Topological phase transitions, Majorana modes, and quantum simulation of the Su–Schrieffer–Heeger model with nearest-neighbor interactions

The Su–Schrieffer–Heeger (SSH) model is one of the simplest topological lattice models and is a promising candidate for quantum simulations of topological systems. In this study we investigate an interacting SSH model with nearest-neighbor interactions using mean field theory and an exact diagonalization method after the Jordan-Wigner transformation. We obtain a rich phase diagram with six different phases. Majorana bound states are found and confirmed by analyzing wave functions and Majorana number. In addition, we propose to use superconducting devices to construct a one-dimensional quantum circuit, where Cooper pair boxes and Josephson junctions are alternately connected in series, to theoretically simulate the interacting SSH model. This approach also provides us a toolkit to simulate correlated topological systems.

Figure 1

(a) Schematic diagram of structure of the interacting SSH model. Blue and green circles are sites on sublattice $A$ and sublattice $B$, respectively. Solid and dashed lines represent intra- and intercell interactions, respectively. (b) Schematic diagram of serial system of CPBs and JJs.

Figure 2

Phase diagrams as functions of $V$ and $V^{\prime }$ with (a) $t=1,t^{\prime }=2$, (b) $t=2,t^{\prime }=1$, (c) $t=1,t^{\prime }=1$, and (d) $t=1,t^{\prime }=3$. Each phase diagram consists of six phases, including intra- and intercell-hopping (Intra- and Inter-CH) phases, intra- and intercell-superconducting-pairing (Intra- and Inter-CSP) phases, a coexisting (CE) phase, and a charge-density-wave (CDW) phase. The number in parentheses gives the number of zero-energy modes in the corresponding finite-length system. The stars are the critical points that are calculated in detail. Triple points are approximated.

Figure 3

For $t=1,t^{\prime }=2,V=3$, (a) total energy, (b) occupation number. and (c-f) order parameters as functions of $V^{\prime }$. Green, blue, and red background colors respectively correspond to the Inter-CSP, Intra-CH, and CDW phases in Fig. 2.

Figure 4

For $t=1,t^{\prime }=2,V=-1$, (a) total energy, (b) band gap, and (c-f) order parameters as functions of $V^{\prime }$. Green, purple, and orange background colors respectively correspond to the Inter-CSP, CE, and Inter-CH phases in Fig. 2. (g) Energy levels and (h-i) spatial distribution of two zero-energy wave functions in a one-dimension system of 100 unit cells with $V^{\prime }=-1$. The two wave functions are purely real. In panels (h) and (i), left and right parts correspond to electron states ($c_{iA/B}$) and hole states ($c_{iA/B}^{†}$), respectively. Red and green bars label the components of the $A$ and $B$ sublattices, respectively.

Figure 5

For $t=1,t^{\prime }=2$, evolution of topological invariant and Majorana number in directions $V=-1$ and $V^{\prime }=-1$.

Figure 6

A system with 12 CPBs is considered. (a) $Y_{ij}$ as a function of $i$ and $j$ for $C_J/C_g=1$. (b) $Y_{1j}$ as a function of $j$ with different $C_J/C_g$. (c) $Y_{12}$, $Y_{13}$ (inset), and their ratio $Y_{12}/Y_{13}$ as functions of $C_J/C_g$.

Figure 7

Spatial distributions of particles for (a) $E_{\mathrm{in}}=1$, $E_{\mathrm{out}}=5$, and (b) $E_{\mathrm{in}}=5$, $E_{\mathrm{out}}=1$ without any nearest-neighbor interactions. The horizontal axis corresponds to 12 qubits (sites). $n_{i\alpha }^{N\mathrm{th}}=n_{i\alpha }(n=N)-n_{i\alpha }(n=N-1)$. For $E_{\mathrm{in}}=1$, $E_{\mathrm{out}}=5$, (c) averaged occupation numbers of the $N\mathrm{th}$ added particle at edges ($n_{1A}^{N\mathrm{th}}$ or $n_{NB}^{N\mathrm{th}}$) and inside the chain (${\overline{n}}_{\mathrm{inside}}^{N\mathrm{th}}$) and (d) averaged intra- and intercell hopping order parameters (${\overline{\delta }}_{\sigma ,\mathrm{in}}$ and ${\overline{\delta }}_{\sigma ,\mathrm{out}}$) as functions of system size $N$. Dashed lines are polynomial fits for $1/N$. ${\overline{n}}_{\mathrm{inside}}^{j\mathrm{th}}=\frac{1}{2N-2}{\sum }_{i\alpha \ne 1A,NB}{n}_{i\alpha }^{j\mathrm{th}}$. Inset of (d): Site-resolved hopping order parameters in the chain with $N=6$.

Figure 8

Spatial distributions of particles with different intercell repulsive interactions $U_{\mathrm{out}}$: $n_{i\alpha }$ for (a) $n=5$ and (b) $n=6$, and (c) $n_{i\alpha }^{6\mathrm{th}}$. In (b) one of two degenerate ground states is considered, and the spatial distribution of the other ground state is opposite to that in (b) (The same applies to following figures.). (d) Averaged intra- and intercell hopping order parameters (${\overline{\delta }}_{\sigma ,\mathrm{in}}$ and ${\overline{\delta }}_{\sigma ,\mathrm{out}}$) as functions of the system size $N$ with $U_{\mathrm{out}}=10$. Inset of (d): Site-resolved hopping order parameters with different $U_{\mathrm{out}}$. The intracell interaction $U_{\mathrm{in}}=0$.

Figure 9

Averaged intra- and interhopping order parameters (${\overline{\delta }}_{\sigma ,\mathrm{in}}$ and ${\overline{\delta }}_{\sigma ,\mathrm{out}}$) as functions of the system size $N$ with $U_{\mathrm{in}}=20$ and $U_{\mathrm{out}}=0$. Inset: Site-resolved hopping order parameters with different intracell repulsive interactions $U_{\mathrm{in}}$.

Figure 10

(a) Spatial distributions of particles, (b) hopping order parameters, and [inset of (c)] pairing order parameters with $U_{\mathrm{in}}=0$ and different intercell attractive interactions $U_{\mathrm{out}}$. (c) Averaged intra- and intercell pairing order parameters (${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{in}}$ and ${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{out}}$) as functions of the system size $N$ with $U_{\mathrm{out}}=-20$.

Figure 11

(a) Spatial distributions of particles, (b) hopping order parameters, and (c) pairing order parameters with different intracell attractive interactions $U_{\mathrm{in}}$ and $U_{\mathrm{out}}=0$. (d) Averaged intra- and intercell pairing order parameters (${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{in}}$ and ${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{out}}$) as functions of the system size $N$ with $U_{\mathrm{in}}=-7$.

Figure 12

(a) Spatial distributions of particles and (c) hopping order parameters with different intracell and intercell repulsive interactions and $U_{\mathrm{in}}=U_{\mathrm{out}}=U$. (b) The difference of particles on $A$ and $B$ sublattices $|n_A-n_B|$ as a function of $U$. (d) Averaged intra- and intercell hopping order parameters (${\overline{\delta }}_{\sigma ,\mathrm{in}}$ and ${\overline{\delta }}_{\sigma ,\mathrm{out}}$) as functions of the system size $N$ with $U=30$.

Figure 13

Averaged hopping order parameters ${\overline{\delta }}_{\sigma ,\mathrm{in}/\mathrm{out}}$ as functions of $E_J$ with (a) $U_{\mathrm{in}}=0,U_{\mathrm{out}}=10$ and (b) $U_{\mathrm{in}}=5,U_{\mathrm{out}}=0$. Insets of (a) and (b): site-resolved hopping order parameters ${\delta }_{\sigma ,i/i+,\mathrm{in}/\mathrm{out}}$ for $E_J=0.1$.

Figure 14

(a) Occupation numbers $n_{1A},n_{1B}$ and (b) averaged pairing order parameters ${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{in}/\mathrm{out}}$ as functions of $E_J$ with $U_{\mathrm{in}}=0$ and $U_{\mathrm{out}}=-20$. Spatial distribution of particles $n_{i\alpha }$ and site-resolved pairing order parameters ${\delta }_{\sigma ,i/i+,\mathrm{in}/\mathrm{out}}$ for $E_J=0.03$ are plotted in the insets of (a) and (b), respectively.

Figure 15

(a) Occupation numbers $n_{1B},n_{2A}$ and (b) averaged pairing order parameters ${\overline{\mathrm{\Lambda }}}_{\sigma ,\mathrm{in}/\mathrm{out}}$ as functions of $E_J$ with $U_{\mathrm{in}}=-7$ and $U_{\mathrm{out}}=0$. Spatial distribution of particles $n_{i\alpha }$ and site-resolved pairing order parameters ${\delta }_{\sigma ,i/i+,\mathrm{in}/\mathrm{out}}$ for $E_J=0.01$ are plotted in the insets of (a) and (b), respectively.

Figure 16

(a) Occupation numbers at $A$ and $B$ sublattices and (b) averaged hopping order parameters ${\overline{\delta }}_{\sigma ,\mathrm{in}/\mathrm{out}}$ as functions of $E_J$ with $U_{\mathrm{in}}=U_{\mathrm{out}}=20$. Inset of (b): site-resolved hopping order parameters ${\delta }_{\sigma ,i/i+,\mathrm{in}/\mathrm{out}}$ for $E_J=0.03$.