Floquet Hopf Insulators

TL;DR

This paper predicts a new three-dimensional Floquet topological insulator called the Floquet Hopf insulator, characterized by two topological invariants, leading to unique edge states and expanding the classification of Floquet phases.

Contribution

It introduces the Floquet Hopf insulator with two distinct topological invariants, linking topological defects in time evolution to novel bulk-boundary phenomena.

Findings

Existence of a Floquet Hopf insulator with two invariants

Topologically protected edge modes at 0- or π-quasienergy

Novel bulk-boundary correspondence in Floquet systems

Abstract

We predict the existence of a novel Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a linking number characterizing the (non-driven) Hopf topological insulator. The second invariant is an intrinsically Floquet $\mathbb{Z}_2$ invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system's time-evolution, subject to a process in which defects at different quasienergy exchange even amounts of topological charge. Their contrasting classifications lead to a measurable physical consequence, namely, an unusual bulk-boundary correspondence where gapless edge modes are topologically protected, but may exist at either 0- or $\pi$-quasienergy. Our results represent a phase of matter beyond the conventional classification of Floquet topological insulators.

Equations

$$\begin{array}{r}U(\mathbf{k},t)=e^{i\varphi }\text{outerproduct}zz+e^{i{\varphi }^{\prime }}\text{outerproduct}{z}^{\prime }{z}^{\prime },\end{array}$$

$$H_F(\mathbf{k})=\mathbf{n}(\mathbf{k})\cdot \mathbf{\sigma }$$

$$h_S=\frac{1}{2}\int d^3\mathbf{k}{ϵ}^{ijk}{\mathcal{A}}_i{\mathcal{F}}_{jk},$$

$$z(\mathbf{k},T)\sim\begin{pmatrix}\sin(k_{x})+i\sin(k_{y})\\ \sin(k_{z})+i\big{[}\sum_{\alpha}\cos(k_{\alpha})-m\big{]}\\ \end{pmatrix},$$

$$h_F=\frac{1}{4\pi }\int dt{d}^3\mathbf{k}{ϵ}^{ijkl}{\partial }_i\Delta {\varphi }_m(\mathbf{k},t){\mathcal{A}}_j{\mathcal{F}}_{kl}\text{ mod 2},$$

$$H_{(h_S,h_F)}(\mathbf{k},t)=\{\begin{array}{ll}\frac{2\pi }{T}{H}_{h_S-h_F}(\mathbf{k})& 0\le t<T/2\\ -\frac{\pi }{T}{H}_{h_S}(\mathbf{k})& T/2\le t<T\end{array},$$

$$h_S$$

$$h_F$$

$$U(\mathbf{k},t)=\xi (\mathbf{k},t)\text{mathbbm}1+i\mathbf{n}(\mathbf{k},t)\cdot \mathbf{\sigma },$$

$$U(\mathbf{k},t;\lambda )=R_{\xi n_z}[\lambda \theta (t)]\{U(\mathbf{k},t)\},$$

Full text

Periodically driven systems host a rich variety of phases of matter, many of which cannot be realized by any static Hamiltonian Moessner and Sondhi (2017); Zhang et al. (2017); Choi et al. (2017); Titum et al. (2016); Maczewsky et al. (2017); Khemani et al. (2016); von Keyserlingk and Sondhi (2016); Kolodrubetz et al. (2018); Li et al. (2018). Prime representatives of this are the so-called Floquet topological insulators (FTIs): non-interacting, driven phases of matter, whose physical properties are characterized by a set of underlying quantized topological invariants Kitagawa et al. (2010); Gu et al. (2011); Lindner et al. (2011); Rudner et al. (2013); Rechtsman et al. (2013); Asbóth et al. (2014); Carpentier et al. (2015); Nathan and Rudner (2015); Fruchart (2016); Roy and Harper (2017). Unlike their non-driven counterparts, the topology of FTIs arises directly from the unitary time-evolution, leading to robustly protected gapless edge modes even when the stroboscopic time-evolution is topologically trivial.

A common pattern has emerged in the classification of Floquet topological insulators, which relates their topological invariants to those of static topological insulators with the same dimension and symmetries. A given FTI is found to possess all the invariants of its static counterpart, plus one additional invariant of identical classification. Intuitively, this is understood by extending the bulk-boundary correspondence to Floquet systems: under periodic modulation, the energy – now, quasienergy – becomes defined only modulo $2\pi$ (in units of the driving frequency) and thus an additional and identically classified edge mode emerges, associated with the bulk gap at quasienergy $\pi$.

This result has been established rigorously in systems described by K-theory Roy and Harper (2017), and explored at great length in the context of specific symmetries and dimensionality Kitagawa et al. (2010); Rudner et al. (2013); Asbóth et al. (2014); Carpentier et al. (2015); Nathan and Rudner (2015); Fruchart (2016). Nevertheless, one could wonder whether these arguments leave room for more unique topology in Floquet phases that escape this stringent bulk-boundary correspondence.

In this Letter we answer the above inquiry in the affirmative, demonstrating a three-dimensional Floquet topological insulator characterized by two distinct topological invariants: a ‘static’ $\mathbb{Z}$ invariant, and a uniquely Floquet $\mathbb{Z}_{2}$ invariant. At the heart of our proposal is the Hopf insulator (HI) Moore et al. (2008); Deng et al. (2013, 2014, 2018); Kennedy (2016); Liu et al. (2017); Yuan et al. (2017); Schuster et al. (2019), a 3D topological insulator (TI) in the absence of symmetries, which exists beyond the standard K-theoretic classification Kitaev (2009); Schnyder et al. (2009) via its restriction to two-band systems. The $\mathbb{Z}$ invariant of our system is precisely the Hopf invariant of this insulator. The $\mathbb{Z}_{2}$ invariant replaces the expected additional integer invariant, and characterizes the same topology that underlies the Witten anomaly in (3+1)D $SU(2)$ gauge theories Witten (1982, 1983); Elitzur and Nair (1984); Bär and Campos (2000); Fukui et al. (2008). In our context, it can be understood both as a twisting number extension of the Hopf invariant, as well as in terms of gapless topological defects of the Floquet evolution. These ‘Hopf’ defects may smoothly exchange even amounts of their topological charge, which leads to the reduced $\mathbb{Z}_{2}$ classification. Physically, the difference in invariants creates an atypical bulk-boundary correspondence, where gapless edge modes are topologically protected but may occur at either [math]- or $\pi$-quasienergy, depending on non-universal properties of the boundary.

We are concerned with non-interacting systems governed by a space- and time-periodic Hamiltonian, written in momentum-space as $H(\mathbf{k},t)=H(\mathbf{k},t+T)$, where $H(\mathbf{k},t)$ is a matrix acting on the internal degrees of freedom that form the two bands of the system. Time-evolution is captured by the unitary operator U(\mathbf{k},t)=\mathcal{T}\big{(}e^{-i\int_{0}^{t}H(\mathbf{k},t^{\prime})dt^{\prime}}\big{)},\,0\leq t<T. Much like static insulators, one can view these unitaries in terms of the band-structures composed by their eigenvectors and eigenphases. For a two-band unitary we write

[TABLE]

where $\phi^{(\prime)}(\mathbf{k},t)$, $\ket{z^{(\prime)}(\mathbf{k},t)}$ depend on time as well as momentum, and the quasienergies $\phi^{(\prime)}(\mathbf{k},t)$ are periodic.

Floquet topological insulators are Floquet-Bloch systems where the unitary is gapped at time $T$. The unitary $U(\mathbf{k},T)$ is titled the Floquet unitary, and is equivalently described by the fictitious, time-independent Floquet Hamiltonian, $H_{F}(\mathbf{k})=-i\log(U(\mathbf{k},T))/T$. Similar to static TIs, two FTIs are in the same phase if one can smoothly interpolate between them without closing the gaps of the Floquet unitary. Much recent work has focused on characterizing these systems, revealing anomalous phases that arise from the topology of the full evolution $U(\mathbf{k},t)$, and not just the Floquet Hamiltonian. Nevertheless, some common classifications have emerged. Focusing on Floquet unitaries with two band gaps for simplicity, in all situations with static analogues (i.e. in the absence of explicitly Floquet symmetries, e.g. time-glide symmetry Morimoto et al. (2017)), the FTI is labelled by two topological invariants, each with the same classification as the static TI of the same dimension and symmetries Kitagawa et al. (2010); Rudner et al. (2013); Asbóth et al. (2014); Carpentier et al. (2015); Nathan and Rudner (2015); Fruchart (2016); Roy and Harper (2017); Morimoto et al. (2017). In Ref. Nathan and Rudner (2015) these invariants are associated with band gaps of the Floquet unitary, and correspond to topological defects, or “singularities”, in the Floquet evolution. The defects are found to be classified by the static topological invariant in a range of systems, establishing the above scheme. Ref. Roy and Harper (2017) defines the invariants differently, yet finds a similar classification for all unrestricted-band systems (where classification by K-theory is appropriate).

Here we find a finer distinction in the classification of FTIs with fixed band number. We decompose the unitary evolution into two components: the evolution over a full period, captured by the Floquet unitary $U(\mathbf{k},T)$, and that within a period, captured by the micromotion unitary, U_{m}(\mathbf{k},t)\equiv U(\mathbf{k},t)\,\big{[}U(\mathbf{k},T)\big{]}^{-t/T}. From this decomposition, one sees that the classification factorizes into two potentially distinct invariants: a ‘static’ invariant classifying the Floquet Hamiltonian $H_{F}(\mathbf{k})$, and an intrisically Floquet invariant classifying the micromotion operator $U_{m}(\mathbf{k},t)$ fn (3). In $d$ space dimensions, the former classifies maps from the $d$D Brillouin zone to the set of gapped Hamiltonians, identical to the scheme for static TIs. The Floquet invariant classifies maps from the ($d$+1)D Floquet Brillouin zone, parameterized by $(\mathbf{k},t)$, to $SU(n)$, for an $n$-band system without symmetries Rudner et al. (2013). These invariants are identical in all cases previously considered. However, for systems with fixed band number they may differ.

We now introduce the Floquet Hopf insulator, a three-dimensional Floquet-Bloch system with two bands and no symmetries. The static invariant is the Hopf invariant of the Floquet Hamiltonian, which we briefly review. The gapped two-band Hamiltonian

[TABLE]

maps the 3D Brillouin zone to the Bloch sphere $S^{2}$. Neglecting “weak” lower-dimensional invariants Moore et al. (2008); Kennedy and Guggenheim (2015), such maps are classified by the homotopy group $\pi_{3}(S^{2})=\mathbb{Z}$, thus possessing an integer topological invariant – the Hopf invariant. Expressed in terms of the eigenvectors $\ket{z(\mathbf{k},T)}$ of the Floquet unitary, it takes the form

[TABLE]

where we define the Berry connection $\mathcal{A}_{i}=\frac{-i}{4\pi}(\innerproduct{z}{\partial_{i}z}-\innerproduct{\partial_{i}z}{z})$ and curvature $\mathcal{F}_{jk}=\frac{-i}{4\pi}(\innerproduct{\partial_{j}z}{\partial_{k}z}-\innerproduct{\partial_{k}z}{\partial_{j}z})$, and $\ket{z(\mathbf{k},T)}$ is related to $\mathbf{n}(\mathbf{k})$ on the Bloch sphere by $\mathbf{n}(\mathbf{k})=\bra{z(\mathbf{k},T)}\boldsymbol{\sigma}\ket{z(\mathbf{k},T)}$. The prototypical model HI is defined by

[TABLE]

which gives Hopf invariant $h_{S}=1$ for $1<m<3$, and $h_{S}=0$ for $|m|>3$ Moore et al. (2008); Deng et al. (2013).

The Hopf invariant has an intriguing visual interpretation as a linking number. To elaborate, consider the pre-image of any $\mathbf{n}^{\prime}$ on the Bloch sphere, i.e. the set of all $\mathbf{k}$ that are mapped to $\mathbf{n}^{\prime}$ by $\mathbf{n}(\mathbf{k})$. This is generically a 1D loop in the Brillouin zone. The topology of the Hopf insulator enters when one considers two such pre-images. In the HI phase, any two pre-images are linked, with a linking number equal to the Hopf invariant. Intriguingly, this linking can be equivalently viewed as a twisting of the Jacobian of $\mathbf{n}(\mathbf{k})$ along a single pre-image Pontrjagin (2007); Kennedy (2016) (see Fig. 1).

We now turn to the Floquet invariant. The micromotion operator maps the 4D Floquet Brillouin zone fn (1) to $SU(2)$, isomorphic to the 3-sphere $S^{3}$. Again neglecting weak invariants, this is classified by the group $\pi_{4}(S^{3})=\mathbb{Z}_{2}$: a parity invariant, different from the integer Hopf invariant! This invariant was previously studied as the foundation of the Witten anomaly in $SU(2)$ gauge theories, where a formula for it was introduced Witten (1983). In terms of the micromotion operator’s eigenvectors $\ket{z_{m}(\mathbf{k},t)}$ and their relative eigenphase $\Delta\phi_{m}(\mathbf{k},t)$, we find sup

[TABLE]

where the Berry connection and curvature are defined analogous to those in Eq. (3), now over space-time indices $\{k_{x},k_{y},k_{z},t\}$. The Floquet invariant also relates to the Jacobian twisting along a 1D pre-image, now in (3+1)D (see Fig. 1). The higher dimensionality leads to the reduced $\mathbb{Z}_{2}$ classification Pontrjagin (2007); sup , familiar from the ‘belt trick’ argument in spin-statistics Kauffman (2001).

Combining the two invariants, we conclude that the Floquet Hopf insulator has a $\mathbb{Z}\times\mathbb{Z}_{2}$ classification. A system with arbitrary $(h_{S},h_{F})$ can be generated by strobing two flat band Hamiltonians according to

[TABLE]

where $H_{h}(\mathbf{k})$ has Hopf invariant $h$ and energies $\pm 1$. To verify the static invariant, note that the total effect of the first evolution is a minus sign, $e^{-i\pi H}=-\mathbbm{1}$. The Floquet unitary is therefore given by $U(\mathbf{k},T)=-e^{i\frac{\pi}{2}H_{h_{S}}}$, whose bands correctly have Hopf invariant $h_{S}$. The Floquet invariant is also verified sup : schematically, the contributions of the two halves of the evolution subtract, giving Floquet invariant $h_{S}-(h_{S}-h_{F})=h_{F}\text{ mod }2$.

It is illuminating to discuss how the Floquet Hopf insulator fits in the context of Ref. Nathan and Rudner (2015). Here one again views the evolution in terms of bands [see e.g. Fig. 2(b)], with particular attention to fixed time-slices. If the unitary $U(\mathbf{k},t)$ is gapped at time $t$, one may define an instantaneous static topological invariant $\mathcal{C}(t)$ from its bands, exactly as one defines the static invariant of the Floquet unitary at time $t=T$. This invariant must be constant throughout each gapped region of the evolution, and can only change at times containing gapless points. Such points are topological defects of the evolution, at which the eigenvectors are degenerate and $\mathcal{C}(t)$ is not defined. They possess a defect charge, equal to the total change in $\mathcal{C}(t)$ across the defect. Further, they come in two varieties, [math]-defects and $\pi$-defects, labelled by the quasienergy at which the gap closes (assuming $\phi=-\phi^{\prime}$ without loss of generality). The total charges of the [math]- and $\pi$-defects are locally conserved, and are thereby identified as the topological invariants of the evolution. As an example, take the Floquet Chern insulator Rudner et al. (2013). The instantaneous Chern number changes at gapless Weyl points  Wan et al. (2011), and the integer charges of the [math]- and $\pi$-Weyl points comprise a $\mathbb{Z}\times\mathbb{Z}$ classification Nathan and Rudner (2015).

Like other topological defects, Hopf topological defects possess an integer charge $h_{0/\pi}$ equal to the change in the instantaneous Hopf invariant across the defect. Two types of Hopf defect exist, each depicted in Fig. 2(a). The first occurs a single gapless point with a quadratic energy degeneracy, exemplified by Eq. (4) at $m=1,\,\mathbf{k}=0$. Interestingly, the Hopf invariant may also change across loops of gapless points. These arise e.g. when adding an on-site splitting $\mu\sigma_{z}$ to Eq. (2), and may occur within a single time-slice or over a range of times. They have linear energy degeneracy and feature a Weyl cone Wan et al. (2011) at each point, with the frame of the Weyl cone rotating by $2\pi\Delta h$ about the loop Liu et al. (2017); sup .

How does conservation of the two integer defect charges reconcile with the correct $\mathbb{Z}\times\mathbb{Z}_{2}$ classification? The answer lies in a smooth deformation that exchanges even charge between the [math]- and $\pi$-defects, such that $(h_{0},h_{\pi})\rightarrow(h_{0}-2,h_{\pi}+2)$. This process has no analogue in previously studied FTIs and keeps both band gaps of the Floquet unitary open, establishing the two configurations as being of the same phase. The total charge $h_{0}+h_{\pi}$ is conserved in this process, while the individual charge $h_{\pi}$ is only conserved mod 2. This suggests the identifications

[TABLE]

The former follows from the definition of defect charge: the invariant at $t=T$ equals the sum of all changes to it throughout the evolution. The Floquet invariant is verified in the Supplemental Material sup .

We explicitly describe the above deformation for the specific case of $(h_{0},h_{\pi})=(1,-1)\rightarrow(-1,1)$, finding a continuous family of evolutions $U(\mathbf{k},t;\lambda)$ with defect charges $(1,-1)$ at $\lambda=0$ and $(-1,1)$ at $\lambda=1$ [see Fig. 2(b)]. Note that the static Hopf invariant, $h_{S}$, vanishes in each configuration. Recall that $SU(2)$ is topologically equivalent to the 3-sphere via the parameterization

[TABLE]

with components $(n_{x},n_{y},n_{z},\xi)\!\in\!\mathbb{R}^{4}$ obeying $\xi^{2}+\mathbf{n}(\mathbf{k})^{2}=1$. The deformation acts as a time-dependent rotation of $U(\mathbf{k},t)$ in the $\xi n_{z}$-plane:

[TABLE]

where the rotation angle $\lambda\theta(t)$ interpolates from [math] at $t=0$ to $\lambda\pi$ at times after the earliest defect (thus preserving the boundary condition $U(\mathbf{k},t=0)=\mathbbm{1}$).

To observe that this interpolates between the two configurations without closing the Floquet gap, we examine five regions of the $\lambda=0$ and $\lambda=1$ evolutions, shown in Fig. 2(c). First we consider the early and late gapped regions of the evolution, whose trivial topology allows their images to be contracted to a point. Throughout the rotation they remain points, and so stay gapped with trivial Hopf invariant. Next, we consider the [math]- and $\pi$- defects. Note that $\xi$ is related to the quasienergy as $\xi=\cos(\phi)$, so [math]-defects correspond to the North pole of the 3-sphere ($\xi=1$) and $\pi$-defects to the South ($\xi=-1$). Upon a $\pi$-rotation in the $\xi n_{z}$-plane the two are interchanged, so the defects’ ordering at $\lambda=0$ is reversed at $\lambda=1$.

To verify that the defect charges $h_{0},h_{\pi}$ have changed sign at the end of the deformation, we turn to the middle region, gapped with an instantaneous Hopf invariant $h(T/2)=1$ at $\lambda=0$. Since the times of the [math]- and $\pi$-defects are interchanged by the deformation, this time-slice must have the same invariant $h(T/2)=1$ at $\lambda=1$ in order to reverse the sign of the defect charges. To show that this occurs, note that the effect of the rotation on the initial eigenvectors, described by $\mathbf{n}(\mathbf{k},T/2)$, is to take $n_{z}\rightarrow-n_{z}$ and leave $n_{x},n_{y}$ unchanged. These are topologically equivalent to the ‘flipped’ eigenvectors $-\mathbf{n}=(-n_{x},-n_{y},-n_{z})$, as the two differ by a continuous rotation about the $n_{z}$-axis. The Hopf invariant of $-\mathbf{n}$ is identical to that of the initial eigenvectors $\mathbf{n}$ fn (2), completing our verification of the deformation.

Critically, this relation does not hold for TIs described by K-theory (e.g. the Chern insulator). Here the invariant is additive between bands, and the requirement that a fully filled system have trivial topology implies that the bands’ invariants sum to zero. For two bands, this implies $\mathbf{n}$ and $-\mathbf{n}$ give opposite invariants. The deformation no longer does anything surprising: although it interchanges the location of the [math]- and $\pi$-defects, the invariant of the intermediate region also changes sign, and so the defect charges remain unchanged.

What allows the seemingly-conserved defect charges to change? Recall how defect charge is rigorously defined: one encloses the defect with a surface of gapped points, and computes the static topological invariant of the surface’s eigenvectors Nathan and Rudner (2015). As shown in Fig. 2(d), during the deformation the defects become loops of gapless points. At some value of $\lambda$, the [math]- and $\pi$-defect loops link such that it is impossible to separately enclose each defect, causing the individual defect charges to be undefined. This defect linking arises directly from the linking of the HI sup . After linking, the defects again have well-defined charges, which may differ from their initial values.

We compute the invariants and defect charges numerically in two scenarios (see Fig. 3). Across a phase transition $(h_{S},h_{F})=(1,0)\rightarrow(0,1)$, both the invariants and defect charges change, following Eq. (7). In contrast, along the deformation Eq. (9), the invariants remain robustly quantized while the defects exchange charge sup .

Like its static counterpart Moore et al. (2008); Deng et al. (2013); Schuster et al. (2019), the Floquet Hopf insulator features gapless edge modes at smooth boundaries between phases with different topological invariants fn (4). An unusual situation occurs at boundaries where the static invariant changes, but the defect charge parities do not. Here, a gap closing is protected by the change in invariant, but may occur at either [math]- or $\pi$-quasienergy, depending on details of the edge region. The anomalous $\mathbb{Z}\times\mathbb{Z}_{2}$ classification is precisely what allows this ambiguity: since the defect charges are only defined up to parity, neither quasienergy individually requires a gap closing, despite the change in topological invariant.

We hope our work opens the door to additional topological phases that escape the typical correspondence between driven and non-driven classifications. It would also be interesting to search for other phases characterized by the $\mathbb{Z}_{2}$ twisting invariant, and to further elucidate its physical signatures. Analogues of the Floquet Hopf insulator with a stabilizing crystalline symmetry Liu et al. (2017), and in the many-body localized Ponte et al. (2015); Khemani et al. (2016); von Keyserlingk and Sondhi (2016); Kolodrubetz et al. (2018) or prethermal Abanin et al. (2015); Kuwahara et al. (2016); Else et al. (2017) regimes, where Floquet phases can remain robust to interactions, are open problems.

Acknowledgments: This work was supported by the DARPA DRINQS program, the Packard foundation and NIST. T.S. acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814. SG acknowledges support from the Israel Science Foundation, Grant No. 1686/18.