Abstract
Bloch oscillations (BOs) are a fundamental phenomenon by which a wave packet undergoes a periodic motion in a lattice when subjected to a force. Observed in a wide range of synthetic systems, BOs are intrinsically related to geometric and topological properties of the underlying band structure. This has established BOs as a prominent tool for the detection of Berryphase effects, including those described by nonAbelian gauge fields. In this work, we unveil a unique topological effect that manifests in the BOs of higherorder topological insulators through the interplay of nonAbelian Berry curvature and quantized Wilson loops. It is characterized by an oscillating Hall drift synchronized with a topologicallyprotected interband beating and a multiplied Bloch period. We elucidate that the origin of this synchronization mechanism relies on the periodic quantum dynamics of Wannier centers. Our work paves the way to the experimental detection of nonAbelian topological properties through the measurement of Berry phases and centerofmass displacements.
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Introduction
The quest for topological quantization laws has been a central theme in the exploration of topological quantum matter^{1,2}, which originated from the discovery of the quantum Hall effect^{3}. In the last decade, the development of topological materials has led to the observation of fascinating quantized effects, including the halfinteger quantum Hall effect^{4} and the quantization of Faraday and Kerr rotations^{5} in topological insulators^{6,7}, as well as halfinteger thermal Hall conductance in spin liquids^{8} and quantum Hall states^{9}. In parallel, the engineering of synthetic topological systems has allowed for the realization of quantized pumps^{10,11}, and revealed quantized Hall drifts^{12,13,14}, circular dichroism^{15}, and linking numbers^{16}.
In this context, Bloch oscillations (BOs)^{17,18,19,20,21,22} have emerged as a powerful tool for the detection of geometric and topological properties in synthetic lattice systems^{23,24,25,26}, hence providing access to quantized observables. Indeed, transporting a wavepacket across the Brillouin zone (BZ) can be used to explore various geometric features of Bloch bands, including the local Berry curvature^{23} and the Wilson loop of nonAbelian connections^{25}. This strategy has been exploited to extract the Berry phase^{27,28}, the Berry curvature^{29,30}, the Chern number^{12,13,14}, and quantized Wilson loops^{31} in ultracold matter and photonics.
The Wilson loop measurement of ref. ^{31} highlighted a fundamental relation between two intriguing properties of multiband systems: the quantization of Wilson loops, a topological property related to the WilczekZee connection^{32}, and the existence of “multiple Bloch oscillations”, which are characterized by a multiplied Bloch period^{31,33,34,35,36,37,38,39}. The effect investigated in ref. ^{31} was eventually identified as an instance of “topological Bloch oscillations”, whose general framework was proposed in ref. ^{40} based on the space groups of crystals and its implications on the quantization of geometric quantities (Zak phases differences). The Bloch period multiplier appears as a topological invariant, protected by crystalline symmetries, thus making multiple BOs genuinely topological. Furthermore, when a Wannier representation of the bands is possible, Zak phases correspond to the positions of charges within the unit cell, namely the Wannier centers. As a consequence, a ZakWannier duality allows to connect BOs to the relative phases acquired by charges within a classical pointcharge picture. More recently, BOs displaying topologically protected suboscillations have also been found in periodically driven systems in the context of quantum walks^{41}.
In this work, we identify a distinct topological effect that manifests in the BOs of higherorder topological insulators (HOTIs). These newly discovered systems belong to the family of topological crystalline insulators^{42,43,44,45,46}, i.e., gapped quantum systems characterized by crystal symmetries; they are characterized by quantized multipole moments in the bulk and unusual topologically protected states (e.g., corner or hinge modes) on their boundaries; see refs. ^{47,48,49,50,51,52,53,54,55,56,57,58,59,60}. Considering the prototypical BenalcazarBernevigHughes (BBH) model^{47,48}, we unveil a phenomenon by which multiple BOs take the form of an oscillating Hall drift, accompanied with a synchronized interband beating, for special directions of the applied force, as summarized in Fig. 1a. Although the Hall motion is attributed to the finite nonAbelian Berry curvature of the degenerate band structure, the interband beating captured by the Wilson loop is shown to be topologically protected by winding numbers. The synchronization of realspace motion and interband dynamics is elucidated through a quantum Rabi oscillation of Wannier centers. Finally, we observe that detached helical edge states are present on specific boundaries, compatible with the special symmetry axes associated with the topological BOs. A topological transition signaled by the sign change of the identified winding numbers and by the corresponding appearence/disapperance of these states is identified.
Overall, our results demonstrate the rich interplay of nonAbelian gauge structures and winding numbers in the topological BOs of HOTI’s, but also establish BOs as a powerful probe for nonAbelian topological properties in quantum matter.
Results
Model and symmetries
We consider the BBH model, as introduced in refs. ^{47,48}. It consists of a square lattice with alternating hopping amplitudes J_{1} and J_{2} in the two spatial directions and a π flux per plaquette, as depicted in Fig. 1b. We have introduced the flux by Peierls phases on the vertical links but other conventions can be used without affecting the results of this work. The model is represented by a chiralsymmetric Hamiltonian of the form
where the 4 × 4 Dirac matrices are written in the chiral basis Γ^{i} = −σ_{2} ⊗ σ_{i} for i = 1, …, 3 and \({\Gamma }^{4}={\sigma }_{1}\otimes {\mathcal{I}}\). This model has twofold degenerate energy bands E(k) = ±ϵ(k) with \(\epsilon ({\bf{k}})=\sqrt{{\left{\bf{d}}({\bf{k}})\right}^{2}}\). The eigenfunctions of the lowest two bands read \(\left{u}_{{\bf{k}}}^{1}\right\rangle =\frac{1}{\sqrt{2}\epsilon }{\left({d}_{1}i{d}_{2},{d}_{3}i{d}_{4},0,i\epsilon \right)}^{T}\) and \(\left{u}_{{\bf{k}}}^{2}\right\rangle =\frac{1}{\sqrt{2}\epsilon }{\left({d}_{3}i{d}_{4},{d}_{1}+\, i{d}_{2},i\epsilon ,0\right)}^{T}\).
The full expressions of the d_{i}(k)’s for the isotropic BBH model is \({d}_{1}({\bf{k}})=({J}_{1}{J}_{2})\sin ({k}_{y}/2)\), \({d}_{2}({\bf{k}})=({J}_{1}+{J}_{2})\cos ({k}_{y}/2)\), \({d}_{3}({\bf{k}})=({J}_{1}{J}_{2})\sin ({k}_{x}/2)\) and \({d}_{4}({\bf{k}})=({J}_{1}+{J}_{2})\cos ({k}_{x}/2)\) and the corresponding energy dispersion is displayed in Fig. 1c. Here, we take the periodicity d = 2a = 1, where a is the lattice spacing. The ordering of the unit cell sites chosen to represent the model in Eq. (1) is indicated in Fig. 1b. Notice that the chosen basis takes into account the geometric shape of the unit cell. As a consequence, the Hamiltonian is not Bloch invariant, namely H(k + G) ≠ H(k), with G a reciprocal lattice vector.
The presence of timereversal symmetry \(\hat{T}\), with \({\hat{T}}^{2}=1\) and chiral symmetry \(\hat{S}\), represented by \({\Gamma }^{0}={\sigma }_{3}\ \otimes \ {\mathcal{I}}\), sets the model into the BDI class^{1}. Moreover, several crystalline symmetries are also present: two noncommuting mirror symmetries with respect to the x and y axis, namely \({\hat{M}}_{x}={\sigma }_{1}\otimes {\sigma }_{3}\) and \({\hat{M}}_{y}={\sigma }_{1}\otimes {\sigma }_{1}\), respectively; and a π/2 rotation symmetry \({C}_{4}=\left(\begin{array}{ll}0&{\mathcal{I}}\\ i{\sigma }_{2}&0\end{array}\right)\). The two mirror symmetries guarantee that inversion (C_{2}) is also a symmetry of the model, with \({\hat{C}}_{2}={\hat{M}}_{x}{\hat{M}}_{y}\). Moreover, the presence of mirror and rotation symmetries allows us to define a pair of mirror symmetries with respect to the diagonal axes of the lattice, \({\hat{M}}_{xy}={\hat{M}}_{y}{\hat{C}}_{4}\) and \({\hat{M}}_{x\bar{y}}={\hat{M}}_{x}{\hat{C}}_{4}\). This is one of the central ingredients allowing for topologically protected BOs, as we will show below.
Owing to the two noncommuting mirror symmetries \({\hat{M}}_{x}\) and \({\hat{M}}_{y}\), the BBH model is a quadrupole insulator that has a quantized quadrupole moment in the bulk, vanishing bulk polarization and corner charges^{47}. The noncommutation of the mirror symmetries also provides a nonvanishing nonAbelian Berry curvature \({\Omega }_{xy}({\bf{k}})={\partial }_{{k}_{x}}{A}_{y}{\partial }_{{k}_{y}}{A}_{x}i[{A}_{x},{A}_{y}]\) of the twofold degenerate lowest (or highest) bands^{61,62}, where A is the nonAbelian Berry connection^{32}. However, the total Chern number of the degenerate bands remains zero due to timereversal. It is then possible to define Wannier functions \({\nu }_{x,{k}_{y}}^{\alpha }\rangle\) and \({\nu }_{y,{k}_{x}}^{\alpha }\rangle\), with α = 1, 2 numbering the bands below the energy gap, which are eigenstates of the position operators \(\hat{P}\hat{x}\hat{P}\) and \(\hat{P}\hat{y}\hat{P}\) projected onto the lowest two bands, respectively^{47,48}. The noncommutation of the mirror symmetries (and therefore of the projected position operators) forces the use of hybrid Wannier functions, namely Wannier states that can only be maximally localized in one direction^{63,64}. Furthermore, it provides a necessary condition to have gapped Wannier bands, namely Wannier centers that are displaced from each other at every momentum k_{x} or k_{y}. In ref. ^{47,48}, the Wannier gap has been exploited to define a winding of the Wannier states (nested Wilson loop) as a condition to have a quantized quadrupole moment in the bulk, which can be revealed from the WannierStark spectrum^{65}.
Winding numbers
We now prove that a nontrivial topological structure captured by novel winding numbers characterizes the BBH model along the diagonal paths of the BZ, \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), which are shown in Fig. 1d. In order to emphasize the generality of these results, we hereby consider a generic Diraclike model [Eq. (1)] without specifying the components of the d(k) vector. We assume that all previously discussed symmetries are satisfied with the additional constraint that each d_{i} function only depends on one component of the momentum k, namely we assume that d_{1} = d_{1}(k_{y}), d_{2} = d_{2}(k_{y}), d_{3} = d_{3}(k_{x}), d_{4} = d_{4}(k_{x}). Such constraint is satisfied by the BBH model. Mirror symmetries impose that d_{1} and d_{3} are odd functions whereas d_{2} and d_{4} are even. We then find that along \({\mathcal{C}}\) (i.e., for k = k_{x} = k_{y}), the diagonal mirror symmetry represented by the operator \({\hat{M}}_{xy}\) requires d_{3}(k) = d_{1}(k) and d_{4}(k) = d_{2}(k), while along \(\bar{{\mathcal{C}}}\) (i.e., for k = k_{x} = −k_{y}), the symmetry operator \({\hat{M}}_{x\bar{y}}\) requires d_{3}(k) = −d_{1}(k) and d_{4}(k) = d_{2}(k). We then conclude that only two components of d are independent and we therefore define the vector \(\tilde{{\bf{d}}}(k)\equiv ({d}_{1}(k),{d}_{2}(k))\). After writing the Hamiltonian in its chiral representation \(\hat{H}({\bf{k}})=\left(\!\!\begin{array}{ll}0&Q({\bf{k}})\\ Q{({\bf{k}})}^{\dagger}&0\end{array}\!\!\right)\), where \(Q({\bf{k}})={d}_{4}({\bf{k}}){\mathcal{I}}+i{d}_{i}({\bf{k}}){\sigma}^{i}\), we obtain the following result
where ε^{12} = −ε^{21} = 1, and where we used the \(\tilde{{\bf{d}}}\) vector of the BBH model in the last step.
We therefore conclude that the quantities \({w}_{{\mathcal{C}}}\) and \({w}_{\bar{{\mathcal{C}}}}\) count how many times the vector \(\tilde{{\bf{d}}}\) winds over the closed paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), respectively. The quantized windings \({w}_{{\mathcal{C}}}\) and \({w}_{\bar{{\mathcal{C}}}}\) are here protected by the crystalline symmetries \({\hat{M}}_{x}\), \({\hat{M}}_{y}\), and \({\hat{C}}_{4}\), as shown in Supplementary Note 1. These symmetries also imply that the two topological invariants are not independent. We point out that similar winding numbers have been introduced in chiralsymmetric onedimensional topological superconductors^{66}.
Finally, the sign change of the winding numbers at the gap closing point J_{1} = J_{2} signals a phase transition. We will show below that the transition corresponds to the appearance of detached helical edge states. Let us now focus on the BOs of the BBH model and the role played by the quantized winding numbers discussed above.
Topological BOs: bandpopulation dynamics
We consider a wavepacket obtained as a superposition of the lowest two bands and centered at k, which we write as \(\left{u}_{{\bf{k}}}(t)\right\rangle ={\eta }_{1}(t)\left{u}_{{\bf{k}}}^{1}\right\rangle +{\eta }_{2}(t)\left{u}_{{\bf{k}}}^{2}\right\rangle\) with \(\eta ={({\eta }_{1},{\eta }_{2})}^{T}\). Owing to the degeneracy of the states \(\left{u}_{{\bf{k}}}^{1,2}\right\rangle\), other parametrizations can be chosen. As shown in Supplementary Note 2, this gauge ambiguity can be removed by weakly breaking timereversal symmetry, thus splitting the two states in energy. Under an applied homogeneous and constant force F, which makes the crystal momentum change linearly in time, \(\dot{{\bf{k}}}={\bf{F}}\), the bands occupation evolves according to^{67}
where the matrix elements of the Berry connection are defined as \({A}_{i}^{\alpha \beta }=i\langle {u}_{{\bf{k}}}^{\alpha } {\partial }_{{k}_{i}} {u}_{{\bf{k}}}^{\beta }\rangle\). Here, the force is assumed to be weak enough so that transitions to upper bands are neglected.
We can formally solve Eq. (3) as \(\eta (t)=\exp (i\mathop{\int}\nolimits_{0}^{t}{\rm{d}}t\ {\epsilon }_{{\bf{k}}})W\ \eta (0)\). The Wilson line operator W is defined as \(W={\mathcal{T}}\exp (i\mathop{\int}\nolimits_{0}^{t}{\rm{d}}t\ {\bf{F}}\cdot {\bf{A}})={\mathcal{P}}\exp (i\mathop{\int}\nolimits_{{{\bf{k}}}_{i}}^{{{\bf{k}}}_{f}}{\bf{A}}\cdot {\rm{d}}{\bf{k}})\), where we have denoted as k_{i} and k_{f} the initial and final momenta of the BO, respectively. For a closed path \({{\mathcal{C}}}_{0}\) with k_{f} = k_{i} + G, where G is a reciprocal lattice vector, the bands population dynamics is determined by the Wilson loop matrix \({W}_{{{\mathcal{C}}}_{0}}={\mathcal{P}}\exp (i{\int}_{{{\mathcal{C}}}_{0}}{\bf{A}}\cdot {\rm{d}}{\bf{k}})\).
Importantly, the winding numbers \({w}_{{\mathcal{C}}(\bar{{\mathcal{C}}})}\) that we have previously introduced appear in the Wilson loops defined along the diagonal paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), as \({W}_{{\mathcal{C}}(\bar{{\mathcal{C}}})}=\exp (i(2\pi /4){w}_{{\mathcal{C}}(\bar{{\mathcal{C}}})}{\sigma }_{1(3)})\), with \({w}_{{\mathcal{C}}(\bar{{\mathcal{C}}})}=\pm\! 1\). From this, we obtain that BOs require four loops in momentum space in order to map the wavefunction back to itself, namely \({[{W}_{{\mathcal{C}}(\bar{{\mathcal{C}}})}]}^{4}={\mathcal{I}}\). Notice that the degeneracy of the bands brings a trivial dynamical phase that does not influence the internal bandpopulation dynamics.
According to the classification of topological BOs discussed in ref. ^{40}, rotational symmetries \({\hat{C}}_{n}\) can quantize BOs with a force applied orthogonal to the rotational symmetry axis. This is (partially) the case here, with \({\hat{C}}_{4}\) providing periodfour BOs. However, \({\hat{C}}_{4}\) symmetry alone is not sufficient to quantize the BOs. Additional symmetries, namely \({\hat{M}}_{x}\) and \({\hat{M}}_{y}\), are required in order to have a protected winding number along the paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), see Supplementary Note 1. In the general framework presented in ref. ^{40}, a “WannierZak” relation is demonstrated when mirror symmetries commute. As a consequence, the Zak phase winding that appears in the Wilson loop has a onetoone correspondence with the position of the Wannier centers. This is well described by independently evolving point charges within a classical picture. In our case, such a direct correspondence is not possible owing to the nonvanishing Berry curvature and we will see below that the physical consequences of this feature appear on the realspace motion of the wavepacket.
Topological BOs: realspace dynamics
Let us now consider the realspace motion of the wavepacket’s centerof mass, which satisfies the following semiclassical equations^{67}
Here, Ω_{xy} denotes the SU(2) Berry curvature, whose components are shown in Fig. 2a for the BBH model; they satisfy the following conditions: \({\Omega }_{xy}^{11}={\Omega }_{xy}^{22}\), \({\rm{Re}}\ {\Omega }_{xy}^{12}={\rm{Re}}\ {\Omega }_{xy}^{21}\) and \({\rm{Im}}\ {\Omega }_{xy}^{12}={\rm{Im}}\ {\Omega }_{xy}^{21}\). One anticipates from the accumulation of Berry curvature near the M point of the BZ that the paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\) may display nontrivial features also in the realspace dynamics and not only in the band population beating discussed above. As we shall explain in detail below, the wavepacket experiences a transverse Hall drift that changes sign after each BO, thus bringing the centerof mass position back to its initial point after two BOs. This behavior is synchronized and tightly connected with the bandpopulation dynamics captured by the Wilson loop.
The twofold degeneracy of the bands allows us to parametrize the evolving state \(\left{u}_{{\bf{k}}}(t)\right\rangle\) on the Bloch sphere as \({\eta }_{1}(t)=\cos \theta (t)\) and \({\eta }_{2}(t)=\sin \theta (t){e}^{i\phi (t)}\). We can therefore rewrite the anomalous velocity as
On the \({\mathcal{C}}\) path, the angle ϕ is a constant of motion, namely \(\dot{\phi }=0\). This means that the Bloch vector is confined to a meridian of the Bloch sphere. Moreover, since \({\rm{Re}}\ {\Omega }_{xy}^{12}=0\) on \({\mathcal{C}}\), only the first and the last term of Eq. (5) are relevant. After one BO, the two bands populations exchange, symmetrically with respect to the M point, as displayed in Fig. 2b [i.e., \({W}_{{\mathcal{C}}}\propto {\sigma }_{1}\)]. Let us consider the case with ϕ = 0 and θ(0) = 0, where only the first term in Eq. (5) matters. The Berry curvature has a node and the band population starts with η_{1}(0) = 1. Near the M point and before crossing it, the occupation of band 1 is larger than the occupation of band 2. The Berry curvature \({\Omega }_{xy}^{11}\) is positive and the Hall displacement in the x direction is therefore negative (see the minus sign in the first of Eq. (4)). Once the path has crossed the M point, the occupations are flipped but so is also the sign of the Berry curvature, thus the Hall displacement continues with the same sign until the wavepacket reaches the Γ point. As the bands occupations have exchanged, a second BO will experience an opposite Hall drift and bring back the wavepacket to its initial position, as shown in Fig. 2c, d. From this, we obtain that the realspace motion is a witness of the nonAbelian band dynamics. For θ(0) ≠ 0, the offdiagonal component of the Berry curvature also contributes, and it fully suppresses the Hall displacement when θ(0) = π/4 since the anomalous Hall velocity vanishes identically: the two bands are equally populated, no band exchange takes place and therefore the positive and negative deflections compensate each other.
On the \(\bar{{\mathcal{C}}}\) path, the Berry curvature has only offdiagonal components and the Hall dynamics is determined by the relative phase ϕ, whereas θ is a constant of motion. In this case, the populations of the two bands do not exchange over time but the relative phase does by an angle π. The transverse displacement as a function of θ(0) and ϕ(0) for the two paths is shown in Fig. 2e.
We have thus found that the centerof mass of the wavepacket displays a periodtwo BO instead of a periodfour one. This difference with respect to the Wilson loop analysis occurs because after two BOs, the wavefunction has picked up an overall phase 2 × (π/2), which does not appear in observables \(\left\langle \hat{O}\right\rangle\), such as for the centerof mass position.
Atomic limit: singleplaquette dynamics
In order to elucidate the role of Wannier functions in the BOs analyzed here and the synchronization between realspace and bandpopulation dynamics, we consider the instructive atomic limit with J_{2} = 0, where we can study the time dynamics of a single plaquette. The lowest energy eigenstates read \(\left{u}^{1}\right\rangle ={(1/2,1/2,0,1/\sqrt{2})}^{T}\) and \(\left{u}^{2}\right\rangle ={(1/2,1/2,1/\sqrt{2},0)}^{T}\).
We construct the position operator \(\hat{{\bf{r}}}={\sum }_{i}({{\bf{r}}}_{i}{{\bf{r}}}_{0})\left{{\bf{r}}}_{i}\right\rangle \left\langle {{\bf{r}}}_{i}\right\) by setting the spatial origin at the plaquette center. We obtain the matrices \(\hat{x}/a={\rm{diag}}(1/2,1/2,1/2,1/2)\) and \(\hat{y}/a={\rm{diag}}(1/2,1/2,1/2,1/2)\). Let us now call \(\hat{P}\) the projector operator on the states \(\left{u}^{1}\right\rangle\) and \(\left{u}^{2}\right\rangle\), from which we can construct the projected position operators \(\hat{P}\hat{x}\hat{P}\equiv {\hat{x}}_{P}={\sum }_{\alpha ,\beta = 1,2}\left{u}^{\alpha }\right\rangle \left\langle {u}^{\alpha } \hat{x} {u}^{\beta }\right\rangle \left\langle {u}^{\beta }\right\) and \(\hat{P}\hat{y}\hat{P}\equiv {\hat{y}}_{P}={\sum }_{\alpha ,\beta = 1,2}\left{u}^{\alpha }\right\rangle \left\langle {u}^{\alpha } \hat{y} {u}^{\beta }\right\rangle \left\langle {u}^{\beta }\right\). It follows that \([{\hat{x}}_{P},{\hat{y}}_{P}]\,\ne\, 0\), whereas \(\{{\hat{x}}_{P},{\hat{y}}_{P}\}=0\). The eigenfunctions of the projected position operators are the Wannier functions \({\nu }_{x,y}\rangle\) and the corresponding eigenvalues are the Wannier centers^{48,61,68}, which read here ν_{x} = ν_{y} = ±a/4.
In the presence of an external tilt (or electric field), the perturbative Hamiltonian governing the dynamics for small values of the force F reads
The projected Hamiltonian reveals how the external force induces a quantum dynamics between the eigenstates of noncommuting position operators, in the form of a Rabi oscillation. This result is in sharp contrast with the classical pointcharge picture introduced in ref. ^{40}, which is valid for commuting position operators. We can now diagonalize \({\hat{H}}_{F}\) and we find the spectrum \(E=\pm\! Fa/2\sqrt{2}\), with \(F=\sqrt{{F}_{x}^{2}+{F}_{y}^{2}}\) . The Rabi period can be easily obtained as \({T}_{R}=2\pi /(Fa/\sqrt{2})\). This solution is general and it does not depend on the direction of the force. Besides, we can always rotate the coordinate system in order to have one axis parallel to the force and one axis orthogonal to it, r^{∥} and r^{⊥}, and reduce the Hamiltonian to \({\hat{H}}_{F}={F}^{\parallel }\ {\hat{r}}_{P}^{\parallel }\). Then, the corresponding time dynamics can be represented by the eigenstates of \({\hat{r}}_{P}^{\perp }\), namely the Wannier functions obtained by diagonalizing \({\hat{r}}_{P}^{\perp }\). As a consequence, we observe a transverse dynamics compared with the direction of the applied force F, as shown in Fig. 3a, b.
However, the Wannier centers dynamics is not directly connected to the BOs and its period does not have to be the same as the Rabi period of the Wannier centers. For example, let us consider a BO with F_{y} = 0. The periodicity of the BO occurs at the discrete times T_{B} = 2πn/dF_{x}, for \(n\in {{\mathbb{Z}}}_{+}\) where d = 2a. There is no solution that satisfies T_{B} = T_{R}. However, if we take F_{x} =±F_{y} we find that n = 2 provides T_{B} = T_{R}. Therefore, a force oriented along the diagonal axes allows to synchronize the Wannier centers dynamics with the BOs, whereas the other directions yield outofsync oscillations that does not bring the wavepacket back to its initial position at integer multiples of the fundamental Bloch period.
Away from the atomic limit, we can still use Wannier functions as a complete basis to express the wavepacket. A direct calculation (see Fig. 3c) shows that along the paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), the Wannier centers remain gapped and their spectrum flat, namely they are equispaced along the entire path. We interpret this fact as a witness that the Wannier centers can be thought as oscillators with the same oscillation frequency (i.e., displacement), as in the atomic limit represented by Eq. (6), thus keeping the same oscillatory motion while changing the momentum \({k}_{{\mathcal{C}}}\) or \({k}_{\bar{{\mathcal{C}}}}\). In conclusion, Wannier centers perform a quantum periodic dynamics where their transverse motion with respect to the applied force is periodic and synchronized with the BO period.
Edge states
The quantized winding numbers \({w}_{{\mathcal{C}}}\) and \({w}_{\bar{{\mathcal{C}}}}\), which we have previously identified along the paths \({\mathcal{C}}\) and \(\bar{{\mathcal{C}}}\), indicate that a topological transition takes place when J_{1} = J_{2}. Here, we show that an open system with edges along the diagonals x ± y displays detached helical edge states. From Fig. 4a, we notice that near the atomic limit J_{2} → 0, the bulk has gapped states at energies \({E}_{b} \sim \pm\! \sqrt{2}{J}_{1}\). The edge displays disconnected single sites at energy E_{s} ~ 0 and trimers, with energies E_{t1} ~ 0 and \({E}_{t2}=\pm\! \sqrt{2}{J}_{1}\). Thus, a pair of zero energy (E_{s} and E_{t1}) modes exists at the edge.
Near the gap closing point, J_{2} = (1 + m)J_{1} with ∣m∣ ≪ 1, we construct an effective continuum theory^{69} for two (pseudo)spins satisfying
These equations provide two independent zeroenergy solutions
where σ_{2}χ_{η} = ηχ_{η} and η = ±1, which are localized at \(x^{\prime} =0\) and exist only for m > 0, namely when J_{1} < J_{2} (see Methods section). To compute the dispersion relation of the edge modes, it is convenient to consider a cylindrical geometry. In this case, we find that the edge modes become helical, see Fig. 4b. An example of such states is shown in Fig. 4c.
Discussion
In this work, we have shown a new type of multiple BOs that is connected to the quantum beating of Wannier centers and we have identified HOTIs as a model where this effect can be observed. By studying the BBH model, we have shown that the Wilson loop imposes periodfour oscillations and the centerofmass motion displays an anomalous Hall displacement over one period of oscillation. We have connected these features to the crystalline symmetries of the model and we have identified quantized winding numbers that protect the topological BOs. Moreover, we have shown that detached helical edge states emerge in an open system with the required symmetries.
Our results can be observed with cold atoms^{70,71}, where flux engineering can be achieved through timedependent protocols^{72,73} and where the staggered hopping amplitudes requires a bipartite lattice^{27,74}. Interferometric and tomographic methods can be exploited to measure the Wilson loop winding^{25,31,75} and realspace cloud imaging makes possible to measure the centerofmass displacement^{76}. A fundamental question concerns the preparation of the initial state, owing to the degenerate nature of the bands. As shown in Supplementary Note 2 the bands can be split by slightly breaking timereversal symmetry. In this case, it is possible to prepare a nondegenerate BoseEinstein condensate (BEC) at the Γ point. When projected onto the eigenstates of the BBH model, this state is peaked at specific values of θ and ϕ. One can then obtain the desired superposition of the two zeromomentum modes (the BEC and the gapped mode) by a coherent coupling through an external driving. The subsequent BOs require that the applied force has a magnitude that is larger than the band separation to effectively recover the band degeneracy during the BOs.
In the context of photonics, our results can be investigated by using optical waveguides^{77}, where it has been recently possible to realize synthetic π flux^{78,79}. In this platform, the input laser profile can be inprinted in order to map the degenerate manifold of states at the Γ point that are parametrized by the angles θ and ϕ. It is then possible to reconstruct the Wilson loop dynamics by measuring the output field phase profile, whereas the Hall displacement is obtained from the spatial profile of the field intensity.
As a perspective of our work, it would be interesting to generalize our results to other two and threedimensional topological crystalline insulators and consider corrections to the semiclassical equations, e.g., involving the quantum metric once an inhomogeneous electric field or a harmonic trap potential are introduced^{80,81}. Finally, given the role played by the initial state in the observation of the anomalous Hall displacement, BOs can be thought as a tool to witness the phenomenology of symmetrybroken condensates where the ground state degeneracy has been removed by interactions (Di Liberto et al., in prepration).
Methods
Berry connection and curvature
For the BBH model introduced in the main text, the corresponding matrix elements of the nonAbelian Berry connection, defined as \({A}_{i}^{\alpha \beta }=i\left\langle {u}_{{\bf{k}}}^{\alpha } {\partial }_{{k}_{i}} {u}_{{\bf{k}}}^{\beta }\right\rangle\), read
The SU(2) Berry curvature, defined as \({\Omega }_{xy}({\bf{k}})={\partial }_{{k}_{x}}{A}_{y}{\partial }_{{k}_{y}}{A}_{x}i[{A}_{x},{A}_{y}]\), reads
Realspace wavepacket dynamics
To validate the semiclassical realspace dynamics, we have numerically simulated the evolution of a realspace wavepacket using a finite size L_{x} × L_{y} lattice. We start by constructing a Gaussian wavepacket centered at k_{0} = Γ = (0, 0) of the form
where μ indicates the sublattice degree of freedom in the unit cell and r_{μ} is its spatial position. In the simulations we take a grid of k_{pts} × k_{pts} points in k space within an interval k ∈ [−3σ, 3σ] × [−3σ, 3σ]. We evolve the state with the realspace Hamiltonian \(\hat{H}\) and calculate the observable \({{\bf{r}}}^{{\rm{num}}}(t)=\left\langle \psi (t) \hat{{\bf{r}}} \psi (t)\right\rangle\), with \(\hat{{\bf{r}}}\equiv {\sum }_{{\bf{r}}}({\bf{r}}{{\bf{r}}}_{0})\ \left{\bf{r}}\right\rangle \left\langle {\bf{r}}\right\).
We also study the exact evolution of the Bloch wave vector, by considering the velocity operator \(\hat{{\bf{v}}}({\bf{k}})=\partial \hat{H}({\bf{k}})/\partial {\bf{k}}={\sum }_{i}{\partial }_{{\bf{k}}}{d}_{i}({\bf{k}}){\Gamma }^{i}\). At each time t the velocity reads \({\bf{v}}({\bf{k}}(t))=\left\langle \psi ({\bf{k}}(t)) \hat{{\bf{v}}} \psi ({\bf{k}}(t))\right\rangle\), where \(\left\psi ({\bf{k}}(t))\right\rangle ={\mathcal{T}}\exp [i\mathop{\int}\nolimits_{0}^{t}{\rm{d}}t\ \hat{H}({\bf{k}}(t))]\left{u}_{\Gamma }(0)\right\rangle\) and k(t) = k(0) + Ft. This method corresponds to solving the Schrödinger equation in k space. We find the displacement by integration \({{\bf{r}}}^{{\rm{exact}}}(t)=\mathop{\int}\nolimits_{0}^{t}{\rm{d}}t\ {\bf{v}}(t)+{{\bf{r}}}_{0}\).
Edge states
Here we derive the effective theory at the edge by considering periodic boundary conditions along the \(y^{\prime}\) direction (see Fig. 5) for J_{1} ≈ J_{2}. In this stripe geometry, we have to double the unit cell to correctly represent the lattice periodicity, which reads \(d^{\prime} =2\sqrt{2}a\). In chiral form, the Hamiltonian reads
where
Here, we are taking units \(d^{\prime} =1\) and we are also using the convention that the lattice points within the unit cell are all sitting in the center of the unit cell. The Hamiltonian H_{s}(k) is therefore in Bloch form.
In order to build an effective theory near zero energy^{69}, let us take J_{2} = (1 + m)J_{1}, with ∣m∣ ≪ 1. We can then split the Hamiltonian into \({\hat{H}}_{{\rm{s}}}({\bf{k}})={\hat{H}}_{{\rm{s}}}({\bf{k}}=0)+{\hat{V}}_{{\rm{s}}}({\bf{k}})\), where \({\hat{V}}_{{\rm{s}}}({\bf{k}})\) is expanded to lowest order in k. The zeroth order term \({\hat{H}}_{{\rm{s}}}({\bf{k}}=0)\) can be diagonalized and we find four eigenvectors \(\left{v}_{0}^{i}\right\rangle\) with energy \(E=\pm\! \sqrt{2}m{J}_{1}\), which we use as a basis for the effective theory, and four highenergy states \(\left{v}_{e}^{i}\right\rangle\) that we neglect. We can then construct the projection operator \({\hat{P}}_{s}={\sum }_{i}\left{v}_{0}^{i}\right\rangle \left\langle {v}_{0}^{i}\right\) to obtain at lowest order
where we have rearranged the order of the components to have the Hamiltonian in blockdiagonal form and we have defined
in units where J_{1} = 1.
We can now substitute \({k}_{x^{\prime} }\to i{\partial }_{x^{\prime} }\) and use m ≪ 1 to obtain the coupled equations
We use standard procedures to solve these equations, namely we take \(\psi (x^{\prime} )\) as an eigenstate of σ_{2}, i.e., we decompose it as \(\psi (x^{\prime} )=\varphi (x^{\prime} ){\chi }_{\eta }\), where σ_{2}χ_{η} = ηχ_{η} with η = ±1. After taking the ansatz \(\varphi (x^{\prime} )\propto {e}^{tx^{\prime} }\), we find that the following algebraic equations must be satisfied
for \({H}_{\uparrow }({k}_{x^{\prime} })\) and \({H}_{\downarrow }({k}_{x^{\prime} })\), respectively. Let us focus on the solution for \({H}_{\uparrow }({k}_{x^{\prime} })\), namely the one with plus sign. We find \({t}_{\uparrow }=\eta \pm \sqrt{14m}\approx \eta \pm (12m)\). For η = −1, we can construct a solution \(\varphi (x^{\prime} )={c}_{1}{e}^{{t}_{\uparrow }^{+}x^{\prime} }+{c}_{2}{e}^{{t}_{\uparrow }^{}x^{\prime} }\) that is exponentially localized for m > 0 and that vanishes at \(x^{\prime} =0\), namely c_{1} = −c_{2}. The solution constructed for η = 1 does not satisfy these requirements for any value of m. A similar reasoning can be repeated for \({H}_{\downarrow }({k}_{x^{\prime} })\), where we have to take the solution with η = 1 in this case and the solution only exists for m > 0. We end up with the two zeroenergy solutions
that are localized at the edge \(x^{\prime} =0\) and that exist for m > 0, namely for J_{2} > J_{1}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that supports the plots within this paper are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge M. Aidelsburger, W. Benalcazar, F. Grusdt, H. M. Price, G. Salerno for fruitful discussions. This work is supported by the ERC Starting Grant TopoCold, and the Fonds De La Recherche Scientifique (FRSFNRS, Belgium).
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G.P. and M.D. initiated the project. M.D. and G.P. performed the analytical calculations with inputs from N.G. M.D. performed the numerical analysis with inputs from N.G. All the authors discussed the results and prepared the manuscript.
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Di Liberto, M., Goldman, N. & Palumbo, G. NonAbelian Bloch oscillations in higherorder topological insulators. Nat Commun 11, 5942 (2020). https://doi.org/10.1038/s4146702019518x
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DOI: https://doi.org/10.1038/s4146702019518x
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