# Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice

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## Abstract

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## 1. Introduction

## 2. Model and Hamiltonian

## 3. Results

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Modified Real-Space-Pfaffian Method

**Test on the Kitaev model (${V}_{n}=V$) [34]**: We considered $t=1$, $\alpha =(\sqrt{5}-1)/2$, $\mathsf{\Delta}=0.5t$, and $L=5$ in all our tests. $V=1.5t$ and $V=2.5t$ were two chosen parameters. We know that the topological boundary is located at ${V}_{c}=2t$. Therefore, $V=1.5t$ corresponds to the topological non-trivial phase and gives $M=-1$; $V=2.5t$ corresponds to the topological non-trivial phase and gives $M=1$; taking $V=1.5t$ and other parameters into h and performing the Schur decomposition, we have $Det\left(U\right)=1$ and $\mathrm{Pf}\left(D\right)=-8.4688$. According to Equation (16), we obtain the topological invariant $M=sgn\left(\mathrm{Pf}\right(h\left)\right)=-1$. Next, we take $V=2.5t$ and other parameters into h. After performing the Schur decomposition, we have $Det\left(U\right)=-1$ and $\mathrm{Pf}\left(D\right)=-3.8469$. According to Equation (16), we obtain $M=1$. If leaving the V changing while other parameters invariable, by means of this method, we can obtain the topological phase diagram of the Kitaev model, as shown in Figure A1.

**Test on the quasiperiodic**

**p**

**-wave superconducting model [44,47]**: This model requires ${V}_{n}=Vcos\left(2\pi \alpha n\right)$, and its topological boundary is located at ${V}_{c}=2t+2\mathsf{\Delta}$. Without loss of generality, we take $\mathsf{\Delta}=0.5t$; thus, the topological boundary is ${V}_{c}=3t$. $V=2.8t$ and $V=3.2t$ are two chosen parameters, corresponding to the topological non-trivial phase ($M=-1$) and topological trivial phase ($M=1$), respectively. We first tested the case with $V=2.8t$. After performing the Schur decomposition, we obtain $Det\left(U\right)=-1$ and $\mathrm{Pf}\left(D\right)=1.6637$. Therefore, the topological invariant is $M=sgn\left(Det\right(U\left)\mathrm{Pf}\right(D\left)\right)=-1$. Next, we take $V=3.2t$ and other parameters into h. After performing the Schur decomposition, we obtain $Det\left(U\right)=-1$ and $\mathrm{Pf}\left(D\right)=-3.9369$, so the topological invariant is $M=sgn\left(Det\right(U\left)\mathrm{Pf}\right(D\left)\right)=1$. If we change the V continuously and keep other parameters invariable, then we can obtain the topological phase diagram, as shown in Figure A2.

**Figure A2.**(Color online) M-$V/t$ topological phase diagram of the quasiperiodic p-wave superconducting model.

**Test on our model (see Equation (1))**: The potential is presented in Equation (2). We considered $b=0.95$ in our test. According to the topological phase diagram in Figure 1, we know the numerical topological boundary is located at about ${V}_{c}=1.5t$. $V=1.2t$ and $V=1.7t$ are two chosen parameters, corresponding to the topological non-trivial phase and topological trivial phase, respectively. We first tested the case with $V=1.2t$. After performing the Schur decomposition, we have $Det\left(U\right)=-1$ and $\mathrm{Pf}\left(D\right)=9.1651$. Therefore, the topological invariant is $M=sgn\left(Det\right(U\left)\mathrm{Pf}\right(D\left)\right)=-1$. Next, we tested the case with $V=1.7t$. In the same way, we performed the Schur decomposition, then we obtain $Det\left(U\right)=-1$ and $\mathrm{Pf}\left(D\right)=-6.9383$. Therefore, in this case, the topological invariant is $M=sgn\left(Det\right(U\left)\mathrm{Pf}\right(D\left)\right)=1$. If we change V continuously and leave other parameters invariable, then we can obtain the topological phase diagram, as shown in Figure A3.

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**Figure 1.**(Color online) b-$V/t$ topological phase diagram for systems with $\alpha =(\sqrt{5}-1)/2$, $\mathsf{\Delta}=0.5t$. $M=-1$ corresponds to the topological non-trivial phase, and $M=1$ corresponds to the topological trivial phase. The blue dashed line denotes the phase boundary.

**Figure 2.**(Color online) The energy gap ${\mathsf{\Delta}}_{g}$ versus V with various b. Other involved parameters are $\alpha =(\sqrt{5}-1)/2$, $\mathsf{\Delta}=0.5t$, and $L=1000$.

**Figure 3.**(Color online) (

**a**) Excitation energy spectrum of the system under the open boundary condition. (

**b**–

**e**) are respectively the spatial distributions of $\mathsf{\Phi}$ and $\mathsf{\Psi}$ for the lowest excitation state with $V=1.5t$ ($V=2t$). Other involved parameters are $L=500$, $b=0.7$, $\alpha =(\sqrt{5}-1)/2$, and $\mathsf{\Delta}=0.5t$.

**Figure 4.**(Color online) The excitation spectrum and IPR as a function of V with $\mathsf{\Delta}=0.5t$ in (

**a**), $\mathsf{\Delta}=1.5t$ in (

**b**), $\mathsf{\Delta}=2.5t$ in (

**c**), and $\mathsf{\Delta}=4.5t$ in (

**d**). Other involved parameters are $b=0.7$, $\alpha =(\sqrt{5}-1)/2$, and $L=500$.

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**MDPI and ACS Style**

Cheng, S.; Zhu, Y.; Gao, X.
Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice. *Symmetry* **2022**, *14*, 371.
https://doi.org/10.3390/sym14020371

**AMA Style**

Cheng S, Zhu Y, Gao X.
Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice. *Symmetry*. 2022; 14(2):371.
https://doi.org/10.3390/sym14020371

**Chicago/Turabian Style**

Cheng, Shujie, Yufei Zhu, and Xianlong Gao.
2022. "Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice" *Symmetry* 14, no. 2: 371.
https://doi.org/10.3390/sym14020371