Competition of first-order and second-order topology on the honeycomb lattice

The first Chern number is the topological invariant which was first discovered in a physical system and linked to an experiment klitzing-80prl494 (also known as the Thouless–Kohmoto–Nightingale–den Nijs invariant thouless-82prl405 ; it characterizes the (anomalous) quantum Hall effect. It is well-established that for quantum Hall systems the Chern number can be measured in standard transport experiments via transverse (Hall) conductance,

A finite Chern number implies broken time-reversal (TR) symmetry (explicitly or spontaneously). For our model (1), TR is explicitly broken by Haldane’s term, i.e., whenever $t^{\prime}\not=0$. A characteristic property of topological phases is the presence of bulk–boundary correspondence. That is, a finite topological invariant causes metallic states, which are topologically protected, to be present at (some of) the edges or surfaces and they must traverse the spectral bulk gap.

Suppose a band is separated from other bands by an energy gap, the Chern number of that band is defined as

where the integral is taken over the first Brillouin zone. $F^{(n)}_{xy}$ is the Berry curvature of the $n$th band,

The Berry connection $\bm{A}^{(n)}(\bm{k})$ of the $n$th band is defined as

where $|n(\bm{k})\rangle$ is the normalised Bloch state of the $n$th band,

Note that the normalised wavefunctions, and therefore the Berry connection is gauge dependent, however the Chern number is not.

The Chern number at a particular filling is then the sum of the Chern numbers of the filled bands,

For instance, for (1) the Chern number at half filling is $C=\sum_{n=1}^{3}c_{n}$.

The integral (12) can be numerically evaluated using the method established in Ref. fukui2005, . We discretize the Brillouin zone into an $N\times N$ grid, defined by reciprocal lattice vectors: $\bm{\mu}_{1}=\bm{b}_{1}/N,\bm{\mu}_{2}=\bm{b}_{2}/N$. At each point in the discretized Brillouin zone $\bm{k}_{\ell}$, the lattice field strength $F_{12}$ is defined as

$F_{12}(\bm{k}_{\ell})$ is purely imaginary, and Im($F_{12}(\bm{k}_{\ell})$) is defined on the principal branch of the logarithm. $U_{i}(\bm{k}_{\ell})$ is the link variable,

where $|n(\bm{k}_{\ell})\rangle$ is the $n$th normalised energy eigenvector at $\bm{k}_{\ell}$. The Chern number of the $n$th band is then

where we sum over the entire discretized Brillouin zone.

By virtue of bulk–boundary correspondence, $C$ can also be read off from ribbon spectra such as those in Fig. 3 (a,c,e). The ribbon spectra are calculated by Fourier transforming the lattice in one direction to momentum space, while keeping the other real space coordinate. The result is a $6N$-dimensional basis system, with one quantum number $k$. For Chern phases with finite Chern number $C$, we expect the difference of right- and left-moving edge modes per edge to be $C$. For all Chern phases considered in this paper, we have checked that the edge mode counting in ribbon geometry matches the calculated Chern numbers. We note that this procedure applies to any other filling.

Phases I-III have a zero Chern number; all other nine phases have a non-zero Chern number, ranging in value from -4 to 5, see Table 1.

We have selected three phases which exemplify the behaviour of the nine Chern phases. Ribbon spectra, spectra in momentum space and Berry curvature for these three phases are the subject of Fig. 3; the corresponding plots for the other six phases are delegated to the Appendix, see Figs. 13 and 14.

Consider the dimer cluster limit $\alpha\rightarrow\infty,\lambda=0$ adiabatically connected to phase I, where the only non-zero bonds are the nearest neighbour bonds, i.e., dimers, between the hexamer. Dimers have a $\mathbb{Z}_{2}$ sublattice symmetry, thus we consider the $\mathbb{Z}_{2}$ Berry phase.

We choose one of the disconnected dimer bonds, and w.l.o.g. label its lattice sites 1 and 2. We may separate the Hamiltonian as

Next we introduce a “twist” on this dimer bond by transforming the operators corresponding to one of the dimer sites,

while leaving the other one unchanged. Thus the Hamiltonian becomes a function of $\theta$:

Since the energy eigenstates $|n(\theta)\rangle$ are often degenerate, we need to compute the trace over the matrix $\langle n(\theta)|\frac{\partial}{\partial\theta}|m(\theta)\rangle$ instead of the standard Berry potential, thus sometimes also referred to as Non-Abelian Berry phase hatsugai2004 ; hirano2008 . The non-Abelian Berry phase accumulated as we ‘twist’ this dimer through $2\pi$ is

where $A(\theta)$ is the non-Abelian Berry connection hirano2008 :

$\Psi(\theta)$ is a multiplet, defined as

i.e., a matrix whose columns are the lowest $M$ normalised energy eigenvectors $|n(\theta)\rangle$,

and $M$ is the number of filled states. For instance, $M=N/2$ for half filling, where $N$ is the number of lattice sites.

If we consider the integration path in Eq. (25) as half of a closed loop from $\theta=0$ to $\theta=2\pi$ and back again, then the condition that the gauge must be regular (single-valued) demands that the Berry loop over the whole closed loop must be equal to an integer multiple of $2\pi$. However, the integral for $\theta$ from $0$ to $2\pi$ is identical to the integral from $2\pi$ back to $0$. The latter integral is the equivalent of the former integral if we were to apply the gauge transformation (22) to the other atom in the dimer $c_{2}$ rather than $c_{1}$. The $\mathbb{Z}_{2}$ symmetry of the dimer forces the two integrals to be identical. Therefore, the Berry phase defined in Eq. (25) must be equal to 0 or $\pi$ modulo $2\pi$, i.e., the $\mathbb{Z}_{2}$ twist Berry phase is quantized hatsugai2011 .

Numerically, we can calculate the non-Abelian Berry phase by breaking up the integration path into $N+1$ discrete values $\theta_{i}$, $\{\theta_{i}=2\pi i/N\,|\,0\leq i\leq N\}$. We can then use a lattice Berry connection so Eqn. (26) at $\theta=\theta_{i}$ becomes

The Berry phase (25) can then be evaluated as

Note that this calculation is gauge-invariant as long as we start and end the integration at the same point, which is the case in our calculation king-smith_vanderbilt_1993 ; hatsugai2004 .

Up to this overall gauge freedom, these Berry phases are well defined and do not change unless the spectral gap, depending on twist angle $\theta$, closes. There are two ways in which the gap may close: (i) the gap closes with no twist and is thus identical to the bulk gap (as it happens for the boundaries of the phases I and II in Fig. 2); (ii) the gap remains open for zero twist but closes for a finite twist angle. In the $\mathbb{Z}_{2}$ twist Berry phase calculation, the latter case only occurred when $\theta=\pi$, where the $\mathbb{Z}_{2}$ twist Berry phase transitions between $\pi$ and 0 away from the phase boundaries through the middle of phase IV (see Fig. 6).

The numerical calculation shown in Fig. 6 has two phases where the $\mathbb{Z}_{2}$ twist Berry phase $\gamma_{2}=\pi$. One coincides with phase I, where the Chern number is zero, so the constant and non-zero $\mathbb{Z}_{2}$ Berry phase characterises the phase. The other phase overlaps some of phase IV, where the Chern number is one. Corner states that sit in the bulk energy gap at $E=0$ can be observed in phase I. Where there is a finite Chern number in the half-filling gap, corner states cannot be discerned from the edge states which are linearly dispersed across the bulk gap. Phases with a finite Chern number and finite $\mathbb{Z}_{Q}$ Berry phase are discussed further in Sec. VI.

The presence of corner states in phase I is dependent on the edge geometry. There are many possible geometries that give corner states, but a requirement is that there is a complete hexamer on a corner that is attached to either one, three, or five other hexamer unit cells mizoguchi2019 ; liu2019 ; zangeneh-nejad2019 ; lee2020 ; li2020 . Consider the limit where inter-hexamer bonding is much stronger than a small (but finite) intra-hexamer bond. Then, all atoms in the hexamer that are connected to a different hexamer in the lattice are involved in forming a strong inter-hexamer dimer, and are practically disconnected from the other atoms in the hexamer. The atoms that are not dimerized in this way then form a free monomer, trimer or pentamer with a zero energy eigenstate. An example geometry is considered in Fig. 7 (which is reminiscent to that shown in Ref. li2020, ) where the 120 degree corner hexamers are connected to three other hexamer unit cells each, so a weakly connected trimer is formed. The inset (b) shows the zero-energy corner states are localised on these corners. Note that isolated trimer (and pentamer) states feature additional energy states different from $E=0$. Where corner hexamers are attached to two other hexamer unit cells, a tetramer is formed, which are the green energy eigenstates in the diagram. These tetramer structures (and dimer structures, like those on the edge of the lattice) do not have zero energy states.

Now consider the decoupled hexamer cluster limit $\alpha=0$. Each individual hexamer has a $\mathbb{Z}_{6}$ symmetry (rotating the labels of the atoms in a hexamer), thus we consider the $\mathbb{Z}_{6}$ Berry phase. The construction is similar to that of the $\mathbb{Z}_{2}$ Berry phase, but instead of considering a dimer in the decoupled dimer limit, we consider a hexamer in the decoupled hexamer limit. W.l.o.g. we choose a hexamer, and split the Hamiltonian $H$ into two parts. $H_{\text{hexamer}}$ contains all bonds between two atoms in the hexamer, and $H-H_{\text{hexamer}}$ contains all other bonds (bonds between hexamer and any atoms not in the hexamer, and bonds entirely not in the hexamer).

We then introduce “twists” onto the hexamer by transforming the operators in $H_{\text{hexamer}}$ as

for $j=1,\ldots,6$ with $\phi_{j}=\Sigma_{i=1}^{j}\theta_{i}$ and $\phi_{6}=0$. The individual $\theta_{i}$ are the “twists” on the nearest neighbour bonds between atoms $i$ and $i+1$ in the hexamer (where $\theta_{6}$ is the twist on the bond between atom 1 and 6). Note $\theta_{6}=-\Sigma_{i=1}^{5}\theta_{i}$, since $\phi_{6}=0$. The transformation also introduces twists on the next nearest neighbour bonds within the hexamer, where the transformation ensures that the net phase accumulated in any closed loop within the hexamer is constant for any amount of “twisting”.

This transformation renders the Hamiltonian a function of $\Theta=(\theta_{1},...,\theta_{5})$. The non-Abelian Berry phase accumulated as we twist the hexamer over a path $L$ in the 5-dimensional parameter space is,

where $A(\Theta)$ is the non-Abelian Berry connection (26).

The path $L$ is any one of the paths $L_{i},\,0\leq i\leq 5$, where $L_{i}$ is defined as

with $E_{0}=E_{6}=(0,0,0,0,0),\,E_{i}=2\pi\hat{e}_{i}$ for $1\leq i\leq 5$, where $\hat{e}_{i}$ is the unit vector in the $i$th direction, and $G=\frac{2\pi}{6}(1,1,1,1,1)$ is the centre of mass of a five dimensional tetrahedron with side lengths of $2\pi$. As implied by the definition of the Berry phase as the integral over any of these paths, these paths are equivalent. This stems from the $\mathbb{Z}_{6}$ symmetry of rotating the labels of the hexamer, which is equivalent to rotating through the $\theta_{i}$.

Since the Berry phase accumulated by taking all 6 paths (which is a closed loop) must be equal to an integer multiple of $2\pi$, and the 6 paths individually are equivalent to each other, then the $\mathbb{Z}_{6}$ Berry phase Eq. (32) must be equal to an integer multiple of $2\pi/6=\pi/3$. The $\mathbb{Z}_{6}$ twist Berry phase is unchanging unless the gap closes, which, similar to the $\mathbb{Z}_{2}$ calculation, happens with either no twist, or when $\Theta=G$, the centre of mass point.

The numerical calculation of the $\mathbb{Z}_{6}$ Berry phase is analogous to the calculation of the $\mathbb{Z}_{2}$ Berry phase, see Fig. 8 (left). We have observed a finite-size dependence of the results. For instance, the $\gamma_{6}=\pi$ phase at $\alpha\approx\frac{1}{0.4},\lambda\approx 0.6$ on the left in Fig. 8, which is calculated on an $18\times 18$ hexamer lattice, is not present when the $\mathbb{Z}_{6}$ Berry phase is calculated on a $17\times 17$ hexamer lattice. The energy gap diagram on the right in Fig. 8 shows that the energy gap is small across this entire phase, which may explain the discrepancy. Similarly, the energy gap is small for the $\gamma_{6}=5\pi/3$ phase next to it, and the size of this phase decreases and increases depending if the size of the lattice is $n\times n$ hexamer unit cells with $n$ odd or even.

There are three phases with $\mathbb{Z}_{6}=4\pi/3$, covering a large part of phase IV (Haldane phase), and phases X and XI. These phases all have finite Chern numbers ($C=1,-1,3$ respectively). Analogous to the $\mathbb{Z}_{2}$ Berry phase, where there was no change in the Berry phase from phase I to the Haldane phase, in the $\mathbb{Z}_{6}$ Berry phase calculation a Berry phase of $\gamma_{6}=\pi$ extends from phase II to the middle of the Haldane phase. This makes a phase with $\gamma_{6}=\pi$ and a Chern number with $C=1$. For further discussion of the interplay of $\mathbb{Z}_{Q}$ Berry phases and Chern numbers, see Sec. VI.

Also note that the gap is closed for any value of $\alpha$ when $\lambda=0$. From Fig. 10 (e), we can also see that the two left top phases both also have a $\mathbb{Z}_{2}$ twist Berry phase of $\pi$.

The lower two phases both have a Chern number of zero, $\mathbb{Z}_{2}$ twist Berry phase of zero, and a $\mathbb{Z}_{6}$ Berry phase of $2\times 2\pi/6=2\pi/3$. This matches the analytic result, which is that the $\mathbb{Z}_{6}$ twist Berry phase is equal to $n\times 2\pi$ in a phase is adiabatically connected to the decoupled hexamer limit, where $n$ is the filling factor araki2020 .

A Chern number of zero and a non-zero $\mathbb{Z}_{6}$ Berry phase lead us to look for isolated corner states in the third-filling gap. The pentamer corner structures which gave corner states at $E=0$ across the whole of half-filling phases II and III also give corner states in the third-filling gap at $\alpha=0$, i.e., in the disconnected hexamer cluster limit. This is not true for other corner structures, i.e., monomers, dimers, trimers, or tetramers. We thus have identified the structure with “pentamer corners” as the one to observe corner states of the HOTI phases at lower fillings.

Spectra in Fig. 12 in the Appendix show that the pentamer corner states persist while the gap remains open. However, it should be noted that corner states can enter a bulk band while the third filling gap is still open, like in Fig. 12 (e) and (g). This is a problem unique to fillings that are not half filling for this system, as a particle-hole symmetry in the Hamiltonian ensures that bands come in positive-negative pairs, and corner states exist at zero energy, so for the corner state to enter the bulk band, the half filling gap must close. A similar behaviour is also seen in the breathing kagome model in a triangular shape ezawa2018-kagome .

As can be seen in Fig. 10 (a), there are 4 unique insulating phases at sixth filling. The two phases at the top of the diagram (for $\alpha>1$) are again both Chern phases. The phase on the top left has a Chern number of -1, as demonstrated in Figs. 5 and 5 (a) and (b), and the phase on the top right has Chern number of +1, see Fig. 3 (c) and (d). Like for third filling, the sixth filling bulk band gap is closed for $\lambda=0$ (not visible in Fig. 10 (a)), so neither of these phases are adiabatically connected to the dimer decoupled cluster limit. Portions of these phases have a $\mathbb{Z}_{2}$ twist Berry phase equal to $\pi$, and where the transition happens within a phases, the twist Hamiltonian has closed at $\theta=\pi$. From Fig. 10 (c), we see that the left top phase also has a $\mathbb{Z}_{6}$ twist Berry phase equal to $\pi$. Note that the portions of the phase with other twist Berry phase values correspond to regions with very small gaps, so are not equal to $\pi$ due to finite size effects. See the discussion for further comments about the interplay between first- and higher-order topology in Sec. VI.

The two lower phases both have a Chern number of zero, see Fig. 5 (b) and (d) for example spectra and Berry curvature calculations for the bottom left phase. As with the two lower phases for third filling, the expected $\mathbb{Z}_{6}$ twist Berry phase of $\frac{1}{6}\times 2\pi=\pi/3$ is seen in the numerical calculations, and persists over the entire phase. Pentamer corner states also are present in the bulk band gap across the phase, as seen in Figs. 9 and 12.

The Chern phases of this model at half filling are notable for their high Chern numbers. Large changes in Chern number can be observed when crossing phase boundaries, with the largest change occurring from phases VIII to IX. These phases have Chern numbers of -4 and 5, respectively, meaning that the Chern number changes by $\Delta C=9$ moving between these phases. The changes in Chern number can be understood by examining the gap closings. Dirac cones are the sources and sinks of the topological invariant bernevig2013 , where the change in Chern number when crossing a phase boundary is usually equal to the number of Dirac cones that form at the gap closing. The change in the Chern number by $\Delta C=9$ occurs at the intersection of two phase boundary lines, with one phase boundary line corresponding to 3 Dirac cones at the three $M$ points, and the other line corresponding to the 2 points symmetrically between each of the 3 unique $M$ and $K$/$K^{\prime}$ points, which gives 6 unique gap closing points on this phase boundary line. When crossing over the intersection of these two phase boundary lines, the two gap closings happen at once, which explains the jump by $\Delta C=9$ of the Chern number.

Aside from the $M$ points and the two points around each of the $M$ points, gap closings can also happen at the $\Gamma$ point, or the $K$ and $K^{\prime}$ points (as mentioned previously). These gap closings represent the creation of 3, 6, 1 and 2 uniquely placed Dirac cones in the Brillouin zone, respectively. All the boundaries between phases fall into one of these categories; for example, towards the left of the phase boundary diagram Fig. 2, the phase boundary between the HOTI phases with Chern number 0 and the Haldane phases with Chern number 1 occurs where the gap closes at $\Gamma$ with one Dirac cone.

There are a few higher-order phase boundary points across the phase boundary diagram. A few of these points are where phase boundary lines cross over each other, i.e., a double phase boundary point, similar to what we discussed above for the case $\Delta C=9$. Another higher-order point is at $\alpha=0,\lambda=1/\sqrt{3}$, where three lines come out of the one point, i.e., a triple phase boundary point. As this point is in the decoupled hexamer limit, the energies are obtained easily through diagonalisation of a $\bm{k}$ independent $6\times 6$ matrix, and the bands are therefore flat. Thus the gap closing occurs everywhere in the Brillouin zone (touching of flat bands), rather than just at specific points.

The corresponding gap closing point does not happen in the opposite limit $\lambda=1/\sqrt{3},\alpha=1/0$ (i.e., $\beta=0$), as the dimers are not decoupled for finite $\lambda$, so the bands are $\bm{k}$ dependent. The point $\alpha=1/0,\lambda=1/\sqrt{3}$ is a gap closing point, however, just at the $K/K^{\prime}$ points. Another higher-order phase boundary point is $\lambda=1/\sqrt{3},\alpha=6$, which is connected to three phase boundary lines (triple phase boundary point). Approaching along $\lambda<1/\sqrt{3}$, the phase boundary line occurs at $M$, and for $\lambda>1/\sqrt{3}$, the phase boundary splits into two – with one phase boundary continuing at $M$, and the other for the two points around $M$ along the $K$ to $K^{\prime}$ line. This higher-order point is the origin of this later line, which is where the zero energy points bifurcate from $M$, moving closer to $K$ with increasing $\lambda$.

The $\mathbb{Z}_{Q}$ Berry phase calculation is straightforward for phases with no Chern number, i.e., the HOTI phases. Without competition from the Chern number, the finite $\mathbb{Z}_{Q}$ Berry phase $=2\pi\nu/Q$, where $\nu=$ filling factor, characterizes a phase adiabatically connected to a decoupled cluster limit, either hexamer or dimer araki2020 . In the absence of dispersive (chiral) edge states, we can engineer incomplete cluster unit cell boundaries to get corner states which remain energetically separated from bulk states within a finite gap, even in lower fillings than half filling.

The story is not so clear for phases with a finite Chern number and $\mathbb{Z}_{Q}$ Berry phase. We have previously identified such phases, see Fig. 8 but also Fig. 10. Not only have we found examples with both finite Chern number and finite Berry phase, but also observed changes of the $\mathbb{Z}_{Q}$ Berry phases within a fully gapped Chern phase. Such changes of the Berry phase happen at gap closings for a finite twist angle (while the bulk gap, i.e., the spectral gap for twist angle $\theta=0$, remains finite). Crucially, none of the Chern phases are adiabatically connected to the decoupled cluster limit. Some of these Chern phases have boundary points that are in a decoupled cluster limit, but not the phases themselves. There are two cases: one is the half-filling Chern phase and $\alpha=0,\lambda=1/\sqrt{3}$, and the sixth and third filling phases with $\alpha=1/0,\lambda=0$. In both these cases, the gap mentioned is closed a that point, so the Chern phases are not adiabatically connected to these phases.

This means that finite $\mathbb{Z}_{Q}$ Berry phase does not characterize a connection to a decoupled cluster limit in a Chern phase. This is not surprising, as the unit cell of a decoupled cluster limit can always be defined in a way that there are no bonds between unit cells. Thus the Bloch matrix is $\bm{k}$-independent and the bands are flat. That implies that phases adiabatically connected to a decoupled cluster limit must have a Chern number of zero. We might also consider that the finite $\mathbb{Z}_{Q}$ Berry phase in the half-filling Chern phases might indicate that lower filling band gaps are still connected to the decoupled cluster limit, however this is not the case, as the extension of the $\mathbb{Z}_{Q}=\pi$ phase in the half-filling phase IV does not overlap with the open gap at lower fillings. It is also meaningless to discuss the presence of in gap corner states in Chern phases, as the bulk-boundary correspondence of the first-order topology requires edge states to traverse the entire bulk gap. Therefore, isolated corner states are not observable.

We are tempted to claim that $\mathbb{Z}_{Q}$ Berry phases in phases which are disconnected from the decoupled cluster limit are not necessarily well-defined; clearly further work is required to clarify this situation. We would like to mention that the $\mathbb{Z}_{Q}$ Berry phase evaluated in Ref. li2020, defined in momentum space is not equivalent to the Berry phase computed and discussed in this work (thus we have not explored it any further here).

Since the $\mathbb{Z}_{Q}$ Berry phase calculation is local invariant (in contrast to the Chern number which is a global quantity), there are also parallels to the way a material responds to local impurities. Following the method described in Refs. slager-15prb, ; diop2020, , we have tested the spectral response to a positive or negative local impurity potential. The prediction slager-15prb ; diop2020 is that a topologically non-trivial phase will show an impurity bound state within the spectral gap, no matter how large the impurity potential might be. In contrast, a trivial phase might feature a fine-tuned in-gap state, but for sufficiently large impurity strength it will have moved into the bulk and not appear within the gap. It was further clarified diop2020 that this characterization does only hold for topological phases which are not caused by a modulation of bond strengths. In particular, higher-order topological insulator phases such as the Kekulé phases do not share this impurity response. Instead, the impurity response can be fully understood by the decoupled cluster limit diop2020 .

In the following, we discuss the impurity response within the Haldane phase for $\alpha\not=1$, i.e., for Haldane phase with finite (anti-) Kekulé distortion. We have tested single site, bond and hexamer impurities. A hexamer impurity consists of six bond impurities sharing the bonds of the same hexagon. Our analysis is consistent with previous findings diop2020 . Everywhere in the Haldane phase, we find one (multiple) in-gap bound state(s) for arbitrary impurity potential strength $V_{0}$ for a site (hexamer) impurity, see Fig. 11 at half filling. In this figure, we have exemplarily shown two parameter points within the Haldane phase; the left panels shows the spectral response for a site impurity, while the right panel the one for a hexamer impurity. The parameters for the left panel of Fig. 11 are chosen such that the lower filling gaps also exhibit a Chern number $C=1$. In contrast, the parameters for the right panel are chosen such that the lower filling gaps possess zero Chern number. As expected, only for a fine-tuned parameter range of $V_{0}$ in-gap bound states are present, but absent for large positive and negative $V_{0}$. In our analysis, we cannot observe any features in the spectral response to local impurities which would explain the finite Berry phases within the Chern phases.