding to (a). The parameters of the curves ${\gamma }_{a,b,c}$ are depicted in (a) by the points $p=a$, $b$, $c$. The parameters $p=a$ and $p=b$ correspond to Figs. 1 and 1. Curve ${\gamma }_c$ can be contracted to point $P$, so that it is topologically trivial according to the explanations in the main text.

Figure 3

(a) Quasienergy spectrum for a finite-sized system with boundaries corresponding to ${\gamma }_a$. There are four midgap states located near $\mathrm{Re}{\epsilon }_i=0$ which we mark with arrows. The imaginary part of two of the midgap states is finite, which renders these states unstable. States not marked by the arrows extend within the bulk and are stable. (b) Time evolution of the occupation of the sites $j=1$, 9, 13. The initial state is the vacuum state. In the inset we depict the wave function ${\mathrm{\Psi }}_j=⟨\mathrm{v}\mathrm{a}\mathrm{c}|{\hat{a}}_j|{\mathrm{\Psi }}_i⟩$ of one of the unstable midgap states which are responsible for the exponential growth as seen in (b). It is strongly confined close to the boundaries. Note that the site number is $j=2m+s$, with $m$, $s$ defined in Eq. (1).