Signatures of lattice geometry in quantum and topological Hall effect

The band structure for B = 0 (i.e., the zero-field band structure, depicted within the structural Brillouin zone in figure 2(b)) has a maximum at $E=+4\,t$, a minimum at $E=-4\,t$, and two energetically degenerate VHSs at ${E}_{\mathrm{VHS}}=0$ (for $t\gt 0$); the latter appear as one pole in the density of states and are referred to as 'the VHS' in the following.

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For $B\gt 0$, the emerging LLs are symmetrically distributed about the VHS, which implies that for odd q one LL shows up exactly at the VHS (q = 13 in figure 3(a)). On top of this, the LLs exhibit q oscillations. The amplitudes of these oscillations are largest for LLs close to the VHS; on the contrary, LLs close to the band edges appear practically dispersionless. The positions (in reciprocal space) of maxima and minima of every second band coincide.

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The energy-resolved Hall conductivity ${\sigma }_{{xy}}$ is zero at energies below the band bottom $E=-4\,t$ of the zero-field band structure (figure 3(b)). With increasing energy, ${\sigma }_{{xy}}$ decreases in steps of ${\sigma }_{0}\equiv {e}^{2}/h$ at each LL, which is readily explained by their Chern numbers of −1. These steps comply with LLs of free electrons (section 2) and are abrupt because the associated LLs are practically dispersionless.

The sizable oscillation amplitudes of the LLs near the VHS at ${E}_{\mathrm{VHS}}=0$ manifest themselves as modulations in ${\sigma }_{{xy}};$ in other words, the jumps are not abrupt. This is explained by the Berry curvature which is inhomogeneously distributed within the Brillouin zone, in contrast to the Berry curvature of free-electron LLs. Nevertheless, the Chern numbers equal −1.

For odd q, the LL closest to the VHS has a Chern number of $q-1$: this causes a sizable jump and a change of sign in ${\sigma }_{{xy}}$. For even q, the two LLs closest to the VHS touch each other and carry a joint Chern number of $q-2$. At even larger energies, ${\sigma }_{{xy}}$ decreases and reaches zero at the top of the band structure ($E=+4\,t$).

The overall shape of the energy-resolved conductivity is antisymmetric, which reflects the symmetric shape of the zero-field band structure. Briefly summarizing at this point, LLs and Hall conductivity show features that are clearly attributed to lattice properties.

To elaborate on the above findings we assume that the LLs can be separated into free-electron and lattice-influenced ones. LLs of the first type are almost dispersionless, possess an almost constant Berry curvature ${{\rm{\Omega }}}_{0}$, and have Chern numbers of −1. The second type shows up close to E_VHS, with oscillations in both energy and Berry curvature; the positions (in reciprocal space) of their extrema coincide with those of their Berry curvature.

Now, we discuss the Berry curvature distributions in detail. For this purpose we determine the fermion character of the electrons at constant-energy cuts of the zero-field band structure.

At low energies (cut γ in figure 2) the dispersion is almost parabolic and the circular Fermi line encloses occupied states. This electron pocket has positive curvature and is associated with a positive effective mass ${m}^{\star }$. With increasing energy, the dispersion deviates more and more from that of free electrons; the constant energy contours become warped but the Fermi lines remain electronlike.

At E_VHS (cut β) the Fermi line is a square; its vanishing curvature implies an infinite effective mass. Hence, the Lorentz force of an external magnetic field leaves the electronic states unaffected, which explains why the LLs close to E_VHS show oscillations that resemble the zero-field band structure (see figures 2(b) and 4(a)).

At higher energies, the Fermi line becomes circular again but holelike and with negative curvature. The band structure exhibits a Lifshitz transition [42] at E_VHS, which is accompanied by a change of the fermion character: from electronlike below the VHS (with a positive effective mass ${m}^{\star }$) to holelike above the VHS (with negative ${m}^{\star }$).

The Berry curvature of LLs at the band bottom and at the top of the bands is almost homogeneous, like those of free-electron LLs. In contrast, LLs close to the VHS exhibit an inhomogeneous Berry curvature. ${{\rm{\Omega }}}^{(z)}({\boldsymbol{k}})-{{\rm{\Omega }}}_{0}$ shows extrema at the band extrema; for $E\lt 0$ (electron pockets), it is negative at the band maxima and positive at the minima. For $E\gt 0$ (hole pockets), this behavior is reversed.

Now, we explain the large Chern number of the LL close to the VHS for odd q. The Berry curvature [43]

$$\begin{eqnarray*}&&{{\boldsymbol{\Omega }}}_{n}({\boldsymbol{k}})={\rm{i}}\displaystyle \sum _{m\ne n}\displaystyle \frac{\langle {u}_{n}({\boldsymbol{k}})| {{\rm{\nabla }}}_{{\boldsymbol{k}}}H({\boldsymbol{k}})| {u}_{m}({\boldsymbol{k}})\rangle \times (n\leftrightarrow m)}{{[{E}_{n}({\boldsymbol{k}})-{E}_{m}({\boldsymbol{k}})]}^{2}}\end{eqnarray*}$$

of band n is dominated by contributions from the adjacent bands. The maxima of band n coincide with the minima of the adjacent band above, its minima coincide with the maxima of the adjacent band below. Viewing these avoided crossings as split Dirac points suggests to describe each avoided crossing by a two-band Hamiltonian

$$\begin{eqnarray*}&&H={\hslash }\omega (-{k}_{x}{\sigma }_{x}+{k}_{y}{\sigma }_{y})+{m}^{\star }{\sigma }_{z},\end{eqnarray*}$$

in which $({k}_{x},{k}_{y})$ is taken relative to the ${\boldsymbol{k}}$ of the respective extremum. The nonzero effective mass ${m}^{\star }$ lifts the linear band crossing at ${\boldsymbol{k}}=0$. Its sign determines the sign of the Berry curvature and, consequently, that of the Chern number $C=\pm \mathrm{sgn}({m}^{\star })/2$ of the avoided crossing. Since the avoided crossings appear in even numbers, the (total) Chern number of the LL is integer.

As argued before, the sign of ${m}^{\star }$ corresponds to the fermion character. Therefore, the fermion character defines the sign of Berry curvature (minus ${{\rm{\Omega }}}_{0}$). This argument fits to our numerical findings: below (above) the VHS, i.e., in the electron (hole) regime with ${m}^{\star }\gt 0$ (${m}^{\star }\lt 0$), energy maxima coincide with minima (maxima) of the Berry curvature. As a result, the Berry curvature contributions of the maxima and minima of the LL oscillations cancel out and the Chern number of −1 is that of free-electron LLs.

The above reasoning does not hold for the LL near the VHS because its Berry curvature is dictated by states below and above the VHS. Therefore, the dispersion minima are electronlike, which leads to a maximum of the Berry curvature. The maxima are holelike and, thus, also coincide with maxima of the Berry curvature (see panels b and c of figure 4). In total, the Berry curvature of this particular LL is positive throughout the BZ. Considering the two-band model for this LL, each of the q minima and q maxima induces a Chern number of $+1/2$. The total Chern number of the LL is thus C = q, from which the Chern number of −1 due to the background ${{\rm{\Omega }}}_{0}$ (free electrons) has to be subtracted.

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In summary, we obtain an outstanding Chern number of $C=q-1$ for the LL at the VHS. A similar reasoning for even q results in $C=q-2$ for the LL pair close to the VHS. All other LLs carry the Chern number −1 of free-electron LLs because their Berry curvature is dictated by states with the same fermion character, with the consequence that minima and maxima contributions due to band oscillations cancel out (see the top and the bottom LL in figure 4(c)).

The above line of argument lends itself to formulate an approximation for deriving Hall conductivities. This rule of thumb only requires knowledge of the zero-field band structure (B = 0).

If a Fermi line encloses an area $\zeta =(j+1/2)\,{\zeta }_{0}$ (j integer, ${\zeta }_{0}=F/q,F$ area of the Brillouin zone) irrespective of the fermion character, a dispersionless LL is formed at the respective energy (figure 5(a)), according to Onsager's quantization scheme. All LLs carry Chern numbers of −1; an exception are the LLs close to the VHS which carry a large Chern number. Recall that at the VHS the Hall conductivity changes sign. This rough picture yields quite a detailed energy dependence of the Hall conductivity. Taking as an example a square lattice, the approximated conductivity (opaque in figure 5(b)) matches the numerically computed one (see figure 3(b)).

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The semiclassical expression [44]

$$\begin{eqnarray}&&{\sigma }_{{xy}}={\sigma }_{0}\left(\displaystyle \frac{{\zeta }_{{\rm{h}}}}{{\zeta }_{0}}-\displaystyle \frac{{\zeta }_{{\rm{e}}}}{{\zeta }_{0}}\right)\end{eqnarray} \tag{ 7 }$$

relates the Hall conductivity with the number of enclosed states and their fermion character (bright in figure 5(b); ${\zeta }_{{\rm{e}}}$ for electrons, ${\zeta }_{{\rm{h}}}$ for holes); it reproduces the overall shape well but lacks quantization [45].

The above construction works well for free-electron LLs. However, it appears questionable for LLs close to the VHS because the Fermi lines are not closed (confer the Lifshitz transition at the constant-energy cut β in figure 2). Anyway, the rule describes the jump at the VHS if the fermion character is taken into account for the enclosed area ζ (taken negative for electrons and positive for holes). This corresponds to a shift of +q at the VHS. Other lattice-induced features are not taken into account, for example the dispersion of the LLs. Still, the proposed approximation estimates well the overall shape of the Hall conductivity.

We now address the effect of the VHS on the topological edge states (TESs). The winding number of a band gap, equation (2), tells how many TESs bridge this gap.

Starting from the band bottom for odd q (q = 13 in figure 6(a)), the Chern number of −1 for each LL decreases the winding number by 1. Consequently, the number of TESs propagating to the left (with negative velocity) increases by 1 per band gap. At the VHS, the TESs cling to the oscillating LLs. The large Chern number in this region 'compensates' all left-propagating TESs and creates $(q-1)/2$ right-propagating TESs (with positive velocity). Approaching the top of the band structure, the number of TESs decreases until it reaches zero.

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The same holds for even q (q = 14 in figure 6(b)), with the exception that the winding number of the band gap at the VHS is zero. Hence, there are either no TESs at all or there are $q/2$ left- and $q/2$ right-propagating TESs. The latter is the case here: edge states from the bottom penetrate the lower band of the pair at E_VHS and edge states from above penetrate the upper band of the pair; then they cling to the other band. The Hall conductivity at E_VHS vanishes although there are edge states.

In the definition of the flux $p/q={\rm{\Phi }}/{{\rm{\Phi }}}_{0},q$ defines the number of atoms in the magnetic unit cell and, therefore, fixes the number of LLs. While we focus on p = 1 in this Paper, a few remarks on the case $p\gt 1$ will contribute to the discussion.

For $p\gt 1$, one observes the formation of LL groups ($p/q=3/16$ in figure 7(b)), which is not described by Onsager's original quantization scheme. In an extended scheme $p/q$ is expressed as a continued fraction [26, 48, 49]

$$\begin{eqnarray*}&&\displaystyle \frac{p}{q}=\displaystyle \frac{1}{{f}_{1}+\tfrac{1}{{f}_{2}+\tfrac{1}{{f}_{3}+\cdots }}},\end{eqnarray*}$$

which establishes a hierarchy of LL groups. f₁ is the number of LL groups of order 1, while the number of groups of higher order can be calculated from the f_i [49]. With an unterminated continued fraction even irrational values of p/q can be calculated.

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In our example

$$\begin{eqnarray*}&&\displaystyle \frac{p}{q}=\displaystyle \frac{3}{16}=\displaystyle \frac{1}{5+\tfrac{1}{3}},\end{eqnarray*}$$

${f}_{1}=5$ tells that the LLs are arranged into 5 groups of first order (figure 7). These LL groups are related to those of the case $p/q=1/5$ (reproduced in panel a). These bands split up in $\{3,3,4,3,3\}$ LLs (of second order). There are no LLs of higher order because the continued fraction is terminated.

The Chern numbers of the first-order LL groups concur with Onsager's quantization scheme [30, 49, 50]; the column $\sum {C}_{n}$ in figure 7(b) can be produced from the approximation given in section 3.1.3. These group Chern numbers are the sums of the Chern numbers of the individual LLs; see the column C_n. Besides explicit calculation, the latter can be obtained from the Diophantine equation [30, 49, 51].

The hierarchy of LLs and their Chern numbers dictate the Hall conductivity; the first-order LL groups are dominating the overall behavior (see opaque ($p/q=3/16$) and transparent ($p/q=1/5$) curves in figure 7(c)).

Consider a two-dimensional lattice which either is subject to a homogeneous magnetic field or hosts a crystal of skyrmions with topological charge N_Sk. Its unit cell comprises n atoms. n determines how the initial b bands of the zero-field band structure are quantized. The zero-field band structure hosts v van Hove singularities at energies ${E}_{\mathrm{VHS}}^{(i)},i\,=\,1,\,\ldots ,\,v$, given by the symmetry of the structural lattice.

The approximation of the conductivity ${\sigma }_{{xy}}(E)$ proceeds as follows.

• 1.  
  The zero-field band(s) is (are) quantized in accordance with Onsager's quantization prescription. The resulting LLs are assumed dispersionless. Their homogeneously distributed Berry curvature yields a Chern number of $-{N}_{\mathrm{Sk}}$ ($+{N}_{\mathrm{Sk}}$) per band in the lower (upper) band block for the THE in a SkX. For the QHE with a uniform field one sets ${N}_{\mathrm{Sk}}=\mathrm{sign}(B)$ and treats only the lower block.
• 2.  
  The conductivity is shifted at ${E}_{\mathrm{VHS}}^{(i)}$ by $\pm {N}_{\mathrm{Sk}}{\sigma }_{0}n/v$ for the lower and upper block, respectively, to account for the lattice influence (${\sigma }_{0}={e}^{2}/h$). For the QHE the shift is $\mathrm{sign}(B){\sigma }_{0}n/v$.

This procedure yields the conductivities

$$\begin{eqnarray}&&{\sigma }_{{xy}}(E)=\mp \displaystyle \frac{{e}^{2}}{h}{N}_{\mathrm{Sk}}{\left\lfloor n{\int }_{-\infty }^{E}D(E^{\prime} )-\displaystyle \frac{1}{v}\displaystyle \sum _{i}^{v}\theta (E^{\prime} -{E}_{\mathrm{VHS}}^{(i)}){\rm{d}}E^{\prime} \right\rfloor}_{o(E)}\end{eqnarray} \tag{ 9 }$$

for the lower (−) and the upper (+) band block. D(E) is the normalized density of states, $\int D(E^{\prime} )\,{\rm{d}}E\,^{\prime} =\,b$, and θ is the Heaviside function. o(E) counts the number of pockets at E and redefines the conventional floor function: $\lfloor x{\rfloor }_{o}$ rounds down in steps of o, while the conventional $\lfloor x\rfloor$ gives the next lower integer of x. Thus, $\lfloor x{\rfloor }_{o(E)}$ accounts for Onsager's quantization scheme, in which the winding number at the next lower E_VHS determines the offset of the integer quotient. An offset of $o/2$ has to be included to account for the 1/2 in the Onsager scheme (LL formation if $\zeta =(j+1/2)\,{\zeta }_{0}$) per electron- or hole-pocket.

Addition. For a SkX, the band energies have to be scaled by $\cos (\overline{{\vartheta }_{{ij}}}/2)$ to adjust the total band width (see also equation (A2) in the appendix). The average angle of neighboring spins can be approximated by $\overline{{\vartheta }_{{ij}}}=3/2\cdot \pi a/\lambda$. Here, λ is the pitch of the spin spirals whose superposition forms the SkX (see [9]). For large skyrmions ($\lambda \gg a$) the scaling factor approaches 1 and the scaling is irrelevant.

As an illustration, we apply equation (9) to the honeycomb lattice. The two atoms in the structural unit cell yield b = 2 bands and a DOS that is symmetric about the VHS (panels a and b of figure 12). Considering q sites in each of the sublattices yields $n=2\,q$. The QHE Hamiltonian then reads in matrix form

$$\begin{eqnarray*}H=t\left(\begin{array}{cccccccc}0 & {h}_{1}^{(+)} & 0 & 0 & ... & 0 & 0 & {g}^{(-)}\\ {h}_{1}^{(-)} & 0 & {g}^{(+)} & 0 & ... & 0 & 0 & 0\\ 0 & {g}^{(-)} & 0 & {h}_{2}^{(+)} & ... & 0 & 0 & 0\\ 0 & 0 & {h}_{2}^{(-)} & 0 & ... & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & ... & 0 & {g}^{(+)} & 0\\ 0 & 0 & 0 & 0 & ... & {g}^{(-)} & 0 & {h}_{q}^{(+)}\\ {g}^{(+)} & 0 & 0 & 0 & ... & 0 & {h}_{q}^{(-)} & 0\end{array}\right),\end{eqnarray*}$$

with

$$\begin{eqnarray*}&&{h}_{j}^{(+)}={({h}_{j}^{(-)})}^{\star }={\phi }_{x}{\phi }_{y}+{({\phi }_{x}^{2})}^{\star }\,\,\,\,\,\,{{\rm{e}}}^{-2\pi {\rm{i}}j\displaystyle \frac{p}{q}},{g}^{(+)}={({g}^{(-)})}^{\star }={\phi }_{x}^{\star }{\phi }_{y},\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{\phi }_{x}\equiv \exp \left({\rm{i}}\displaystyle \frac{{{ak}}_{x}}{2\sqrt{3}}\right),\,\,\,\,\,\,{\phi }_{y}\equiv \exp \left({\rm{i}}\displaystyle \frac{{{ak}}_{y}}{2}\right).\end{eqnarray*}$$

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To check the validity of the approximation we compare the Hall conductivities of the THE and of the QHE—computed from the above Hamiltonian—with that produced by equation (9) for n = 72 and for large coupling m. Since in all cases ${\sigma }_{{xy}}(E)$ is antisymmetric to the block center we show the THE data only for the lower half and QHE data only for the upper half of the band block.

All three data sets agree well, even the quantization plateaus are reproduced in a wide energy range (figure 12(c)). The quantum Hall conductivity (blue) matches the approximation (gray). Deviations show up close to the VHS at which sizable modulations indicate dispersive LLs; recall that the LL dispersion is not taken into account by the approximation. The jump itself is reproduced best for large n. For small n the jump may be shifted in energy because it is not pinned to the VHS (see the triangular lattice in section 3.2). In the approximation, however, the jump is introduced artificially at E_VHS, what explains the deviation.

Larger deviations appear for the THE (red). Again, the plateaus are reproduced by the approximation but not in great detail. The inhomogeneous emergent field of the skyrmion texture creates band bendings even for the former dispersionless LLs of the QHE far off the VHSs. Therefore, the conductivity shows small modulations at the plateau edges. In addition, lattice effects are significant at E_VHS, as for the QHE; the jump is quite broad, spread over several levels.

The THE data have been scaled by $\cos (\overline{{\vartheta }_{{ij}}}/2)=\cos (\pi /12)$ (equation (A2) in the appendix), as explained in the previous section (see 'Addition'). This scaling is insignificant for skyrmions of size n = 72, as can be seen by the minimal shift of E_VHS (compare the sign change of the gray curve in the lower and upper halves of the block in figure 12(c)).

Summarizing, the approximation works well at energies apart of VHSs: both the quantization steps and the change of sign are reproduced for QHE and THE. Close to VHSs the approximation of the THE improves with skyrmion size.