Topological router induced via long-range hopping in a Su-Schrieffer-Heeger chain

We show how to implement the various kinds of topological routers via introducing the specific long-range hopping in the one-dimensional modulated Su-Schrieffer-Heeger chain. We find that the long-range hopping with hopping amplitudes identical to those of the intercell hopping between the first site and other odd sites holds a distinctive topological channel (gap state), by which the particle initially prepared at the last site can appear at the first site and all of the even sites with approximately equal probabilities. This extraordinary performance indicates that, from the perspective of treating the last site as the input port and treating the first site and other even sites as output ports, the present system can be naturally equivalent to a distribution device with multiple output ports, i.e., the topological router. We show that the number of the output ports for the present topological router can be flexibly tuned via reducing the long-range hopping terms. Especially, the system experiences a phase transition with the increasing of the long-range hopping amplitudes, in which the original gap state becomes a topological channel to implement the topological router with the output ports only at all even sites. We stress that both the two kinds of topological routers are protected by the energy gap and hence are robust to the mild disorder added into the system. Moreover, together with the construction of the topological interface, we find that the topological routers can own the output ports in a symmetrical way about the interface site. The topological routers may play the pivotal role in the large-scale entanglement distributions, which significantly expands and supplies the potential applications of the topological materials in quantum information processing.

Figure 1

The diagrammatic sketch of the modulated Su-Schrieffer-Heeger (SSH) model with the long-range hopping. The size of the SSH chain is $L=2N+1$ with $N$ unit cells, in which each unit cell contains two sublattice sites $a_n$ and $b_n$. The intra- and intercell coupling between two adjacent sites is $J_1$ and $J_2$. Note that the SSH chain has the special long-range hopping between the first site $a_1$ and all other $a$-type sites $a_{n=2,...,N+1}$ with the amplitude $T_{n=1,...,N}$.

Figure 2

Energy spectrum, distribution of gap state, and diagrammatic sketch of the localized state. (a) The energy spectrum of the SSH chain when $T_1=T_2=...=T_N=J_2$. The energy spectrum has a gap state (red line). The minimal energy space between the gap state and the up energy band is labeled as ${\mathrm{\Delta }}_E$. (b) The distribution of the gap state versus the periodic parameter $\theta$, in which the gap state is mainly localized at the last site within $\theta \in [0,0.5\pi ]$ while it is mainly localized at sites $a_1$, $b_1$, $b_2$, ..., and $b_N$ uniformly within $\theta \in [0.5\pi ,\pi ]$. Note that the distribution of the gap state when $\theta \in [\pi ,1.5\pi ]$ ($\theta \in [1.5\pi ,2\pi ]$) has the same form as the distribution in $\theta \in [0.5\pi ,\pi ]$ ($\theta \in [0,0.5\pi ]$). (c) The diagrammatic sketch of the distribution for the gap state when $\theta \in [0.5\pi ,\pi ]$. When $\theta \in [0.5\pi ,\pi ]$, the intra- and intercell coupling satisfy $J_1<J_2$, implying the weak bond $J_1$ and strong bond $J_2$. Under the limiting of the weak bond, the SSH chain is divided into several components in the unit of three sites, such as $a_1$, $a_2$ ($a_3,...,a_{N+1}$), and $b_1$ ($b_2,...,b_N$). When the long-range hopping satisfies $T_{n=1,...,N}=J_2$, the strong bonds $T_n$ and $J_2$ generate a domain wall (see Fig. 1.8 in Ref. [49]), leading the sites $a_1$ and $b_1$ ($b_2,...,b_N$) to be isolated. These isolated sites just correspond to the distribution of the gap state within $\theta \in [0.5\pi ,\pi ]$. The other parameter takes $L=2N+1$ with $N=6$. The unit is $J=1$.

Figure 3

Fidelity, evolution process of the initial state, and diagrammatic sketch of the topological router. (a) The fidelity between the evolved final state and the ideal final state ${|\mathrm{\Psi }\rangle }_{f,1}$ versus the ramping speed $\mathrm{\Omega }$ and on-site disorder strength $W$. Note that here we only focus on the process of the state transfer and hence artificially abandon the phase information of the evolved state in the process of calculating the fidelity. The disorder is randomized 100 times and the average is taken. (b) The process of the evolution for the initial state ${|\mathrm{\Psi }\rangle }_{i,1}$ when the on-site disorder is added into the system, in which the parameters satisfy $\mathrm{\Omega }=5\times {10}^{-4}J$ and $W=0.25J$. (c) The fidelity between the evolved final state and the ideal final state ${|\mathrm{\Psi }\rangle }_{f,1}$ versus the ramping speed $\mathrm{\Omega }$ and NN disorder strength $W$. (d) The process of the evolution for the initial state ${|\mathrm{\Psi }\rangle }_{i,1}$ when the NN disorder is added into the system with $\mathrm{\Omega }=5\times {10}^{-4}J$ and $W=0.25J$. (e) If we treat the last site $a_{N+1}$ as the input port and treat the sites $a_1$, $b_1$, $b_2$, ..., and $b_N$ as output ports, the present process of the state transfer is naturally equivalent to a router in form. The other parameter takes $L=2N+1$ with $N=6$. The unit is $J=1$.

Figure 4

(a) The varying of minimal energy space ${\mathrm{\Delta }}_E$ with the increasing of the long-range hopping amplitude. (b) The energy spectrum around the gap under the large long-range hopping amplitude condition with $T=8J$. The energy spectrum still has a gap state. (c) The distribution of the gap state. (d) The fidelity between the evolved final state and the state ${|\mathrm{\Psi }\rangle }_{f,1}^{\prime }$ versus the ramping speed $\mathrm{\Omega }$ and $T$. The fidelity exhibits a clear dividing line around $T=2.4J$. (e) The diagrammatic sketch of the distribution for the gap state when $\theta \in [0.5\pi ,\pi ]$. Compared with the case shown in Fig. 2, the strong enough long-range hopping amplitudes $T_{n=1,...,N}=T{J}_2$ lead the two sites $a_1$ and $a_2$ ($a_3,...,a_{N+1}$) to be paired, further leading the sites $b_1,b_2,...,b_N$ to be isolated. The other parameter takes $L=2N+1$ with $N=6$. The unit is $J=1$.

Figure 5

The diagrammatic sketch of the modulated Su-Schrieffer-Heeger (SSH) model with the interface. The size of the SSH chain is $L=2N+1$ with $N$ unit cells ($N\in \mathrm{even}$). The SSH chain has the intra- and intercell coupling $J_1$ ($J_2$) and $J_2$ ($J_1$) before (after) the interface site $a_{N/2+1}$. Note that the SSH chain has the long-range hopping between the first site $a_1$ and all other $a$-type sites $a_{n=2,...,N/2+1}$ with the amplitude $T_{n=1,...,N/2}$ before the interface site $a_{N/2+1}$ and possesses the the long-range hopping between the last site $a_{N+1}$ and all other $a$-type sites $a_{n=N/2+1,...,N}$ with the amplitude $T_{n=N/2+1,...,N}$ after the interface site $a_{N/2+1}$.

Figure 6

Fidelity and evolution process of the initial state ${|\mathrm{\Psi }\rangle }_{i,2}$. (a) The fidelity between the evolved final state and the ideal final state ${|\mathrm{\Psi }\rangle }_{f,2}$ versus the ramping speed $\mathrm{\Omega }$ and parameter $T$. (b) The fidelity between the evolved final state and the ideal final state ${|\mathrm{\Psi }\rangle }_{f,2}^{\prime }$ versus the ramping speed $\mathrm{\Omega }$ and parameter $T$. (c) The process of the evolution for the initial state ${|\mathrm{\Psi }\rangle }_{i,2}$ when parameters satisfy $\mathrm{\Omega }={10}^{-5}J$ and $T=1.1J$. (d) The process of the evolution for the initial state ${|\mathrm{\Psi }\rangle }_{i,2}$ when parameters satisfy $\mathrm{\Omega }={10}^{-5}J$ and $T=8J$. The other parameter takes $L=2N+1$ with $N=6$. The unit is $J=1$.

Figure 7

(a) The minimal energy space ${\mathrm{\Delta }}_E$ versus the size of the lattice $L$ when $T_1=T_2=...=T_N=J_2$. (b) The fidelity of the topological router versus the ramping speed $\mathrm{\Omega }$ corresponding to different size of the lattice. The unit is $J=1$.

Figure 8

Energy spectrum and distribution of gap state. (a) The energy spectrum of the SSH chain when $T_1=T_2=...=T_{N-1}=J_2$. The energy spectrum has a gap state (red line). (b) The distribution of the gap state in (a) versus the periodic parameter $\theta$, in which the gap state is mainly localized at the last site within $\theta \in [0,0.5\pi ]$ while it is mainly localized at sites $a_1$ and $b_{n=1,...,N-1}$ uniformly within $\theta \in [0.5\pi ,\pi ]$. (c) The energy spectrum of the SSH chain when $T_1=T_2=...=T_{N-2}=J_2$. The energy spectrum has a gap state (red line). (d) The distribution of the gap state in (c) versus the periodic parameter $\theta$, in which the gap state is mainly localized at the last site within $\theta \in [0,0.5\pi ]$ while it is mainly localized at sites $a_1$ and $b_{n=1,...,N-2}$ uniformly within $\theta \in [0.5\pi ,\pi ]$. The other parameter takes $L=2N+1$ with $N=6$. The unit is $J=1$.