Majorana bound states in the finite-length chain
Recent experiments investigating edge states in ferromagnetic atomic chains on superconducting substrate are analyzed. In particular, finite size effects are considered. It is shown how the energy of the Majorana bound state depends on the length of the chain, as well as on the parameters of the m...
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irk-123456789-1279642018-01-01T03:03:33Z Majorana bound states in the finite-length chain Zvyagin, A.A. Низкоразмерные и неупорядоченные системы Recent experiments investigating edge states in ferromagnetic atomic chains on superconducting substrate are analyzed. In particular, finite size effects are considered. It is shown how the energy of the Majorana bound state depends on the length of the chain, as well as on the parameters of the model. Oscillations of the energy of the bound edge state in the chain as a function of the length of the chain, and as a function of the applied voltage (or the chemical potential) are studied. In particular, it has been shown that oscillations can exist only for some values of the effective potential. 2015 Article Majorana bound states in the finite-length chain / A.A. Zvyagin // Физика низких температур. — 2015. — Т. 41, № 8. — С. 806–811. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 71.10.Pm, 74.20.–z http://dspace.nbuv.gov.ua/handle/123456789/127964 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Zvyagin, A.A. Majorana bound states in the finite-length chain Физика низких температур |
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Recent experiments investigating edge states in ferromagnetic atomic chains on superconducting substrate are
analyzed. In particular, finite size effects are considered. It is shown how the energy of the Majorana bound state
depends on the length of the chain, as well as on the parameters of the model. Oscillations of the energy of the
bound edge state in the chain as a function of the length of the chain, and as a function of the applied voltage (or
the chemical potential) are studied. In particular, it has been shown that oscillations can exist only for some values
of the effective potential. |
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Article |
| author |
Zvyagin, A.A. |
| author_facet |
Zvyagin, A.A. |
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Zvyagin, A.A. |
| title |
Majorana bound states in the finite-length chain |
| title_short |
Majorana bound states in the finite-length chain |
| title_full |
Majorana bound states in the finite-length chain |
| title_fullStr |
Majorana bound states in the finite-length chain |
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Majorana bound states in the finite-length chain |
| title_sort |
majorana bound states in the finite-length chain |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2015 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/127964 |
| citation_txt |
Majorana bound states in the finite-length chain / A.A. Zvyagin // Физика низких температур. — 2015. — Т. 41, № 8. — С. 806–811. — Бібліогр.: 14 назв. — англ. |
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Физика низких температур |
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AT zvyaginaa majoranaboundstatesinthefinitelengthchain |
| first_indexed |
2025-07-09T08:06:25Z |
| last_indexed |
2025-07-09T08:06:25Z |
| _version_ |
1837155892336787456 |
| fulltext |
© A.A. Zvyagin, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8, pp. 806–811
Majorana bound states in the finite-length chain
A.A. Zvyagin
Max-Planck-Institut für Physik komplexer Systeme, Noethnitzer Str. 38, D-01187, Dresden, Germany
B.I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: zvyagin@ilt.kharkov.ua
Received May 15, 2015, published online June 23, 2015
Recent experiments investigating edge states in ferromagnetic atomic chains on superconducting substrate are
analyzed. In particular, finite size effects are considered. It is shown how the energy of the Majorana bound state
depends on the length of the chain, as well as on the parameters of the model. Oscillations of the energy of the
bound edge state in the chain as a function of the length of the chain, and as a function of the applied voltage (or
the chemical potential) are studied. In particular, it has been shown that oscillations can exist only for some va-
lues of the effective potential.
PACS: 71.10.Pm Fermions in reduced dimensions;
74.20.–z Theories and models of superconducting state.
Keywords: Majorana edge states, topological superconductivity.
Majorana fermions have attracted a great attention due to
their main features, comparing to usual Dirac's fermions.
Majorana fermions are particles, identical to their own anti-
particles. The combination of a pair of Majorana fermions
can form a Dirac fermion. Also, Majorana fermions satisfy a
non-Abelian statistics, different from the Fermi–Dirac one.
It permitted [1 to point out potential use of Majorana fermi-
ons as qubits (elementary cells of quantum computers) in
fault-tolerant topological computations. Their another use-
fulness in quantum computation is caused by the fact that
spatially separated pairs of Majorana fermions can encode
information as a highly nonlocal qubit, minimizing that way
the decoherence of the quantum computer. This is why, the
search for Majorana fermions is among the most prominent
tasks for physicists.
For conventional superconductors with s-wave pairing,
superpositions of electrons and holes carrying opposite spin
are different from Majorana's construction. Majorana fermi-
ons can emerge in special superconductors, in which elec-
trons and holes with the same value of spin can be paired.
Kitaev has proposed to use zero-energy Majorana bound
states at the opposite edges of a superconducting chain for
the construction of a composite nonlocal qubit [2]. He has
considered the simplest model of one-dimensional (1D)
spinless fermions with pairing. 1D systems are important for
the potential use in the quantum computation because:
(i) their quantum features are enhanced (comparing to sys-
tems of higher dimensionality) due to the 1D peculiarity in
the density of states, and (ii) theorists can obtain exact (non-
perturbative), often analytic results in 1D. The latter can be
used for comparison of experimental data with theoretical
predictions, very important for probabilistic quantum com-
putation.
For the realization of Kitaev's scenario several 1D su-
perconducting systems were proposed: low-dimensional
topological insulators [3], quantum wires with the strong
spin-orbit coupling in the external magnetic field in the
vicinity of the standard s-wave superconductor [4], and
ferromagnetic chains on the surface of such a superconduc-
tor [5]. Experimental attempts to implement the semicon-
ductor wire proposal were successful to observe the zero-
bias peak in the tunneling spectroscopy studies of hybrid
superconductor–semiconductor nanostructures [6]. That
detected peak was explained as the manifestation of the
Majorana edge states at zero-energy in a quantum wire.
However, observed zero-bias peaks can be also explained
as the manifestation of the disorder, or the Kondo reso-
nance [7]. Moreover, the disadvantage of tunneling studies
in semiconductor wires is that they lack the ability to spa-
tially resolve zero bias peak features together with the ze-
ro-energy of the observed quantum state — to demonstrate
that they are really connected with the edge Majorana
states. This is why, ferromagnetic chains in the vicinity of
the s-wave superconductor (e.g., on its surface) promise
Majorana bound states in the finite-length chain
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 807
more advantages in observation and manipulation with
edge Majorana modes [5].
Recently zero-energy edge states were observed in fer-
romagnetic iron atomic chains (from 30 up to 500 Å) on
the surface of superconducting lead using spectroscopic
imaging technique [8]. Excitations in the bulk of chains
were gapped due to the proximity effect of the s-wave su-
perconducting lead substrate, and the strong spin-orbit
coupling in Pb yielded the effective p-wave pairing. The
spin-orbit coupling (necessary for the transformation of the
s-wave pairing of the substrate to the effective p-pairing in
the chain [9]) was estimated to be 100 meV in Ref. 8,
comparing to the much smaller value in semiconducting
quantum wires ( 0.05 meV) [6]. Zero bias peaks disap-
peared over a distance of 10 Å from the edges of studied
magnetic chains. The Kondo resonance as the explanation
of the zero-energy states was excluded: The application of
the external magnetic field (up to 0.1 T) suppressed super-
conductivity, but did not suppress the zero bias peak, as
was expected from the theory of the Kondo effect. Also, in
accordance with Anderson's theorem, defects of the sub-
strate did not produce in-gap states. Finally, spin-polarized
scanning tunneling microscope measurements did not
show substantial magnetization changes at the edges of
chains, which excludes the suppression of the pairing at the
edges of chains, hence producing zero-energy peaks.
Despite such a great progress in the observation of
(Majorana) edge modes in ferromagnetic atomic chains on
a superconducting surface, some loose ends remain to con-
tend with [10]. In fact, it was clear from the estimates
[2,11] that Majorana edge fermions in finite-length (of the
size L) quantum chains have to hybridize to form a con-
ventional Dirac fermion with the energy proportional to
exp( / ),L where is the coherence length of the super-
conducting chain. Obviously, for large lengths L and
small enough coherence lengths, ,L the energy of
such a bound state goes to zero. However, in the experi-
ment [8] the energy gap of the induced superconductivity
was very small, about 0.2 meV, comparing to the band-
widths of order of 0.2–0.7 eV. That implies (together with
the not so pronounced gap structure) long coherence
lengths. In this situation, the opposite case L can per-
sist, which means that Majorana edge states strongly hy-
bridize to form a Dirac fermion with essentially nonzero-
energy. The short-distance physics of open superconduct-
ing chains with edge Majorana modes is not yet fully un-
derstood [10].
Then, the goal of the present study follows: to investi-
gate analytically how the energy of bound edge states in
open superconducting chains depends on the length. In the
present work we consider first the Kitaev chain model of
the finite length. We show how the energy of the bound
state depends on the length of the chain, as well as on the
parameters of the model, like the bandwidth, the pairing
amplitude, and the potential. Depending on the value of the
potential, the wavenumbers of the bound state can be either
totally imaginary, or complex. The real part of the wave-
number implies standing wave, which magnitude decays
exponentially (caused by the imaginary part of the wave-
number) from the distance of the edge. Hence, oscillations
of the energy of the bound edge state in the Kitaev chain as
a function of the length of the chain have to exist. Im-
portantly, such oscillations can exist only for some region
of the values of the effective voltage. Oscillations of the
energy of such a bound state as a function of the applied
voltage (or change of the chemical potential) can be also
observed. We have mapped the effective Hamiltonian of
the ferromagnetic metallic chain in the vicinity of the s-
wave superconducting substrate to the Kitaev model with
modified pairing amplitude and effective potential, and
checked whether predicted effects of the finite size of the
chain can be observed in potential experiments on such
ferromagnetic atomic chains.
To start with, let us consider the model, which de-
scribes experiments on metallic magnetic chains situated
on the s-wave superconductor substrate. The Hamiltonian
of such a system can be written in the form [8]
† †
, eff
, , , , , ,
=m j jj j
j j j
t c c U c c
†
,,
, , ,
ˆˆ ( )R jj j soj
j j
it z d s c c L s +
† †
in
, , ,
( h.c.) ,
j j
j j
c c Jm s (1)
where
†
jc ( )jc creates (destroys) an electron belong-
ing to the -orbital (for the magnetic metallic chain for
3d orbitals we have three bands with the magnetic quan-
tum numbers 0, 1 and 2) with the spin projection = ,
at the site j ( ,j j denote nearest and, possibly, next-
nearest neighbors), ,t are the hopping integrals in the
Slater–Koster approximation for the tight-binding Hamil-
tonian, J is the exchange integral, m is the magnetization
per site of the ferromagnetic chain (it is directed perpen-
dicular to the chain), s is the spin operator, L is the opera-
tor of the orbital moment, so is the on-site spin-orbit
coupling constant for 3d orbitals of the magnetic ion of
the chain, Rt is the Rashba spin-orbit coupling due to the
hybridization between 3d orbitals of ions of the metallic
magnetic chain and p-orbitals of the superconducting sub-
strate, in describes the proximity-induced pairing poten-
tial caused by the interaction of the electrons of the metal-
lic chain with the s-wave superconducting substrate, and,
finally, effU defines the effective on-site potential. Only
minority spin bands for Fe chains [8] cross the Fermi level
with the hopping integrals of order of 0.6702; 0.5760 and
0.1445 eV. Then, for each band, for which the topological
A.A. Zvyagin
808 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
superconductivity is expected to exist, we can write the
following approximate Hamiltonian
†
appr = ([ ]( [ jj
j
w U c c
†
1,,
1
( ) h.c.
2
y jjw i c c
† † †
in 1
) ( h.c.) ,] )z jj j j
J c c c c (2)
where w is the effective width of the band for the consid-
ered minority spin band (one of the mentioned above val-
ues of , ),t U is the sum of the applied external voltage
and internal potential (e.g., the on-site scalar potential in-
duced by the magnetic atoms in the chain), and is the
effective spin-orbit coupling constant (both on-site and
Rashba terms can contribute). The Hamiltonian is similar
(up to the re-definition of constants) to the Hamiltonian of
the quantum wire with the strong spin-orbit coupling in the
external magnetic field (here the effective molecular field
/2J of the magnetic chain plays the role of the external
field), and with the proximity-induced pairing in , con-
sidered [4] in the long-wave limit. First, let us diagonalize
the in-independent part of the Hamiltonian apprH by the
unitary transformation (turning spins about the x axis).
After such a turn the effective p-wave pairing between
electrons with the same projection of the spin takes place,
which is proportional to in / .J Then, as it was pointed
out [12], we can project out the upper unoccupied band,
which is legitimate for inJ (it is the case for the ex-
periment in [8], see below). The effective Hamiltonian for
the remaining band (in which edge Majorana fermions can
exist) has the form of Eq. (3) with eff in= 2 /J
and eff = ( / 2) .V V J U w Hence, the most im-
portant part of the effective Hamiltonian, which describes
the behavior of the electrons of the ferromagnetic metallic
chain in the vicinity of the s-wave superconductor with the
strong spin-orbit coupling, is equivalent to the Hamiltonian
of the Kitaev chain with the renormalized effective poten-
tial and effective pairing amplitude.
Now let us turn to the Kitaev chain model [2]. The
Hamiltonian of the model can be written as [9]
1
† † † †
1 1
=1 =1
1
= h.c. ,
2
[ ]
L L
K j jj j jj
j j
wd d d d V d d (3)
where w and are the hopping and pairing amplitudes,
V is the effective potential (the combination of the chemi-
cal potential and the applied external and internal poten-
tials), L is the length of the chain (in units of inter-atomic
distances), and the operators jd
†
( )jd destroy (create) a
(Dirac) fermion in the site .j Here we consider the case of
real (the situation with complex values of can be
studied in a similar way). Suppose the following relation
holds: , > 0w and 0V with w (other cases can be
studied in a similar way). The ground state of the open
chain is two-fold degenerate in the limit ,L see
Refs. 2, 13. In one of two degenerate sets of the ground
state eigenstates edge Majorana states persist [2,13], which
can form a nonlocal Dirac fermion with zero-energy. Here
we are looking for such a bound state for the finite-length
chain.
The details of the solution of the stationary Schrödinger
equation with the Hamiltonian K for eigenstates for the
finite open Kitaev chain are similar to the one for spin-1/2
chains [14]. The Hamiltonian K can be diagonalized,
i.e., presented in the diagonal form
†
= constK k kk
k
, (4)
where †
k
( )k creates (destroys) a fermion with the
wavenumber k (that parametrizes all eigenfunctions), and
0k is the energy of the eigenstate with the wave-
number k. For the cyclic chain for any L any excitation
has a gap, which for 2 2( )/V w w is equal to
2 2 2
2 2
= ,
w V
Gap
w
(5)
and for 2 2( )/V w w is equal to
= | | .Gap w V (6)
The original Kitaev's Hamiltonian gets its diagonal form
with the help of the unitary (Bogolyubov) transformation
†
= ,k kj j kj j
j
u d v d
† †
= ,kj kj jjk
j
u d v d (7)
where the coefficients of the Bogolyubov transformation
are
2 = , 2 =kj kj kj kj kj kju v . (8)
For the chain consisting of L sites k and k are L-com-
ponent vectors. Those vectors satisfy the equations
† 2 †ˆ ˆ ˆ= , = ,k k k k k kCC C
ˆ = .k k kC (9)
Here the matrix elements of Ĉ are
, , , 1 , 1= .
2 2
n m n m n m n m
w w
C V (10)
There exist exactly L roots of Eqs. (9) for wavenumbers,
which parametrize eigenvalues and eigenfunctions. We know
that eigenstates for the open chain are standing waves. Then
the general form for the components of the L-component
eigenvector k (analogously for )k can be looked for (for
=1, , )j N as
Majorana bound states in the finite-length chain
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 809
1 1 2 2
1 2 3 4= e e e e
ik j ik j ik j ik j
kj A A A A (11)
with complex wavenumbers 1,2 .k The coefficients 1,2,3,4A
are normalization constants. Notice that for the odd values
of L for = 0V we have ˆdet = 0,C so that there exists a
zero eigenvalue of the matrix †ˆ ˆ .CC Obviously, the wave
number 1,2k also satisfy the equation
1 2
= .k k (12)
Real values of 1,2k describe the band eigenstates and com-
plex ones describe bound (edge) states. Two sets of
eigenfunctions for the considered open chain satisfy
Eqs. (9). For example, for even L and V = 0 the compo-
nents jk are nonzero (they are proportional to
sin( ) sin(2 )/ sin[2( 1) ],jk k L k where = 0,2,4, , )j L
for even sites of the chain and they are zero for odd sites of
the chain for the first set of eigenstates. For jk nonzero
components are related to the odd sites of the chain (pro-
portional to sin[( ) ] sin(2 )/ sin[2( 1) ],L j k k L k where
= 0,2, , )j L for the first set in this case. For the second
set the situation is opposite: even components of jk and
odd components of jk are nonzero, while other compo-
nents are zero. This is why, if bound eigenstate exists, then
its wave function decays exponentially with distances from
each edge of the chain for the first and the second set of
eigenfunctions. The energy of the bound eigenstate is zero
in the thermodynamic limit ,L hence two sets of
eigenfunctions are degenerate. It is, in general, not true for
the finite L.
Let us concentrate on bound (edge) states. The wave
functions of these states decay exponentially with the dis-
tance from edges. After some standard calculations one can
show that for large enough L there are 1L real solutions
for wavenumbers, which parametrize all eigenstates, and
one complex solution (bound state) for .V w (For >V w
one has L real solutions for wavenumbers of band
eigenstates.) We study long enough Kitaev's chain > 4L
(in what follows we will consider only the main correc-
tions for large ).L For relatively small values of the effec-
tive potential, 0 < V F the wavenumbers of the edge
bound states have nonzero real part k. Here for 1L one
has for the critical value of the potential V,
4
2 2
2
= .
8( 1) ( )
L
w
F w
wL w w
(13)
In the main order in L for the energy of the bound state
one gets (cf. Ref. 11)
( 1)/2
= | sin [( 1) ] | ,
L
b
w
G L k
w
(14)
where
2 2
2 2 2 1/2
2 ( )
=
| |
w V
G
w w V
, (15)
and the value of k is determined from cos = / .k V F Notice
that the nonzero real part of the wavenumber of the bound
state, 0,k exists only for 0.V For the limiting cases
= 0,V and for 0V and = ,w the real part is zero,
= 0.k
On the other hand, for F V w wavenumbers for the
edge bound states are only imaginary. For this region of
parameters we have (cf. Ref. 11)
1
2 2 2
= sinh[( 1) ] ,
( ) exp ( )
L
b
V V w
G L p
w p
(16)
where the value of p is determined from cosh p
1 2cosh[( )/2] = / .p p V F One can check that for =V F
the first solution for the bound state transforms to the se-
cond one. One can see from Eqs. (14) and (16) that the
energy of the (Majorana) bound state is equal to zero (i.e.,
the exact degeneracy of two ground eigenstates persists),
= 0,b only for ,L while for long, however finite
lengths of the chain, L, the eigenvalue for the bound state
become nonzero according to the law exp( / ),b L cf.
Refs. 2, 11. The correlation length depends on the val-
ues of the parameters of the Hamiltonian (the hopping and
pairing amplitudes, w and ) and on the effective poten-
tial V . It turns out that for V F the correlation length
= ln[( ) / ( )]w w does not depend on the value of
the effective potential, while for large values of the latter,
for ,V F the correlation length is equal to
2 2 2= 2 ln[( )/ ],p w V V w i.e., it becomes
V-dependent. We emphasize that for small enough values
of the applied potential V F for each value of the length
of the chain L there exists the discrete set of the values of
V at which the exact degeneracy, = 0b takes place.
Namely, for
= cos[ /( 1)] , =1,2, , /2 ,nV F n L n L (17)
the energy of the bound state is zero, and the exact degen-
eracy of two sets of ground state eigenstates is restored for
the finite-length Kitaev chain. It would be interesting to
check experimentally the possibility of those values of the
effective potential for large enough quantum chains, in
which zero-energy Majorana bound states exist. Notice the
main differences in our results comparing to the ones, ob-
tained for quantum semiconductor chains [11]: Oscillations
of the bound state energy exist only if the effective poten-
tial is smaller than F, and in that range of parameters the
correlation length does not depend on eff ,V unlike Ref. 11.
Now, the exchange integral for Fe chains on Pb sub-
strate was approximately 2.4 eV. The on-cite spin-orbit
coupling constant was 60so meV for chains and
A.A. Zvyagin
810 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
0.6 eV for zigzag chains, comparing to the Rashba cou-
pling = 0.05Rt eV. Finally, the pairing in was estimat-
ed as 2–3 meV, see Ref. 8. The topological superconduc-
tivity (i.e., the case with the Majorana topological num-
ber is equal to –1, for which Majorana edge states can
exist) is expected for many values of the chemical poten-
tial [8], in particular for = 2.4J eV we can use the val-
ues of | | roughly between 0.1 and 2.5 eV. This is why,
we can use our results for the finite-length Kitaev chain
for the estimation of the effect of the finite length for the
energy of the Majorana bound state in the ferromagnetic
chain with the induced p-wave superconductivity [8].
Figure 1 shows how the energy of the bound state de-
pends on the length of the chain (here we used Veff =
= 0.1 meV). Let us analyze how the energy of the bound
state depends on the value of the effective potential for the
fixed length of the chain. The value of the effective poten-
tial Veff can be changed not only by application of the ex-
ternal voltage (with relatively small energies), but also by
changing the chemical potential. The results are presented
in Fig. 2. Large values of Veff are possible if one takes into
account that it is determined as eff = ( /2) .V J U w
We see the clear difference between the values of the ener-
gy of the bound states in the finite chain for small and
large values of the effective potential. Calculated values of
the bound state energy can be observed in experiments
using, e.g., the scanning tunneling spectroscopy.
Summarizing, we have considered the situation of re-
cent experiments on magnetic atomic chains on s-wave
superconducting substrate that observed Majorana edge
states for chains, and have shown that the Hamiltonian of
the Kitaev chain can describe the experimental situation
for the renormalized values of parameters. It has been
shown how the energy of Majorana bound state depends on
the size of the chain for large enough chain lengths. For
small enough values of the effective potential such energy
oscillates with the size of the system and with the value of
the potential. For larger value of the potential there are no
oscillations. Oscillations are caused by the real part of the
wavenumber of the bound state. We have pointed out that
for small values of the effective potential there exists a set
of such values at which the energy of the Majorana edge
mode is zero, i.e., the degeneracy of the ground state is
restored. Real part of the wavenumber for the bound state
becomes zero for large values of the potential. According
to our estimates we expect that mentioned finite-length
effects for the Majorana bound states can be observed in
experiments (like scanning tunneling spectroscopy) in such
magnetic chains on a superconducting substrate. We em-
phasize that finite-size devices with Majorana edge states
are very important for practical applications for possible
arrays of topological qubits. The effect of the applied ef-
fective voltage is also important from the viewpoint of
perspective of manipulations with such sets of topological
qubits.
Support from the Institute for Chemistry of the V.N.
Karasin Kharkov National University under grant 4-15-13
(state registration number 0112U007569) is acknowledged.
Fig. 1. The dependence of the energy of the bound state b in the
Kitaev chain on the length of the chain L (in units of inter-atomic
distances in the chain). The following parameters are used: the
bandwidth w = 0.1445 eV, eff = 0.2 meV, and the effective po-
tential V = 0.1 meV.
Fig. 2. The dependence of the energy of the bound state b in the
Kitaev chain for the length of the chain L = 50 (in units of inter-
atomic distances in the chain) on the effective potential Veff. Oth-
er parameters are the same as in Fig. 1.
Majorana bound states in the finite-length chain
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