Topology and localization of a periodically driven Kitaev model

Periodically driven quantum many-body systems support anomalous topological phases of matter, which cannot be realized by static systems. In many cases, these anomalous phases can be many-body localized, which implies that they are stable and do not heat up as a result of the driving. What types of anomalous topological phenomena can be stabilized in driven systems, and in particular, can an anomalous phase exhibiting non-Abelian anyons be stabilized? We address this question using an exactly solvable, stroboscopically driven 2D Kitaev spin model, in which anisotropic exchange couplings are boosted at consecutive time intervals. The model shows a rich phase diagram which contains anomalous topological phases. We characterize these phases using weak and strong scattering-matrix invariants defined for the fermionic degrees of freedom. Of particular importance is an anomalous phase whose zero flux sector exhibits fermionic bands with zero Chern numbers, while a vortex binds a pair of Majorana modes, which as we show support non-Abelian braiding statistics. We further show that upon adding disorder, the zero flux sector of the model becomes localized. However, the model does not remain localized for a finite density of vortices. Hybridization of Majorana modes bound to vortices form “vortex bands,” which delocalize by either forming Chern bands or a thermal metal phase. We conclude that while the model cannot be many-body localized, it may still exhibit long thermalization times, owing to the necessity to create a finite density of vortices for delocalization to occur.

Figure 1

(a) Driving protocol of the 2D Kitaev model. We use a 2D $L\times W$ honeycomb lattice with Bravais vectors ${\vec{a}}_x$ and ${\vec{a}}_y$ and two sublattices labeled as $A$ and $B$. The driving protocol involves consecutive boosts of $J_x,J_y$, and $J_z$. When $J_{s}T/4=\pi /2$ and $J_u=0$ Majorana operators return to their initial positions after two driving periods, $2T$. In a finite system, this will lead to the formation of chiral edge states at the boundaries. (b) The phase diagram of the static Kitaev model consists of three gapped $A$ phases and a gapless $B$ phase. The driving strength $J_s$ acts as additional axis in this diagram.

Figure 2

(Top) Phase diagram mapped by determining gap closings at quasienergies $\varepsilon =0$ and $\varepsilon =\pi$. Each phase is labeled by its strong and weak topological invariants at $\varepsilon =0,\pi ,({\mathcal{W}}_0,{\nu }_{x,0},{\mathcal{W}}_{\pi },{\nu }_{x,\pi })$, computed using the scattering matrix formalism (see Appendix pp1). The invariants are: $(-1,-1,-1,1)$ in phase $A,(-1,-1,0,1)$ in phase $B,(0,-1,-1,1)$ in phase $C,(1,-1,0,1)$ in phase $D,(0,-1,3,1)$ in phase $E$, and $(3,-1,0,1)$ in phase $F$. The pattern of topological phases repeats when increasing $J_s$ due to the periodicity of the system in driving strength, Eq. (17). (Bottom) Band structures of the system in an infinite ribbon geometry (infinite along ${\vec{a}}_x$, with $W=40)$. The color scale denotes the wavefunction amplitude on the bottom 20 lattice sites, meaning that bulk states are plotted in green, whereas modes on the top and bottom boundaries are plotted in red and blue, respectively. From left to right, the panels show the band structure computed for values of $J_s$ and $J_u$ marked by orange symbols in the top panel: $(J_{u}T/4,J_{s}T/4)=(0.225,0)▵,(0,\pi /4)\square ,(0,\pi /2)⬠$, and $(0.225,\pi /2)⬡$.

Figure 3

Dynamics and quasienergy spectrum in the anomalous phase with $J_u=0\text{and}{J}_s=\pi /2$. On the left, we show the evolution of two Majorana operators $c_A,c_B$. During the first three parts of the period, labeled 1 to 3, the Majorana operators are transferred to a neighboring site, as shown by the arrows. After a full period of the time evolution, $c_A,c_B$ are swapped. On the right, a pair of vortices is introduced by flipping the sign of two bonds (marked with thick lines). States on plaquettes with an even number of flipped bonds (blue ellipses) have energies $\varepsilon =\pm \pi /2$. Each vortex is modeled as a plaquette with an odd number of flipped bonds (yellow ellipses), and binds two Floquet-Majorana states, at quasienergies $\varepsilon =0$ and $\pi$.

Figure 4

The disorder term $V_{\mathbf{r}}\left(t\right)$ acts only on the localized orbitals of each plaquette and does not lead to delocalization. States in the flat bands repel in quasienergy depending on the magnitude of $V_{\mathbf{r}}\left(t\right)$ (here denoted by the width of the diagonal lines), but remain localized on their respective plaquettes. Floquet-Majorana states with energies $\varepsilon =0,\pi$ formed at vortices (yellow ellipses) also remain unaffected.

Figure 5

The average two-terminal conductance of a cylindrical system of $L\times L$ unit cells, with periodic boundary conditions along ${\vec{a}}_x$, is plotted as a function of quasienergy. We use $J_{s}T/4=1.45,J_{u}T/4=0.1$, and a disorder strength $\delta VT/4=1.0$. The conductance is peaked close to $\varepsilon =0.6$, but decreases with increasing system size at all values of quasienergy, indicating that bulk states are localized. The inset shows a logarithmic-linear plot of the average conductance versus system size for a fixed value $\varepsilon =0.6$. The solid line is a fit to an exponential decay $aexp(-bL)$, with $a\simeq 0.42$ and $b\simeq 0.08$. Each point is obtained by averaging over 1000 independent realizations of disorder.

Figure 6

Quasienergy spectrum of a $24\times 40$ supercell, which is repeated infinitely many times in the ${\vec{a}}_x$ direction, such that $k_x$ is a good quantum number. We use a high density ${\rho }_v=0.5$ of randomly distributed vortices, obtained by randomly flipping the sign of each bond with a probability $1/2$, setting $J_{u}T/4=-0.12$ and $J_{s}T/4=\pi /2$. Bulk states are shown in gray, and edge modes in blue. Only states belonging to one edge are plotted.

Figure 7

Average conductance as a function of system size at $\varepsilon =0,J_{s}T/4=\pi /2$, and $J_{u}T/4=-0.12$, for an $L\times L$ lattice with periodic boundary conditions in the ${\vec{a}}_x$ direction. Each point is obtained by averaging over 1000 independent vortex configurations at fixed density ${\rho }_v=0.5$. The conductance shows a positive scaling with system size, consistent with a thermal metal phase in which $G\propto lnL$ (solid line).

Figure 8

Setup discussed in Sec. 5b, demonstrating the presence of delocalized states in a region with a finite density of unpaired fluxes. The middle region of a system in the anomalous phase is occupied by a finite density of fluxes, while the outer region is flux free. Depending on microscopic details, the inner region hosts either bands with a nonzero Chern number (in which case a chiral edge state appears at the boundary between the inner and outer regions), or a thermal metal phase.

Figure 9

Schematic representation of the Hamiltonians $H_{x/y,i}\left(k\right)$ of the Floquet-Kitaev model during the four steps of the driving protocol, for a ribbon infinite along $a_x$ (top row) or $a_y$ (bottom row). For both geometries, the unit cell of the ribbon consists of 6 sites, shown as white and black circles $(A$ and $B$ sublattices, respectively). Adjacent unit cells are shown in light gray. During each step of the time-evolution, nearest-neighbor hoppings may take the value $J_u$ (green dashed lines) or $J_u+J_s$ (red solid lines). From the structure of the hoppings it follows that the Hamiltonians of the x ribbon are identical to those of the y ribbon, but differ in their order. Steps 2 and 4 are identical, $H_{x,2}\left(k\right)=H_{y,2}\left(k\right)$ and $H_{x,4}\left(k\right)=H_{y,4}\left(k\right)$, whereas steps 1 and 3 are interchanged, $H_{x,1}\left(k\right)=H_{y,3}\left(k\right)$ and $H_{x,3}\left(k\right)=H_{y,1}\left(k\right)$.

Figure 10

For $J_u=0$ phase diagram is $2\pi$-periodic in $J_{s}T/4$. For $J_{s}T/4=\pi /5+\delta$ spectrum returns to itself as $\delta =2\pi n$. The band structures are computed in an infinite ribbon geometry (infinite along ${\vec{a}}_x$, width $W=40)$ and the color scale indicates the wavefunction amplitude on the bottom 20 lattice sites.