Non-Hermitian Boundary State Engineering in Anomalous Floquet Topological Insulators

TL;DR

This paper demonstrates how non-Hermiticity in Floquet topological insulators allows independent engineering of boundary states, enabling enhanced and directional boundary transport beyond Hermitian constraints.

Contribution

It introduces non-Hermitian boundary state engineering in Floquet topological insulators, breaking Hermitian bulk-boundary constraints and enabling tailored boundary transport.

Findings

Non-Hermitian systems can modify boundary states independently of the bulk.

Boundary transport can be enhanced or made directional through non-Hermiticity.

Experimental relevance shown for photonic waveguide lattices.

Abstract

In Hermitian topological systems, the bulk-boundary correspondence strictly constraints boundary transport to values determined by the topological properties of the bulk. We demonstrate that this constraint can be lifted in non-Hermitian Floquet insulators. Provided that the insulator supports an anomalous topological phase, non-Hermiticity allows us to modify the boundary states independently of the bulk, without sacrificing their topological nature. We explore the ensuing possibilities for a Floquet topological insulator with non-Hermitian time-reversal symmetry, where the helical transport via counterpropagating boundary states can be tailored in ways that overcome the constraints imposed by Hermiticity. Non-Hermitian boundary state engineering specifically enables the enhancement of boundary transport relative to bulk motion, helical transport with a preferred direction, and chiral transport in the same direction on opposite boundaries. We explain the experimental relevance of our findings for the example of photonic waveguide lattices.

Equations

$$H_{\mathsf{B}}=\left(\begin{array}{cc}-\mathrm{i}\gamma & J\\ J& -\mathrm{i}\gamma \end{array}\right),$$

$${\iota }_r=-4{\gamma }_r,{\iota }_b=-4{\gamma }_b.$$

$${\breve{\iota }}_r={\iota }_r/2-2{\breve{\gamma }}_r,{\breve{\iota }}_b={\iota }_b/2-2{\breve{\gamma }}_b.$$

$$({\mathsf{TRS}}^{\ast }):SH(t)S^{-1}$$

$$({\mathsf{TRS}}^t):SH(t)S^{-1}$$

$$({\mathsf{TRS}}^{\ast }):$$

$$({\mathsf{TRS}}^t):$$

$$({\mathsf{TRS}}^{\ast }):$$

$$({\mathsf{TRS}}^t):$$

$$({\mathsf{TRS}}^{\ast }):$$

$${\breve{\iota }}_r^{\bigsqcup }={\breve{\iota }}_b^{\sqcap }=-2{\gamma }_{\ast },{\breve{\iota }}_r^{\sqcap }={\breve{\iota }}_b^{\bigsqcup }=-6{\gamma }_{\ast },$$

$$({\mathsf{TRS}}^t):$$

$$({\mathsf{TRS}}^{\ast ,t}):\mathrm{Re}\{\epsilon (\mathbf{k})\}$$

$$({\mathsf{TRS}}^{\ast }):\mathrm{Im}\{\epsilon (\mathbf{k})\}$$

$$({\mathsf{TRS}}^t):\mathrm{Im}\{\epsilon (\mathbf{k})\}$$

Full text

Topological states of matter have proven to be a research topic where fundamental theoretical insights lead almost inevitably to state-of-the-art practical applications Hasan and Kane (2010); Qi and Zhang (2011); Ozawa et al. (2019). A key feature of topological systems is directional transport via chiral boundary states, which is protected by topological invariants and thus impervious to the imperfections of real-world implementations Klitzing et al. (1980); Thouless et al. (1982); König et al. (2007); Wang et al. (2009); Stützer et al. (2018). In combination with fundamental symmetries Chiu et al. (2016); Schnyder et al. (2008); Roy and Harper (2017); Yao et al. (2017), especially time-reversal symmetry (TRS) in topological insulators Kane and Mele (2005); Fu and Kane (2006), topology even protects bidirectional helical transport via counterpropagating boundary states König et al. (2007); Maczewsky et al. (2018). Such symmetry-protected topological phases emerge in a variety of physical systems Haldane and Raghu (2008); Jotzu et al. (2014); Slobozhanyuk et al. (2016); Ningyuan et al. (2015); Yang et al. (2015); Yan et al. (2015); Lee et al. (2018); Klembt et al. (2018); Höckendorf et al. (2018, 2019), where they give rise to a wide spectrum of experimentally observable phenomena Hsieh et al. (2008); Zhang et al. (2009); Rechtsman et al. (2013); Zeuner et al. (2015); Lustig et al. (2019). The recent discovery of anomalous topological phases in periodically driven (i.e., Floquet) insulators demonstrates the singular relevance of topological concepts also in systems far from equilibrium Kitagawa et al. (2010); Rudner et al. (2013); Nathan and Rudner (2015); Maczewsky et al. (2017); Mukherjee et al. (2017); Höckendorf et al. (2017).

Only very recently, the notion of topological phases has been extended to non-Hermitian systems Gong et al. (2018); Zhou and Lee (2019); Kawabata et al. (2019). The perception of the role of topology in this context is still changing rapidly through theoretical investigation and classification Shen et al. (2018); Yao and Wang (2018); Yao et al. (2018); Kunst et al. (2018); Lee and Thomale (2019); Harari et al. (2018); Rui et al. (2019); Lee et al. (2019); Ghatak and Das (2019); Borgnia et al. (2019); Yoshida et al. (2019) as well as experimental exploration Bandres et al. (2018); Zeuner et al. (2015); Weimann et al. (2016); Helbig et al. (2019); Lin et al. (2019) of non-Hermitian topological phases, which includes investigation of the interplay of non-Hermiticity and Floquet dynamics Zhou and Gong (2018); Zhou and Pan (2019). Intriguingly, topology is expected to protect transport even against damping and dissipation Rui et al. (2019).

In the present work we introduce the topological concept of boundary state engineering (BSE) that combines the specific aspects of non-Hermitian and anomalous Floquet topological phases, and has no counterpart in systems with a Hermitian or static Hamiltonian. The concept underlying BSE is illustrated in Figure 1, where we sketch the spectrum of the Floquet propagator $U\equiv U(T)$, obtained as the solution of the Schrödinger equation $\mathrm{i}\mkern 1.0mu\partial_{t}U(t)=H(t)U(t)$ after one period of a time-periodic Hamiltonian $H(t)=H(t+T)$.

For a Hermitian system, with real Floquet quasienergies $\varepsilon$, the spectrum $\{e^{-\mathrm{i}\varepsilon}\}$ of $U$ lies on the unit circle 111Note that we measure quasienergies normalized to time step unity.. In regular topological phases, as they appear in systems with a static Hamiltonian, any boundary state, viewed as a continuous curve $k\mapsto e^{-\mathrm{i}\varepsilon(k)}$ parametrized by momentum $k$, connects two different bulk bands. Anomalous Floquet topological phases, in contrast, possess boundary states that wind around the unit circle Rudner et al. (2013). Thinking in terms of the quasienergy $\varepsilon$, this possibility results from the periodicity $\varepsilon\mapsto\varepsilon+2\pi$.

In a non-Hermitian system, the spectrum of $U$ can move away from the unit circle. Regular boundary states have to remain attached to the bulk bands, since otherwise the continuous dependence on momentum would be violated. Anomalous boundary states, however, can detach from the bulk bands and thus be manipulated independently. This new freedom is exploited in BSE.

BSE is indeed a topological concept: Since the propagator $U$ is invertible, its spectrum cannot move through the origin, and the winding number is preserved. Therefore, an anomalous boundary state, which winds around the origin, retains this property during non-Hermitian BSE and remains topologically protected.

From our discussion it is evident that BSE requires the combination of non-Hermiticity with anomalous Floquet topological phases. Non-Hermiticity arises naturally in optical settings such as photonic waveguide systems Zeuner et al. (2015), since coupling involves losses due to the bending of the waveguides (see Fig. 2). The idealized Hamiltonian for the symmetric coupling of two lossy waveguides,

[TABLE]

involves a coupling parameter $J$ and damping $\gamma$. The associated propagator $U_{\mathsf{B}}=\exp[-\mathrm{i}H_{\mathsf{B}}]$, over a time step $\delta t\equiv 1$, is an $\mathrm{SU}(2)$ rotation, modified by the attenuation ($\gamma>0$) or amplification ($\gamma<0$) factor $e^{-\gamma}$. At perfect coupling $J=\pi/2$ we have $U_{\mathsf{B}}=-\mathrm{i}{}e^{-\gamma}\hat{\sigma}_{x}$, with the Pauli matrix $\hat{\sigma}_{x}$. Amplitude is swapped between the two coupled sites, but changes as $e^{-\gamma}$.

Regarding the experimental relevance of our theoretical considerations, it is useful to allow for a shift $\sigma(t)\in\mathbb{C}$ of the Hamiltonian, where we map $H(t)\mapsto H(t)+\sigma(t)$, and thus $U\mapsto\Gamma\,U$ with $\Gamma=\exp(-\mathrm{i}\int_{0}^{T}\sigma(t)\mathrm{d}t)$. Through the shift, loss and gain become relative terms, and weak loss can be interpreted as (pseudo-) gain relative to strong loss. The physical content of $H(t)$ or $U$ remains unchanged: Measuring normalized intensities, of the form $I(\mathbf{r})=|\psi(\mathbf{r})|^{2}/\max_{\mathbf{r}^{\prime}}|\psi(\mathbf{r}^{\prime})|^{2}$, the factor $\Gamma$ in $U$ cancels.

Concerning the second aspect of BSE, the anomalous Floquet topological phases, we resort to the idea of a driving protocol Rudner et al. (2013). To critically assess the potential of BSE in complex situations, we will use a driving protocol with fermionic TRS that supports counterpropagating boundary states. This protocol belongs to a class of universal protocols for symmetry-protected topological phases, which are investigated in Ref. Höckendorf et al. (2019).

One period of the protocol concatenates $n=6$ steps, as given in Fig. 3 in a pictorial representation. The protocol takes place on a (finite or infinite) square lattice, which is composed of a “red” and “blue” sublattice. Due to fermionic TRS, the Hamiltonian obeys the relation $SH(t)S^{-1}=H(T-t)^{*}$, with a unitary symmetry operator $S$ that fulfills $SS^{*}=-1$ Schnyder et al. (2008). Here, it is $S=\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}$ if we identify the “red” and “blue” sublattice with the up and down component of a pseudo-spin $\nicefrac{{1}}{{2}}$.

Fermionic TRS is essential for the $\mathbb{Z}_{2}$ phases of topological insulators Kane and Mele (2005). As can be seen in Fig. 3, the driving protocol indeed supports a symmetry-protected $\mathbb{Z}_{2}$ topological phase, with counterpropagating boundary states whose intersection at momentum $k=0$ is protected by Kramers degeneracy. This phase is an anomalous Floquet phase, since the boundary states connect the same bulk band at quasienergies separated by $2\pi$. This phase has been explored experimentally in Ref. Maczewsky et al. (2018).

In general, the driving protocol supports topological phases on a continuous parameter manifold. Here, we consider a minimal parameter set, with the two parameters $J$ (for diagonal couplings in steps $1,3,4,6$) and $J^{\prime}$ (for horizontal couplings in step $2,5$). We use $J=1.5$, $J^{\prime}=0.4$ in all plots. The minus sign $\pm J^{\prime}$ between steps $2,5$ shown in Fig. 3 is required for fermionic TRS, but negative couplings can be replaced by positive couplings to facilitate the experimental implementation Maczewsky et al. (2018).

Since the $\mathbb{Z}_{2}$ phase in Fig. 3 is an anomalous topological phase, it is a candidate for BSE. To understand the possibilities arising in this situation we first address the analytically tractable case of perfect coupling ($J=\pi/2$, $J^{\prime}=0$), before returning to general parameters.

At perfect coupling, the “red” and “blue” sublattice are decoupled (since $J^{\prime}=0$). As depicted in Fig. 4, states in the bulk move in a circular clockwise (counterclockwise) orbit on the “red” (“blue”) sublattice. At a boundary, which in Fig. 4 is oriented horizontally with respect to Fig. 3 and lies at the bottom of the (half-infinite) lattice, states propagate either to the right (red sublattice) or the left (blue sublattice). The propagation direction does not depend on the precise position of the boundary, but is prescribed by the bulk-boundary correspondence. Here, with counterpropagating boundary states, transport is protected by topology and fermionic TRS.

According to these patterns of motion, perfect coupling gives rise to a four-fold degenerate dispersionless bulk band at quasienergy $\varepsilon=0$, and two counterpropagating boundary states with linear dispersion $\varepsilon_{r}(k)=\pi+k$, $\varepsilon_{b}(k)=\pi-k$. As has been seen in Fig. 3, these features survive qualitatively for general parameters.

Non-Hermiticity is now introduced into the driving protocol in the following way. In the bulk, we assume uniform losses (as in the type B configuration in Fig. 2) for the diagonal couplings of the red (losses with $\gamma_{r}$) or blue (losses with $\gamma_{b}$) sublattice in steps $1,3,4,6$. The bulk bands thus acquire imaginary quasienergies

[TABLE]

Here and below, a $\iota$ variable denotes the imaginary part $\iota\equiv\operatorname{Im}\varepsilon$ of a quasienergy. Positive (negative) $\iota$ implies gain (loss) according to $e^{-\mathrm{i}\varepsilon t}=e^{\iota t}\times e^{-\mathrm{i}t\operatorname{Re}\varepsilon}$. For the horizontal couplings in step 2, 5 we may assume identical losses, which can be absorbed into the shift of the Hamiltonian and will not be listed explicitly.

At the boundaries, losses $\breve{\gamma}_{r}$, $\breve{\gamma}_{b}$ occur at the isolated sites that, in the respective step of the protocol, do not couple to other sites (as in the type A configuration in Fig. 2). For example, consider the state starting at a “filled red” site in the central top panel in Fig. 4. Through one period of the protocol, this state moves by two sites in steps $1,3$ (incurring bulk losses $-2\gamma_{r}$), and remains at an isolated red site during steps $4,6$ (incurring boundary losses $-2\breve{\gamma}_{r}$). Working out the respective patterns for all four states depicted in the top row of Fig. 4, we see that the boundary states acquire imaginary quasienergies

[TABLE]

Free choice of the four loss parameters $\gamma_{r},\gamma_{b},\breve{\gamma}_{r},\breve{\gamma}_{b}$ allows for free placement of boundary states relative to the bulk, as we had anticipated with the concept of BSE introduced in Fig. 1. In particular, the boundary states are detached from the bulk bands if $\iota_{r}\neq\breve{\iota}_{r}$ or $\iota_{b}\neq\breve{\iota}_{b}$.

The free assignment of losses $\gamma$ (or imaginary parts $\iota$) is not compatible with TRS. To restore TRS, we can now impose two independent conditions in extension of the Hermitian case, namely

[TABLE]

In comparison to, e.g., Ref. Kawabata et al. (2019), these conditions incorporate the shift of the Hamiltonian through the functions $\xi_{*}(t)=\sigma(t)-\sigma(T-t)^{*}$, $\xi_{t}(t)=\sigma(t)-\sigma(T-t)$. For a constant shift $\sigma(t)\equiv\sigma$, we have $\xi_{*}\equiv\mathrm{i}\operatorname{Im}\sigma$ and $\xi_{t}\equiv 0$.

Similar to the coupling parameters $J,J^{\prime}$, where fermionic TRS requires a minus sign between steps $2,5$, non-Hermitian TRS imposes constraints

[TABLE]

on the loss parameters, which are equivalent to

[TABLE]

Note that $\mathsf{TRS}^{t}$, but not $\mathsf{TRS}^{*}$, implies equal damping of counterpropagating boundary states.

While in a uniform system the parameters $\gamma_{r,b}$ can be assumed to be identical throughout the bulk, it is essential for BSE that the losses $\breve{\gamma}_{r,b}$ may very well depend on the boundary. For the schematic plot in the lower part of Fig. 4, we consider an infinite horizontal strip with different losses at the “top” ($\breve{\gamma}_{r}^{\sqcap}$, $\breve{\gamma}_{b}^{\sqcap}$) and “bottom” ($\breve{\gamma}_{r}^{\sqcup}$, $\breve{\gamma}_{b}^{\sqcup}$) boundary. The central part of this plot, along the $\gamma_{*}$ and $\gamma_{t}$ abscissa, uses a parametrization that results in

[TABLE]

for the bulk and boundary states. $\mathsf{TRS}^{*}$ or $\mathsf{TRS}^{t}$ symmetry is preserved for all values of the respective parameter $\gamma_{*}$ or $\gamma_{t}$, and bulk motion is suppressed in favor of boundary transport for $\gamma_{*,t}>0$. The remaining figures will use this parametrization of the infinite horizontal strip.

General assignment of losses allows for more complex manipulation of boundary states, as schematically depicted along the “general parameters” abscissae. As long as TRS is preserved, the $\iota$ values in Fig. 4 obey the relations of Eq. (6). Without TRS, boundary states can be manipulated freely and independently on each boundary, as shown towards the left axis of the plot.

The schematic plot in Fig. 4 illustrates a few particularly noteworthy features of BSE. First, without TRS, the damping of bulk and boundary states can be chosen entirely freely. Second, even with TRS, the damping of boundary states relative to bulk states can be chosen freely, suppressing or enhancing either boundary transport or bulk motion. Third, the properties of boundary transport are not dictated by the properties of the bulk, as a strict bulk-boundary correspondence would demand. For example at the right axis of the plot, boundary transport occurs predominantly via “red” states, opposite to the predominant motion in the bulk via “blue” states.

In the general situation, away from perfect coupling, both the real and imaginary part of the quasienergies depend on momentum. TRS implies the constraints 222See the supplemental material for a detailed derivation.

[TABLE]

on the quasienergy spectrum $\{\varepsilon(\mathbf{k})\}$ at momentum $\mathbf{k}$, which generalize Eq. (6). Here, Eq. (8b) includes an imaginary shift, which drops out of Eq. (8c).

In Fig. 5, we observe the Kramers-like crossing of $\operatorname{Re}\varepsilon(k)$ according to Eq. (8a). For $\mathsf{TRS}^{t}$ the two boundary states have to cross at the same $\operatorname{Im}\varepsilon(k)$ and are thus truly degenerate, while for $\mathsf{TRS}^{*}$ they are separated by their imaginary part. This difference suggests that in the first case $\mathsf{TRS}^{t}$ is required to protect the boundary states, while in the second case they are robust against breaking of $\mathsf{TRS}^{*}$. Indeed, if we break TRS by adding detunings 333For details on TRS breaking and preserving detunings, see the supplemental material., we observe an avoided crossing in Fig. 5 only for broken $\mathsf{TRS}^{t}$ but not for broken $\mathsf{TRS}^{*}$.

Evidently, when counterpropagating boundary states have been separated via BSE, TRS is no longer required for their protection. However, now a preferred direction of transport exists due to the different damping of the boundary states. Only if the damping is equal, as in the $\mathsf{TRS}^{t}$ case in Fig. 5, true bidirectional transport without a preferred direction can be observed. In this scenario, TRS is still required for the protection of transport.

The new freedom introduced by BSE is thus two-fold: It allows us to modify boundary transport relative to the bulk motion, and to selectively modify transport on different boundaries. Fig. 6 provides a visual demonstration of the potential of such modifications. The point of reference is the Hermitian case in panel (B), with bidirectional helical transport via symmetry-protected counterpropagating boundary states (cf. Fig. 3). When protected by $\mathsf{TRS}^{t}$, the counterpropagating boundary states survive the transition into the non-Hermitian regime, but now BSE allows us to suppress bulk motion in favor of boundary transport (panel (C)). In the $\mathsf{TRS}^{*}$ case, we can additionally suppress either one of the two states on each boundary, which we do in such a way that transport on opposite boundaries takes place in the same (and not opposite) direction (panel (D)). Such modifications are a unique feature of BSE: In a Hermitian system, they are prohibited by the bulk-boundary correspondence, while in a non-Floquet system boundary states remain attached to the bulk bands (cf. Fig. 1) and thus cannot be selectively amplified or suppressed.

To assess the full extent of topological protection in the present situation we have to examine the robustness of transport for imperfect boundaries and under the influence of disorder. In panels (E), (F) in Fig. 6 we use serrated boundaries, and include disorder 444Disorder is both spatial and temporal, the exact form is specified in the supplemental material. The disorder strength amounts to $0.2$, or $\approx 13\%$ of the coupling $J$. that preserves (breaks) TRS in the lower (upper) half of each strip. In panel (E), $\mathsf{TRS}^{t}$ indeed protects the scatter-free bidirectional boundary transport, while breaking $\mathsf{TRS}^{t}$ leads to visible back-scattering. In panel (F), even disorder that breaks $\mathsf{TRS}^{*}$ does not lead to appreciable back-scattering because of the suppression of one of the two boundary states. Effectively, this situation realizes chiral transport with a preferred direction, which, in contrast to helical bidirectional transport, is protected by topology but does no longer require TRS.

In conclusion, BSE opens up new avenues to control topological transport in non-Hermitian Floquet systems. In conjunction with fermionic TRS to protect counterpropagating boundary states, BSE enables the selective enhancement of individual topological transport channels, and thus the effective manipulation of (bi-) directional boundary transport. The potential for applications is immediate, and photonic waveguides, which are intrinsically non-Hermitian systems, are a natural platform to explore this potential. Experiments should also investigate the robustness of the different transport phenomena described here, thereby extending our analysis of the influence of disorder and symmetry breaking. Not least, the surprising new possibilities of BSE highlight the importance of further theoretical research regarding the status of topological invariants and the bulk-boundary correspondence in non-Hermitian Floquet systems.