# Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders

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## Abstract

**:**

## 1. Introduction

## 2. Majorana Fermions in the Kitaev Chain

## 3. Ladder of Two Kitaev Chains

**Figure 5.**Low energy part of the ladder energy spectra as a function of $\mu $, evaluated for different values of ${\Delta}_{1}$ and ${t}_{1}$. From the left to the right and from the top to the bottom, the parameters are: $({\Delta}_{1},{t}_{1})=(0,0.3),(0.09,1.2),(0.5,0.6),(0.8,0.4)$.

## 4. Quantum Transport through a Normal/Kitaev Chain/Superconductor Device

## 5. Quantum Transport through a Normal/Kitaev Ladder/Superconductor Device

**Figure 11.**N-KL-SC device: zero-temperature differential conductance (in the unit of $\frac{2{e}^{2}}{h}$) as a function of the energy. Different panels, namely (

**a**,

**e**,

**i**), are obtained by changing the linking position, given by n, between the normal lead and the nearest Kitaev wire of the ladder. The blue and purple modes in panels (

**b**–

**d**,

**f**–

**h**,

**l**–

**n**) represent the modulus squared of the resonant modes on the upper and lower chain of the ladder (shifted by a convenient vertical offset) evaluated at energy values corresponding to the sub-gap conductance peaks. From the left to the right and from the top to the bottom, we have set the following parameters: (

**b**): $n=1$, $E=5\xb7{10}^{-5}$. (

**c**): $n=1$, $E=4.5\xb7{10}^{-3}$. (

**d**): $n=1$, $E=6.5\xb7{10}^{-3}$. (

**f**): $n=61$, $E=5\xb7{10}^{-5}$. (

**g**): $n=61$, $E=4.5\xb7{10}^{-3}$. (

**h**): $n=61$, $E=6.5\xb7{10}^{-3}$. (

**l**): $n=121$, $E=5\xb7{10}^{-5}$. (

**m**): $n=121$, $E=4.5\xb7{10}^{-3}$. (

**n**): $n=121$, $E=6.5\xb7{10}^{-3}$. The remaining model parameters have been fixed as: $\Delta =0.02$, ${t}_{N}={t}_{S}=0.2$, $t=1$, $\mu =0.5$, ${\mu}_{S}={\mu}_{N}=0$, ${\Delta}_{1}=0.09$, ${t}_{1}=0.6$.

## 6. Disorder Effects in a Normal/Kitaev Ladder/Superconductor Device

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Tight Binding Bogoliubov–de Gennes Equations

#### Appendix A.1. Tight Binding Bogoliubov–de Gennes Equations for the Normal Lead

#### Appendix A.2. Tight Binding Bogoliubov–de Gennes Equations for the Superconducting p-Wave Lead

## Appendix B. Conductance Lowering Effects in Branched Quantum Waveguides

**Figure A2.**N-KC-SC device. Zero-temperature differential conductance of the model depicted in Figure A1. The model parameters have been fixed as: $\Delta =0.02$, ${t}_{S}=t=1$, $\mu =0.5$, ${\mu}_{S}={\mu}_{N}=0$, while ${t}_{N}=0.1$, $0.4$, $0.8$, 1 has been used in obtaining the different plots.

**Figure A3.**Transparent limit of the N-KL-SC device. Zero-temperature differential conductance as a function of energy. The model parameters have been fixed as: $\Delta =0.02$, $t={t}_{S}=1$, $\mu =0.5$, ${\mu}_{S}={\mu}_{N}=0$, ${\Delta}_{1}=0.09$, ${t}_{1}=0.6$, while ${t}_{N}=0.1$, $0.4$, $0.8$, 1 has been used in obtaining the different plots.

## Appendix C. Charge Neutrality of Quasi-Majorana Modes

**Figure A4.**Site-dependent charge density ${\rho}_{n}=|{f}_{n}{|}^{2}-{\left|{g}_{n}\right|}^{2}$ as a function of the position along the Kitaev chain device discussed in Figure 8 of the main text. Curves in (

**a**,

**c**,

**e**) correspond to the zero-bias conductance peak in Figure 8 ($E=5\xb7{10}^{-5}$), while the remaining panels correspond to the satellite conductance peak ($E=6\xb7{10}^{-3}$); (

**a**) is computed by setting the model parameters as done in Figure 8b. The average charge density is given by $\overline{\rho}\approx -0.05$; (

**b**) corresponds to Figure 8c with $\overline{\rho}\approx 5$; (

**c**) corresponds to Figure 8e with $\overline{\rho}\approx -0.6$; (

**d**) corresponds to Figure 8f with $\overline{\rho}\approx 6$; (

**e**) corresponds to Figure 8h with $\overline{\rho}\approx -0.25$; (

**f**) corresponds to Figure 8i with $\overline{\rho}\approx 2.5$. Strong deviations from the charge neutrality condition provide evidence of an important contamination of topological properties. Hybridized modes with prevalent electron-like (hole-like) character are defined by $\overline{\rho}>0$ ($\overline{\rho}<0$).

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**Figure 1.**A schematic representation of the ideal Kitaev chain model. Unpaired Majorana zero modes ${c}_{1}$ and ${c}_{2L}$ are localized at the system edges and do not enter the Hamiltonian H. The remaining Majorana modes, namely ${c}_{2j}$ and ${c}_{2j+1}$, recombine into ordinary fermionic excitations ${\tilde{a}}_{j}$.

**Figure 2.**Energy bands of the Kitaev chain in the non-topological phase $\mu =3$ (

**a**), at the phase transition point $\mu =2$ (

**b**), and inside the topological region $\mu =0.5$ (

**c**); (

**d**) represents the modulus squared of Majorana zero-modes wave functions in the topological phase (

**c**). Energy is expressed in units of the hopping amplitude t, while the remaining model parameters have been fixed as $L=100$ and $\Delta =2$.

**Figure 3.**Topological phase diagram of the ladder in the $({t}_{1},\mu )$ plane, given by the winding number for different values of ${\Delta}_{1}$, (${\Delta}_{1}=0,0.09,0.5,0.8$ from top left to bottom right) and for $\Delta =0.8$, $t=1$. The orange, blue and green regions are respectively the regions with 2, 1 and 0 Majorana modes per edge. The black line represents the cut on which we take the spectra in Figure 5.

**Figure 4.**Topological phase diagram of the ladder in the $({\Delta}_{1},\mu )$ plane given by the winding number. The two plots are realized for $\Delta =0.8$, $t=1$ and ${t}_{1}=0.6$ (left panel) or ${t}_{1}=2.1$ (right panel).

**Figure 6.**Topological phase diagram of the ladder in the (${t}_{1}$, $\mu $) plane. The orange, blue and green regions are respectively the regions with 2, 1 and 0 MZMs at one end. The fixed parameters for both the panels are $t=1$, ${t}_{1}=0.6$, while we set $\Delta =0.02$ and ${\Delta}_{1}=0.09$ for the left panel and $\Delta ={\Delta}_{1}=0.09$ for the right panel.

**Figure 7.**(

**a**): Schematic of a tunnel conductance measurement setup where the normal tip position can be changed along the nanowire; (

**b**): tight binding model of the N-KC-SC device. The black chains represent the normal and superconducting p-wave leads with hopping amplitude t, chemical potential ${\mu}_{N}$, ${\mu}_{S}$ and pairing $\Delta $, respectively. The vertical finite line represents the Kitaev chain with parameters: $\mu ,t,\Delta $. The couplings parameters between the leads and the Kitaev nanowire are given by ${t}_{N}$, ${t}_{S}$, ${\Delta}_{S}$.

**Figure 8.**N-KC-SC device: Zero-temperature differential conductance (in the unit of $\frac{2{e}^{2}}{h}$) as a function of the energy in the sub-gap regime. (

**a**,

**d**, and

**g**) are obtained by setting different linking positions, specified by n, between the normal lead and the Kitaev wire. (

**b**,

**c**,

**e**,

**f**,

**h**,

**i**) represent the modulus squared ${\left|\Phi \right|}^{2}\equiv |{f}_{n}{|}^{2}+{\left|{g}_{n}\right|}^{2}$ of the resonant modes along the Kitaev chain evaluated at energy values corresponding to the resonant sub-gap peaks in the conductance. From the left to the right and from the top to the bottom, the parameters used are: (

**b**): $n=1$, $E=5\xb7{10}^{-5}$. (

**c**): $n=1$, $E=6\xb7{10}^{-3}$. (

**e**): $n=61$, $E=5\xb7{10}^{-5}$. (

**f**): $n=61$, $E=6\xb7{10}^{-3}$. (

**h**): $n=121$, $E=5\xb7{10}^{-5}$. (

**i**): $n=121$, $E=6\xb7{10}^{-3}$. The remaining model parameters have been fixed as: $\Delta =0.02$, ${t}_{N}={t}_{S}=0.2$, $t=1$, $\mu =0.5$, ${\mu}_{S}={\mu}_{N}=0$.

**Figure 9.**N-KC-SC device: Differential conductance (in units of $\frac{2{e}^{2}}{h}$) as a function of the energy. Different plots are obtained by setting ${t}_{N}=0.1,0.4,0.8,1$, while fixing the remaining parameters as: $\Delta =0.02$, ${t}_{S}=0.2$, $t=1$, $\mu =0.5$, ${\mu}_{S}={\mu}_{N}=0$, $n=2$.

**Figure 10.**N-KL-SC device. A Kitaev ladder (central region) coupled to a movable normal lead and to a superconducting p-wave lead.

**Figure 12.**N-KL-SC device in the presence of disorder. (

**a**,

**e**,

**i**) show the zero-temperature differential conductance (in the unit of $\frac{2{e}^{2}}{h}$) as a function of the energy for a normal lead attached to the first site of the first Kitaev wire of the ladder and for the three different values of $\delta $, related to the variance of the random on-site potential; (

**b**–

**d**,

**f**–

**h**,

**l**–

**n**) represent the modulus squared of the resonant modes of the upper and lower chain of the ladder evaluated at energy values corresponding to the sub-gap conductance peaks. Model parameters have been fixed as done in first line of Figure 11.

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**MDPI and ACS Style**

Maiellaro, A.; Romeo, F.; Perroni, C.A.; Cataudella, V.; Citro, R. Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders. *Nanomaterials* **2019**, *9*, 894.
https://doi.org/10.3390/nano9060894

**AMA Style**

Maiellaro A, Romeo F, Perroni CA, Cataudella V, Citro R. Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders. *Nanomaterials*. 2019; 9(6):894.
https://doi.org/10.3390/nano9060894

**Chicago/Turabian Style**

Maiellaro, Alfonso, Francesco Romeo, Carmine Antonio Perroni, Vittorio Cataudella, and Roberta Citro. 2019. "Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders" *Nanomaterials* 9, no. 6: 894.
https://doi.org/10.3390/nano9060894