ALBERTA-THY-7-22 Majorana Modes of Giant Vortices Logan Gates1and Alexander A. Penin1y

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ALBERTA-THY-7-22

Majorana Modes of Giant Vortices

Logan Gates1, ∗and Alexander A. Penin1, †

1Department of Physics, University of Alberta, Edmonton, Alberta T6G 2J1, Canada

We study Majorana zero modes bound to giant vortices in topological superconductors or topo-

logical insulator/normal superconductor heterostructures. By expanding in inverse powers of a large

winding number n, we ﬁnd an analytic solution for asymptotically all nzero modes required by the

index theorem. Contrary to the existing estimates, the solution is not pinned to the vortex boundary

and is composed of the warped lowest Landau level states. While the dynamics which shapes the

zero modes is a subtle interference of the magnetic eﬀects and Andreev reﬂection, the solution is

very robust and is determined by a single parameter, the vortex radius. The resulting local density

of states has a number of features which give remarkable signatures for an experimental observation

of the Majorana fermions in two dimensions.

Majorana quasiparticle excitations in various con-

densed matter systems are in a spotlight of theoretical

and experimental studies for over a decade [1–3]. A

renown example of the Majorana quasiparticles are the

zero-energy states bound to the vortices in a topological

superconductor [4, 5] or on the interface between a topo-

logical insulator and a normal superconductor [6]. The

giant vortices of large winding number nare of particular

interest since they host multiple zero modes [7] and can

be used to study highly nontrivial systems of interacting

Majorana states such as the Sachdev-Ye-Kitaev model

[8]. The vortices with n > 1 have already been observed

in mesoscopic semiconductors [9] and can be engineered

in the specially designed heterostructures. Unambiguous

identiﬁcation of the zero modes in the vortex core poses a

great challenge for the modern experimental techniques

[10, 11], and the measurement of their spatial distribu-

tion is a promising method to distinguish the true Ma-

jorana states [12]. Thus, the search and design of the

systems with the signature spatial properties of the zero

modes as well as the theoretical evaluation of their shape

and the local density of states are of primary interest.

Though the existence and stability of the zero modes are

predicted by the index theorem [13, 14] and can be veri-

ﬁed through a qualitative analysis of the ﬁeld equations,

such an approach is too coarse to catch subtle dynami-

cal eﬀects which signiﬁcantly aﬀect the structure of the

solution. On the other hand, the brute-force numerical

simulations may be insuﬃcient to identify the univer-

sal properties and characteristic features of the solution,

hence a form of quantitative analytic approach is manda-

tory. A systematic analysis in this case is complicated by

nonintegrable nonlinear nature of the vortex dynamics.

In this Letter we present such an analysis for the case

of the giant vortices. It is based on a novel method of

the asymptotic expansion in inverse powers of the vortex

winding number [15, 16] and provides the analytic solu-

tion for almost all of the nzero modes required by the

index theorem. Our prediction for the density of states

has a number of remarkable properties which have been

overlooked before and can be used to get compelling ex-

perimental evidence of the Majorana vortex states.

The equations for the Majorana zero modes can be

inferred from the Jackiw-Rossi theory of charged mass-

less two-component Dirac fermion in 2 + 1 dimensions

described by the Lagrange density [7]

LJR =i¯

ψ/

Dψ +1

2¯

ψψcφ+¯

ψcψφ∗,(1)

where /

D=γµDµ,Dµ=∂µ+iAµis the gauge covariant

derivative, the Dirac matrices reduce to the Pauli ma-

trices γµ= (σ3, iσ2,−iσ1), ψc=−iσ1ψ∗is the charge

conjugate spinor, and φis a scalar ﬁeld of charge 2 rep-

resenting the pair potential. For the static zero-energy

states the ﬁeld equations for the spinor components read

D±ψ±+φψ∗±= 0 ,(2)

where the chiral derivatives are D±=D1±iD2. We

are interested in the solution of Eq. (2) in the back-

ground of the axially symmetric Abrikosov vortex [17] of

the winding number n, which implies the following ﬁeld

conﬁguration in polar coordinates φ(r, θ) = f(r)einθ ,

Aθ=−na(r)/2, Ar= 0, with f(0) = a(0) = 0 and

f(∞) = f∞,a(∞) = 1. Then the negative chirality

equation does not have a normalizable solution and the

nzero modes of positive chirality can be written as fol-

lows

ξ+

l=1

√2eilθψ+

l+ei(n−1−l)θψ+

n−1−l,

η+

l=i

√2eilθψ+

l−ei(n−1−l)θψ+

n−1−l,

(3)

where 0 ≤l≤n/2−1 for even nand 0 ≤l≤(n−1)/2,

η+

(n−1)/2= 0 for odd n. The partial wave amplitudes

satisfy the following equations

d

dr −l

r+na

2rψ+

l+fψ+

n−1−l= 0 ,

d

dr −n−l−1

r+na

2rψ+

n−1−l+fψ+

l= 0 .

(4)

After identiﬁcation of ψland ψn−1−lwith the compo-

nents of the Nambu spinor, and of fwith the pair po-

tential the above system reproduces the Bogoliubov-de-

Gennes equations for the Majorana vortex zero modes of

arXiv:2210.04908v1 [cond-mat.mes-hall] 10 Oct 2022

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ALBERTA-THY-7-22MajoranaModesofGiantVorticesLoganGates1,andAlexanderA.Penin1,y1DepartmentofPhysics,UniversityofAlberta,Edmonton,AlbertaT6G2J1,CanadaWestudyMajoranazeromodesboundtogiantvorticesintopologicalsuperconductorsortopo-logicalinsulator/normalsuperconductorheterostructures.Byexpandingininver...

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ALBERTA-THY-7-22 Majorana Modes of Giant Vortices Logan Gates1and Alexander A. Penin1y

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