Extensions to the Su-Schrieffer-Heeger Model: Linear chains and their topological properties

The trimer chain (see Fig. 2) is composed of three sublattices $A$, $B$, and $C$ arranged in a diamond chain with staggered hopping potentials $v$ and $w$ accounting for the intracell and intercell hopping energies, respectively.The single particle Hamiltonian for this trimer chain is given by

where $m$ is the cell index, $v$ and $w$ are the hopping potentials, and $\rm{h.c.}$ denotes hermitian conjugation. This gives the matrix elements $H_{ij\alpha\beta}=\braket{i,\alpha}{\hat{H}}{j,\beta}$, where the indices $i,j=1,2,\ldots,N$ denote the cell number and $\alpha,\beta=A,B,C$ denote the sublattice index. We also set the on-site potential to zero, i.e., $H_{i=j,\alpha=\beta}=0$. Furthermore, electron-electron interactions are neglected.

For the bulk, we set the boundary condition to be periodic, i.e., the Born-von Karman condition. We consider an effectively infinitely long chain because of the periodic boundary condition. This also implies that the bulk now is translationally invariant and Bloch’s theorem holds [1]. Our bulk Hamiltonian with the periodic boundary condition reads,

To account for the periodicity, we introduce terms for the hopping between sites $1A$ and $NB$, $1A$ and $NC$, and vice versa.

The generalized trimer chain is a generalization of the trimer chain introduced in Sec. II.1. This model differentiates intracell and intercell hopping energies for $AC$, i.e., site $A$ to site $C$ in the same unit cell, and $AB$, i.e., site $A$ to site $B$ in the same unit cell, and $BA$, i.e., site $B$ to site $A$ in the neighboring unit cell, and $CA$, i.e., site $C$ to site $A$ in the neighboring unit cell, hopping energies (see Fig. 3). The single-particle Hamiltonian for this model is

For the bulk part of the chain, we model it as a chain with periodic boundary conditions. The Hamiltonian for the bulk is

The additional terms in the Hamiltonian account for the periodicity of the bulk.

Similar to the work of Bercioux et al. [7], we calculated the band structures of a tripartite lattice with infinite and finite unit cells. Lattice I of their work is similar to our trimer chain with intercell hopping in the $A$, $B$, and $C$ sublattices. Our generalized trimer chain generalizes the tripartite system which includes their Lattice II as a special case when $v_{1}=w_{2}$ and $v_{2}=w_{1}$. To highlight the novelty of our work, we will show the eigenstates of the finite chains in the topological nontrivial and trivial regime. We will also calculate the winding number to establish the bulk-boundary correspondence of our three model chains.

The hexagonal chain forms a resemblance to a 1D graphene armchair ribbon, as shown in Fig. 4. It consists of six sublattices and staggered hopping potentials $v$ and $w$ for the intracell and intercell hopping, respectively. The single-particle Hamiltonian for this hexagonal chain is

where $m$ is the unit cell index. The elements of the Hamiltonian matrix are given by $H_{ij\alpha\beta}=\braket{i,\alpha}{\hat{H}}{j,\beta}$ where $i,j=1,2,\ldots,N$ and $\alpha,\beta=A,B,C,D,E,F$. The Hamiltonian for the bulk part of the chain modeled using periodic boundary conditions is

Considering the bulk Hamiltonian of the trimer chain, Eq. (2), the energy eigenvalues of the system can be determined using exact diagonalization. The Schrödinger equation for the bulk Hamiltonian in matrix form is given by

This gives the energy eigenvalues

where $k$ is the wave number taking up values in the first Brillouin zone [1]. There are three eigenvalues in Eq. (17). The first eigenvalue (zero energy) has energy coinciding with the Fermi Energy, i.e., $E_{0}(k)=0=E_{f}$ [1]. By varying the magnitude of the hopping parameters $v$ and $w$, a dispersion relation shown in Fig. 5 can be obtained.

The dispersion relation consists of five different scenarios concerned with the different magnitudes of the hopping parameter. Fig. 5(a) describes the system with $v=1$ and $w=0$. Examining the figure, it can be seen that there exists a flat band situated at the Fermi level $E=0$. This is due to $E_{0}(k)$ which has a constant value and is independent on the wavenumber $k$. This characteristic can be seen in all the configurations indicating that all of the configuration of the system is a conductor [10]. Furthermore, two flat bands can also be observed at regions $E=\pm\sqrt{2}v$. The same observations can be seen in Fig. 5(e) with flat bands situated at $E=\pm\sqrt{2}w$ for $v=0$ and $w=1$. The case for $v>w$, Fig. 5(b), and $v<w$, Fig. 5(d), show sinusoidal variations of the energy with respect to the wave number. A defined gap between the uppermost and the lowest energy band can be seen. Lastly, for the case where $v=w$, Fig. 5(c), the uppermost and the lowest energy band crossed in the region $k=\pm\pi$ indicating an intersection, at a point, between the two bands.

To consider the characteristics of the boundaries of the chain, we examine a finite trimer chain. For the extreme cases where $v=0$ or $w=0$ the chain breaks down into trimers. For the trivial case $w=0$, it can be observed that all of the sites belong to a corresponding trimer. However, for the topological case where $v=0$, there are sites that do not belong to a corresponding trimer. Since fermions in these sites do not hop, then these edge sites host zero-energy states given by $\hat{H}\ket{1,A}=\hat{H}\ket{N,B}=\hat{H}\ket{N,C}=0$.

To calculate for the eigenvalues of the finite trimer chain, we diagonalize the Hamiltonian given by

with

where for $N=10$ unit cells, the Hamiltonian is a $30\times 30$ matrix. By varying the hopping amplitude $v$ of the finite trimer chain, the eigenvalue spectra can be obtained (see Fig. 6). As observed from Fig. 6(a), zero-energy states span throughout the graph independent of the values of $v$. This corresponds to the zero-energy eigenvalue that we calculated earlier for the bulk (see Eq. (17)). Notice that these states with zero energy are degenerate. In Fig. 6(a), there are $10$ states having the same zero energy. Furthermore, in the region where $v<w$, there exists two additional zero-energy states that reside at the edges of the chain. These results are consistent with those found by Bercioux et al. [7]. This region is defined as the topologically nontrivial phase as opposed to the region where $v>w$, which we define as the topologically trivial phase. As we will see later in this section, these regions have different topological invariants that distinguishes their topological phase. Additionally, a topological phase transition can be observed in the region where the hopping potentials are equal. Fig. 6(b) and (c) show the probability density $|\Psi|^{2}$ for the corresponding unit cells. Fig. 6(b) shows an arbitrary state where $E\neq 0$ wherein the probability density extends throughout the whole chain. In contrast, Fig. 6(c) shows a state where $E=0$. As seen in this plot, the probability density is localized on both the left and right edges of the chain. These states are defined as edge states, where the wave function is localized on both edges of the chain and exponentially decaying into the bulk. These edge states are typical characteristics of a one-dimensional topological insulator [1].

Following the same procedure as in the previous subsection, we solve for the energy eigenvalues of the generalized trimer chain using exact diagonalization. The Schrödinger equation, using the bulk Hamiltonian in matrix representation, is given by

This gives the energy eigenvalues

Notice that when we set $v_{1}=v_{2}$ and $w_{1}=w_{2}$, we will arrive at the same energy eigenvalues as those for the trimer chain in Eq. (17). Shown in Fig. 7 are the plots of the energy eigenvalues of the generalized trimer chain with respect to the wave number $k$. As seen from the plots, different combinations of the hopping potentials lead to the same conducting characteristics due to the flat band situated along the Fermi level. This band is due to the $E_{0}(k)$ that we obtained. Furthermore, as expected, the configurations where $v_{1}=v_{2}$ and $w_{1}=w_{2}$ resemble the dispersion relations of the trimer chain shown in Fig. 5(c). A visible gap between the uppermost and lowest energy band can be seen in Fig. 7(c) in the case where one hopping parameter is zero ($v_{1}=0$) while the rest are equal. It can be generalized that this band gap will remain open until we reach the limit where $v_{1}=w_{1}$ or $v_{2}=w_{2}$ or both (see Fig. 7(d) and (e)).

To determine the characteristics of the finite generalized trimer chain, we consider a lattice composed of $10$ unit cells. We find that when only one of the hopping potentials is nonzero and the rest are set to zero, the chain breaks down into dimers and the isolated sites would host zero-energy states. To calculate for the eigenvalues of the generalized trimer chain, we diagonalize the Hamiltonian given by

with

where for a $N=10$ unit cell of the generalized trimer chain, the Hamiltonian is a $30\times 30$ matrix. Fig. 8(a) shows the eigenvalue spectra of the finite $N=10$ generalized trimer chain by varying the three parameters $v_{2}=w_{1}=w_{2}$ simultaneously while maintaining $v_{1}=1$. Also, the topological nontrivial phase in the region $v<w$, which we found earlier in the trimer chain (see Fig. 6(a)), motivated us to explore the generalized case where $\left(v_{1}\neq v_{2}\right)<\left(w_{1}=w_{2}\right)$. This case is shown in Fig. 8(b) where we vary the value of $v_{1}$ in the range $[0,3]$ while maintaining $v_{2}=0.5$ and $w_{1}=w_{2}=1$. As noticed from Fig. 8(b), the states which form the edges in the trimer chain are now gapped in the region where $v_{1}<v_{2}$ and $v_{1}>v_{2}$. This gap only closes in the trimer chain limit where $\left(v_{1}=v_{2}\right)<\left(w_{1}=w_{2}\right)$.

Fig. 8(c) shows a representative probability density located at non-zero energy levels. Fig. 8(d) shows the probability density of one of the degenerate states in the generalized trimer chain. Lastly, we examined the eigenstates in the region where $v_{1}<v_{2}$, of which a representative probability density is shown in Fig. 8(e). Although these states are non-zero energy gapped states, it displays a unique characteristic where the probability density is only localized in one of the edges. They are known as chiral edge states [11], which resemble only half of the full edge states of the original SSH model. As we will show later in this section, these chiral edge states do not share the same winding number as the gapless zero-energy edge states observed in the trimer chain.

To establish the bulk-boundary correspondence of the trimer and generalized trimer chains, we express their bulk momentum-space Hamiltonian in terms of the basis states. Consider the following traceless and hermitian matrices, which are four of the matrices known as Gell-Mann matrices representing the basis of the group SU(3) [12],

The bulk momentum-space Hamiltonian of the trimer chain can then be expressed as,

in which the new basis $\Lambda_{x}$ and $\Lambda_{y}$ are linear combinations of the Gell-Mann matrices,

where the normalization coefficients ensure that the basis states follow the same trace orthonormality condition as that of the Gell-Mann matrices, ${\rm Tr}\left[\Lambda_{i}\Lambda_{j}\right]=2\delta_{ij}$ [12]. On the other hand, for the generalized trimer chain, the bulk momentum-space Hamiltonian can be expressed as,

We can see that Eq. (74) reduces to Eq. (72) in the limit $v_{1}=v_{2}$ and $w_{1}=w_{2}$. To reduce the parameters imposed by the 4-dimensional vector of the bulk momentum-space Hamiltonian of the generalized trimer chain, we set $w_{1}=w_{2}=w$, i.e., effectively reducing the free parameters from four to three. The bulk momentum-space Hamiltonian of the generalized trimer chain now reads,

Note that the Hamiltonian in Eq. (75) satisfies chiral symmetry with the corresponding chiral operator

where $\Gamma H\Gamma^{\dagger}=-H$ and $\Gamma^{2}=1$.

To calculate for the topological invariant winding number, we project the bulk momentum-space Hamiltonians into their corresponding basis states. As for the Hamiltonian of the trimer chain, Eq. (72), it has resemblance with that of the bulk momentum-space Hamiltonian of the original SSH dimer. As such, we expect its trajectory in the $\Lambda_{x}\,\Lambda_{y}$ space to look like the trajectory of the SSH Hamiltonian in $dx\,dy$ space. From these trajectories, we can calculate the winding number $\nu$ by counting the number of times the trajectory orbits around the origin, i.e., around $\Lambda_{x}=\Lambda_{y}=0$. We determine the following winding number $\nu$ for the trimer chain,

The winding number is a useful tool in predicting the existence of gapless localized edge states (states where the wave function of the system is localized on both edges of the chain [1]). As shown in Eq. (83), for the case where $v>w$, we found that $\nu=0$, thus indicating a trivial topology of the system characterized by the absence of localized edge states. On the other hand, the case where $v<w$ indicates a non-trivial topology where $\nu=1$ indicating the presence of a pair of gapless localized edge states. Furthermore, the case where $v=w$ leads to an undetermined $\nu$ indicating that the trajectory of $H$ is in direct contact with the origin. This case entails a topological phase transition, which shows that the only way to change the winding number of the trimer chain from $\nu=0$ to $\nu=1$, or vice versa, is by closing the gap between the highest and lowest band of the trimer chain.

Extending our analysis to the generalized case, we plot the trajectory of the bulk-momentum space Hamiltonian of the generalized trimer chain, Eq. (75) in $\lambda$-space, as shown in Fig. 9(a). Here we are interested to see if the winding number changes or remains the same for different variations in the upper, $v_{1}$, and lower, $v_{2}$, intracell hopping parameters. We are particularly interested in this case due to the non-trivial topological nature of the trimer chain in the $v<w$ regime, as we have shown earlier in this section. As such, we would like to generalize the conditions for the existence of gapless localized edge states in the $\left(v_{1}\neq v_{2}\right)<w$ regime in terms of their winding numbers. As observed from Fig. 9(a) we determined the following winding numbers,

where $w=1$ for all cases. From this result, we can see that only in the trimer chain limit, i.e., $v_{1}=v_{2}$, is the winding number $\nu=1$ where we have shown a localized edge state in Fig. 6(c). It is also the region where the gap closes at $E=0$ between the pair of positive and negative energy band in Fig. 8(b). The other configurations, on the other hand, in Eq. (87), with $v_{1}>v_{2}$ and $v_{1}<v_{2}$, demonstrate a chiral edge state emerging from a non-zero energy state, which is quite different from the usual gapless localized edge states where the probability densities are localized on both edges of the chain [11].

Solving for the bulk energy eigenvalues of the hexagonal chain requires us to solve Schrödinger’s equation given by

The energy eigenvalues of the hexagonal chain with varying hopping parameters can be calculated by numerically diagonalizing the matrix in Eq. (106). We do this using the GNU Octave computational tool software. The eigenvalues are shown in Fig. 10.

We see that Fig. 10(a) to 10(c) show an insulating system due to the presence of a band gap. Furthermore, in Fig. 10(d) where $v<w$, a conical band structure can be seen at $k=0$. This band structure displays gapless and linear conducting bands which are typical for a semi-metal [13]. Additionally, when $v=0$, see Fig. 10(e), the chain shows conducting characteristics due to the flat band situated along the Fermi energy, $E_{f}=0$, independent of the wave number $k$. The eigenvalue spectra can be determined by diagonalizing the Hamiltonian given by

with

where for a $N=10$ unit cells hexagonal chain, the Hamiltonian $H$ is a $60\times 60$ matrix.

Shown in Fig. 11(a) is the eigenvalue spectra obtained by plotting the energy eigenvalues with $w=1$ and slowly tuning $v$ in the range $[0,3]$. Furthermore, Fig. 11(b) and 11(c) show a representative extended state and an edge state of the finite hexagonal chain. A pair of zero energy edge states (one is shown in Fig. 11(c)) show up when $v$ is varied. Additionally, we observe a topological phase transition around $v=0$. The representative extended state seen in Fig. 11(b) shows an arbitrary non-zero energy state. In this figure, it can be seen that the probability density is extended across the chain indicating the conducting characteristics of a trivial topology. In contrast, the edge state shown in Fig. 11(c) indicates that the probability density is localized at the edges of the chain revealing the nontrivial topology of the system.

To establish the bulk-boundary correspondence of the hexagonal chain, we express the bulk momentum-space Hamiltonian in terms of its basis states:

where $\zeta_{1}$, $\zeta_{2}$, and $\zeta_{3}$ are

These matrices satisfy the orthonormality condition ${\rm Tr}\left[\zeta_{i}\zeta_{j}\right]=2\delta_{ij}$. They are 3 of the 35 matrices representing the basis of the group SU(6). Furthermore, we define the chiral symmetric operator,

where $\Gamma H\Gamma^{\dagger}=-H$ and $\Gamma^{2}=1$. This $6\times 6$ chiral operator ensures that the energies of the system form chiral symmetric pairs.

The trajectories of the bulk momentum-space Hamiltonian of the hexagonal chain in $\zeta$-space, Eq. (128) with varying hopping parameters are shown in Fig. 9(b). The trajectory of Eq. (128) forms a cylindrical plot with radius $w$ and height $v$. Thus, increasing the value of $w$ and $v$ will increase the radius and height, respectively. The same as the procedure introduced in the previous section, we determine the winding number to be

where we keep the value of $w=1$. As seen from the calculation of the winding number, we expect a pair of localized edge states in the region $v=0$ where the winding number $\nu=1$. An example of this edge state is shown in Fig. 11(c).