# Topological Edge States of a Majorana BBH Model

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Topological Phase

## 4. Topological Properties of Finite Size Systems

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Non-Local Fermionic Correlations

#### Appendix A.1. Covariance Matrix

#### Appendix A.2. Fermionic Correlations in Degenerate Ground State Manifolds

## References

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**Figure 1.**(

**a**) Lattice geometry in the Majorana basis, the green plaquette identifies the translational invariant unit cell, while pink and brown circles are the Majorana modes a and b associated with a complex fermionic mode. w and v are, respectively, the intracell and intercell couplings, a dashed line indicates a negative coupling. (

**b**) Fully dimerized limit of the lattice model depicted in panel (

**a**) with $w=0$ and $v\ne 0$. The mapping to complex fermionic modes is shown in panel (

**c**) for generic w, v. Indeed, in (

**c**), the following identities hold: $\mu =w$, $t=\Delta =-v/2$ and ${\Delta}_{1}$ is staggered, with $-iw$ associated with continuous red lines and $-iv$ associated to dashed red lines.

**Figure 2.**(

**a**) Topological phase diagram of a 2D Majorana Benalcazar–Bernevig–Hughes (BBH) model in the parameter space v, w. Blue and yellow regions identify the topological and trivial phases, respectively. The red horizontal line is the cut at which we evaluate the lowest energy spectrum in Figure 3 panel (

**a**). Panel (

**b**) shows the spectrum along the $MX\Gamma M$ path in the Brillouin zone in a trivial (topological) phase $w=2$, $v=1$ ($w=1$, $v=2$) and panel (

**c**) shows the spectrum at the phase transition point $w=v=1$.

**Figure 3.**(

**a**) Low energy spectrum of a strip following the horizontal red cut of Figure 2 panel (

**a**). Here, w is in units of v. In panel (

**b**), we show the square modulus of the lowest four energy modes corresponding to a gap closing point (red point of panel (

**a**)) and in the gapped phase (green point of panel (

**a**)). The sizes of the strip for all the plots are $L=100$ and $N=50$.

**Figure 4.**Scheme of the lowest energy many body states in a strip with finite size and in a topological point of the phase diagram. ${t}_{p}$ denotes the even, odd parity sector, ${\delta}_{0}$ and ${\delta}_{1}$ are the energy gaps between the ground states and the first excited states, ${E}_{0}$ and ${E}_{1}$ are the energy separation between the second and the first excited states, $\u03f5$ is the energy separation between the ground states of the two parity sectors and the subscripts 0 and 1 denote the parities.

**Figure 5.**Plots of the energy gaps at varying the sizes of the Majorana BBH strip L and N, as in the panel labels. In panel (

**a**,

**b**), the length is fixed $L=68$ and the behavior of ${\delta}_{0}$, ${\delta}_{1}$, $\u03f5$ (panel (

**a**)) and ${E}_{0}$, ${E}_{1}$ (panel (

**b**)) are plotted for different values of the width $N=4$, 6, 8, 10, 14. The same plots obtained by fixing $N=14$ and considering $L=16$, 24, 32, 48, 56, 61, 68 are reported in panels (

**c**,

**d**). The model parameters have been fixed as $w=0.1$, $v=1$.

**Figure 6.**Fermion correlations $<{c}_{1,1}^{\u2020}{c}_{m,l}>$ (panels (

**a**,

**c**)) and $<{c}_{L,N}^{\u2020}{c}_{m,l}>$ (panels (

**b**,

**d**)) for a strip with $L=100$, $N=30$ (panels (

**a**,

**b**)) and $L=30$, $N=100$ (panels (

**c**,

**d**)). The lattice sites indices are m and l, and the parameters have been fixed in a topological point: $w=0.1$, $v=1$.

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Maiellaro, A.; Citro, R.
Topological Edge States of a Majorana BBH Model. *Condens. Matter* **2021**, *6*, 15.
https://doi.org/10.3390/condmat6020015

**AMA Style**

Maiellaro A, Citro R.
Topological Edge States of a Majorana BBH Model. *Condensed Matter*. 2021; 6(2):15.
https://doi.org/10.3390/condmat6020015

**Chicago/Turabian Style**

Maiellaro, Alfonso, and Roberta Citro.
2021. "Topological Edge States of a Majorana BBH Model" *Condensed Matter* 6, no. 2: 15.
https://doi.org/10.3390/condmat6020015