Transport Induced Dimer State from Topological Corner States

In the single corner-state setup shown in Fig.2(a), left and top leads are attached to the central region. Edge states along zigzag boundaries of the $\lambda$=0 regions are shown in blue. Single corner state arises at the $C_{2}$ corner with an exponentially decaying wavefunction. The dependence of the transmission $T$ on the Zeeman field strength for different energies are plotted in Fig.2(b) showing two transmission plateaus with $T$=2 and $T$=1. For a particular incident energy, there is a critical strength of the Zeeman field beyond which the transmission drops. When the incident energy is further away from zero, the $T=2$ plateau is more easily destroyed by the Zeeman field. These transmission plateaus usually manifest the signature of quantized transport. However, the $T$=2 transmission plateau is contributed from two distinct conducting channels, one from the edge-state channel and the other from the corner-state-mediated resonant tunneling channel. To understand the physical origin of transmission plateaus, we plot the partial LDOS for open systems in Fig.2(c) and Fig.2(d). Given $\lambda$=0.2 and $E=10^{-13}$ for $T=2$, the incoming electron tunnels through the flake via the corner state at $C_{2}$. Due to the dressing of incoming electrons, the corner state becomes delocalized and its wavefunction expands into the classically forbidden bulk region. Two conducting channels are clearly seen in Fig.2(c): one connects edge states along zigzag boundaries of both leads as labeled by the red arrow near $C_{1}$, the other is formed through the corner state at $C_{2}$. When the Zeeman field is increased further, the corner-state wavefunction shrinks towards the $C_{2}$ corner until it is completely localized and the corresponding conducting channel is closed. Thus, only one edge-state channel survives in Fig.2(d) resulting in the $T$=1 plateau.

Now we study the typical resonant feature. The transmission vs the incident energy is shown in Fig.S3(a) of the supplemental material, which exhibits extremely sharp peaks near $E=0$. The $T=2$ transmission peak is easily destroyed by either changing the incident energy or increasing the Zeeman field strength, showing its resonant nature. Meanwhile, the $T=1$ plateau is robust against both $\lambda$ and $E$, since it originates from the edge state connecting left and top leads at the $C_{1}$ corner in Fig.2(c). To further confirm the resonant nature, we define $\Delta E$ as the half-width energy at half-maximum of the resonant peak ($T=1.5$). Apparently, the stronger the Zeeman field, the smaller the $\Delta E$. A stronger Zeeman field leads to more localized corner states, and it is easier to close the resonant channel through the $C_{2}$ corner. $\Delta E$ can also be obtained by solving the eigenvalue problem of the effective Hamiltonian including the effect of both leads. The imaginary part of the eigenvalue is identified as the lifetime of the resonant state, which is equivalent to $\Delta E$. Numerical results presented in the supplemental material show that this is indeed the case.

As shown in Fig.3(a), the diamond-shaped flake is connected to narrower leads around the $C_{3}$ corner. Both leads are far away from $C_{1}$ and $C_{2}$ to ensure weak impact on the corner states. The zero-energy eigenfunction of this isolated system with $\lambda=0.1$ shows the precursor of two corner states at both $C_{1}$ and $C_{2}$. Transmission as a function of the Zeeman field is displayed in Fig.3(b). At $\lambda=0$, there are two conducting channels due to the edge states giving rise to $T=2$. When the Zeeman field is applied and increased, $T$ decreases from $T=2$ and quickly reaches the $T$=1 plateau, which indicates the closing of one conducting channel. Remarkably, with further increasing of $\lambda$, multiple resonant peaks of $T=2$ emerge. As discussed in the supplemental material, the second sharp peak is actually double peaks and the influence of lead width is also evaluated. In view of resonant transport, these integer transmission peaks of $T=2$ manifest the existence of two perfect propagating channels in the system. In the following, we will examine the nature of this resonant tunneling and explain why multiple resonant peaks appear.

We focus on the setup with lead width $W_{L}$=11 and plot partial LDOS of the open system in Fig.3(c-f), which correspond to four typical Zeeman field strengths labeled as $\lambda_{1}$ to $\lambda_{4}$ in Fig.3(b). When $\lambda$=0, two edge states of quantum spin-Hall nature dominate the transport, which leads to $T=2$. For a small $\lambda_{1}$=0.024, two conducting channels are still visible in Fig.3(c). But the channel along boundary 2 is not transmissive enough so that it can only sustain a $T=1.5$ transmission. For a larger strength $\lambda_{2}$=0.2 at the $T$=1 plateau, only the channel along boundary 1 survives and the other channel is destroyed (Fig.3(d)). Nevertheless, electrons incident from the left lead can still tunnel through the barrier $U_{1}$ labeled in Fig.3(a) and reach the upper corner $C_{1}$ with large partial LDOS, which shows the dynamic signature of the dressed corner state. However, it can not reach the lower corner $C_{2}$ leaving the static corner state completely isolated. Fig.3(d) shows the distinction between static and dynamic corner states located at $C_{2}$ and $C_{1}$, respectively. When $\lambda$ is increased to $\lambda_{3}$, Fig.3(e) shows the dynamic feature of corner states at both $C_{1}$ and $C_{2}$, which expand and grow inside the bulk due to the dressing of incoming electrons. Counter-intuitive phenomenon happens at $\lambda_{4}$. In Fig.3(f), dynamic corner states at $C_{1}$ and $C_{2}$ bridge the spatial gap inside the system and form a dimer state, which leads to a new resonant channel inside the bulk instead of along the boundaries of the flake. As a result, a resonant peak with $T$=2 appears, which is sharp and sensitive to both $\lambda$ and $E$. Numerical results verify that, all the resonant peaks in Fig.3(b) originate from the same mechanism: the dimer-state-mediated resonant tunneling. The nature of the dimer state has been studied in detail in the supplemental material. To identify the dimer state, the following observations are in order: (1) the dressing of incoming electrons leads to the binding of two corner states; (2) the distribution of the dimer wavefunction should be symmetrical in order to mediate the resonant tunneling; (3) comparing Fig.3(f) with Fig.S5(b) the binding of two corner states is closely related to the sharpness of the resonant peak.

The existence of multiple resonant peaks with respect to the Zeeman field strength, instead of a single one, can be understood with a simple model. We note that in the presence of the Zeeman field, there are three spatial barriers in Fig.3(a) for an incident electron to overcome in resonant tunneling. $U_{1}$ is the effective potential between the leads and the corner states, and $U_{2}$ is that in the bulk separating $C_{1}$ and $C_{2}$. The evolution of $U_{1}$ and $U_{2}$ follows that of the corner state. As the Zeeman field is increased, the edge state shrinks along the boundary and delocalizes away from it so that the strength of $U_{1}$ increases while $U_{2}$ decreases. In another word, the Zeeman field strength controls the barrier variation in this system, which is confirmed by partial LDOS shown in Fig.3. We construct a triple-barrier model based on a one-dimensional tight-binding chain sdatta , where barrier strengths $U_{1}$ and $U_{2}$ are simultaneously varied with a relative amount $\Delta U$ (see Fig.4(a)). The tight-binding chain has $35$ lattice sites and all three barriers with width $5$ sites are evenly spaced. When $U_{1/2}=1$, the resonant level for this triple-barrier system is found to be $E_{res}$=0.1828 as shown in Fig.4(b). By fixing $E_{res}$ and tuning the effective potentials with $U_{1}=1+\Delta U$ and $U_{2}=1-\Delta U$, the transmission with respect to $\Delta U$ shows two resonant peaks with $T=1$ as shown in Fig.4(c). These peaks are similar to the multiple resonant peaks in Fig.3(b). Here the relative variation $\Delta U$ plays a similar role as the Zeeman field strength $\lambda$ in Fig.3(b). The evolution of the resonant level $E_{res}$ of the triple-barrier system is plotted in Fig.4(d). The procedure of calculating $E_{r}$, which is the real part of the complex eigenenergy, is demonstrated in the supplemental material. We see that $E_{r}$ crosses the incident energy twice, which gives rise to two resonant peaks in Fig.4(c). Hence these two resonant peaks are mediated by the same resonant state showing the re-entrance effect. Similarly, by varying the Zeeman field strength in the double corner-state setup, multiple resonant tunneling processes are mediated by the same dimer state.

In the setup of Fig.5(a), the diamond-shaped central flake with nonzero Zeeman field is connected to left and right leads. The lead Hamiltonian follows the conventional Kane-Mele model defined in Eq.(3), and spin-helical edge states propagate in both zigzag leads when the electron energy is inside the bulk gap. In Fig.5(a), we plot the zero-energy eigenfunction distribution for the isolated system, and blue dots along zigzag boundaries of those $\lambda=0$ regions illustrate the topological edge states. The edge states disappear in the $\lambda=0.2$ region due to the breaking of time-reversal symmetry. Topological corner states are absent in this setup due to two reasons: (1) when the leads are directly coupled to $C_{1}$ and $C_{2}$ of the central flake, inside the $\lambda\neq 0$ region there are no corners or boundary intersections where corner states could reside; (2) in the presence of leads, corner states will leak out. We have confirmed that, when a narrow lead of width $W_{L}=1$ is directly connected to the corner, the corresponding corner state will be destroyed. Besides, the eigenfunction distribution in the $\lambda=0.2$ region is zero, which indicates that electrons can not cross this region. To confirm this point, the transmission as a function of the Zeeman field strength is plotted in Fig.5(b). It is found that the transmission rapidly decreases from $T=2$ to zero with the increasing of $\lambda$, and $T$ is hardly affected by the variation of electron energies. $T=2$ corresponds to the propagation of two in-gap edge modes, and they are quickly destroyed by the Zeeman field inside the central flake. In Fig.5(c), we plot the zero-energy partial LDOS for electrons incident from the left lead, which shows only nonzero density of states in the left lead due to the spin-helical edge states.

Two setups are depicted in panels (a) and (d) of Fig.6, where left and bottom leads are connected to the central region. The zero-energy eigenfunction shows no sign of corner states in Fig.6(a), which is similar to Setup I. In Fig.6(b), the transmission of this system rapidly declines from $T=2$ to $T=1$ when the Zeeman field is applied, and slightly decays upon further increasing of $\lambda$. Single conducting channel is found in Fig.6(c) labeled by the red arrow, which connects the left and down leads near the $C_{3}$ corner. In the presence of strong Zeeman fields, the $C_{3}$ corner with an acute angle does not host electronic states, which is responsible for the slow decay of transmission from $T=1$ in this setup. Compared with the single corner-state setup in Fig.2, though no corner state is found at the $C_{1}$ corner, it can still support certain electronic states and result in a robust $T$=1 plateau against the Zeeman field as shown in Fig.2(b). The single conducting channel in Fig.6(c) can be easily destroyed by considering narrower leads and exposing the $C_{3}$ corner in vacuum. Such a setup is presented in Fig.6(d), where the eigenfunction distribution at the $C_{3}$ corner is depleted by the in-plane Zeeman field. In Fig.6(e) and (f), it is found that the transmission of this setup directly drops to zero at large Zeeman fields, which is consistent with the partial LDOS that the propagating channel is closed.

To demonstrate the resonant feature of the $T=2$ transmission plateau for the single corner-state setup shown in Fig.2, we plot the dependence of transmission of this setup on the incident energy in Fig.7(a). Note that the energy is on the logarithmic scale in order to illustrate the extremely shape peak near $E=0$. The $T=2$ transmission peak is easily destroyed by either changing the incident energy or increasing the Zeeman field strength, showing its resonant nature. Meanwhile, the $T=1$ plateau is robust against both $\lambda$ and $E$, since it originates from the edge state connecting left and top leads at the $C_{1}$ corner in Fig.2(c). To further confirm the resonant nature, we define $\Delta E$ as the half-width energy at half-maximum of the resonant peak ($T=1.5$). Apparently, the stronger the Zeeman field, the smaller the $\Delta E$. A stronger Zeeman field leads to more localized corner states, and it is easier to close the resonant channel through the $C_{2}$ corner. It is well known that in resonant tunneling, electrons can tunnel through the system if the incident energy $E$ is in line with the resonant level $E_{res}$. $E_{res}$ can be calculated by solving the eigenvalue problem of an effective Hamiltonian $H_{eff}=H+\Sigma^{r}$ for the open system, which satisfies Datta1997

The eigenenergies of this effective Hamiltonian are complex, i.e., $\epsilon_{\gamma}=E_{r}+iE_{i}$. The real part $E_{r}$ corresponds to the resonant level $E_{res}$, which is zero in the zero-energy corner-state-mediated resonant tunneling process. The corresponding imaginary part $E_{i}$ is equivalent to $\Delta E$, the half-width energy at half-maximum of the resonant peak. This well-established theory has been verified in various tunneling systems 1983Transmission ; 1984Physics ; 1985Effect ; 1992Transmission . In Fig.7(b), we plot $\Delta E$ as well as $E_{i}$ against the Zeeman field strength for the $T=2$ peak in Fig.7(a). $\Delta E$ is measured at the transmission peak and $E_{i}$ is obtained through diagonalizing the effective Hamiltonian $H_{eff}$. Clearly, $\Delta E$ perfectly matches with $E_{i}$ in a wide range of $\lambda$, which further consolidates the resonant tunneling characteristic of the $T=2$ transmission peak.

For the double corner-state setup shown in Fig.8(a), narrower left and bottom leads are connected to the diamond-shaped flake around the $C_{3}$ corner to ensure weak impact on the corner states, which are clearly seen at $C_{1}$ and $C_{2}$. We evaluate the influence of lead width on the transmission of this setup. Transmission as a function of the Zeeman field strength for two lead widths are displayed in Fig.8(b). These two systems share similar features: the transmission drops from $T=2$ in the presence of a Zeeman field, and reaches the $T=1$ plateau with the increasing of $\lambda$, and finally exhibits multiple $T$=2 resonant peaks when the Zeeman field is further enhanced. For a narrower lead of width $W_{L}=7$, the multiple resonant peaks are wide and close to each other. While for the lead width $W_{L}=11$, single peak is separated from double peaks shown in the inset. A larger Zeeman field is required to detect the first $T$=2 resonant peak for $W_{L}=11$. Dynamic details of the double resonant peaks are revealed in the following.

The double transmission peaks with respect to the Zeeman field strength is highlighted in Fig.9(a), where two sharp peaks is separated by a dip. The transmission is $T$=2 for $\lambda_{5}$ and $\lambda_{7}$, and $T$ drastically drops to exactly $T$=1 at $\lambda_{6}$ forming the dip. The partial LDOS in Fig.9(b) shows that, at $\lambda_{5}$, a symmetrical dimer state mediates a resonant tunneling channel, which crosses almost the entire bulk of the diamond-shaped flake. In contrast, the precursor of the dimer state has asymmetrical distributions, which is exemplified at $\lambda_{6}$ where this delicate dimer state is ”destroyed” by slight increasing of the Zeeman field strength. In Fig.9(c), it is obvious that the resonant channel is closed at $\lambda_{6}$ and single edge-state channel (red arrow) is conducting. Remarkably, the dressing of incoming electrons (albeit asymmetrical) around both corner states is still present, which is preparing for the next resonance. However, the wavefunction becomes asymmetrical, which is typical for the precursor of the dimer state. When the Zeeman field is lightly enhanced to $\lambda_{7}$, the dimer state emerges again and leads to the second $T=2$ resonant peak. These numerical results vividly demonstrate the dynamic nature of the dimer state induced by electron-dressed corner states. The same dimer state repeatedly emerges in the resonant window when the system is tuned by the Zeeman field, which gives rise to multiple resonant peaks in the transmission spectrum. Finally, we note that the total DOS characterizing the bonding of the dimer state is just the dwell time of incoming electrons, which is proportional to the sharpness of the resonance. A sharper resonance leads to tighter bonding of the dimer state. To summarize, three features of the dimer state are identified: (1) the dressing of incoming electrons bridges two corner states; (2) the scattering wavefunction symmetrically distributes around two corner states; (3) a sharper resonance gives tighter bonding of the dimer state.

To evaluate the size effect of the diamond-shaped flake, we consider two system sizes in the double corner-state setup, which are $20a\times 20a$ and $50a\times 50a$, respectively. The lead width is fixed at $W_{L}=11$. The corresponding transmission and partial LDOS results are shown in Fig.10. Combined with numerical results for the system size $30a\times 30a$, we find that all systems have multiple resonant peaks at large Zeeman field strengths, which corresponds to the dimer state induced by electron-dressed corner states. With the increasing of system size, the resonant peaks become narrower. At the same time, a relatively smaller Zeeman field strength is required to observe the first resonant peak for a larger system. Therefore, the dimer-state-mediated resonant tunneling is a common feature in the double corner-state setup, regardless of the system size.