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# Unraveling materials Berry curvature and Chern numbers from real-time evolution of Bloch states

Contributed by Angel Rubio, January 18, 2019 (sent for review October 10, 2018; reviewed by Hideo Aoki, Mei-Yin Chou, and Erio Tosatti)

## Significance

It was established by Thouless, Kohmoto, Nightingale, and den Nijs in 1982 that the topology of the solid-state wavefunctions leads to quantization of transverse electrical conductivity of an insulator. This recognition has led to the development of the new field of topological materials characterized by symmetry-protected quantum numbers. Here, we propose a general and computationally efficient framework enabling one to unveil and predict materials-topological invariants in terms of physical observables, such as the bulk time-dependent current. We show how the quantized charge and spin Hall effect appears even for materials with a non-Abelian Berry phase. This dynamical approach is not necessarily restricted to density functional theory, but can be extended to other schemes and to other methods dealing with correlations explicitly.

## Abstract

Materials can be classified by the topological character of their electronic structure and, in this perspective, global attributes immune to local deformations have been discussed in terms of Berry curvature and Chern numbers. Except for instructional simple models, linear response theories have been ubiquitously used in calculations of topological properties of real materials. Here we propose a completely different and versatile approach to obtain the topological characteristics of materials by calculating physical observables from the real-time evolving Bloch states: The cell-averaged current density reveals the anomalous velocities that lead to the conductivity quantum. Results for prototypical cases are shown, including a spin-frozen valley Hall and a quantum anomalous Hall insulator. The advantage of this method is best illustrated by the example of a quantum spin Hall insulator: The quantized spin Hall conductivity is straightforwardly obtained irrespective of the non-Abelian nature in its Berry curvature. Moreover, the method can be extended to the description of real observables in nonequilibrium states of topological materials.

- time-dependent density functional theory
- Berry curvature
- quantum spin Hall effect
- topological insulator

When the Hamiltonian of a system is subject to adjustable periodic parameters, the eigenstates can acquire a nontrivial gauge-independent phase over the adiabatic evolution in the parameter space (1, 2). In various areas of physics the presence and importance of such geometrical phases have been recognized and characterized through the formulation attributed to Berry (1) and Chern and Simons (3). In particular, the geometrical phase resulting from a variation of the Bloch vector (k) of a periodic Hamiltonian (

In addition to the anomalous charge Hall conductivity, various other transport properties can reflect the effects of the Berry curvature, such as the valley Hall or spin Hall effect and others (7, 9⇓–11). Except for instructionally designed simple models, the calculation of Berry curvature and the Chern number for real materials requisitely involves a perturbative approach within linear response theory, which in most cases requires the Wannierization technique to cope with the fine-grid integral over the BZ (12, 13). In the linear response formulation, a monochromatic field (

This work is motivated by the question of whether the Berry curvature can be extracted from physical observables, such as the total current or others, that can be obtained from the time-evolving states of a solid (

## Theoretical Framework and Computation Method

The computation method consists of the use of a spatially uniform electric field (E field) in the form of a time-dependent vector potential: *SI Appendix*.

Once the time-evolving solid states (*D*-dimensional unit cell (e.g., the area of the 2D unit cell is denoted by **2**. If the parameters for the Hamiltonian, such as the E-field strength, are suitably chosen to make the adiabatic evolution, up to a first-order variation of the wavefunctions, the velocity expectation can be written as (2)

Up to now, most first-principles topological studies of real materials have used the Wannierization method using maximally localized Wannier functions (MLWFs) to compute the Berry curvature and to derive the topological numbers (12). From the standpoint of equality of Thouless et al., the topological nontrivialness of the insulators can be discussed in terms of physical quantity, such as charge or spin Hall conductivity (28, 29). However, such trials have mostly been hampered by the nonintuitiveness of spin Hall current, particularly when the Hamiltonian is furnished with a measurable spin-orbit coupling (SOC) of real materials (16, 17, 29). Here, as the dynamical approach, noted in Eq. **1**, provides a better computational versatility in dealing with experimentally accessible observables, we focus on the direct evaluation of the physical gauge-invariant velocity operator, as given in the left-hand side of Eq. **3**. This method requires only the occupied orbitals and thus avoids the large sum over states of the response formula (13). The required fine sampling of the BZ is here realized by the adiabatic sweep with a fine time step. Furthermore, even when a strong field perturbs the system, expectation values can be evaluated, which enables us to capture the Berry curvature effects in the nonadiabatic or nonequilibrium regime.

## Results and Discussion

### Trivial Atomic Insulator.

To demonstrate the efficiency in obtaining topological characteristics of materials from real-time evolving band states, we first consider the time evolution of a trivial insulator. As a representative example of an atomic insulator, here we devised a Gedanken system by placing a He atom in the simple cubic (SC) lattice with a given fixed lattice constant of *A*, *Inset*. The valence band is derived from the atomic *x* direction in Fig. 1*A*. In this system, simultaneous presence of time reversal and inversion symmetry enforces that *x* direction, *A* in units of *B* up to 200 fs. This model of an atomic insulator by construction possesses a large band gap of *B* demonstrates that these evolving Bloch states (

The time profile of the velocity of the state starting from the Γ point (denoted by the circles in Fig. 1 *A* and *B*) is shown in Fig. 1*C*. It is noteworthy that the calculated transverse velocities always remain at zero (*SI Appendix*. The calculated longitudinal velocity, using the formula given in the left-hand side of Eq. **3**, is indeed consistent with the instantaneous curvature of the band at k: *C*, originates from the periodic nature of the energy band in the reciprocal cell, which can be compared with the Bloch oscillation (30).

### Valley Hall System: Inversion Symmetry-Broken Graphene.

We now investigate a system which possesses a locally nonvanishing Berry curvature, but the band Chern number is absent owing to the presence of time-reversal symmetry. The simplest example in this perspective can be achieved from the spin-frozen bands of the graphene by intentionally breaking its inversion symmetry. To that end, we performed standard DFT+U calculation by adding an asymmetric U potential of *A*, *Upper*). As a result, the Dirac cones in the K and K′ valleys develop a band gap of *A*, *Upper*. The Berry curvature of this artificially inversion-broken graphene band is calculated and presented in Fig. 2*A*, *Lower* (7). Onto the self-consistently converged ground state, we applied a constant and uniform static E field along the *x* direction (*SI Appendix*). The evolutions of three selected k points near the K valley (K and *B*, and the time profiles of the band energies [the valence band maximum (VBM) and the conduction band minimum (CBM)] are summarized in Fig. 2*C*. Note that the state starting from the *C*) arrives at the top of the K valley at *B*.

The time evolution of the longitudinal velocities *D*. The calculated velocities coincide with the instantaneous slope of the band energy dispersion: For example, the states starting from the exact K point have zero initial velocity. It is noteworthy that, after around 70 fs, the velocities of all three states reveal a similar linear trend. After this time, all three states move down the linear surface of the Dirac cone. We also performed the same calculation for the point in the K′ valley, which showed the same longitudinal behavior as that in the K valley: Fig. 2*D*, *Inset* shows the velocity of the state starting from K′ + *δ*, which is almost identical to that starting from K + *δ*. On the other hand, the transverse velocities of the states in the K valley are contrastingly different from those of the states in the K′ valley. Fig. 2*E* shows that the transverse velocities of the states starting from *δ* points have opposite sign. This demonstrates that the spin-frozen bands of the inversion-broken graphene can indeed reveal the valley Hall transport: The carriers in the K valley deflect with an opposite anomalous velocity from that in the K′ valley (7). In Fig. 2 *C* and *D*, we demonstrate that the two states arrive at the top of the valley at around 46 fs, leading to zero-longitudinal velocity. At this point the instantaneous velocities consist only of the anomalous velocity given by *A*, *Lower*) (13).

This example thus suggests that the real-time propagation of the KS equations can indeed be used to evaluate the Berry curvature, Hall conductivity, and consequently the band Chern number, by choosing a reasonable parameter, such as a modestly weak E field. To make the time-evolving states follow well the adiabatic energy surfaces, the strength of the E field needs to be sufficiently weak compared with the band gap (*C*–*E* (*F*, we compare the simulation results with four different strengths of the E field:

### Quantum Anomalous Hall Insulator.

Our method of time propagation can also be directly applied to the QAHI case. Since the first experimental realization of the QAH phase on a magnetized topological insulator (14), various studies ensued to find a new material that can preserve the intrinsic quantum Hall conductivity in an elevated temperature. Several works, in this perspective, have focused on the possible transitions of a 2D hexagonal lattice from the QSH to the QAH (21, 31). One of the most intriguing examples in this direction is the single layer of Sn (32), named stanene, with various hydrogen or halogen coverages: The full coverage of halogens reveals a QSH phase, whose band structures can be described with the Bernevig–Hughes–Zhang (BHZ) model Hamiltonian, while the ideal half coverage on one side results in a QAH phase (31). This half-hydrogenated Sn (HHS) is depicted in Fig. 3*A*. Before the evaluation of the time-evolving state, we calculated the static ground-state band structure and also the Berry curvature using MLWFs, as summarized in Fig. 3*B*. In agreement with the previous study, HHS exhibits a highly localized peak in the Berry curvature (

To evaluate the charge Hall conductivity from the expectation value of the velocity operator, as noted in Eqs. **2** and **3**, we applied a static E field in the *x* direction and performed the time propagation of the uniformly sampled Bloch states. Since the band gap in this case is quite small, to achieve a reasonable adiabatic quality, we chose smaller strengths of the E field which was turned on gradually over a 25-fs period. The transverse current is calculated from the time-evolving Bloch states, and the Hall conductivity is given by*C*, which shows that, after the initial turning-on period (*SI Appendix*.

### Quantum Spin Hall Insulator.

We now extend our example of the time-evolution study to a QSHI. As an example, we calculated the single-layer bismuthane in which the hydrogenated Bi atom locates in the hexagonal lattice (21, 33), as shown in Fig. 4*A*. When the SOC is intentionally turned off, the band structure reveals the typical Dirac cone structure of a honeycomb bipartite lattice, as depicted by the dashed line in Fig. 4*B*. When we include the SOC of Bi atoms, the bands near the Dirac cones open a gap of 0.8 eV at K and K′ points, presented as solid lines (Fig. 4*B*). The optimized geometry of the layer has small buckling but preserves the inversion symmetry, and thus the spinor states in the K and the K′ valley constitute the time-reversal and inversion partners, enforcing the spin-up and spin-down states to be degenerated over the whole BZ (16, 21). Results for a similar system without inversion symmetry are presented in *SI Appendix*. In Fig. 4*B*, the valence bands are labeled from A to D, where the subscript up and down arrows (e.g., *z* direction, but has varying textures depending on the k points as a result of the strong SOC.

We computed the electron dynamics of this QSHI system by gradually ramping the E field over 10 fs toward a static value of *SI Appendix*. Using the obtained spin current, the spin Hall conductivity is evaluated as

Fig. 4*C* depicts the time profile of the spin Hall conductivity and the charge Hall conductivity.

Remarkably, even though *SI Appendix*). On the other hand, the charge Hall conductivity consistently remained at zero, which is a natural outcome of the time-reversal symmetry of the system. As depicted in Fig. 4*C*, *Inset*, the longitudinal current of the insulator is summed to zero, but the Hall current of each spin sector was directed oppositely, as a result producing a finite spin Hall current.

The band-resolved contribution of each doubly degenerate valence band (denoted by A, B, C, and D in Fig. 4*B*) to the spin Hall conductivity is presented in Fig. 4*D*. We observe that the QSH phase of bismuthane is dominated solely by the second valence band (B), while the effects of the others (A, C, and D) are marginal. This can be explained by the structure of the band inversions, which can be inferred from the comparison between the bands with and without SOC. Bands C and D are inverted at K and K′, and the VBM band (A) has two points of band inversion: one with the CBM band at K and K′ and the other with the band B at Γ. This double inversion renders the band A topologically trivial, and the full occupation of the inverted pair of C and D bands makes their topology (the spin Chern number) cancel each other. As a result, the band B, which is doubly degenerate and is solely inverted from the band A at Γ, remains as a single source of the quantized spin Hall state.

Here, we discuss more in depth the structure of the Berry curvature of the QSHI state. The degeneracy in the valence bands requires the Berry curvature to be defined as a matrix constructed in the degenerate subspace, putting in evidence its non-Abelian character (2, 36). The components of the Berry curvature matrix (Ω) and of the spin matrix (**4** and **5**) is advantageous over other methods: The spin Hall current, as presented in Fig. 4*C*, does not require any additional cost or separate treatment due to the degeneracy. Moreover, since the expectation value of an operator is independent of the unitary rotation within the degenerate subspace, the spin current can be written in terms of the velocity given by the diagonalized basis at each k point:**6** is valid regardless of whether the spin is well defined as a good quantum number throughout the whole BZ or varies over k points (component-wise detailed derivation is provided in *SI Appendix*). This suggests that the spin Hall conductivity, as written in Eq. **5**, can be used to identify the spin Chern number, exactly in the same way as introduced in the simplified Kane–Mele model (16, 17), even when a strong SOC demotes the spin from the status of a good quantum number of the system (18, 29). In *SI Appendix*, we present the same results for an inversion symmetry-broken bismuthane, in which the degeneracy is modestly lifted, but the calculated spin Hall conductivity converges well into the quantized value, identifying its QSH phase. This strategy to deal with the 2D quantum spin Hall conductivity can be applied to the time-reversal planes in the BZ of a 3D topological insulator, such as (*SI Appendix*).

## Conclusions and Outlook

In summary, we demonstrated here that the calculated total time-dependent current from the real-time propagation of the Kohn–Sham Bloch states provides a completely alternative method to explore the topological character of solids. Results for exemplary cases were presented, including a trivial atomic insulator, a valley Hall system, and a quantum anomalous Hall system. On a prototypical example of the quantum spin Hall insulator, we discussed that this direct evaluation of the physical observables can serve as a natural platform for an adequate description of the non-Abelian Berry curvature. We highlight that the concept suggested here is not necessarily limited to the DFT-based single-particle scheme. One-body physical observables (such as charge or spin current), derivable from the time-evolving many-body states, can be used to gauge the anomalous behaviors rooted in the geometrical phase structures of the quantum mechanical wavefunctions. Moreover, the flexibility of the proposed computation scheme is clearly advantageous to deal with general time-dependent perturbations (38⇓–40). To illustrate this feature, in *SI Appendix*, we present transiently emerging Hall current in a graphene nanoribbon when a circularly polarized driving force is applied as an externally driven time-reversal breaking mechanism (41, 42).

## Acknowledgments

We acknowledge financial support from the European Research Council (ERC-2015-AdG-694097) and Grupos Consolidados Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU) (IT578-13). The Flatiron Institute is a division of the Simons Foundation. S.A.S. gratefully acknowledges the support from the Alexander von Humboldt Foundation. D.S. and N.P. acknowledge the support from the National Research Foundation of Korea (NRF) through the Basic Research Laboratory (NRF-2017R1A4A1015323) and the Basic Science Research Program (NRF-2016R1D1A1B03931542).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: angel.rubio{at}mpsd.mpg.de or noejung{at}unist.ac.kr.

Author contributions: N.P. and A.R. designed research; D.S., S.A.S., H.H., U.D.G., and J.K. performed research; D.S., S.A.S., and H.H. analyzed data; and D.S., H.H., N.P., and A.R. wrote the paper.

Reviewers: H.A., University of Tokyo; M.-Y.C., Academia Sinica; and E.T., International School for Advanced Studies, CNR-IOM (Istituto Officina dei Materiali - Consiglio Nazionale delle Ricerche) DEMOCRITOS, and International Center for Theoretical Physics.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816904116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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