# Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm

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## Abstract

**:**

## 1. Introduction

## 2. The Quadratic Model

#### 2.1. Hamiltonian

#### 2.2. Tamm States

#### 2.3. Majorana States

#### 2.4. Competition between Tamm and Majorana States

## 3. Effective Low-Energy Model

#### 3.1. Linearization

#### 3.2. Qualitative Interpretation of the Results

#### 3.3. Linear Model with One Boundary and $\Delta =0$

#### 3.4. The Wavefunction of the Bound State for $\phi =3\pi /2$

## 4. Majorana Polarization

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of the energy spectrum for different values of the amplitude of the periodic potential: $B=0$ (black); $B={\mu}_{0}/2$ (green); $B={\mu}_{0}$ (blue); $B=2{\mu}_{0}$ (red). The case $\phi =0$ is shown in panel (

**a**) for the first 200 eigenstates, while $\phi =\pi /2$ in panel (

**b**) for the first 100 eigenstates.

**Figure 2.**Plot of ${E}_{\nu}/{\mu}_{0}$ for $0\le \nu \le 7$ (from black to red) as a function of $\phi $ (panel

**a**) and probability density of the $\nu =0$ state $|{\psi}_{0}{\left(x\right)|}^{2}$ (units ${\mathrm{L}}^{-1}$) as a function of x for $\phi =\pi /2$ (panel

**b**) and $\phi =0$ (panel

**c**). In all panels, the amplitude of the periodic potential is set at $B={\mu}_{0}/2$.

**Figure 3.**Probability density of the zero-energy Majorana state $|{\psi}_{0}{\left(x\right)|}^{2}$ (units ${\mathrm{L}}^{-1}$) as a function of x for $B=0$ and $\Delta ={\mu}_{0}/2$.

**Figure 4.**Probability density of the zero-energy state $|{\psi}_{0}{\left(x\right)|}^{2}$ (units ${\mathrm{L}}^{-1}$) as a function of x for different values of the p-wave pairing strength: $\Delta =0$ (

**a**); $\Delta =B/2$ (

**b**); $\Delta =B$ (

**c**); $\Delta =2B$ (

**d**); $\Delta =8B$ (

**e**). In all panels, $B={\mu}_{0}/4$ and $\phi =\pi /2$.

**Figure 5.**Density plot of the spectral weight $w(n,E)$ (see text) as a function of the index n labeling the eigenstates of a particle in a box of length L and energy E (units ${\mu}_{0}$) for different values of B and $\Delta $: $B=\Delta =0$ (

**a**); $B={\mu}_{0}/4$, $\Delta =0$ (

**b**); $B=\Delta ={\mu}_{0}/4$ (

**c**); $B={\mu}_{0}/4$, $\Delta =2B$ (

**d**).

**Figure 6.**Plot of ${\u03f5}_{j}\left(k\right)$ (units ${v}_{F}/L$) as a function of k (units $\pi /L$) for: $B=3{v}_{F}/L$, $\Delta =0$ (

**a**); $B=3{v}_{F}/L$, $\Delta =3{v}_{F}/L$ (

**b**); $B=3{v}_{F}/L$, $\Delta =6{v}_{F}/L$ (

**c**). In all panels, red lines represent ${\u03f5}_{2}\left(k\right)$; black lines correspond to ${\u03f5}_{1}\left(k\right)$.

**Figure 7.**Plot of the probability amplitude of particle and hole components of the zero-mode wavefunction (units ${\mathrm{L}}^{-1}$) as a function of x: ${\left|u\left(x\right)\right|}^{2}$ (

**a**); ${\left|v\left(x\right)\right|}^{2}$ (

**b**). Here, $B=\Delta =30{v}_{F}/L$ and ${k}_{F}=30\pi /L$.

**Figure 8.**Plot of the Majorana polarization ${P}_{M}$ as a function of $\Delta /B$ for ${k}_{F}=30\pi /L$ and $B=30{v}_{F}/L$.

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**MDPI and ACS Style**

Vigliotti, L.; Cavaliere, F.; Carrega, M.; Ziani, N.T.
Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm. *Symmetry* **2021**, *13*, 1100.
https://doi.org/10.3390/sym13061100

**AMA Style**

Vigliotti L, Cavaliere F, Carrega M, Ziani NT.
Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm. *Symmetry*. 2021; 13(6):1100.
https://doi.org/10.3390/sym13061100

**Chicago/Turabian Style**

Vigliotti, Lucia, Fabio Cavaliere, Matteo Carrega, and Niccolò Traverso Ziani.
2021. "Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm" *Symmetry* 13, no. 6: 1100.
https://doi.org/10.3390/sym13061100