Transfer matrix approach to the persistent current in quantum rings: Application to
hybrid normal-superconducting rings
Andrea Nava1 , Rosa Giuliano1 , Gabriele Campagnano2, and Domenico Giuliano1

arXiv:1607.01707v2 [cond-mat.str-el] 15 Nov 2016

1

Dipartimento di Fisica, Universit` della Calabria Arcavacata di Rende I-87036, Cosenza,
a
Italy and I.N.F.N., Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy
2
CNR-SPIN, Monte S. Angelo-Via Cintia, I-80126, Napoli,
Italy and Dipartimento di Fisica, Universit` di Napoli “Federico II”,
a
Monte S. Angelo-Via Cintia, I-80126 Napoli, Italy
(Dated: April 14, 2018)
Using the properties of the transfer matrix of one-dimensional quantum mechanical systems, we
derive an exact formula for the persistent current across a quantum mechanical ring pierced by a
magnetic flux Φ as a single integral of a known function of the system’s parameters. Our approach
provides exact results at zero temperature, which can be readily extended to a finite temperature
T . We apply our technique to exactly compute the persistent current through p-wave and s-wave
superconducting-normal hybrid rings, deriving full plots of the current as a function of the applied
flux at various system’s scales. Doing so, we recover at once a number of effects such as the crossover
in the current periodicity on increasing the size of the ring and the signature of the topological phase
transition in the p-wave case. In the limit of a large ring size, resorting to a systematic expansion
in inverse powers of the ring length, we derive exact analytic closed-form formulas, applicable to a
number of cases of physical interest.
PACS numbers: 73.23.Ra, 74.78.Na, 74.45.+c, 73.23.-b

I.

INTRODUCTION

Due to the zero-resistance of a superconductor, once a current is induced for instance by an applied magnetic
flux Φ, in a superconducting ring, it is expected to last in principle forever. The existence of such a persistent
current in superconducting rings was already predicted at the discovery of superconductivity in 19111,2 , and later
experimentally demonstrated3 . Remarkably, as predicted in 1983 by B¨ttiker, Landauer and Imry, superconductivity
u
is not a necessary mechanism to have persistent currents. Indeed, even a normal ring threaded by a magnetic flux can
host a persistent current, provided that the temperature is low enough to suppress inelastic scattering from phonons
and other electrons and that the size of the ring is short enough compared to the phase coherence length4 . Because
of the gauge invariance, both for superconducting and normal rings, the persistent current I[Φ] must be a periodic
function of the magnetic flux Φ. This prediction has been eventually verified in a number of experiments5,6 . However,
while in the superconducting case the period is equal to the flux quantum appropriate for a superconductor, with
current carried by Cooper pairs, Φ∗ = h/(2e), in the normal case such a period is doubled, and equal to Φ0 = h/e.
0
A large amount of literature about persistent current in a normal mesoscopic ring has addressed a number of
issues such as the effect of disorder in the ring with consequent possible halving in the period of the current7,8 , the
role of the spin degree of freedom9 , the consequences of the electron-electron interaction, with and without impurity
scattering10 , and the presence of spin-orbit interaction11 (for a recent comprehensive review on electron transport in
mesoscopic rings see, for instance, [12]). Moreover, the issue of how the current is affected by collective fluctuations in
quasi one-dimensional superconducting rings with a weak link has been considered13 , together with the possibility of
using small superconducting rings, realized with pertinent Josephson junction networks, as high-coherence quantum
devices14–16 .
Recent progresses in the fabrication of nanostructures made it possible to engineer hybrid devices where superconductivity is induced by proximity effect only in a section of the ring17 (Hybrid Rings (HRs)). This enables one
to explore a number of physical regimes. For instance, one may think of looking at the persistent current across
the ring varying the lengths of the two regions, while keeping the length of the normal region ℓN shorter than the
phase coherence length. In this way, one may monitor the crossover between the normal mesoscopic regime, in which
the length of the superconducting region ℓS is shorter than the superconducting coherence length ξ0 , towards the
complementary Josephson-junction regime, in which ℓS ≫ ξ0 18 . Accompanied to the above crossover, one is also
expected to see a crossover of the period of the persistent current, respect to the magnetic flux, between Φ0 (current
carried by electrons), and Φ∗ (current carried by Cooper pairs)18,19 .
0
A special type of HR has recently attracted great interest, in view of the possibility of designing a non ambiguous
way of probing Majorana fermions (MFs) in condensed matter systems. MFs have been predicted by Kitaev to emerge
as localized end modes in a spinless p-wave one-dimensional superconductor in its topological phase20 . It has been
proposed that such a system can be realized by inducing superconductivity by proximity effect in a semiconducting

2
quantum wire with a sizable spin-orbit coupling (e.g., an InAs wire) when subject to an external magnetic field21,22 .
A zero-bias peak in a tunneling spectroscopy measurement has been claimed as an evidence for the existence of the
localized MF23 , but alternative possible explanations of the experimental data of Ref.[23] have been provided, leaving
still as a debated question whether a MF has been really detected, or not24,25 . Other proposals to detect MFs have
been presented, for instance, using a quantum switch made with two quantum dots coupled to MFs26 or by means of
a local flux measurement in a topological rf-SQUID with a frustrating π-junction27 or, ultimately, of the analysis of
the persistent current in metallic rings interrupted by a Coulomb blockaded topological superconducting segment28 .
When using I[Φ] as a tool to monitor the emergence of MFs, it is of the utmost importance to disentangle effects
related to MF physics from ”spurious” effects, which are expected to appear in nontopological phases, as well. For
instance, the crossover of the period of I[Φ] vs. Φ from Φ0 to Φ∗ , can be either attributed to MFs, or can be regarded as
0
a simple crossover of a hybrid ring toward the mesoscopic regime, taking place when the length of the superconducting
regions becomes longer than the corresponding superconducting coherence length18,29 . Therefore, it is extremely useful
to recover an exact (or pertinently approximated) formula for I[Φ], allowing us to make rigorous predictions on its
dependence on the system parameters, on the length of the superconducting and nonsuperconducting regions, etc.
Nevertheless, even after a number of simplifications: considering a ballistic system, using a non self-consistent model
for the superconducting region30 , or its corresponding lattice version for a s-wave31, or for a p-wave superconductor20 ,
computing I[Φ] is typically still quite a challenging task. In fact, the ”standard” approach to the problem consists in
computing the current as
I[Φ] = e∂Φ F [Φ; T ]

,

(1)

with F [Φ; T ] being the system’s free energy at applied flux Φ and temperature T . At T = 0 Eq.(1) yields
I[Φ] = e∂Φ EGS [Φ] = e∂Φ

En

,

(2)

En <0

where the sum is taken over energies of the occupied single-quasiparticle states. To compute these energies, one has
to solve the secular equation for the energy eigenvalues at nonzero Φ with periodic boundary conditions over the
whole ring. In general, the resulting set of equations looks quite formidable and hard to deal with, which requires
resorting to various approximations, such as retaining only the low-energy part of the spectrum19 , or employing
various approximations for the single-quasiparticle energies as a function of Φ in various regions of the spectrum18 .
In this paper we present a technique that, under very general assumptions such as the ones listed above, allows for
exactly expressing I[Φ] as a single integral of a pertinently constructed function of the system’s parameters. At T = 0,
our approach is based on first writing Eqs.(2) in terms of a single integral over an appropriate path in the complex
energy plane, and on eventually deforming the integration path to the imaginary axis. In this respect, our method can
be regarded as an adapted version of the technique developed in Refs.[32–35] to compute the dc Josephson current
across an SNS-junction in terms of the single-quasiparticle S-matrix of the junction, which has been successfully
applied to a number of cases of interest, such as the Josephson current through a chaotic Josephson junction36 , the
critical supercurrent in the quantum spin-Hall effect37 , and the current across a long Josephson junction35,38 . In fact,
our method combines the idea of deforming the integration path to the imaginary axis with the idea of using the
transfer matrix, rather than the S-matrix, to recover the quasiparticle scattering dynamics of the system. Indeed,
while in principle the transfer matrix and the S matrix can be used on an equivalent footing to derive the transport
properties of a mesoscopic system39 , the former approach comes out to be more appropriate for systems with periodic
boundary conditions, such as quantum rings (see, for instance, Ref.[40] for an example of application of transfer
matrix approach to the persistent current through a disordered, normal mesoscopic ring). Besides providing an exact
formula allowing to easily compute I[Φ] by numerically integrating a known function at fixed Φ, in the large-ring
limit, our technique is also suitable for a systematic expansion in inverse powers of the system’s length, which is the
counterpart for a HR of the expansion in inverse powers of the length of a long SNS junction discussed in Ref.[35] for a
single-channel system and generalized in Ref.[38] to the multi-channel case. In this limit the formula for I[Φ] is greatly
simplified, allowing for the derivation of analytic closed-form formulas for the current in a number of cases of interest.
As an example of the effectiveness of our technique, we consistently extend the results of Refs.[28,29,41] by deriving
exact plots of I[Φ] vs. Φ through p-wave and s-wave superconducting-normal hybrid rings, which, by comparing the
plots in the two cases with each other, allows us to highlight the features strictly related to the emergence of MFs in
the p-wave case.
Remarkably, while in this paper we do not account for a number of features that can be in principle important
in HRs, such as quantum phase slips42 , electronic interaction effects10 , disorder7,8 , etc., as we outline among the
concluding remarks, it should not be difficult to pertinently address them within our formalism. In fact, we plan to
treat some of this issues in forthcoming publications, as a natural development of this work.
The paper is organized as follows:

3

FIG. 1: An example of the ring geometry used in the article: nontrivial scattering happens in the shaded region whereas in the
remaining part one can write the wave function in terms of simple scattering states (cfr. main text)

• In section II we introduce the transfer matrix approach which we use to compute the persistent current;
• In section III we introduce the lattice model Hamiltonian for a hybrid superconducting-superconducting ring,
both for the p-wave and for the s-wave superconductor;
• In section IV we compute the persistent current in some relevant reference model, such as a superconducting
ring interrupted by a normal weak link, and in the case of a hybrid normal-superconducting ring. Specifically,
we recover both cases as particular limit of the model introduced in section III: the former one by making one
of the two superconducting region shrink to zero, the latter one by simply setting to zero the superconducting
gap in one of the two regions;
• In section V we consider the limit of large ring size in the HR both in the s-wave, and in the p-wave case.
In particular, we first show how one recovers the results of Refs.[35,38] in the limit of infinite length for the
superconducting region, therefore, we discuss the complementary limit of a short-superconducting and a longnormal region;
• In section VI we outline how our results can be extended to a temperature T finite, but well below the superconducting gap. Specifically, this allows us to recover within our formalism the finite-T formula at Eq.(1);
• In section VII we provide our main conclusions and discuss further possible developments of our work;
• In appendix A we review the derivation of the sub-gap states in an open Kitaev chain, of finite length ℓS , which
is functional to the discussion of the results of section IV.
II.

THE TRANSFER MATRIX APPROACH TO THE PERSISTENT CURRENT

Our technique to compute the persistent current through mesoscopic normal/superconducting rings is based on a
combination of the approach to the Josephson current across an SNS junction based on the analytical continuation
of the quasiparticle S-matrix to the imaginary axis32–35 with the transfer matrix (TM) approach to transport in
mesoscopic systems39 , particularly well-suited to account for periodic boundary conditions in quantum rings. The
key feature of our approach is that it eventually leads to an exact, closed-form formula for the groundstate energy
of the system at a given Φ and, therefore, for the persistent current across the ring. In order to illustrate its main
features, in this section we review the main steps leading to our final formula for the groundstate energy in the
case of a normal, mesoscopic ring. Nevertheless, as we discuss in the following, the case of a HR containing one, or
more, superconducting sections is quite a straightforward generalization of the derivation we provide in this section.
To treat the p-wave and the s-wave case on the same footing we choose to perform our derivation within a lattice
one-dimensional model Hamiltonian, which, in the superconducting region, corresponds to Kitaev’s Hamiltonian in
the p-wave case20 , and to the lattice one-dimensional Hamiltonian of Ref.[31] in the s-wave case.

4
As an introduction to the technique presented in this article, let us consider a normal (non superconducting)
ring with ℓ sites as depicted in Fig.1. We can ideally divide the ring into three regions, within the shaded region
non-trivial scattering processes may happen, while in the remaining part the system is described by the asymptotic
lattice Hamiltonian of the form Has = −J j {c† cj+1 + c† cj } − µ j c† cj , with cj being the single-fermion lattice
j
j+1
j
annihilation operator for spin-less fermions.
Let us for the moment imagine that the ring is open at the last site ℓ and that j1 and j2 are limiting the region of
the ring with non trivial scattering. The wave function at energy E in the two ”asymptotic” regions for j < j1 and
for j > j2 is
uj ∼ A< eikj + A< e−ikj , (j < j1 )
+
−

uj ∼ A> eikj + A> e−ikj , (j > j2 ) ,
+
−

(3)

E = −2J cos(k) − µ .

(4)

with

Now, by definition the transfer matrix between sites ja < j1 and jb > j2 , M[E; ja , jb ], relates the solution at j = ja
to the solution at j = jb , that is
˜
˜
ujb = A+ eikja + A− e−ikja

,

(5)

with
˜
A+
˜−
A

= M[E; ja , jb ]

A<
+
A<
−

.

(6)

Upon considering the closed geometry, we have to impose periodic boundary conditions (PBCs) on the solution in
Eq.(3). Going through Eq.(5), this is accounted for by means of the secular equation
M[E; 1, ℓ]

A<
+
A<
−

A<
+
A<
−

=

.

(7)

As a result, Eq.(7) leads to the secular equation for the allowed values of the energy E:
det{M[E; ℓ] − I} = 0 .

(8)

(In Eq.(8) we suppressed the dependence of M[E; ja , jb ] on ja since, as expected because of the periodic boundary
conditions, it depends only on the distance between the sites, which is equal to ℓ.) Besides constituting an alternative
way of presenting the eigenvalue equation for a single particle on the ring, Eq.(8) also provides an efficient way to
compute the groundstate energy of the system, EGS , defined as the sum of all the negative (if measured with respect
to the chemical potential) single-particle energy eigenvalues En , that is
EGS =

En .

(9)

En <0

In fact, single-particle negative energy eigenvalues are just the zeros of det{M[E; ℓ] − I} lying at the negative part of
the real axis, if one regards det{M[E; ℓ] − I} as a function of the complex variable E. To sum over all of them, we
adapt the approach of Refs.[35,38], namely, we first of all note that the energy eigenvalues are the poles (with residues
all equal to 1) of the function Ψ[E; ℓ], defined as
Ψ[E; ℓ] =

∂E det{M[E; ℓ] − I}
= ∂E ln det{M[E; ℓ] − I} .
det{M[E; ℓ] − I}

(10)

We therefore introduce the integration path Γ depicted in Fig.2a) and compute EGS as
EGS =

1
2πi

Γ

dE EΨ[E; ℓ] = −

1
2πi

Γ

dE ln det{M[E; ℓ] − I} .

(11)

On noting that Ψ[E; ℓ] must have no singularities outside of the real axis, we deform the integration path as illustrated
in Fig.2, so to eventually compute the integral over the imaginary axis as

5

a)

11
00
11
00
11
00

b)

Im(E)

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

11
00
11
00
11
00

c)

Im(E)

11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0
1
0

Re(E)

11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

Im(E)

11
00
11
00
11
00

11
00
11
00
11
00

Re(E)

1
0
1
0
1
0

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

1
0
1
0
1
0

Re(E)

FIG. 2: Sequence of deformations in the integration path Γ eventually allowing to express I[Φ] as an integral over the imaginary
axis:
a) The path Γ obtained as the union of small circles, each one surrounding one, and only one, negative (with respect to the
Fermi level) energy eigenvalue;
b) The integral over Γ can be deformed to an integral over just one closed path, surrounding all the negative energy eigenvalues;
c) The integral over the red-dashed arc is assumed to be equal to 0 in the infinite-radius limit. Only the integral over the
imaginary axis (to which the solid-blue line can be continuously deformed) is left.

EGS = −

1
2π

∞
−∞

dω ln det{M[iω; ℓ] − I} .

(12)

Equation (12) is the heart of our article. When the ring is pierced by a magnetic field the transfer matrix M and
hence the spectrum takes an additional dependence on the total flux Φ. We can calculate the (zero-temperature)
persistent current as
I[Φ] = e∂Φ EGS [Φ] = −

e
2π

∞
−∞

dω ∂Φ ln det{M[iω; ℓ; Φ] − I} .

(13)

Equation (13), together with the corresponding finite-T generalization which we discuss in the following sections, is
the key result of our paper and its main point of novelty: it provides an exact formula for I[Φ] which can be readily
implemented, once one has derived the TM of the ring. Remarkably, while, for the sake of the presentation, in this
section we relied on a normal mesoscopic ring, Eq.(13) does certainly apply equally well to the case of superconducting
and/or hybrid rings, once one has constructed the appropriate TM for the system. Importantly enough, our approach
allows for circumventing the main difficulty in computing the persistent current working out the single-particle energy
levels, using them to compute EGS [Φ], and eventually employing the final result to derive the current. Besides a few,
oversimplified remarkable exceptions, extracting the single-quasiparticle energy levels from the TM (or by means of
an equivalent approach) is quite a difficult task to achieve, which can only be addressed either by means of exact
numerical diagonalization of the Hamiltonian (with the corresponding limitations on the system size), or within
approximate methods based on truncating the full Hilbert space to a small number of low-energy (sub-gap) states,
which are expected to carry the largest part of the current. While both these techniques are expected to suffer of
strong limitations in their range of applicability, our Eq.(13) is universally applicable and, once one has constructed
the TM, does not require to go through any additional step, besides explicitly computing the integral at the right-hand
side.
Our Eq.(13) is the analog, for a superconducting ring, of the equation giving the Josephson current across an SNS
junction in terms of the single-particle S-matrix, analytically continued to the imaginary axis32,34 . In this respect, it
is expected by analogy to have a wide range of applicability, either as it stands, as an exact, closed form formula for
I[Φ], or as a starting point to account, for instance, for the effect of disorder (in analogy to what is done in Ref.[36]
for an SNS junction), or of the electronic interaction. As an example of application of our technique, in the following
we derive full plots of the persistent current in hybrid superconducting-normal rings, both in the case of p-wave, and
of s-wave superconductivity, which allows us to highlight the key feature that, in the former case, can be definitely
attributed to the emergence of MFs at the interfaces.
As a convention on the units of measurement, in the following we shall measure the flux Φ in units of /(2e), so
that a period Φ∗ corresponds to a 2π-periodicity, while a period Φ0 corresponds to a 4π-periodicity.
0

6

FIG. 3: Sketch of the system described by the model Hamiltonian in Eq.(14).

III.

MODEL HAMILTONIAN FOR A HYBRID SUPERCONDUCTING-NORMAL RING

To model a hybrid superconducting-normal ring, we resort to an effective lattice model Hamiltonian with positiondependent parameters. Using a lattice model Hamiltonian allows, on one hand, to easily introduce p-wave pairing
both in real space and in momentum space, on the other hand, it provides a natural mean to regularize divergences
which would arise in the continuum model when summing over the energies of the occupied levels to compute EGS [Φ]
(and, therefore, I[Φ]). To highlight special features of p-wave HRs, in the following, when possible, we systematically
compare the results obtained in p-wave systems to the ones obtained in s-wave systems. Therefore, in this section we
derive the transfer matrix which we shall eventually use in the following to compute the persistent current in both
cases.
A.

Model Hamiltonian for a p-wave nonhomogeneous superconducting ring

We here consider the system sketched in Fig.3: a hybrid ring made of two homogeneous regions, either superconducting (p-wave or s-wave), or normal, each one characterized by different parameters. In the p-wave case, referring
to the one-dimensional Kitaev Hamiltonian20 for a spinless p-wave superconductor, calling 1 and 2 the two regions,
we assume that the normal hopping amplitude, the pairing gap and the chemical potential are respectively given by
w1 , ∆1 , µ1 and by w2 , ∆2 , µ2 . Moreover, we assume that regions 1 and 2 are coupled at their endpoints via a normal
hopping term, with hopping amplitude τ . In a ring configuration, we also assume that a magnetic flux Φ pierces the
ring. By means of an appropriate canonical redefinition of the lattice fermion operators, it is possible to account for
Φ
the applied magnetic flux in terms of a phase factor e±i 4 , symmetrically ascribed to the two hopping terms between
the two regions. As a result, we eventually present the model Hamiltonian as

j=1

{c† cj+1 + c† cj } − µ1
j
j+1

−

j=1

{d† dj+1 + d† dj } − µ2
j+1
j

τ {[c† dℓ2
1

c† cj + ∆1
j
j=1

+

i
d† cℓ1 ]e 4 Φ
1

+

[d†2 c1
ℓ

+

j=1

{cj cj+1 + c† c† }
j+1 j

ℓ2 −1

ℓ2

ℓ2 −1

− w2

ℓ1 −1

ℓ1

ℓ1 −1

H = −w1

d† dj + ∆2
j
j=1
i
c†1 d1 ]e− 4 Φ }
ℓ

j=1

.

{dj dj+1 + d† d† }
j+1 j
(14)

In Eq.(14) we have set the lengths of the two arms of the ring respectively at ℓ1 and at ℓ2 and used cj , c† to denote the
j
lattice annihilation/creation operators at site j within region 1, and dj , d† to denote the lattice annihilation/creation
j
operators at site j within region 2, with standard anticommutation relations {dj , d†′ } = {cj , c†′ } = δj,j ′ , all the other
j
j
anticommutators being equal to 0. Taken in the appropriate limit, the model Hamiltonian in Eq.(14) is suitable to
describing a number of systems of physical interest, such as a p-wave superconducting ring interrupted by a weak link41 ,
a hybrid p-wave-normal metal ring, and a half-topological, half-non-topological superconducting ring29 . Moreover, as
we discuss in the following, taken in the limit of long-superconducting section, the superconducting-normal hybrid
ring is mapped onto the effective model for a Josephson junction made with topological superconductors43,44

7
In the following, we will be mostly focusing onto the ∆2 → 0 limit. To construct the transfer matrix, we start from
the Bogoliubov-de Gennes (BdG) equations for the single-quasiparticle wavefunction at a given energy E. To do so,
we consider a generic energy eigenmode ΓE which, in terms of the single-fermion lattice operators on the ring, is given
by
ℓ2

ℓ1
j=1
(1)

{[uj ]∗ cj + [vj ]∗ c† } +
j

(2)

(2)

(1)

(1)

ΓE =

j=1

{[uj ]∗ dj + [vj ]∗ d† } ,
j

(15)

(2)

uj
uj
and
being the single-quasiparticle wavefunction in region-1 and in region-2. On imposing the
(1)
(2)
vj
vj
canonical commutation relation
with

[ΓE , H] = EΓE ,

(16)

we therefore obtain the BdG equations for the wavefunctions. Within the homogeneous regions, these are given by
(1)

Euj

(1)

Evj

(1)

(1)

(1)

(1)

(1)

= −w1 {uj+1 + uj−1 } − µ1 uj + ∆1 {vj+1 − vj−1 }
(1)

(1)

(1)

= w1 {vj+1 + vj−1 } + µ1 vj

(1)

(1)

− ∆1 {uj+1 − uj−1 } ,

(17)

for 1 < j < ℓ1 , and
(2)

Euj

(2)

Evj

(2)

(2)

(2)

(2)

(2)

= −w2 {uj+1 + uj−1 } − µ2 uj + ∆2 {vj+1 − vj−1 }
(2)

(2)

(2)

= w2 {vj+1 + vj−1 } + µ2 vj

(2)

(2)

− ∆2 {uj+1 − uj−1 } ,

(18)

for 1 < j < ℓ2 . At the interfaces between the two regions, instead, the BdG equations yield
(1)

(1)

(1)

(2)

i

(1)

Eu1 = −w1 u2 − τ e− 4 Φ uℓ2 − µ1 u1 + ∆1 v2
(1)

(1)

i

(1)

(2)

(1)

Ev1 = w1 v2 + τ e 4 Φ vℓ2 + µ1 v1 − ∆1 u2
(2)

(2)

(1)

i

(2)

(2)

Euℓ2 = −w2 uℓ2 −1 − τ e 4 Φ u1 − µ2 uℓ2 − ∆2 vℓ2 −1
(2)

(2)

(1)

i

(2)

(2)

Evℓ2 = w2 vℓ2 −1 + τ e− 4 Φ v1 + µ2 vℓ2 + ∆2 uℓ2 −1
(2)

(2)

(2)

(1)

i

(2)

Eu1 = −w2 u2 − τ e− 4 Φ uℓ1 − µ1 u1 + ∆2 v2
(2)

(2)

i

(2)

(1)

(2)

Ev1 = w2 v2 + τ e 4 Φ vℓ1 + µ1 v1 − ∆2 u2
(1)

(1)

i

(2)

(1)

(1)

Euℓ1 = −w1 uℓ1 −1 − τ e 4 Φ u1 − µ1 uℓ1 − ∆1 vℓ1 −1
(1)

(1)

i

(2)

(1)

(1)

Evℓ1 = w1 vℓ1 −1 + τ e− 4 Φ v1 + µ1 vℓ1 + ∆1 uℓ1 −1 .

(19)

According to Eqs.(17,18), within the homogeneous regions, we write the wavefunctions as superpositions of the
solutions of the homogeneous BdG equations, that is, we set
uj
vj

=

u(1)
v (1)

eik1 j ,

(20)

uj
vj

=

u(2)
v (2)

eik2 j ,

(21)

in region 1, with 1 ≤ j ≤ ℓ1 , and

in region 2, with 1 ≤ j ≤ ℓ2 . At a given energy E, the amplitudes

u(a)
v (a)

(a = 1, 2) are determined by solving the

equations
0 = {E + 2wa cos(ka ) + µa }u(a) − 2i∆a sin(ka )v (a)

0 = 2i∆a sin(ka )u(a) + {E − 2wa cos(ka ) − µa }v (a) ,

(22)

8
with k1 , k2 determined by the dispersion relations
E 2 = [2w1 cos(k1 ) + µ1 ]2 + 4∆2 sin2 (k1 ) = [2w2 cos(k2 ) + µ2 ]2 + 4∆2 sin2 (k2 ) .
1
2
(a)

(a)

Solving Eq.(23) at a given energy, we define the particle-like momenta kp
(a)
cos(kp ) = −
(a)

cos(kh ) = −

(23)

and the hole-like momenta kh , as
(a)

wa µa
1
−
2 − ∆2 )
2(wa
2
a

E 2 − [∆w ]2
2
wa − ∆2
a

1
wa µa
+
2
2(wa − ∆2 ) 2
a

E 2 − [∆w ]2
2
wa − ∆2
a

(a)

,

(24)

with
∆(a) = ∆a
w

4−

µ2
a
2
wa − ∆2
a

.

(a)

(25)
(a)

(a)

(Note that Eqs.(24) do in principle hold also for |E| < ∆w ). At given kp , kh , one determines the wavefunctions
(a) (a)
(a) (a)
(up , vp ) and (uh , vh ), defined as solutions of Eqs.(22) with ka respectively equal to kp and kh . Taking the most
general linear combinations of wavefunctions at the same energy, one finds that a generic wavefunction at energy E
within region-a (=1,2) takes the form
(a)

uj
(a)
vj

(a)

= A(p,+)

(a)

up
(a)
vp

(a)

eikp

j

(a)

+ A(p,−)

(a)

up
(a)
−vp

(a)

e−ikp

j

(a)

uh
(a)
−vh

(a)

+ A(h,+)

(a)

e−ikh

j

(a)

+ A(h,−)

(a)

uh
(a)
vh

(a)

eikh

j

. (26)

The transfer matrix is fully determined once one recovers the relations between the amplitudes
(a)
(a)
(a)
(a)
A(p,+) , A(p,−) , A(h,+) , A(h,−) in the two regions. These are determined by the interface conditions obtained from
Eqs.(19). In a compact notation, these are given by
 (b) 
 (a) 
A(p,+)
A(p,+)
 (b) 
 (a) 
 A(p,−) 
 A(p,−) 
(27)
Σa [Φ] · [Υ(a) (E)] ·  (a)  = Ωb [Φ] · [Υ(b) (E)] · Tb (E; ℓb ) ·  (b)  ,


A
A
 (h,+) 
 (h,+) 
(b)
(a)
A(h,−)
A(h,−)
with b = 2(1) if a = 1(2), and the matrix Σa [Φ] defined as
 i
τe4Φ
0
0 0
i
 0 τ e− 4 Φ 0 0
Σa [Φ] = 
 0
0
wa ∆a
0
0
∆a wa

the matrix [Υ(a) (E)] defined as



(a)

(a)

(a)

(a)

(a)






,

(28)

(a)

(a)

(a)

e−ikh uh
eikh uh
e−ikp up
eikp up
(a) (a)
(a) (a)
(a) (a)
(a)
 ikp (a)
−ikp
−ikh
e
vp −e
vh eikh vh
vp −e
[Υ(a) (E)] = 
(a)
(a)
(a)
 u(a)
up
uh
uh
p
(a)
(a)
(a)
(a)
vp
−vp
−vh
vh

the matrix Ωa [Φ] given by


wa −∆a
0
0
 −∆a wa
0
0 

i
Ωa [Φ] = 
−4Φ
 0
0 
0 τe
i
0
0
0
τe4Φ


,

and, finally, the transfer matrix for a homogeneous region of length ℓ being given by







,

(29)

(30)

9




Ta (E; ℓ) = 


(a)

eikp
0
0
0

ℓ

0
(a)

e−ikp
0
0

0
0

ℓ

(a)

e−ikh
0



0
0
0

ℓ

(a)

eikh

As a result, the transfer matrix for the full ring takes the form

ℓ






.

(31)

−1
Mp−wave [E; Φ; ℓ1 ; ℓ2 ] = [Υ(2) (E)]−1 ·Σ−1 [Φ]·Ω1 [Φ]·[Υ(1) (E)]·T1 (E; ℓ1 )·[Υ(1) (E)]−1 ·Σ1 [Φ]·Ω2 [Φ]·[Υ(2) (E)]·T2 (E; ℓ2 ).
2
(32)
Equation (32) is the key ingredient we need to compute the persistent current in the various regimes of interest. As
stated above, we now derive the analog of Eq.(32) in the case of a ring made of s-wave superconducting regions.

B.

Model Hamiltonian for an s-wave nonhomogeneous superconducting ring

In the case of a system made of two s-wave superconducting regions with parameters respectively given by w1 , ∆1 , µ1
and by w2 , ∆2 , µ2 , and connected to each other with hopping amplitude τ , the corresponding (spinful) Hamiltonian
is given by

σ

j=1

c† cj,σ + ∆1
j,σ

{c† cj+1,σ + c†
j,σ
j+1,σ cj,σ } − µ1

σ j=1

−

σ

σ

j=1

{d† dj+1,σ + d†
j+1,σ dj,σ } − µ2
j,σ

τ {[c† dℓ2 ,σ
1,σ

j=1

+

i
d† cℓ1 ,σ ]e 4 Φ
1,σ

+

d† dj,σ + ∆2
j,σ
σ

[d†2 ,σ c1,σ
ℓ

+

{cj,↑ cj,↓ + c† c† }
j,↓ j,↑

ℓ2

ℓ2

ℓ2 −1

− w2

ℓ1

ℓ1

ℓ1 −1

H = −w1

j=1

j=1
i
c†1 ,σ d1,σ ]e− 4 Φ }
ℓ

{dj,↑ dj,↓ + d† d† }
j,↓ j,↑

.

(33)

Going through the same steps as in the p-wave case, we eventually find that the transfer matrix of the ring is now
given by
˜
˜
˜
˜
˜
˜
˜
˜
Ms−wave [E; Φ; ℓ1 ; ℓ2 ] = [Υ(2) (E)]−1 ·Σ−1 [Φ]·Ω1 [Φ]·[Υ(1) (E)]·T1 (E; ℓ1 )·[Υ(1) (E)]−1 ·Σ−1 [Φ]·Ω2 [Φ]·[Υ(2) (E)]·T2 (E; ℓ2 ).
2
1
(34)
˜
˜
The Σa [Φ], Ωa [Φ] matrices in Eq.(34) are simply obtained by setting ∆a to 0 respectively in Eq.(28) and in Eq.(30).
˜
The matrix [Υ(a) (E)] is given by
 (a) (a)
(a) (a)
(a) (a) 
(a) (a)
eikp up e−ikp up e−ikh uh eikh uh
(a)
(a)
(a)
(a)
 ikp (a) −ikp (a) −ikh (a) ikh (a) 
e
vh 
vp e
vh e
vp e
˜
(35)
[Υ(a) (E)] = 
 ,
(a)
(a)
(a)

 u(a)
up
uh
uh
p
(a)

vp

(a)

vp

(a)

(a)

vh

vh

with u(a) , v (a) determined as nontrivial solutions of the algebraic system

0 = {E + 2wa cos(ka ) + µa }u(a) − ∆a v (a)

0 = −∆a u(a) + {E − 2wa cos(ka ) − µa }v (a) ,

(36)

for k1 , k2 solving the secular equation
E 2 = [2w1 cos(k1 ) + µ1 ]2 + ∆2 = [2w2 cos(k2 ) + µ2 ]2 + ∆2 ,
1
2
(a)

(a)

and kp , kh

(37)

defined by
(a)
cos[kp ] = −
(a)

cos[kh ] = −

µa
−
2wa

E 2 − ∆2
a
2
4wa

µa
+
2wa

E 2 − ∆2
a
2
4wa

.

(38)

10

FIG. 4: Graphical representation of the factorizability of the transfer matrix: for the specific system depicted in figure, the
transfer matrix is given by M = T2 · M1 · T1 · M2 · T2 (from right to left).

Besides the differences in the form of the matrices appearing in Eqs.(32,34), an important point to stress is that
both matrices are block-factorizable, with the factorization corresponding to the possibility of regarding the system
as a sequence of homogeneous regions separated by interfaces. Indeed, the TM for a one-dimensional system comes
out to be simply the ordered product of the matrices corresponding to the homogeneous regions and of the ones
corresponding to the interfaces, taken in the appropriate sequence. From this respect, the matrices corresponding to
each homogeneous region and to each interface are sort of ”building blocks” of the global transfer matrix (see Fig.4
for a sketch of the factorizability of the matrix).
IV.

CALCULATION OF THE PERSISTENT CURRENT

In this section we compute the persistent current in a number of cases of interest. To highlight the feature related
to the emergence of MFs at the SN-interfaces, it is worth comparing the results obtained in p-wave systems with
the ones obtained in s-wave systems. Therefore, in the following we perform the calculation in both cases, by using
Eq.(13), with the transfer matrix computed according to Eq.(32) (p-wave case), or to Eq.(34) (s-wave case). To keep
in touch with the results of Ref.[29], we begin with the calculation of the current in the case of a superconducting
ring interrupted by a weak link though, at variance with the discussion of Ref.[29], we will not assume fermion parity
conservation. As stated above, for comparison, we also compute the current in the case of a ring made with an s-wave
superconductor.
A.

Persistent current across a superconducting ring interrupted by a weak link

A p-wave superconducting ring interrupted by a weak link can be physically realized at a semiconducting quantum
wire with a sizeable spin-orbit coupling (e.g., an InAs wire) deposited onto a bulk superconducting ring pierced
by a magnetic flux Φ. The combined effect of spin-orbit coupling, Zeeman spin splitting and proximity-induced
superconductivity from the bulk superconductor underneath has been shown to make the wire effectively behave as a
one-dimensional p-wave superconductor21,22 . As for what concerns a concrete proposal of an experimental realization
of the p-wave HR, we refer to Refs.[28,29]. Specifically, we assume that the weak link is actually realized as a physical
interruption of the superconducting ring with a tiny insulating layer, which cuts the current within the superconductor,
thus allowing the persistent current to only flow across the semiconducting nanowire. In fact, among other advantages,
this geometry allows for recovering as only detectable current the one flowing through the semiconducting wire, which
is what we are eventually interested in. An important point to stress is that it is typically difficult to keep the ring
perfectly isolated from the substrate, so to avoid fermion parity non-conserving relaxation processes [29,45]. Moreover,
in order for the grand-canonical like description of our system we employ here to be reliable, one has to think of a
ring in contact with a substrate, working as an electronic reservoir. Therefore, we should not expect fermion parity
to be preserved here. For this reason, though one might in principle account for fermion parity conservation by
implementing some pertinent adapted version of the approach presented in Ref.[37], throughout all our paper we
assume fermion parity not to be conserved which, nevertheless, does not affect the possibility of probing emergent
MFs by means of an appropriate persistent current measurement.
In Fig.5 we provide a sketch of the system we are considering here, that is, a homogeneous ring interrupted by a
weak link. The corresponding model Hamiltonian can be recovered from Eq.(14) in the p-wave case and from Eq.(33)
in the s-wave case, by setting to zero the length of one of the two regions. In the former case, it is given by

11

FIG. 5: Sketch of the superconducting ring interrupted by a weak link as introduced and discussed in Ref.[29]. A semiconducting
ring (depicted as a solid black line) is deposited on top of a bulk superconducting ring, interrupted by a tiny insulating layer
(light blue sector). The superconductor induces superconductivity within the semiconducting wire by proximity effect. All the
current circulating across the system is due to tunneling effect within the semiconducting wire.

−

j=1

ℓ−1

ℓ

ℓ−1

Hp−w = −w

{c† cj+1 + c† cj } − µ
j
j+1

i
τ {c† cℓ e 2 Φ
1

+

i
c† c1 e − 2 Φ }
ℓ

c† cj + ∆
j
j=1

j=1

{cj cj+1 + c† c† }
j+1 j

,

(39)

while in the latter case it can be presented as

−

σ

σ j=1

ℓ

ℓ1

ℓ−1

Hs−w = −w

{c† cj+1,σ + c†
j,σ
j+1,σ cj,σ } − µ

i
τ {c† cℓ,σ e 2 Φ
1,σ

+

i
c† c1,σ e− 2 Φ }
ℓ,σ

c† cj,σ + ∆
j,σ
σ j=1

j=1

{cj,↑ cj,↓ + c† c† }
j,↓ j,↑

.

(40)

Let us note that, in this case, the flux is fully ”loaded” on the single hopping term. The transfer matrix derived from
Eqs.(39,40) can therefore be simply expressed in terms of the ones we provide in section III B by simply setting one
of the two lengths (say ℓ2 ) to 0. We therefore obtain
Mp−w [E; ℓ; Φ] = [Υ(E)]−1 · Σ−1 [2Φ] · Ω[2Φ] · [Υ(E)] · T[E; ℓ] ,

(41)

˜
˜
˜
˜
Ms−w [E; ℓ; Φ] = [Υ(E)]−1 · Σ−1 [2Φ] · Ω[2Φ] · [Υ(E)] · T[E; ℓ] .

(42)

and, similarly

To compute the current, let us start with the ring described by the TM in Eq.(41). In Fig.6, we plot I[Φ] vs. Φ for
the values of the parameters reported in the caption, particularly for a chemical potential µ = 0. At zero chemical
potential, the p-wave superconductor lies well within the topological region, with two MFs γL , γR localized at its
endpoints. This allows us to provide a simple interpretation of the curves we draw in Fig.6 at different values of the
length ℓ of the superconductor. The key parameter here is the ”hybridization length” ℓM between γL and γR , which
we estimate according to the derivation of the Appendix A. At µ = 0 and for the values of the parameters we used,
from Eq.(A13) we obtain ℓM ∼ 5. Therefore, when considering the largest ring (ℓ = 40), we may safely neglect the
overlap between γL and γR across the superconducting region and accordingly describe the low-energy excitations
of the system by approximating the fermion operators in the tunneling term of the total Hamiltonian [second row
of Eq.(39)] by means of the truncated mode expansion in Eq.(A19). As a result, we obtain the effective low-energy
Hamiltonian Hp;Ef f for the ring, given by
Hp;Ef f ≈ −ǫ0 cos

Φ
2

2Γ† Γ − 1

,

(43)

with ǫ0 ∝ τ and the Dirac fermion operator Γ related to γL , γR by means of Eqs.(A22). We now use Eq.(43) as
a main reference to discuss the behavior of the current for large-ℓ. In fact, while one should, in principle, consider

12

I[Φ]
0

−2π

0

2π

FIG. 6: Plot of the persistent current I[Φ] vs. Φ (in units of Φ0 ) for the p-wave mesoscopic ring with a weak link described by
Eq.(39). The parameters are chosen so that µ = 0, ∆ = 0.2 and τ = 0.5 (in units of w), ℓ = 4 (full red curve), ℓ = 8 (dashed
green curve), ℓ = 40 (dot-dashed blue curve). The estimated Majorana hybridization length in this case is ℓM ≈ 5.

the contributions arising from all the populated single-quasiparticle states at any energy (which is exactly done in
the calculation we performed based on our TM approach), based upon arguments similar to the ones provided in
Refs.[35,38], in the large-ℓ limit we expect that the result for I[Φ] can be safely recovered by taking into account
only low-energy states of the system. Now, for −π < Φ < π, Eq.(43) tells us that the ground state has the Γ-level
populated. As Φ crosses π, there is a crossing between the filled- and the empty-Γ state which, in the absence of
constraints on fermion parity conservation, makes the system ”jump” from the populated to the empty Γ-fermion
state, with the corresponding jump in the current evidenced at Φ = π in the blue curve of Fig.6, corresponding to
ℓ = 40. By symmetry Φ → −Φ, an analogous jump is observed at Φ = −π. The total current is periodic, with period
equal to 2π, due to the sequential level crossings at Φ = 2πk + π, with integer k 22 .
At variance, as ℓ = 4, the hybridization between the MFs across the topological superconductor is no longer negligible.
As a result, at low energy the system is described by an effective Hamiltonian such as the one in Eq.(A24), with a
modulation with Φ of the energy splitting between the empty- and the filled-state, which never closes (avoided level
crossing). In this case, the persistent current is only supported by Cooper pair tunneling across the weak link, which
restores a 4π-periodicity in Φ22 . Again, this is consistent with our plot in Fig.6 for ℓ = 4, with the intermediate
case ℓ = 8 lying in between the two ”asymptotic” cases. As a further check, we report in Fig.7 the plots of I[Φ]
generated by keeping w = 1, ∆ = 0.2, τ = 0.5 and varying µ, with ℓ = 16 (Fig.7 a) and ℓ = 40 (Fig.7 b) (note that
ℓM ∼ 5, as estimated above). From the plots we draw for µ = 0.0, 0.9, 1.5, we see that, increasing µ toward the critical
value µ = 2 at which the topological phase transition takes place20 , effectively corresponds to increasing ℓM . This
is expected from the results of the Appendix A, where we show that the hybridization between γL and γR scales as
e−ℓ/ℓM . Thus, again our results appear to be consistent with the low-energy dynamics of our system as inferred from
appendix A and from the discussion reported in Ref.[22].
By contrast, we now discuss the current across an s-wave ring. Despite the lack of low-energy MFs in such a system, the
crossover in the periodicity of I[Φ] from 4π to 2π is known to take place as the length ℓS of the superconducting region
crosses over from values lower than the coherence length ξ0 to values higher than ξ0 19 . Such a crossover corresponds
to a crossover in the ”physical nature” of I[Φ]: from a 4π-persistent current in a mesoscopic, effectively normal,
system to a 2π-periodic current, analogous to the Josephson supercurrent in an SNS-junction18,19 . In Fig.8, we plot
the exact results for I[Φ] obtained using our TM-approach for the system parameters w = 1, ∆ = 0.2, µ = 0, τ = 0.5
and for various values of ℓ. In our specific case, having as model Hamiltonian the one in Eq.(40), as we are setting to
1 the lattice step, an acceptable estimate for ξ0 is ξ0 ∼ 2w/∆. For the numerical values of the parameters we chose
to generate Fig.8, this implies ξ0 ∼ 10. Such an estimate is definitely consistent with our results: on increasing ℓ
from ℓ = 4 to ℓ = 40, we ultimately see a crossover in the periodicity of I[Φ] definitely similar to the one we found
for the p-wave superconducting ring with a weak link, though without the jumps in the current due to the Γ-fermion
level crossings. To conclude this section, let us stress once more that our technique does provide us with the exact
result for I[Φ] at a generic value of the system parameters, whether the superconductor is p-wave, or s-wave, etc. To
recover the final result one just needs to construct the appropriate TM and to numerically compute a single integral
for various values of Φ. Using the standard approach, based on the solution of the secular equations for the allowed

13

a)

b)

I[ Φ ]

−2π

0

2π

−2π

0

2π

FIG. 7: Plot of the persistent current I[Φ] vs. Φ (in units of Φ0 ) for the p-wave mesoscopic ring with a weak link described by
Eq.(39). The parameters are chosen so that ∆ = 0.2 and τ = 0.5 (in units of w). The curves are plotted for various values of
µ at fixed ℓ. Specifically:
a) ℓ = 16, µ = 0.0 (full red curve), µ = 0.9 (dashed green curve), µ = 1.5 (dot-dashed blue curve);
b) ℓ = 40, µ = 0.0 (full red curve), µ = 0.9 (dashed green curve), µ = 1.5 (dot-dashed blue curve).

I[Φ]

−2π

0

2π

FIG. 8: Plot of the persistent current I[Φ] vs. Φ (in units of Φ0 ) for the s-wave mesoscopic ring with a weak link described by
Eq.(40). The parameters are chosen so that ∆ = 0.2 and τ = 0.5 (in units of w). The curves are plotted for various values of ℓ
at µ = 0. Specifically: ℓ = 4 (full red curve), ℓ = 16 (dashed green curve), ℓ = 40 (dot-dashed blue curve). The crossover from
a 4π periodicity for ℓ < ξ0 ∼ 10 (see text) to a 2π-periodicity for ℓ > ξ0 is apparent.

values of the momenta at fixed Φ, and eventually taking the derivative with respect to Φ to obtain the current is, in
general, much less straightforward and, typically, exact results cannot be provided and different approximations must
be implemented to attack different regimes such as the short-ring, or the long-ring limit (see Ref.[18] for a careful and
valuable discussion about this point). At variance, as we are showing here, our approach applies to any specific case,
with potentially no limitations at all. It allows us to recover the full plots of the persistent current at various system
scales, which we show in this section and in the following one: an original result that complements and extends the
analysis of Ref.[29], where the analysis of the current was performed by just considering how the relative weight of
the first two harmonics (in Φ) varies, as a function of the system parameters. In the following, to discuss a further
application of our technique, we consider a hybrid ring, made by a p-wave superconducting segment of length ℓS and
a normal segment of length ℓN : this can be regarded as a generalization of the ring with a weak link which, as we are
going to discuss, opens the way to a number of interesting physical effects.

14

FIG. 9: Sketch of the hybrid N-S ring with the two regions (red and blue, respectively) separated by two a weak links. A
possible practical realization of such a system is the same as discussed before for the superconducting ring interrupted by a
weak link.

a)

b)

I[Φ]

−2π

I[Φ]

0

2π

−2π

0

2π

FIG. 10: Plot of I[Φ] vs. Φ for a hybrid normal - superconducting ring.
a) Plot of I[Φ] vs. Φ for a hybrid normal-p-wave superconducting ring for w1 = w2 = 1, ∆ = 0.2, τ = 0.5, µ = 0, which
corresponds to ℓM ≈ 6, ξK,M ≈ 16, for ℓS = ℓN = 4 (solid red curve), ℓS = ℓN = 16 (dashed green curve), ℓS = ℓN = 40
(dot-dashed blue curve). The crossover to a sawtooth behavior is evident for ℓ = 40;
b). Plot of I[Φ] vs. Φ for a hybrid normal-s-wave superconducting ring for w1 = w2 = 1, ∆ = 0.2, τ = 0.5, for ℓS = ℓN = 4
(solid red curve), ℓS = ℓN = 16 (dashed green curve), ℓS = ℓN = 40 (dot-dashed blue curve). There is no crossover in the
functional form of I[Φ], but a mere scaling of I[Φ] that is ∼ ℓ−1 .
N

B.

Persistent current across a hybrid N-S ring

In this section we discuss the persistent current across a hybrid ring, composed of a superconducting segment of
length ℓS and of a normal segment of length ℓN . Such a system can be regarded as a generalization of the ring
interrupted by a weak link discussed in Ref.[29], in which one induced superconductivity by proximity only in a part
of the ring, leaving a finite normal region of length ℓN . In Fig.9 we provide a sketch of the system we discuss here.
Again, for comparison, we consider both cases of a p-wave and of an s-wave superconducting region. The corresponding
model Hamiltonian can then be obtained from H in Eq.(14) by setting ℓ1 → ℓS , ℓ2 → ℓN , ∆1 → ∆, ∆2 = 0 and from
H in Eq.(33), taken in the same limit. To spell out the behavior of I[Φ] in the various regimes of interest, let us first
focus onto the p-wave case. Specifically, we compute I[Φ] at given τ and ∆ for w1 = w2 ≡ w and for various values
of ℓS = ℓN ≡ ℓ. To further simplify the calculation we restrict ourselves to the particle-hole symmetric case, µ = 0.
In Fig.10a), we plot I[Φ] vs. Φ for ℓ = 4, 16, 40, with the values for the system’s parameters chosen as in the caption.
The behavior of I[Φ] depends on the system size in relation to the length scales determined by the system parameters.
At µ = 0 the p-wave superconductor lies within its topological phase, with corresponding localized MFs emerging at
its endpoints. Taking again ℓM as a reference length scale, when ℓ ≤ ℓM , the two MFs are hybridized into a Dirac
mode Γ. As a result, the MFs have no significant effects on the current, which is 4π-periodic, consistently with the

15
expected behavior of the system as a mesoscopic normal ring22 . On increasing ℓ, when ℓ > ℓM , the hybridization
between the MFs becomes negligible and, accordingly, in the absence of fermion parity conservation, I[Φ] becomes
2π-periodic, with jumps at Φ = π + 2πk. In addition to the periodicity, also the shape of I[Φ] depends on ℓ. This
is due to the Kondo-like hybridization between the MFs and the excitation modes within the normal region of the
ring, which takes place when ℓ ≥ ξK,M , with the Kondo-Majorana hybridization (KMH) length ξK,M ∼ [(2w)2 /τ 2 ]
44
. At the onset of KMH, I[Φ] is expected to cross from a discontinuous sinusoidal dependence on Φ to a sawtoothlike shape44 . Physically, this can be understood by recalling that, as ℓ becomes large, the systematic cancellation
of contributions from high-energy states makes only low-energy states in the normal region next to the Fermi level
contribute to I[Φ]. The physical processes at the SN-interfaces that determine these states can be inferred from
Fig.11, where, as a function of ℓS , we plot the scattering coefficients across the superconducting regions corresponding
to normal reflection at the SN-interfaces and to normal transmission across the superconducting region, as well as the
coefficients corresponding to Andreev reflection (AR) at the interfaces and to ”crossed Andreev reflection” (CAR)
across the superconducting region41,45 . As it clearly appears from Fig.11, as soon as ℓS > ℓM , all the coefficients
drop to 0 but the one corresponding to AR, which saturates to 1. This evidences that, as ℓ > ℓS , AR is the only
physical process that takes place at low energy, which implies the sawtooth behavior in I[Φ] evidenced in Fig.10a).
By comparison, in Fig.10b), we plot I[Φ] vs. Φ for the same values of the various parameters as in Fig.10a), but for
an s-wave superconductor. Here we see that, on increasing ℓ, the current still shows the crossover from a 4π-periodic
curve to a 2π-periodic curve, but that the absence of low-energy Majorana modes eventually hybridizing with the
modes in the normal region as ℓ ≥ ξK,M yields no crossover in the functional form of I[Φ] from a sinusoidal to a
sawtooth behavior. The only relevant additional feature that takes place on varying ℓ is, indeed, the expected scaling
of I[Φ] ∼ ℓ−1 [see discussion in the next section]. Thus, the crossover in the functional form of I[Φ] can actually
be regarded as a direct evidence for the existence of low-energy Majorana modes at the endpoints of the topological
superconductor in the p-wave hybrid ring.
To further substantiate the above picture, in Fig.12a) we again plot I[Φ] vs. Φ for the system parameters numerically
set as discussed in the caption. In drawing Fig.12, we hold ℓN = ℓS fixed at 40 (with ∆ and w chosen so that ℓM ∼ 6),
as well as the chemical potential within the normal region µN = 0. At variance, we vary the chemical potential within
the superconducting region µS starting from µS = 0 till µS /w ∼ 1.95 (after which the loss of numerical precision
appears not to give us reliable results). As highlighted by Kitaev20 , as µS /w = 2 the p-wave superconductor undergoes
a (topological) quantum phase transition, characterized by the disappearance (for µS /w > 2) of the localized MFs
at the endpoints of the superconductor. On approaching the phase transition from within the topological region,
the closer the system is to the quantum critical point, the larger is the effective ℓM . While the actual numerical
estimate of ℓM as a function of e.g. µS /w at fixed ∆ can in principle be provided from Eq.(A5) of appendix A, here
we just focus on the consistency of our exact results with the expectation one gets from the above discussion. In
fact, the estimated KMH-length for the system used to derive I[Φ] in Fig.12a) is ξK,M ≈ 16 [see the discussion in
the caption of Fig.10a), which is drawn at the same values of w and τ ]. In Fig.12a) we plot I[Φ] for ℓS = ℓN = 40,
we see full KMH in the normal region for µN = 0, as evidenced by the sawtooth behavior of the current and by
the corresponding 2π-periodicity in Φ. On increasing µS towards the critical value corresponding to the topological
quantum phase transition, the nonnegligible hybridization between the MFs across the superconducting region is
expected to compete with KMH and eventually to suppress it (in fact, this can be regarded as a ”Majorana analog”
of the competition between Kondo effect and RKKY-interaction in the two-impurity Kondo model46–49 , just as the
KMH can be regarded as the Majorana analog of the onset of the Kondo cloud in a Kondo system44 ). Consistently
with the expectation, we see that, on increasing µS , the sawtooth is smoothed (with a sizable reduction in the critical
current) and clearly evolves back towards a restoration of the 4π-periodicity that characterizes the regime ℓS ≤ ℓM
(see discussion above). For comparison, in Fig.12b) we draw similar plots generated in the s-wave case. No particular
changes in the functional form of the current appear, except the reduction in the value of the current at a given Φ.
In our view, this result does actually enforce the reliability of a persistent current measurement as a tool to detect
the presence of MFs at the endpoints of a p-wave superconductor in the topological phase.
It is important to stress once more that, while our approach allowed for readily studying the additional consequences
of KMH in the case of a hybrid ring with an extended normal region, due to the increasing complexity of the system,
this would be hardly doable within an alternative approach, without resorting to some ”ad hoc” approximations
and possibly washing out some relevant physical effects. This enforces once more the importance of having an exact
analytical formula, that applies independently of the specific values of the system parameters. Besides the possibility
of exactly computing the current in a number of different physical systems by simply evaluating the integral in Eq.(13)
for different values of Φ, our approach also provide a remarkable tool to write, for large enough systems, I[Φ] in a
power series of the inverse system size which, as we are going to discuss in the following, greatly simplifies the various
calculations, by even providing explicit analytic results for the current, in some simple cases.

16

1

0
2

lS

32

FIG. 11: Plot of the normal and Andreev reflection coefficients at the SN-interface and of the normal transmission and CAR
coefficients for the p-wave superconductor-normal hybrid ring discussed in section IV B as a function of ℓS . To generate the
plot we have chosen the system’s parameters so that w1 = w2 = 1, ∆ = 0.2, τ = 0.5, which corresponds to ℓM ≈ 6. The various
curves correspond to the coefficients as
- Full blue curve: Andreev reflection coefficient at the SN-interface;
- Dashed blue curve: Normal reflection coefficient at the SN-interface;
- Full red curve: Normal transmission across the superconducting region;
- Dashed red curve: Crossed Andreev reflection across the superconducting region.
Apparently, for ℓS > ℓM all the processes are suppressed, except the Andreev reflection at the SN-interface, with the corresponding coefficient saturating to 1.

V.

THE LARGE-RING LIMIT

As the ring size goes large, one may recast the integral formula for I[Φ], Eq.(13), in an expansion in inverse powers
of the length that gets large. As we discuss in the following, this leads to a number of simplifications in the calculation
of the current, similar to the ones implemented in Refs.[35,38], even leading, in some cases, to a closed-form analytic
formula for I[Φ] vs. Φ at given system parameters. In the following, we discuss a few examples of calculation of the
persistent current in the large-size limit, also showing how a number of known results can be easily recovered within
our formalism, once the appropriate limit is taken.
A.

The limit of long superconducting region

The limit of long superconducting region is defined by sending ℓ2 → ∞ in the system described by the model
Hamiltonian in Eq.(14) (p-wave case), or by the Hamiltonian in Eq.(33) (s-wave case), after setting ∆1 = 0, so that
region-1 is normal, and by keeping ℓ1 finite. In this limit one expects to recover the results for a the Josephson current
across an SNS-hybrid junction. To show that this is, in fact, the case, we start by rewriting det{Mp−wave [E; Φ; ℓ1 ; ℓ2 ]−
I} as
−1
det{Mp−wave [E; Φ; ℓ1 ; ℓ2 ] − I} = c det{T2 (E; ℓ2 )

−[Υ(2) (E)]−1 · Σ−1 [Φ] · Ω1 [Φ] · [Υ(1) (E)] · T1 (E; ℓ1 ) · [Υ(1) (E)]−1 · Σ−1 [Φ] · Ω2 [Φ] · [Υ(2) (E)] ,
2
1

(44)

with c being an over-all factor independent of Φ and, similarly, by rewriting det{Ms−wave [E; Φ; ℓ1 ; ℓ2 ] − I} as
det{Ms−wave [E; Φ; ℓ1 ; ℓ2 ] − I} = c′ det{T−1 (E; ℓ2 )
2
˜ (2) (E)]−1 · Σ−1 [Φ] · Ω1 [Φ] · [Υ(1) (E)] · T1 (E; ℓ1 ) · [Υ(1) (E)]−1 · Σ−1 [Φ] · Ω2 [Φ] · [Υ(2) (E)]} ,
˜
˜
˜
˜
˜
˜
˜
−[Υ
2
1

(45)

with, again, c′ being a constant independent of Φ. As a next step, we define the matrix Mp as
−1
Mp = [Υ(2) (E)]−1 · Σ2 [Φ] · Ω1 [Φ] · [Υ(1) (E)] · T1 (E; ℓ1 ) · [Υ(1) (E)]−1 · Σ−1 [Φ] · Ω2 [Φ] · [Υ(2) (E)]
1

,

(46)

17

a)

b)

I[Φ]

I[Φ]

0

0

−2π

0

2π

−2π

0

2π

FIG. 12: Plot of I[Φ] vs. Φ for a hybrid normal - superconducting ring.
a) Plot of I[Φ] vs. Φ for a hybrid normal-p-wave superconducting ring for w1 = w2 = 1, ∆ = 0.2, τ = 0.5, µN = 0, which
corresponds ξK,M ≈ 16, ℓS = ℓN = 40 and
- µS /w = 0.0 (solid blue curve)
- µS /w = 1.5 (dashed green curve)
- µS /w = 1.9 (dot-dashed red curve)
- µS /w = 1.95 (dot-dot-dashed black curve)
b) Same as in a), but for a hybrid normal-s-wave superconducting ring for w1 = w2 = 1, ∆ = 0.2, τ = 0.5, µN = 0,
ℓS = ℓN = 40 and
- µS /w = 0.0 (solid blue curve)
- µS /w = 1.0 (dashed green curve)
- µS /w = 1.5 (dot-dashed red curve)
- µS /w = 1.9 (dot-dot-dashed black curve)
Panel a) shows a clear smoothing of the sawtooth dependence of I[Φ] on Φ towards a sinusoidal plot as µS /w increases towards
the critical value µS /w = 2, at which Majorana fermions disappear.

for the p-wave hybrid ring, and
˜
˜ −1
˜
˜
˜
˜
˜
˜
Ms = [Υ(2) (E)]−1 · Σ2 [Φ] · Ω1 [Φ] · [Υ(1) (E)] · T1 (E; ℓ1 ) · [Υ(1) (E)]−1 · Σ−1 [Φ] · Ω2 [Φ] · [Υ(2) (E)] ,
1

(47)

for the s-wave hybrid ring. From Eqs.(46,47), we see that the current in the p-wave and the s-wave hybrid ring,
Ip,s [Φ], can respectively be written as
Ip,s [Φ] = −

1
2πi

Γ

dE ∂Φ ln[Ψp,s [E; Φ]] = −

1
4πi

Γ

dE ∂Φ ln det[T−1 (ǫ; ℓ2 ) − Mp,s ] − [ln det[T−1 (ǫ; ℓ2 ) − Mp,s ]]∗
2
2

,

(48)
where we have used the reality of the persistent current to go through the last step in Eq.(48). In order to systematically
take the ℓ2 → ∞-limit, we recall that one has eventually to deform the integrals over Γ in Eq.(48) into integrals over
the imaginary axis, which corresponds to E → iω. Along the imaginary axis, from the dispersion relations for particleand hole-like excitations within the superconducting region, one obtains that the corresponding momenta are defined
by
(2)
cos[kp ] = −
(2)

cos[kh ] = −
in the p-wave case, and

µ2 w2
i
2 − ∆2 ) − 2
2(w2
2

ω 2 + ∆2
w
2
w2 − ∆2
2

i
µ2 w2
+
2
2(w2 − ∆2 ) 2
2

ω 2 + ∆2
w
2
w2 − ∆2
2

,

(49)

18
(2)
cos[kp ] = −
(2)

cos[kh ] = −

i
µ2
−
2w2
2

ω 2 + ∆2
2
2
w2

µ2
i
+
2w2
2

ω 2 + ∆2
2
2
w2

,

(50)

in the s-wave case. To solve Eqs.(49) we therefore set
(2)
kp =

π
π
(2)
∗
+ qp , kh = + qp ,
2
2

(51)

with
sin[qp ] =

µ2 w2
i
+
2
2(w2 − ∆2 ) 2
2

ω 2 + ∆2
w
2
w2 − ∆2
2

,

(52)

while, to solve Eqs.(50), we set
(2)
kp =

π
π
(2)
∗
+ qs , kh = + qs ,
2
2

(53)

with
sin[qs ] =

µ2
i
+
2w2
2

ω 2 + ∆2
2
2
w2

.

(54)

−1
From the explicit formula for T2;(p,s) (E → iω; ℓ2 ) along the imaginary axis in the p-wave and in the s-wave case,
respectively given by


(2)
0
0
0
i−ℓ2 e−iqp,s ℓ2
(2)


0
0
0
iℓ2 eiqp,s ℓ2


T−1 (E → iω; ℓ2 ) = 
(55)
 ,
(2) ∗
2;(p,s)


0
0
0
iℓ2 ei[qp,s ] ℓ2
(2) ∗
0
0
0
i−ℓ2 e−i[qp,s ] ℓ2

we may readily compute the integrals in Eq.(48) in the limit ℓ2 → ∞, obtaining
Ip,s [Φ] =

1
2π

∞
−∞

dω ∂Φ ln Gp,s (E → iω)

.

(56)

with Gp,s (E) = M(p,s);(2,2) (E)M(p,s);(4,4) (E) − M(p,s);(2,4) (E)M(p,s);(4,2) (E) and assuming (as done in Ref.[35,38])
that
• All the poles of G(E) lie over the real axis;
• G(E) is real if E lies over the real axis (and does not coincide with a pole of G),
Eq.(56) yields the dc-Josephson current in the infinite-ℓ2 limit, in which the ring can be regarded as an idealized
model for an SNS-junction. In the specific case of s-wave superconductors, Eq.(52) has been derived in Ref.[35] for
a single-channel junction starting from the S-matrix approach to effectively one-dimensional SNS-junctions32 , and
generalized in Ref.[38] to a multi-channel junction. In fact, a comparison between Eq.(56) and Eqs.(7,9) of Ref.[35]
also clarifies the identification between Mp,s in Eqs.(46,47) and the transfer matrix for the whole SNS-junction, as
introduced and discussed in Ref.[35] for the s-wave case. After resorting to the effective SNS-junction model, at a
second stage one may implement the technique developed in Refs.[35,38] to write I[Φ] in a systematic expansion in
inverse powers of ℓ1 . Basically, one considers that, because one has


(1)
0
0
0
iℓ1 eiqn ℓ1
(1)


0
0
0
i−ℓ1 e−iqn ℓ1


(57)
T1 (E → iω; ℓ2 ) = 
 ,
(1) ∗
−ℓ1 −i[qn ] ℓ1


0
0
0
i e
(1) ∗
0
0
0
iℓ1 ei[qn ] ℓ1

19
with
(1)
sin[qn ] =

µ1
i ω
+
2w1
2 w1

,

(58)

then, only low-|ω| regions do effectively contribute to the integral in Eq.(56). As a result, one may first of all
(1)
approximate qn ≈ q + iσ(ω), with
¯
sin(¯) =
q
σ(ω) =

µ1
2w1

ω
2w1 cos(¯)
q

,

(59)

then, in integrating Eq.(56), one may rescale ω → ωℓ1 and eventually set the rescaled ω at 0 in all the contributions
to the argument of the integral in which ω appears divided by ℓ1 as ω/ℓ1 . Going along this procedure, one may
compute the leading contribution to the current (O(ℓ−1 )) by trading the original model Hamiltonian for a reduced
1
boundary model, such as the one presented in Ref.[31] for the s-wave superconductors and the one used in Ref.[44]
for the p-wave superconductor, which allows for recovering simple, closed-form analytical formulas for I[Φ].
B.

The limit of long normal region

We now discuss the complementary limit of a long normal region, with ℓS less than, or comparable to, the coherence
length of the superconducting region ξ0 . We first discuss the general formula and then consider the case of a hybrid
s-wave superconducting ring as a specific example. In order to address the large-ℓN -limit, let us first of all rewrite
det{Mp−wave (s−wave) [E; Φ; ℓ1 ; ℓ2 ] − I} as
det{Mp−wave (s−wave) [E; Φ; ℓ1 ; ℓ2 ] − I} = cp,s det{T1 (E; ℓ1 ) − Kp,s [E; Φ; ℓ2 ]} ,

(60)

with cp,s being constants independent of Φ, and
Kp [E; Φ; ℓ2 ] = [Υ(1) (E)]−1 · Ω−1 [Φ] · Σ2 [Φ] · [Υ(2) (E)] · T−1 (E; ℓ2 ) · [Υ(2) (E)]−1 · Ω−1 [Φ] · Σ1 [Φ] · [Υ(1) (E)]
1
2
2
˜ (1) (E)]−1 · Ω−1 [Φ] · Σ2 [Φ] · [Υ(2) (E)] · T−1 (E; ℓ2 ) · [Υ(2) (E)]−1 · Ω−1 [Φ] · Σ1 [Φ] · [Υ(1) (E)] . 61)
˜
˜
˜
˜
˜
˜
˜
Ks [E; Φ; ℓ2 ] = [Υ
(
1
2
2
Equations (61) correspond to the standard identification we have employed so far, that is, region-1 has to be identified
with the normal region and region-2 with the (either p-wave, or s-wave) superconducting region. Therefore, ℓ1 = ℓN .
Now, in order to recover the large-ℓN -limit, we strictly follow the derivation of Refs.[35,38], that is, once we have
deformed the integration path to the imaginary axis, we assume that only low-ω(= −iE) regions do effectively
contribute the integral in Eq.(13). This allows us first of all to approximate the inverse dispersion relation within the
normal region as
kp,h ≈ kF ± iλ(ω)
with −2w1 cos(kF ) = µ1 and λ(ω) =
Eq.(13) as

ω
2w sin(kF ) .

Ip,s [Φ] = −

,

(62)

On substituting Eqs.(62) into Eqs.(61), we may eventually rewrite

2ew1 sin(kF )
∂Φ
2π ℓ1

∞
0

dz
ln{Ξp,s [z; Φ; ℓ2 ]}
z

,

(63)

with

 z 0


 0 z −1
Ξp,s [z; Φ; ℓ2 ] = det 
 0 0

 0 0

  −ikF ℓ1
0
0
0
e
0 0
0
0
0
eikF ℓ1
0 0  
−
z 0  
0
0
0
eikF ℓ1
−1
0 z
0
0
0
e−ikF ℓ1









 · Kp,s [E = 0; Φ; ℓ2 ]





.

(64)

To illustrate the effectiveness of our simplified Eqs.(63,64) we now discuss the application to the case of a hybrid
s-wave superconducting ring. For simplicity, we make the assumptions w1 = w2 ≡ w and µ1 = 0. As a result, using
Eqs.(63,64), we obtain the simplified expression for the current

20
I[Φ] = −

2ew1 sin(kF )
∂Φ
2π ℓ1

∞
0

∂Φ a[Φ; ℓ1 ](z 3 + z) + ∂Φ b[Φ; ℓ2 ]z 2
+ 1 + a[Φ; ℓ2 ](z 3 + z) + b[Φ; ℓ2 ]z 2

dz
z

z4

,

(65)

with a[Φ; ℓ2 ], b[Φ; ℓ2 ] being long, though straightforward to derive, functions of the matrix elements of Ks [E = 0; Φ; ℓ2 ].
Eventually, by means of simple manipulations Eq.(65) can be expressed as a closed-form formula only of the four roots
zj [Φ; ℓ2 ] (j = 1, . . . , 4) of the polynomial equation z 4 + 1 + a[Φ; ℓ2 ](z 3 + z) + b[Φ; ℓ2 ]z 2 = 0, which take the generic
form
zj [Φ; ℓ2 ] = −
+ vj

1
2

−2 +

a[Φ; ℓ2 ]
+ uj
4

8 + a2 [Φ; ℓ2 ] − 4b[Φ; ℓ2 ]
4

a3 [Φ; ℓ2 ] + 8a[Φ; ℓ2 ] − 4a[Φ; ℓ2 ]b[Φ; ℓ2 ]
a2 [Φ; ℓ2 ]
− b[Φ; ℓ2 ] + uj
4
2 8 − a2 [Φ; ℓ2 ] − 4b[Φ; ℓ2 ]b[Φ; ℓ2 ]

with uj , vj = ±1. Taking into account that

4
j=1 zj [Φ; ℓ2 ]

,

(66)

= 1, one eventually obtains from Eq.(65)
4

I[Φ] = −

2ew1 sin(kF )
ln2 [zj [Φ; ℓ2 ]]}
∂Φ {
4πℓ1
j=1

.

(67)

Just as in the case of a long SNS-junction35,38 , Eq.(67) only involves data at the Fermi level. This is an additional
example of the remarkable simplifications to which our approach leads, in the large ring limit.
VI.

FINITE-TEMPERATURE RESULTS

It is not difficult to generalize our derivation to a system at temperature T finite, though much lower than the
critical temperature for the superconducting part of the ring. In fact, at finite T , Eq.(13) generalizes to Eq.(1), with
F [Φ; T ] =

Ef (E)

,

(68)

E

and f (E) = [1 + eE/T ]−1 being the Fermi distribution function (having set the Boltzmann constant k = 1. Now, the
˜
integration path Γ in Eq.(11) must be replaced with the integration path Γ obtained as the union of small circles Γn ,
˜
each one surrounding once one, and only one, energy eigenvalue. As illustrated in Fig.13, Γ can be deformed to a
path obtained as the union of small circles, each one surrounding once one, and only one, pole of the Fermi function,
1
that is, a fermionic Matsubara frequency times the imaginary unit i, iωm = 2πT i m + 2 . As a result, the finite-T
current can be presented as
I[Φ; T ] = −eT

ωm

∂Φ {det[M[iωm ; Φ] − I]}

,

(69)

that is the appropriate generalization of Eq.(13) to the finite-T case.
VII.

DISCUSSION AND CONCLUSIONS

In this paper we have presented a technique to exactly compute the zero-temperature persistent current across an
HR pierced by a magnetic flux Φ as a single integral of a known function of the system’s parameters. Our approach
makes use of the properties of the transfer matrix of the ring, which allows us to circumvent technical difficulties
associated with the secular equation for the energy eigenvalues of the system. A straightforward generalization of
the zero-temperature formalism allows us to also compute the current in a ring at a temperature T finite, though
much lower than the superconducting gap. While in general one may readily numerically compute the integral/sum
yielding the current at a given value of the flux Φ, a remarkable simplification takes place in the limit of a large ring
size, where resorting to a systematic expansion in inverse powers of the ring length allows for deriving the current in
analytic closed-form formulas, applicable to a number of cases of physical interest.
As an example of application of our technique, we exactly compute the persistent current through a p-wave
superconducting-normal ring as well as in an s-wave superconducting-normal ring. As a side result, we recover

21

a)

1
0
1
0
1
0

Im(E)

Im(E)

Im(E)

b)

c)

11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

11
00
11
00

1
0
1
0
1
0

11
00
11
00
11
00

11
00
11
00
11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0
1
0
1
0
1
0

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

Re(E)

11
00
11
00
11
00

11
00
11
00
11
00

11
00
11
00
11
00

11
00
11
00
11
00
1
0
11
00
1
0
11
00
1
0
11
00
11
00
11
00

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

Re(E)

11
00
11
00
11
00

11
00
11
00
11
00

11
00
11
00
11
00

11
00
11
00
11
00
1
0
11
00
1
0
11
00
1
0
11
00
11
00
11
00

1
0
1
0

11
00
11
00
11
00
11
00
11
00
11
00
11
00

1
0
1
0
1
0

Re(E)

11
00
11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00

1
0
1
0
1
0

1
0
1
0
1
0

11
00
11
00
11
00

˜
FIG. 13: Sequence of deformations in the integration path Γ eventually allowing to express I[Φ; T ] as a sum over the fermionic
Matsubara frequencies ωm = 2πT m + 1 :
2
˜
a) The path Γ obtained as the union of small circles, each one surrounding one, and only one, energy eigenvalue;
˜
b) The integral over Γ can be deformed to an integral over the union of the two closed path run through counterclockwise;
c) The integral over the two closed path in b) is equal to the integral over a closed path surrounding the poles of the Fermi
function (iωm , displayed as blue full circles in the figure), run through clockwisely. The corresponding over-all - sign is eventually
canceled by the -1 at the residue of the Fermi function at iωm .

at once the crossover in the current periodicity, the effects of localized MFs in the p-wave case (including the signature of the topological phase transition), the large-size limit, etc., which were previously approximatively derived by
means of various approximation schemes, different from case to case28,29,35,38 .
Throughout all the paper, we grounded our approach on a description of the ring in terms of a noninteracting lattice
model Hamiltonian for the normal part of the ring, in terms of a non-self consistent mean field Hamiltonian for the
superconducting part. The system is assumed to be ”clean”, that is, no disorder effects are taken into account and
no interaction between the electrons is assumed. As a further development of our research, we plan to generalize our
approach, so to introduce the effect of disorder by means of a pertinently chosen impurity potential and by eventually
ensemble averaging over the disorder realization50,51 , and the effect of the interaction, by adopting a pertinently
adapted version of the fermion renormalization group approach introduced in Refs.[52–54] for weakly interacting
systems and generalized in Ref.[55] to the strongly interacting case. This might potentially provide an useful method
to also spell out the dynamics of fractionalized excitations in correlated systems56–58 . Moreover, by simply sending to
infinity the length of the superconducting part of the ring, our approach provides an exact formula for the Josephson
current across an SNS junction, which can be readily applied to cases hard to deal with using alternative techniques,
such as in the case of the anomalous Josephson effect in nanowires59–61 .
These, and other potentially interesting generalizations of our approach, do actually lie beyond the range of this
paper, and we plan to address them as a future development of our work.
We thank S. K. Maiti for a very useful correspondence exchange during the preparation of this work. We acknowledge
insightful discussions with L. Lepori and A. Tagliacozzo.
Appendix A: Exact wavefunctions for sub-gap states at a finite-length Kitaev chain

In this section, we discuss in detail the derivation of the exact wavefunction of sub-gap states in an open Kitaev
chain, of finite length ℓS . To do so, we employ a simplified and pertinently adapted version of the solution of the
Kitaev model with open boundary conditions62,63 . While, for finite-ℓS , we find a single Dirac fermion level, which can
be either empty, or filled, with a corresponding finite energy gap EM , as ℓS → ∞, the states become degenerate in
energy at the Fermi level, and appropriate linear combinations of the corresponding wavefunctions become localized
at the two endpoints of the chain, eventually corresponding to the two zero-energy Majorana solutions of Kitaev’s
model20 . The starting point is the dispersion relation in Eqs.(23) which, for sub-gap solutions (|E| < ∆w ) yields the
allowed values of the (complex conjugate) particle- and hole-momenta defined by

22
cos(kp ) = −
cos(kh ) = −

∆2 − E 2
w
w2 − ∆2

i
wµ
−
2(w2 − ∆2 ) 2
2(w2

∆2 − E 2
w
w2 − ∆2

wµ
i
+
2)
−∆
2

.

(A1)

Eqs.(A1) are readily solved by setting kp = π − qR + iqI and kh = (kp )∗ = π − qR − iqI , with
cos(qR ) cosh(qI ) =
sin(qR ) sinh(qI ) =

wµ
− ∆2 )

2(w2
1
2

∆2 − E 2
w
w2 − ∆2

.

(A2)

As a result, the general formula for a subgap solution will be given by
uj
vj

= (−1)j A(p,+)
+ A(h,+)

u∗
v∗

u
−v

u −iqR j −qI j
e
e
+ A(p,−)
v

eiqR j e−qI j + A(h,−)

u∗
−v ∗

eiqR j eqI j

e−iqR j eqI j

,

(A3)

with u, v solutions of the algebraic system
{E + 2w cos(kp ) + µ}u − 2i∆ sin(kp )v = 0
2i∆ sin(kp )u + {E − 2w cos(kp ) − µ}v = 0 .

(A4)

The actual energy eigenstates are determined as nontrivial solutions such as the one in Eq.(A3) satisfying the boundary
u0
uℓ+1
conditions on the ”ghost sites” given by
=
= 0. It is therefore straightforward to verify that this
v0
vℓ+1
implies the secular equation for the energy eigenvalues given by
[ℑm(uv ∗ )]2 sinh2 [qI (ℓ + 1)] = [ℜe(uv ∗ )]2 sin2 [qR (ℓ + 1)] .

(A5)

Eqs.(A3,A4,A5) can be used to estimate the energy of the sub-gap levels and the corresponding wavefunction for any
value of µ. Indeed, we use them to numerically estimate the energy gap and to accordingly infer the overlap scale
between the localized MFs at given values of the system’s parameters. Specifically, of one is only interested in the
low-energy physics of a finite-size one-dimensional topological superconductor coupled to normal conductors at each
of its endpoints, the whole topological superconductor can be traded for an effective Hamiltonian involving only the
low-energy sup-gap degrees of freedom discussed above, with parameters effectively determined by the actual system
parameters. To illustrate how the procedure works, let us focus onto the simple case µ = 0, ℓ even. In this case, kp
and kh are simply given by
π
+ iλ(E)
2
π
kh = − iλ(E) ,
2
kp =

(A6)

with
1
2

sinh[λ(E)] =

4∆2 − E 2
w2 − ∆2

.

(A7)

As a result, Eq.(A3) can now be presented as
uj
vj

= A(p,+)

u
v

+ A(h,+)

u∗
v∗

ij e−λ(E)j + A(p,−)

u
−v

i−j e−λ(E)j + A(h,−)

i−j eλ(E)j

u∗
−v ∗

ij eλ(E)j ,

(A8)

with
up
vp

1
=√
2

i

e 2 sgn(E)ϑ
i
isgn(E)e− 2 sgn(E)ϑ

;

uh
vh

1
= √
2

i

−isgn(E)e− 2 sgn(E)ϑ
i
e 2 sgn(E)ϑ

,

(A9)

23
and
ϑ = atan

2w sinh[λ(E)]
|E|

.

(A10)

It is straightforward, though tedious, to show that the secular equation for the sub-gap energy eigenvalues is now
given by
(ℓ + 1)λ(E) = ± sinh−1

2w
sinh[λ(E)]
E

,

(A11)

which is solved by setting
|E| = ǫ = 2∆

cosh[λ(E)]
∆
≈ 2∆e−ℓλ(E) ≈ 2∆ exp −ℓ sinh−1 √
2 − ∆2
cosh[(ℓ + 1)λ[E]]
w

.

(A12)

Therefore, from Eq.(A12) we readily estimate that the hybridization length scale between the MFs, ℓM , is given by
−1

∆
sinh−1 √
2 − ∆2
w

ℓM ∼

.

(A13)

From Eq.(A3) we may therefore construct the wavefunctions corresponding to the positive- and to the negative-energy
sub-gap solutions. As a result, one obtains for the positive- sub-gap energy solution of the BdG equations
uj
vj

i

ξ

= c e2
+

i

e2ϑ
i
ie− 2 ϑ

e2ϑ
i
−ie− 2 ϑ

ξ

ij e−jλ(E) + ie− 2

i

i

−ie− 2 ϑ
i
−e 2 ϑ

ξ

+ e2

i−j ejλ(E)

−ie− 2 ϑ
i
e2ϑ

ξ

i−j e−jλ(E) − ie− 2

ij ejλ(E)

,

(A14)

with
c=
and ξ(E) = sinh−1
equations

2w
E

1
4

2 sinh[λ(E)]
sinh[ξ(E) − λ(E)]

,

(A15)

sinh[λ(E)] . Similarly, one obtains for the negative-energy sub-gap solution of the BdG

uj
vj

i

e− 2 ϑ
i
−ie 2 ϑ

ξ

−

= −c e 2

i

i

ξ

+ e2

e− 2 ϑ
i
ie 2 ϑ

ξ

ij e−jλ(E) − ie− 2

i−j ejλ(E)

i

ie 2 ϑ
i
−e− 2 ϑ

ξ

i−j e−jλ(E) + ie− 2

ie 2 ϑ
i
e− 2 ϑ

ij ejλ(E)

.

(A16)

From Eqs.(A14,A16), we therefore find that the eigenmodes corresponding to the ± solutions are respectively given
by
ℓ

Γ+ =
j=1

{[uj ]∗ cj + [vj ]∗ c† }
+
+ j

ℓ

Γ− =
j=1

{[uj ]∗ cj + [vj ]∗ c† } ,
−
− j

(A17)

which shows that, as expected, one recovers the relation
Γ+ = Γ† ≡ Γ .
−

(A18)

On inverting Eqs.(A17) and truncating the mode expansion of the real space lattice operators by retaining low-energy
modes only, we therefore get

24
cj ≈ [uj ]+ Γ + [vj ]∗ Γ†
+

c† ≈ [vj ]+ Γ + [uj ]∗ Γ† .
+
j

(A19)

Now, on calling {dj , d† } and ℓN the length of the normal part of the ring, the fermion operators on the normal side,
j
one may rewrite the tunneling contribution to the Hamiltonian in Eq.(14) as
i

i

Hτ = −τ {[c† dℓN + d† cℓN ]e 4 Φ + [d†N c1 + c† d1 ]e− 4 Φ } .
1
1
ℓ
ℓ

(A20)

Using the truncated expansions in Eqs.(A19) and the explicit form of the wavefunctions evaluated at j = 1, ℓ, one
eventually approximates Eq.(A20) as
i

i

i

i

Hτ = tL {γL [e 4 Φ dℓN − e− 4 Φ d†N ]} + itR {γR [e− 4 Φ d1 + e 4 Φ d† ]} ,
1
ℓ

(A21)

with
π

π

γL = e−i 4 Γ + ei 4 Γ†
π

π

γR = −i{e−i 4 Γ − ei 4 Γ† } .

(A22)

and tL = tR = Υτ , with
π

u1,+ = −e−i 4 Υ
π

v1,+ = −e−i 4 Υ
π

uℓ,+ = −e−i 4 Υ
π

vℓ,+ = e−i 4 Υ

.

(A23)

Finally, to recover the energy bias between the Dirac modes, we add a term of the form
HΓ = 2ǫ{2Γ† Γ − 1} = −2ǫiγL γR .

(A24)

In general, tL , tR are smooth functions of µ. The dependence of ǫ on µ can be inferred by numerically solving Eq.(A5),
as we did in the main text, to also derive the dependence of ℓM on the chemical potential.

1
2
3
4
5
6

7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

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