Dynamical topological order parameters far from equilibrium

We introduce a topological quantum number—coined dynamical topological order parameter (DTOP)—that is dynamically defined in the real-time evolution of a quantum many-body system and represented by a momentum space winding number of the Pancharatnam geometric phase. Our construction goes conceptually beyond the standard notion of topological invariants characterizing the wave function of a system, which are constants of motion under coherent time evolution. In particular, we show that the DTOP can change its integer value at discrete times where so called dynamical quantum phase transitions occur, thus serving as a dynamical analog of an order parameter. Interestingly, studying quantum quenches in one-dimensional two-banded Bogoliubov–de Gennes models, we find that the DTOP is capable of resolving if the topology of the system Hamiltonian has changed over the quench. Furthermore, we investigate the relation of the DTOP to the dynamics of the string order parameter that characterizes the topology of such systems in thermal equilibrium.

Figure 1

Left panel: Color plot of the Pancharatnam geometric phase ${\varphi }_k^G\left(t\right)$, see Eq. (8), for chemical-potential quenches $\mu =0\to \mu =3$ in the Kitaev chain as a function of lattice momentum and time. Time is measured in units of the critical time $t_c$ where the first dynamical quantum phase transition (DQPT) occurs. The critical momentum $k_c$ at which nonanalyticities occur is marked with a black dotted line. Right panel: Rate function $g\left(t\right)=-N^{-1}{\mathrm{Re}log\left[\right|\mathcal{G}\left(t\right)|}^2]$ of the Loschmidt echo $\mathcal{L}\left(t\right)={\left|\mathcal{G}\left(t\right)\right|}^2$, see Eq. (1), and the dynamical topological order parameter ${\nu }_D\left(t\right)$, see Eq. (9), as a function of time. The real-time nonanalyticities in $g\left(t\right)$, occurring at odd multiples of $t_c$ (red dashed lines), define the DQPTs. ${\nu }_D\left(t\right)$ changes its value only at the DQPTs thus serving as a dynamical order parameter.

Figure 2

Left panel: Color plot of the PGP ${\varphi }_k^G\left(t\right)$ as a function of lattice momentum and time for $\lambda =1.3$. Time is measured in units of the first critical time $t_c^{\left(1\right)}=\pi /2$. The critical momenta $k_c^{\left(1\right)}$ and $k_c^{\left(2\right)}$ at which DQPTs appear as discontinuities in the PGP is marked with black dotted lines. Right panel: Rate function $g\left(t\right)$ and DTOP ${\nu }_D\left(t\right)$ as a function of time for $\lambda =1.3$. The DQPTs occur at odd multiples of the critical times $t_c^{\left(1\right)}$ and $t_c^{\left(2\right)}$.

Figure 3

Decay of the string order parameter for the same chemical potential quench as in Fig. 1 but for a model also including a next-to-nearest neighbor hopping with amplitude $j$. The time axis is rescaled by $t^{*}$ where DQPTs occur and the DTOP changes its quantized value. The inset shows a comparison of the low-energy time scale $t_{\mathrm{gap}}\left(j\right)=\mathrm{\Delta }{\left(j\right)}^{-1}$ with $\mathrm{\Delta }\left(j\right)$ the gap, and $t_c\left(j\right)$. The simulations are done for a lattice system with up to $N=100$ sites, where we find that the data are converged with respect to system size.