Topological phase transitions in finite-size periodically driven translationally invariant systems

It is known that, in the thermodynamic limit, the Chern number of a translationally invariant system cannot change under unitary time evolutions that are smooth in momentum space. Yet a real-space counterpart of the Chern number, the Bott index, has been shown to change in periodically driven systems with open boundary conditions. Here we prove that the Bott index and the Chern number are identical in translationally invariant systems in the thermodynamic limit. Using the Bott index, we show that, in finite-size translationally invariant systems, a Fermi sea under a periodic drive that is turned on slowly can acquire a different topology from that of the initial state. This can happen provided that the gap-closing points in the thermodynamic limit are absent in the discrete Brillouin zone of the finite system. Hence, in such systems, a periodic drive can be used to dynamically prepare topologically nontrivial states starting from topologically trivial ones.

Figure 1

(a) Chern number phase diagram in the driving frequency $\mathrm{\Omega }$ and $eaA$ plane for ${\widehat{H}}_F$ obtained from a high-frequency expansion to $O\left({\mathrm{\Omega }}^{-1}\right)$ (dashed lines) and from numerically exact calculations (solid lines). The Chern number is computed using the Bott-index formula in Eq. (2) in finite commensurate systems that are sufficiently large so that the result does not change (within machine precision) with increasing system size. (b) Critical value of the magnitude of the electric field $ea{A}_L^{*}$ for the first topological transition when $\mathrm{\Omega }=7J$ [black dot in (a)] in systems with $L_1\times L_2$ lattice sites plotted as a function of $L_1=L_2\equiv L$. (c) Scaling of the critical value for incommensurate systems. We plot $\mathrm{\Delta }\left(ea{A}_L^{*}\right):=ea(A_L^{*}-A_{\infty }^{*})$ vs $L$ for lattices with $L=3\iota +1$ and $L=3\iota +2$ ($\iota \in \mathbb{Z}$), as well as fits to $\mathrm{\Delta }\left(ea{A}_L^{*}\right)=\gamma L^{-2x}$ for $L\ge 10$. The fits yield $x\approx 1.07$ and 0.98, respectively.

Figure 2

Bott index of the time-evolving wave function $\left|\mathrm{\Psi }\right(t)\rangle$, and overlap between $\left|\mathrm{\Psi }\right(t)\rangle$ and the ground state of the instantaneous Floquet Hamiltonian $|{\mathrm{\Psi }}_{GS}^F\left(t\right)\rangle$ as a function of $eaA\left(t\right)$ (bottom labels) at stroboscopic times $t$ (top labels). We also show how the lower band of the instantaneous Floquet Hamiltonian is occupied in the time-evolving state. Specifically, we plot the occupation at the $K^{\prime }$ point, the lowest occupation of any $\mathbf{k}$ state in the lower band excluding the $K^{\prime }$ point, as well as the average occupation of the $\mathbf{k}$ states in the lower band. (a) Results for a $150\times 150$ (commensurate) lattice. (b) Results for a $151\times 151$ (incommensurate) lattice. Note that the latter does not contain the $K^{\prime }$ point, so no result is reported for its occupation. In both lattices, the magnitude of the electric field is ramped up linearly from 0 to $eaA=1$ in 1000 driving periods for $\mathrm{\Omega }=7J$ [corresponding to the black dot in Fig. 1].

Figure 3

Berry curvature in the static, Floquet, and time-evolved Fermi seas of finite systems. The discrete $\mathbf{k}$ space is displayed in coordinates of the primitive reciprocal lattice vectors with unit lattice constants. $K$ and $K^{\prime }$ are located at $(k_1,k_2)=\left(2\pi /3,4\pi /3\right)$ and $\left(4\pi /3,2\pi /3\right)$, respectively. (a) Berry curvature of the static, topologically trivial, ground state in a lattice with $L=900$. The signs are positive around $K$ and negative around $K^{\prime }$. (b) Berry curvature of a Floquet, topologically nontrivial, ground state ($eaA=1$ and $\mathrm{\Omega }=7J$) in a lattice with $L=900$. All signs around $K$ and $K^{\prime }$ are positive. (c) Berry curvature about $K^{\prime }$ in a time-evolved commensurate system with $L=150$. Note the accumulation of negative Berry curvature at $K^{\prime }$. The four data points in the center forming a $2\times 2$ square are negative, opposite to all the surrounding points. (d) Berry curvature about $K^{\prime }$ in a time-evolved incommensurate system with $L=151$. All signs around $K^{\prime }$ are positive. In (c) and (d), we show results from the time evolution of an initial topologically trivial Fermi sea after the magnitude of the electric field is ramped up linearly from $eaA=0$ to 1 (with $\mathrm{\Omega }=7J$) in 8000 driving periods. Here the ramp is 8 times slower than that in Fig. 2 to make the Berry curvature about $K^{\prime }$ indistinguishable (in the scale of these plots) between (b) and (d).

Figure 4

Collapse of the final occupation of momentum states near $K^{\prime }$ to the Landau-Zener prediction (straight line). $\beta$ and $V$ are extracted from ${\widehat{H}}_F\left[A\left(t\right)\right]$ before and after the gap closing. In all simulations, the magnitude of the electric field is turned on linearly from $eaA=0$ to 1. We show results for several ramping rates (six), lattice sizes (ten, in which $L$ ranges from 600 to 16384), and values of $\mathbf{k}$ (about ten for each lattice size and ramping rate).