Simulation of Majorana Modes in Two-Dimensional

Graphene Nanostrip using Kwant

Abstract
Majorana bound states (MBS) are zero-energy quasiparticle excitations localized at the bound-
aries of topological superconductors. Their emergence is dictated by the bulk-boundary corre-
spondence, which ensures that a topologically nontrivial phase cannot transition to a trivial phase
without closing the energy gap at the interface. Due to their non-Abelian statistics and inherent
robustness against local perturbations, MBS are promising candidates for fault-tolerant quantum
computation. Recent advances suggest that graphene-based systems, with their high tunability
and unique electronic properties, could serve as viable platforms for realizing topological super-
conductivity.
In this study, we investigate the emergence of Majorana zero modes (MZMs) in a two-dimensional
(2D) graphene nanostrip using a tight-binding Hamiltonian approach. By incorporating proximity-
induced superconductivity, Rashba spin-orbit coupling, and a Zeeman field, we simulate the system
using the Kwant package to compute energy spectra and probe the existence of zero-energy Majo-
rana modes. Our findings aim to provide theoretical insights into the feasibility of graphene-based
platforms for realizing robust topological superconducting phases, offering potential pathways to-
wards scalable and fault tolerant quantum computation.

1 Introduction
A majorana fermion is a charge neutral particle satisfying the Dirac equation and which, contrarily to
ordinary Dirac fermions, coincides with its own antiparticle.

γα† = γα (1)

In condensed matter, Dirac and Majorana fermions emerge as low energy excitations realized in, e.g.,
superfluid Helium-3, graphene[1], topological insulators, and superconductors. In condensed matter
systems, majorana bound states are zero-energy, charge-neutral, particle hole symmetric and spatially
separated quasiparticle excitations exponentially localised at the boundaries of a topological supercon-
ductor. In order to do an adiabatic transformation from topological state to trivial state, the system
should go through a quantum phase transition and energy gap must close.
Due to nonlocality, charge-neutrality and topologically protected, majorana modes are robust, i.e.,
their existence remains unscathed as long as the energy scale of perturbations does not exceed the
particle hole energy gap. Only a strong enough perturbation which can close the particle hole gap can
destroy the modes.
Like conventional fermions, majorana operators obey anticommutation relations. However, unlike
ordinary fermions, Majorana modes are their own antiparticle. Additionally, while pauli exclusion
principle prohibits two identical fermions from occupying same state, this restriction does not apply to
majorana modes – similar to bosons. Majorana modes obey non-Abelian statistics, where the exchange
of two modes leads to transformation that depend on the order of exchanges.
In a topological quantum computer, quantum information is encoded nonlocally in spatially separated
Majorana modes and manipulated through exchange operations, which serve as quantum gates. Their
non-Abelian statistics and inherent protection from decoherence due to robustness against perturba-
tions make them well-suited for fault-tolerant quantum computation [2].
One of the most extensively studied platforms for realising MFs is a Rashba spin orbit coupled one-
dimensional(1D) semiconductor nanowire with proximity induced superconductivity and zeeman field.
A widely studied platform for realizing Majorana fermions is a 1D semiconductor nanowire with Rashba

1
spin-orbit coupling with proximity induced superconductivity and zeeman field where the system en-
ters a topological phase, hosting Majorana bound states localized at its ends.
In this project, we study a two-dimensional graphene nanostrip with proximity-induced supercon-
ductivity, Rashba spin-orbit coupling, and a Zeeman field using numerical simulations with Kwant, a
widely used tool for quantum transport studies. We begin by analyzing the 1D Kitaev chain, simulating
it with Kwant to identify the topological phase regime[4, 5]. Next, we extend our study to the tight-
binding Oreg-Lutchyn Hamiltonian, explicitly formulating it using the Bogoliubov–de Gennes (BdG)
formalism and the Nambu spinor representation. We then adapt this framework to model graphene.
Through simulations, we explore the existence of a topological phase, examining the influence of system
parameters on its emergence and stability.

2 Research Methodology
We begin with simulating the Kitaev chain model in 1D using Kwan whose tight binding Bogoliubov
Hamiltonian is represented as,
N    N −1   
1 X †  µ 0 cn 1 X †  t −∆ cn+1
H= c cn − c cn + h.c.
2 n=1 n 0 −µ c†n 2 n=1 n ∆ −t c†n+1

where µ is the chemical potential, t is the hopping amplitude, ∆ is the p-wave superconducting pairing
term, and c†n (cn ) represents the creation (annihilation) operator at site n.
Theoretically, a topological superconductor with spatially separated Majorana end modes can be
realized in a 1D lattice of spinless fermions with p-wave superconducting pairing. However, this
model is impractical since true spinless fermions do not exist, superconductivity in 1D is unstable
against quantum fluctuations, and spin-triplet p-wave superconductors are rare in nature. A more
realistic approach is to consider a heterostructure consisting of a two-dimensional topological insulator
coupled to a conventional superconductor. There are 3 main ingredients required to obtain topological
protected majorana modes: time-reversal symmetry breaking, proximity induced pairing and spin
orbit coupling[2]. Engineering a spinless topological superconductor in a real system is challenging
since true spinless electrons do not exist. In an open chain, Majorana Kramers pairs can form, but
they are vulnerable to perturbations. To obtain robust, unpaired, and spatially separated Majorana
modes, time-reversal symmetry must be explicitly broken to lift Kramers degeneracy. Here, It is done
by applying an external magnetic field.
In one-dimensional systems, quantum fluctuations suppress superconducting phase transitions at finite
temperatures. This limitation can be mitigated by coupling to a bulk 3D superconductor, where
electron tunneling across the interface induces superconducting correlations via the proximity effect.
The last ingredient is the Spin-orbit coupling, which is crucial for maintaining the particle-hole gap in
the topologically nontrivial phase. It splits electrons into distinct energy bands, and in the lowest band,
those crossing the Fermi level share the same spin, preventing conventional spin-singlet pairing. This
would typically close the particle-hole gap, hindering Majorana mode formation. However, spin-orbit
coupling generates a momentum-dependent effective magnetic field, tilting electron spins in opposite
directions for opposite momenta. This tilt allows superconducting pairing within the same band,
keeping the particle-hole gap open even at finite magnetic fields. Bringing all the ingredients, the tight
binding Hamiltonian is represented as:
N −1 h  
i c
c†n↑ c†n↓ · (−t − iα̃σy ) n+1↑ + h.c.
X
H=
cn+1↓
n=1

N h  
i c
c†n↑ c†n↓
X
+ · (b · σ + 2t − µ) n↑
cn↓
n=1
N   
X 1 cn↑
cn↓ · − ∆eiϕ iσy
 
+ cn↑ + h.c.
2 cn↓
n=1

where [c†n↑ , c†n↓ )]

2
σy is the Pauli matrix, b · σ corresponds to the Zeeman term with b being the Zeeman field, µ is the
chemical potential, and ∆eiϕ iσy represents the superconducting pairing term with phase ϕ.

3 Expected Results
This project aims to investigate Majorana zero modes (MZMs) in one-dimensional (1D) topological
superconductors and extend the analysis to graphene-based systems. We begin with the Kitaev chain
model, where the existence of MZMs is expected at the system boundaries in the topological phase. The
topological regime will be identified by analyzing the energy spectrum as a function of key parameters
such as the chemical potential. A closing of the bulk energy gap or a transition between trivial
and topological states will indicate phase transitions. In the identified topological phase, we expect
Majorana modes to appear at zero energy, localized at the chain’s edges.
We then consider the Oreg-Lutchyn Hamiltonian, which incorporates Zeeman splitting, spin-orbit
coupling, and superconducting pairing. The energy dispersion will be examined under variations of
Zeeman field strength, superconducting pairing, chemical potential and Rashba spin orbit coupling
strength to determine the conditions under which a topological superconducting phase emerges. Once
the topological regime is established, we will simulate the system on a discrete lattice and verify the
emergence of MZMs at the boundaries.
After benchmarking our results for a 1D nanowire, we will extend the model to a graphene nanostrip.
We aim to determine whether a topological phase exists in this system and, if so, to confirm the
presence of localized MZMs. This study will contribute to understanding the feasibility of graphene-
based platforms for realizing topological superconductivity and potential applications in fault-tolerant
quantum computing.

4 References
References
[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic
properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
[2] P. Marra, Majorana nanowires for topological quantum computation: A tutorial, Graduate School
of Mathematical Sciences, The University of Tokyo, and Department of Physics, Keio University
(2022).
[3] A. Yu. Kitaev, Unpaired Majorana fermions in quantum wires, arXiv:cond-mat/0010440v2 [cond-
mat.mes-hall] (2000).
[4] S. Huang, Introduction to Majorana Zero Modes in a Kitaev Chain, arXiv:2111.06703v2 (2021).

[5] A. Yazdani, F. von Oppen, B. I. Halperin, and A. Yacoby, Hunting for Majoranas,
arXiv:2306.12473v2 (2023).