Nontrivial Topological Phases on the Stuffed Honeycomb Lattice

Investigation of nontrivial topological phases on various two-dimensional lattice models has been increasing rapidly in the recent times. These studies are motivated by the search of novel topological phases within new tight-binding models. These models are primarily characterized by a set of special energy bands those are separated by bulk energy gap but joined by quasi-continuous edge state energies. These are known as topological insulators (TI) which are distinguished by a topologically invariant integral number such as Chern number ($C_{n}$) TKNN , $Z_{2}$ invariant, Kane1 etc.

It is established that the emergence of topological phases is the handiwork of specific phase factors which are incorporated into the tight-binding models through the hopping terms. In case of multipartite lattices, this phase factor appearing in the momentum space representation of the Hamiltonian is coupled with the hopping term between same or different sublattices. For instance, it arises in the one-dimensional bipartite Su-Schrieffer-Hegger model SSH only through the nearest neighbor (NN) hopping between two different sublattices. In this case, it depends on the Bloch wave vector, k. For the two-dimensional bipartite honeycomb lattice, an additional phase coupled with the next-nearest-neighbor (NNN) hopping between the same sublattice plays the crucial role to generate the nontrivial topological phases Haldane . Here, the sign of this phase is opposite for the two different sublattices. The net flux passing through a unit cell due to this gauge field is zero. The resulting Hamiltonians in these two cases break the Time Reversal Symmetry (TRS) due to the presence of this phase factor for which the topological energy bands are characterized by $C_{n}$. In other words, this phase factor drives the system into an Integer Quantum Hall Effect (IQHE) regime.

On the other hand, $Z_{2}$ topological phase appears in the honeycomb lattice due to the presence of spin-orbit coupling (SOC) for which the resulting Hamiltonian does not break the TRS Kane1 . Here, the system is characterized by $Z_{2}$ invariant. However, if the Hamiltonian commutes with the $z$-component of spin operator, $S_{z}$, then the Hamiltonians for up and down spins break the TRS separately. In this case, resulting energy bands are characterized by spin-Chern numbers.

For a trivial insulator, the topological invariant is zero, while for a nontrivial TI, it can assume nonzero values. For example, there is nonzero $C_{n}$ for Chern insulators. Also, the number of edge states is proportional to the value of $C_{n}$, which is known as the ‘bulk-boundary correspondence’ Hatsugai1 ; Hatsugai2 ; Kane2 . Due to this correspondence, the edge states which appear in a strip geometry become topologically protected. Thus, topological invariants are helpful for the classification of the topological phases of a system. A brief review on various TIs is available in the article Das .

Although the Chern insulating phase appears for TRS breaking Hamiltonians, the converse is not true. We cannot readily obtain nontrivial topological phase for all TRS breaking Hamiltonians. So, several attempts have been made in search of this phase in various multi-band systems. A number of two-dimensional lattices with nontrivial topological phase have been found and they are Lieb Franz , kagome Wen , checkerboard Das-sharma1 , square octagon Kargarian , diceWang ; Gong2 , star lattice Chen , etc. In addition, different topological flat-band models have also been proposed, where Fractional Quantum Hall Effect (FQHE) can be realized as those flat-bands carry nonzero Chern number Das-sharma2 .

Recently, nontrivial bands have been realized in optical lattices of ultracold atoms Wu by tuning the strengths of both NN and NNN hopping amplitudes Weiss and thus realization of artificial gauge field has been made possibleSpielman ; Aidelsburger . This has opened a new path of obtaining Chern insulator by varying the hopping parameters of the model tight-binding Hamiltonian in various two-dimensional lattice-systems Bergholtz ; Gong1 ; Beugeling .

In this work, we search for nontrivial topological phase on the stuffed honeycomb lattice, which is a three-band system by itself but actually interpolates the triangular (Bravais) and two-band honeycomb (non-Bravais) lattices. Being a two-band system honeycomb model is capable to exhibit nontrivial topological phase in the presence of phase-coupled hopping terms, while triangular lattice fails to do so under the same situation for obvious reason. Therefore, emergence of new topological phase in the interpolating regime will be of great interest. Eventually it turns out that addition of extra sublattice does lead to the emergence of new topological phase in the resulting system. In support of this claim, another tight-binding model on stuffed square lattice is hereby introduced which is capable to host new topological phase.

The antiferromagnetic (AFM) Heisenberg model on this lattice was studied previously in search of new spin-liquid phase. However, in that case, triangular, honeycomb as well as the intermediate stuffed honeycomb lattices were found to host similar kind of spin-liquid phase Sahoo . Interpolating lattices are capable to exhibit nontrivial spin chirality, which may lead to anomalous Hall Effect Nagaosa . S o, adding another basis to Haldane’s honeycomb model could provide the platform to explore novel topological phases in tight-binding regime. With this motivation, we formulated a three-band tight-binding model on this lattice in the presence of NN and NNN hopping terms. An additional phase coupled with NNN hopping has been considered to incorporate nontriviality in the otherwise trivial system. The sign of this phase depends on the direction of hopping. This phase breaks the TRS in the momentum space representation of the Hamiltonian. Also, the net flux of the gauge field passing through a unit cell is zero. The system is shown to exhibit distinct TI phases characterized by different sets of Chern numbers. Transition between those phases are driven by the modulation of hopping strengths. Chiral edge states are simultaneously obtained in the finite system.

The plan of the paper is as follows. In section II, stuffed honeycomb lattice is described and the tight-binding Hamiltonian is formulated. This is followed by section III, where behavior of the relevant physical quantities to study topological properties is presented. Finally, in section IV, discussions and conclusions on this work are summarized.

Stuffed honeycomb lattice originates as a result of coupling between one honeycomb lattice and one triangular lattice which is shown in Fig. 1(a). The honeycomb lattice is composed of two triangular lattices with site indices $A$ and $B$, while the site index of the additional triangular lattice is $C$. So, essentially it is tripartite and composed of three interpenetrating identical triangular lattices. The resulting lattice is decomposed in this manner to define the hopping parameters for this tight-binding model in a comfortable way. We consider three different species of spinless fermions each for three sublattices to introduce the Hamiltonian

	$\displaystyle H\!\!$	$\displaystyle=$	$\displaystyle\!\!-\sum_{\langle ij\rangle}\left(t_{1}\;A_{i}^{\dagger}B_{j}+t_{2}\;(A_{i}^{\dagger}C_{j}+B_{i}^{\dagger}C_{j})+H.c\right)$		(1)
		$\displaystyle-$	$\displaystyle\sum_{\langle\langle ij\rangle\rangle}\left(t_{ij}\;e^{i\phi_{ij}}\;({A_{i}}^{\dagger}A_{j}+{B_{i}}^{\dagger}B_{j}+{C_{i}}^{\dagger}C_{j})+H.c\right)$
		$\displaystyle+$	$\displaystyle\sum_{i}\epsilon_{i}\left({A_{i}}^{\dagger}A_{i}+{B_{i}}^{\dagger}B_{i}+{C_{i}}^{\dagger}C_{i}\right),$

where $i$ is the site index. $\langle\cdot\rangle$ and $\langle\langle\cdot\rangle\rangle$ indicate the summations over NN and NNN pairs. ${\alpha_{i}}^{\dagger}(\alpha=A,B,C)$ is the fermion creation operator at site $i$. $t_{1}$ is the intra-honeycomb NN hopping amplitude and $t_{2}$ is the NN hopping amplitude between the honeycomb and triangular sublattices (Fig. 1(a)). $t_{ij}$ is the NNN hopping amplitude irrespective of sublattices but $t_{ij}$ = $t_{3}\,(-t_{3})$ for solid (dashed) lines as shown in Fig. 1(b). The direction of the phases $\phi_{ij}=\pi/2$ is indicated by the arrow in Fig. 1(b). $\epsilon_{i}$ is the onsite energy.

This model will become the honeycomb one when both $t_{2}$ and $t_{3}$ across the $C$-$C$ bonds vanish. Similarly, it interpolates separately to triangular and dice lattices at $t_{2}$=$t_{1}$ and at $t_{2}$=$\infty$, respectively Sahoo . The unit cell is defined by the primitive vectors $\mathbf{a_{1}}=\sqrt{3}\left(0,1\right)$ and $\mathbf{a_{2}}=\sqrt{3}\left(\sqrt{3}/2,-1/2\right)$, where the NN distance is taken to be unity. NN sites are connected by the vectors $\mathbf{\delta_{1}},\mathbf{\delta_{2}}$ and $\mathbf{\delta_{3}}$ where $\mathbf{\delta_{1}}=\left(1,0\right)=1/3\left(\mathbf{a_{1}}+2\mathbf{a_{2}}\right),\;\mathbf{\delta_{2}}=\left(-1/2,\sqrt{3}/2\right)=1/3\left(\mathbf{a_{1}}-\mathbf{a_{2}}\right)$ and $\mathbf{\delta_{3}}=\left(-1/2,-\sqrt{3}/2\right)=-1/3\left(2\mathbf{a_{1}}+\mathbf{a_{2}}\right)$. All those vectors are shown in Fig. 1(a). The corresponding reciprocal lattice vectors are $\mathbf{b_{1}}=\left(2\pi/3,2\pi/\sqrt{3}\right)$ and $\mathbf{b_{2}}=\left(4\pi/3,0\right)$ which span the hexagonal first Brillouin zone.

Hamiltonian in the momentum space is written by invoking periodic boundary conditions along both $\mathbf{a_{1}}$ and $\mathbf{a_{2}}$ directions,

$$H\left(\textbf{k}\right)=\sum_{k}{\psi_{k}}^{\dagger}h\left(\textbf{k}\right)\psi_{k},$$

(2)

where $\textbf{k}=(k_{x},k_{y})$, $\psi_{\textbf{k}}=\left(A_{\textbf{k}},B_{\textbf{k}},C_{\textbf{k}}\right)$ is a $3$-component spinor and $h(\textbf{k})$ is a $3\times 3$ matrix. This $h(\textbf{k})$ can be expressed in terms of eight Gell-Mann matrices, $\lambda_{i}$ as

$$h\left(\textbf{k}\right)=\sum_{i}h_{i}\,\lambda_{i}+a\,I_{3},$$

(3)

with

	$\displaystyle h_{1}$	$\displaystyle=t_{1}\big{[}\cos\left(\frac{k_{1}+2k_{2}}{3}\right)+\cos\left(\frac{k_{1}-k_{2}}{3}\right)$		(4)
		$\displaystyle+\cos\left(\frac{2k_{1}+k_{2}}{3}\right)\big{]},$
	$\displaystyle h_{2}$	$\displaystyle=-t_{1}\big{[}\sin\left(\frac{k_{1}+2k_{2}}{3}\right)+\sin\left(\frac{k_{1}-k_{2}}{3}\right)$
		$\displaystyle-\sin\left(\frac{2k_{1}+k_{2}}{3}\right)\big{]},$
	$\displaystyle h_{3}$	$\displaystyle=2t_{3}f(k_{1},k_{2}),$
	$\displaystyle h_{4}$	$\displaystyle=h_{1}\left(\frac{t_{2}}{t_{1}}\right)=h_{6},$
	$\displaystyle h_{5}$	$\displaystyle=-h_{2}\left(\frac{t_{2}}{t_{1}}\right)=-h_{7},$
	$\displaystyle h_{8}$	$\displaystyle=-\frac{4}{2\sqrt{3}}t_{3}f(k_{1},k_{2})+\sqrt{3}\epsilon,$
	$\displaystyle a$	$\displaystyle=\frac{2}{3}t_{3}f(k_{1},k_{2}),$

where $I_{3}$ is the $3\times 3$ identity matrix, $k_{1}=\textbf{k}\cdot\mathbf{a_{1}}=\sqrt{3}k_{y}$, $k_{2}=\textbf{k}\cdot\mathbf{a_{2}}=3/2k_{x}-\sqrt{3}/2k_{y}$ and $f(k_{1},k_{2})=\big{[}\cos\left(k_{1}+\pi/2\right)+\cos\left(k_{2}+\pi/2\right)+\cos\left(k_{1}+k_{2}+\pi/2\right)\big{]}$. $\lambda_{i}$ are shown in the Appendix A. We have written the Hamiltonian matrix in terms of $k_{1}$ and $k_{2}$ for ease of calculation. Evidently, TRS is broken by the phase $\phi_{ij}=\pi/2$ since $h(\textbf{k})\neq h^{\ast}(-\textbf{k})$. Hopping parameters $t_{2}$ and $t_{3}$ as well as $\epsilon_{i}$ are measured with respect to $t_{1}$ which is taken as $1.0$ throughout the article. The value of $\epsilon_{i}$ is taken as $\epsilon$ ($-2\epsilon$) for honeycomb (triangular) sublattice. This special distribution of values of $\epsilon_{i}$ is necessary to open up gaps between the otherwise gapless energy bands.

Signs of hopping terms are taken in such a way that one among the three terms $h_{n}(n=2,5,7)$ has sign opposite to that of the remaining two terms. This reverses the bloch phase of one of the off-diagonal elements of the Hamiltonian with respect to the other two (see Appendix A). We note that in the case of SU(3)-invariant models such a situation is a necessary criterion for generating the nontrivial topology Das2 .

Analytic expressions for the eigenvalues of $h(\textbf{k})$ are given by

$$E_{m}(\textbf{k})=2\sqrt{\frac{-p}{3}}\cos{\left(\frac{1}{3}\arccos{\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)}-\frac{2\pi m}{3}\right)},$$

(5)

$m=0,1,2$; where the real parameters are

	$\displaystyle p$	$\displaystyle=-\frac{4}{3}h_{3}^{2}(\textbf{k})+2h_{3}\,\epsilon-({t_{1}}^{2}+2{t_{2}}^{2}){|f(\textbf{k})|}^{2}-3\,{\epsilon}^{2},$		(6)
	$\displaystyle q$	$\displaystyle=\frac{16}{27}\,{h_{3}}^{3}(\textbf{k})-\frac{4}{3}\,{h_{3}}^{2}(\textbf{k})\,\epsilon-t_{1}{t_{2}}^{2}(f^{3}(\textbf{k})+{f^{\ast}}^{3}(\textbf{k}))$
		$\displaystyle-2({t_{1}}^{2}-{t_{2}}^{2})\left(\frac{h_{3}(\textbf{k})}{3}-2\,\epsilon\right){|f(\textbf{k})|}^{2}.$

Here, $f(\textbf{k})=(h_{1}(\textbf{k})-ih_{2}(\textbf{k}))/t_{1}$. The dispersion relations are shown in Fig. 2 (a) and (b) for two sets of $(t_{2},t_{3},\epsilon)$ where nonzero Chern numbers are found.

In order to study the topological properties, Chern numbers, Hall conductance at zero temperature $(\sigma_{H})$ and edge states of this model have been calculated.

A three-band tight-binding model on the stuffed honeycomb lattice has been proposed where an additional phase associated with the NNN hopping terms is found to break the TRS by yielding a Chern insulating phase in the presence of NN hopping. Although the same phase coupled with the NN hopping breaks TRS, it does not give rise to any nontrivial topological phase. The phase is chosen in such a way that the net flux of gauge field per unit cell vanishes. This system exhibits two different TI phases, Chern insulating and semi-metallic with nonzero Chern numbers in the parameter regimes. Also, this one undergoes quantum phase transition between two topological phases driven by the hopping parameters. Hall conductance exhibits prominent IQHE plateaus. The emergence of topologically protected chiral edge states in a strip configuration with open boundary condition is also found when $C_{n}\neq 0$.

This study reveals the fact that additional triangular sublattice leads to the generation of one additional topological phase in the resulting stuffed honeycomb lattice. It is established that topological phases could be induced within the trivial systems by means of introducing phase dependent hopping terms Haldane . So, more additional topological phases may be obtained by choosing those phases and hoppings in different ways. Topological phases with higher Chern numbers could be realized within a system by means of either exposing the systems to polarized light Arghya or invoking distant-neighbor hopping terms Piechon ; Moumita . Addition of extra sublattice may pave another route for the engineering of new topological phases wherever it is applicable. Thus, a desired topological phase may be realized in any system by combining those procedures along with the incorporation of additional sublattice. In this context, a tight-binding model is formulated on a body-centered stuffed square lattice which is found to harbour new topological phase. A brief description of the system as well as its topological properties is available in Appendix B.

The picture of stuffed honeycomb lattice is brought into light in the context of AFM compound, LiZn₂Mo₃O₈ Flint . In this material the molecular cluster Mo₃O₈ forms a triangular lattice, however, its spin-liquid property is explained in terms of an effective spin-1/2 Heisenberg model built on the stuffed honeycomb lattice. Likewise, magnetic properties of the partially frustrated triangular AFM material, RbFeBr₃, was explained before in terms of distorted triangular lattice, which is essentially the stuffed honeycomb lattice Adachi . However, no material is available at this moment where this electronic model could be realized for the verification of its topological properties.

AS acknowledges the CSIR fellowship, no. 09/096(0934) (2018), India. AKG acknowledges BRNS-sanctioned research project, no. 37(3)/14/16/2015, India.