arXiv:cond-mat/0010440v2 [cond-mat.

mes-hall] 30 Oct 2000

Unpaired Majorana fermions in quantum wires

Alexei Yu. Kitaev∗
Microsoft Research
Microsoft, #113/2032, One Microsoft Way,
Redmond, WA 98052, U.S.A.
kitaev@microsoft.com
27 October 2000

Abstract
Certain one-dimensional Fermi systems have an energy gap in the
bulk spectrum while boundary states are described by one Majorana
operator per boundary point. A finite system of length L possesses two
ground states with an energy difference proportional to exp(−L/l0 )
and different fermionic parities. Such systems can be used as qubits
since they are intrinsically immune to decoherence. The property of a
system to have boundary Majorana fermions is expressed as a condi-
tion on the bulk electron spectrum. The condition is satisfied in the
presence of an arbitrary small energy gap induced by proximity of a
3-dimensional p-wave superconductor, provided that the normal spec-
trum has an odd number of Fermi points in each half of the Brillouin
zone (each spin component counts separately).

Introduction
Implementing a full scale quantum computer is a major challenge to mod-
ern physics and engineering. Theoretically, this goal should be achievable
∗
On leave from L. D. Landau Institute for Theoretical Physics

1
due to the possibility of fault-tolerant quantum computation [1]. Unlim-
ited quantum computation is possible if errors in the implementation of each
gate are below certain threshold [2, 3, 4, 5]. Unfortunately, for conventional
fault-tolerance schemes the threshold appears to be about 10−4 , which is
beyond the reach of current technologies. It has been also suggested that
fault-tolerance can be achieved at the physical level (instead of using quan-
tum error-correcting codes). The first proposal of these kind [6] was based
on non-Abelian anyons in two-dimensional systems. The relation between
quantum computation, anyons and topological quantum field theories was
independently discussed in [7]. A mathematical result about universal quan-
tum computation with certain type of anyons has been obtained recently [8],
but, generally, this approach is still undeveloped. In these paper we describe
another (theoretically, much simpler) way to construct decoherence-protected
degrees of freedom in one-dimensional systems (“quantum wires”). Although
it does not automatically provide fault-tolerance for quantum gates, it should
allow, when implemented, to build a reliable quantum memory.
The reason why quantum states are so fragile is that they are sensitive
to errors of two kinds. A classical error, represented by an operator σjx , flips
the j-th qubit changing |0i to |1i and vice versa. A phase error σjz changes
the sign of all states with the j-th qubit equal to 1 (i. e. j-th spin down, if
the qubits are spins) relative to the states with the j-th qubit equal to 0.
It is generally easy to get rid of one type of errors, but not both. However,
the following method of eliminating the classical errors is worth considering.
Let each qubit be a site that can be either empty or occupied by an electron
(with spin up, say, the other spin direction being forbidden). Let us denote
the empty and the occupied states by |0i and |1i, respectively. (Such sites are
not exactly qubits because electrons are fermions, but they can be also used
for quantum computation [9]). Now single classical errors become impossible
because the electric charge is conserved. Even in superconducting systems,
the fermionic parity (i. e. the electric charge (mod 2)) is conserved. Two
classical errors can still happen at two sites simultaneously, but this would
require that an electron jumps from one site to the other. Such jumps can be
avoided by placing the “fermionic sites” far apart from each other, provided
the medium between them has an energy gap in the excitation spectrum.
Obviously, this method does not protect from phase errors which are now
described by the operators a†j aj . To the contrary, different electron configu-
rations will have different energies and thus will pick up different phases over

2
time. Even without actual inelastic processes, this will produce the same
effect as decoherence. However, a simple mathematical observation suggests
that the situation could be improved. Each fermionic site is described by a
pair of annihilation and creation operators aj , a†j . One can formally define
Majorana operators
aj − a†j
c2j−1 = aj + a†j , c2j = (j = 1, . . . , N) (1)
i
which satisfy the relations
c†m = cm , cl cm + cm cl = 2δlm (l, m = 1, . . . , 2N). (2)
If the operators c2j−1 and c2j belonged to different sites then the phase error
a†j aj = 21 (1 + ic2j−1 c2j ) would be unlikely to occur. Indeed, it would re-
quire interaction between the two “Majorana sites” which could be possibly
avoided. Note that a single Majorana operator c2j−1 or cj can not appear
as a term in any reasonable Hamiltonian because it does not preserve the
fermionic parity. Thus an isolated Majorana site (usually called a Majorana
fermion) is immune to any kind of error!
Unfortunately, Majorana fermions are not readily available in solid state
systems. The goal of this paper is to construct Hamiltonians which would
give rise to Majorana fermions as effective low-energy degrees of freedom.
Surprisingly, this can be done even with non-interacting electrons. (Some
interaction is actually needed to create superconductivity, but it can be ef-
fectively described by terms like ∆aj ak ). The general idea is quite simple.
An arbitrary quadratic Hamiltonian can be written in the form
iX
H= Alm cl cm (A∗lm = Alm = −Aml ). (3)
4 l,m
Its ground state can be described as “pairing” of Majorana operators: nor-
mal mode creation and annihilation operators ã†m , ãm , which are certain lin-
ear combinations of cl , come in pairs. (In this sense, an insulator and a
superconductor represent different types of pairing). In some cases, most
Majorana operators are paired up with an energy gap while few ones (lo-
calized at the boundary or defects) remain “free”. For example, unpaired
Majorana fermions exist on vortices in chiral 2-dimensional p-wave super-
conductors [10, 11]. We will show that Majorana fermions can also occur at
the ends of quantum wires.

3
1 A toy model and the qualitative picture
We are going to describe a simple but rather unrealistic model which exhibits
unpaired Majorana fermions. It attempts to catch two important properties
which seem necessary for the phenomenon to occur. Firstly, the U(1) sym-
metry aj 7→ eiφ aj , corresponding to the electric charge conservation, must
be broken down to a Z2 symmetry, aj 7→ −aj . Indeed, if a single Majorana
operator can be localized, symmetry transformation should not mix it with
other operators. So we should consider superconductive systems. The par-
ticular mechanism of superconductivity is not important; we may just think
that our quantum wire lies on the surface of 3-dimensional superconductor
(see fig. 1). The second property is less obvious and will be fully explained
in Sec. 2. Roughly speaking, the electron spectrum must strongly depend on
the spin. Here we will simply assume that only one spin component (say, ↑)
is present. 1

b′ b′′






θ

Figure 1: A piece of “quantum wire” on the surface of 3-dimensional super-

conductor.

Consider a chain consisting of L ≫ 1 sites. Each site can be either empty

or occupied by an electron (with a fixed spin direction). The Hamiltonian is
X 
H1 = −w(a†j aj+1 + a†j+1 aj ) − µ(a†j aj − 12 ) + ∆aj aj+1 + ∆∗ a†j+1 a†j . (4)
j

Here w is a hopping amplitude, µ a chemical potential, and ∆ = |∆|eiθ the

induced superconducting gap. It is convenient to hide the dependence on the
phase parameter θ into the definition of Majorana operators:
θ θ θ θ
c2j−1 = ei 2 aj + e−i 2 a†j , c2j = −iei 2 aj + ie−i 2 a†j (j = 1, . . . , L). (5)
1
It appears that only a triplet (p-wave) superconductivity in the 3-dimensional sub-
strate can effectively induce the desired pairing between electrons with the same spin
direction — at least, this is true in the absence of spin-orbit interaction.

4
In terms of this operators, the Hamiltonian becomes
i X 
H1 = −µc2j−1 c2j + (w + |∆|)c2j c2j+1 + (−w + |∆|)c2j−1 c2j+2 . (6)
2 j
Let us start with two special cases.
a) The trivial case: |∆| = w = 0, µ < 0. Then H1 = −µ j (a†j aj − 12 ) =
P
i
(−µ) j c2j−1 c2j . The Majorana operators c2j−1 , c2j from the same
P
2
site j are paired together to form a ground state with the occupation
number 0.
b) |∆| = w > 0, µ = 0. In this case
X
H1 = iw c2j c2j+1 . (7)
j

Now the Majorana operators c2j , c2j+1 from different sites are paired
together (see fig. 2). One can define new annihilation and creation
operators ãj = 21 (c2j + ic2j+1 ), ã†j = 21 (c2j − ic2j+1 ) which span the sites
PL−1 †
j and j + 1. The Hamiltonian becomes 2w j=1 (ãj ãj − 12 ). Ground
states satisfy the condition ãj |ψi = 0 for j = 1, . . . , L − 1. There are
two orthogonal states |ψ0 i and |ψ1 i with this property. Indeed, the
Majorana operators b′ = c1 and b′′ = c2L remain unpaired (i. e. do not
enter the Hamiltonian), so we can write
− ib′ b′′ |ψ0 i = |ψ0 i, −ib′ b′′ |ψ1 i = −|ψ1 i. (8)

c1
 c2 
c3 c4 c2L−1 c2L
  c1
 c2 
c3 c4 c2L−1 c2L
 
s s s s ... s s s s s s ... s s
     

a) b)

Figure 2: Two types of pairing.

Note that the state |ψ0 i has an even fermionic parity (i. e. it is a superposition
of states with even number of electrons) while |ψ1 i has an odd parity. The
parity is measured by the operator
Y
P = (−ic2j−1 c2j ). (9)
j

5
 These two cases represent two phases, or universality classes which exist in
the model. A subtle point is that both phases have the same bulk properties.
In fact, one phase can be transformed to the other (and vice versa) by mere
permutation of Majorana operators,

cm 7→ cm+1 . (10)

Such a local transformation (operator algebra automorphism) is usually con-

sidered as “equivalence” in the study of lattice models. 2 Yet the boundary
properties of the two phases are clearly different: only the phase (b) has
unpaired Majorana fermions at the ends of the chain. This is due to the fact
that the operators c2j−1 , c2j belong to one physical site while c2j , c2j+1 do not.
We may put it this way: one can not cut a physical site into two halves; if
one could, both types of boundary states would be possible in both phases.
Also note that the transformation (10) can not be performed in a con-
tinuous fashion, starting from the identity transformation. From the math-
ematical perspective, it means that one should have different definitions for
“weak” and “strong” equivalence of lattice models . We will not touch such
abstract matters here.
Now we want to study the model at arbitrary values of w, µ and ∆. Let
us begin with some generalities. Let N be the total number of fermionic
sites in the system, for now N = L. The Hamiltonian (6) has the general
form (3). Hence it can be reduced to a canonical form
N N
i X
ǫm b′m b′′m = ǫ(ã†m ãm − 21 )
X
Hcanonical = (ǫm ≥ 0). (11)
2 m=1 m=1

Here b′m , b′′m are real linear combinations of c2j−1 , c2j with the same com-
mutation relations whereas ãm = 12 (b′m + ib′′m ), ã†m = 12 (b′m − ib′′m ). More
2
Nonlocal transformations can change the physical properties
Q of the model even more
dramatically. The Jordan-Wigner transformation c2j−1 7→ σjx j−1 z y Qj−1 z
k=1 σk , c2j 7→ σj k=1 σk
takes our model to a spin chain with xx and yy interactions and a z-directed magnetic
field. Unlike (10), the Jordan-Wigner transformation is well defined at the ends of the
chain. However, this mathematical procedure falls apart in the physical context, as far as
perturbations are involved. Indeed, the phase (b) has now an order parameter hσ x i = 6 0.
External fields will interact with the order parameter breaking the phase coherence between
|ψ0 i and |ψ1 i.

6
specifically,
     
b′1 c1 0 ǫ1
b′′1 c2  −ǫ1 0
     
    
.. .. ..
     
. =W . , W AW T =  . ,
     
  
     

 b′N 


 c2N −1 


 0 ǫN 

b′′N c2N −ǫN 0
(12)
T T
where W is a 2N × 2N real orthogonal matrix (W W = W W = I) whose
rows are eigenvectors of A. The numbers ǫm ≥ 0 are one-particle excitation
energies. However, it is more convenient to deal with a “double spectrum”
{ǫm , −ǫm } since the matrix A has eigenvalues ±iǫm .
The bulk spectrum (energy vs. momentum) is given by
q
ǫ(q) = ± (2w cos q + µ)2 + 4|∆|2 sin2 q, −π ≤ q ≤ π. (13)
We may conjecture that the phases (a) and (b) extend to connected domains
in the parameter space where the spectrum has a gap. The signs of µ and w
seem not to be important, so we actually expect that the phase (a) occurs at
2|w| < |µ| while the phase (b) occupies the domain 2|w| > |µ|, ∆ 6= 0. (The
phase boundary is given by the equation 2|w| = |µ| while ∆ = 0, 2|w| > |µ|
is a line of normal metal phase inside the domain (b)).
To verify the conjecture, we need to find boundary modes. They corre-
spond to eigenvectors of A localized near the ends of the chain. Due to the
spectrum symmetry ǫ 7→ −ǫ, zero eigenvalues can occur in a general position.
If exist, such zero modes should have the form
q
′ j ′ j
b′ = j (α+ x+ + α− x− ) c2j−1 −µ ± µ2 − 4w 2 + 4|∆|2
P
x± =
b′′ =
P ′′ −j
j (α+ x+
′′ −j
+ α− x− ) c2j 2(w + |∆|)
(14)
We will consider two cases corresponding to the expected existence domains
of the two phases.
a) If 2|w| < |µ|, we have |x+ | > 1, |x− | < 1 or |x+ | < 1, |x− | > 1.
′ ′ ′′ ′′
Therefore, only one of the coefficients α+ , α− (or α+ , α− ) can be non-
zero, depending on whether the mode is to be localized at the left or at
the right end of the chain. This makes it impossible to satisfy boundary
conditions. So the supposed zero modes (14) do not exist.

7
 b) If 2w > |µ|, ∆ 6= 0, we find that |x+ |, |x− | < 1. Hence b′ is localized
near j = 0 whereas b′′ is localized near j = L. There are also boundary
′ ′ ′′ −(L+1) ′′ −(L+1)
conditions α+ + α− = 0, α+ x+ + α− x− = 0, but they can
be satisfied too. The zero modes b′ , b′′ are actually the same as the
unpaired Majorana fermions discussed before. If −2w > |µ|, ∆ 6= 0
then b′ and b′′ change places. Thus the unpaired Majorana fermions
exist in the whole expected domain of the phase (b).

The above analysis is exact in the limit L → ∞. If the chain length L

is finite, there is a weak interaction between b′ and b′′ . (For definiteness, we
will always assume that b′ is at the left end of the chain whereas b′′ is at the
right end). This interaction is described by an effective Hamiltonian
i ′ ′′
Heff = tb b , t ∝ e−L/l0 , (15)
2
   
where l0−1 is the smallest of ln |x+ | and ln |x− | (note that both logarithms
   

have the same sign). Thus the energies of the ground states |ψ0 i and |ψ1 i
(see eq. (8)) differ by t. Note that it is not obvious any more which state
of the two is even and which is odd. In the case −2w > |µ|, the parity
is proportional to (−1)L . (This factor is the parity of the bulk part of the
chain).
The effective Hamiltonian (15) still holds if we include small electron-
electron interaction (a four-fermion term) into (4). Indeed, the physical
meaning of t is an amplitude for a fermionic quasiparticle to tunnel across the
chain. In a long chain, this amplitude vanishes as e−L/l0 if the bulk spectrum
has a gap.
Finally, we will discuss a role of the phase parameter θ (∆ = eiθ |∆|).
According to eq. (5), the Majorana operators c2j−1 , c2j are multiplied by
−1 when θ changes by 2π. The physical parameter ∆ is the same at θ and
θ+2π, of course, but the ground states should undergo certain transformation
as θ changes to θ + 2π adiabatically. Note that the transformation cm 7→
−cm also occurs if one conjugates cm by the parity operator P . Within the
effective Hamiltonian approach, P is the same as s(L)(−ib′ b′′ ) (s(L) = ±1).
Hence the adiabatic change of the superconducting phase by 2π results in
the unitary transformation

V = s(L)(−ib′ b′′ ) : V |ψ0 i = |ψ0 i, V |ψ1 i = −|ψ1 i. (16)

8
This is equivalent to transfer of an electron between the ends of the chain.
Some physical consequences of this result will be mentioned in Sec. 3.

2 A general condition for Majorana fermions

Let us consider a general translationally invariant one-dimensional Hamilto-
nian with short-range interactions. It has been mentioned that the neces-
sary conditions for unpaired Majorana fermions are superconductivity and a
gap in the bulk excitation spectrum. The latter is equivalent to the quasi-
particle tunneling amplitude vanishing as e−L/l0 . Besides that, it is clear
that there should be some parity condition. Indeed, Majorana fermions at
the ends of parallel weakly interacting chains may pair up and cancel each
other (i. e the ground state will be non-degenerate). So, provided the en-
ergy gap, each one-dimensional Hamiltonian H is characterized by a “Ma-
jorana number” M = M(H) = ±1: the existence of unpaired Majorana
fermions is indicated as M = −1. The Majorana number should satisfy
M(H ′ ⊕ H ′′ ) = M(H ′ )M(H ′′), where ⊕ means taking two non-interacting
chains.
Remarkably, the Majorana number reveals itself even if the chain is closed
into a loop. This is handy as it eliminates the need to study boundary modes.
Let H(L) be the Hamiltonian of a closed chain of length L ≫ l0 . (H itself is
a template which is used to generate H(L) for any L). We claim that
P (H(L1 + L2 )) = M(H) P (H(L1)) P (H(L2)), (17)
where P (X) denotes the ground state parity of a Hamiltonian X (assuming
that the ground state is unique).

L1 L2 L1 L2
eeeeee ee eee eeeeee eeeee
b′1 b′′1 b′2 b′′2 b′1 b′′1 b′2 b′′2

a) b)

Figure 3: Reconnecting closed chains.

The following argument justifies eq. (17). An open chain of length L

can be described by an effective Hamiltonian which only includes boundary

9
modes. If M(H) = −1, there are Majorana operators b′ , b′′ associated with
the ends of the chain. The parity operator P (see eq. (9)) can be replaced
by s(L)(−ib′ b′′ ), where s(L) = ±1. Thus the fermionic parity of |ψα i is
s(L) (−1)α , α = 0, 1. If we close the chain, the effective Hamiltonian is
Heff (L) = 2i u b′′ b′ . (We have chosen to write b′′ b′ in this order because b′′
precedes b′ in the left-to-right order on the loop, where they are next to each
other). The parameter u represents direct interaction between the chain ends
(unlike t from eq. (15)), so u does not depend on L. The ground state of the
closed chain is |ψ1 i if u > 0 and |ψ0 i if u < 0. Hence

P (H(L)) = −s(L) sgn u.

Now let us take two chains, one of length L1 , the other of length L2 .
There are two ways to close them up, see fig. 3. Both cases can be described
by effective Hamiltonians:
i i
Heff (L1 )⊕Heff (L2 ) = u (b′′1 b′1 +b′′2 b′2 ), Heff (L1 +L2 ) = u (b′′1 b′2 +b′′2 b′1 ).
2 2
It follows that

P (H(L1 )) P (H(L2)) = s(L1 ) s(L2 ), P (H(L1 + L2 )) = − s(L1 ) s(L2 ).

So the equation (17) holds for M = −1. It also obviously holds for M = 1
because in this case there are no boundary modes to worry about.
Computing the Majorana number in general (especially for strongly cor-
related systems) may be a difficult task. However, the computation can be
carried through for any system of non-interacting electrons. Consider a peri-
odic chain of L unit cells with n fermionic sites (i. e. 2n Majorana operators)
per cell, which totals to N = nL fermionic sites. We will index the Majorana
operators as clα , where l = 1, . . . , L, α = 1, . . . , 2n. The Hamiltonian is
i XX  
H = Bαβ (m − l) clα cmβ Bαβ (j)∗ = Bαβ (j) = −Bβα (−j) .
4 l,m α,β
(18)
We assume that the chain forms a loop, so m − l should be taken (mod L).
Eq. (18) is a special case of (3), so we will first find P (H) for the general
quadratic Hamiltonian (3), assuming that the matrix A is not degenerate.
The canonical form of this Hamiltonian (11) has an even ground state |0i.

10
The transformation (12) can be represented
 P as conjugation
 by the parity-
1
preserving unitary operator U = exp 4 l,m Dlm cl cm if W has the form
W = exp(D) for some real skew-symmetric matrix D, i. e. if det W = 1.
Otherwise, the transformation (12) changes the parity. Hence

P (H) = sgn det W = sgn Pf A. (19)

We remind the reader that the Pfaffian Pf is a function of a skew-

symmetric matrix such that (Pf A)2 = det A. It is defined as follows
1 X
Pf A = sgn(τ ) Aτ (1),τ (2) · · · Aτ (2N −1),τ (2N ) . (20)
2N N! τ ∈S2N

(Here S2N is the set of permutations on 2N elements). For example,

0 a12 a13 a14

 
 −a12 0 a23 a24 
Pf  
= a12 a34 + a14 a23 − a13 a24 .
−a13 −a23 0 a34
 
 
−a14 −a24 −a34 0

In eq. (19) we have used this property of the Pfaffian:

Pf(W AW T ) = Pf(A) det(W ). (21)

Now we are to compute the Pfaffian of the matrix B from eq. (18). First,
we use the Fourier transform,
k
eiqj Bαβ (j),
X
B̃αβ (q) = q = 2π (mod 2π), k = 0, . . . , N − 1. (22)
j L

The matrix B̃(q) has these symmetries:

B̃ † (q) = −B̃(q) = B̃ T (−q). (23)

The spectrum ǫ(q) is a continuous real 2n-valued function on a circle (real

numbers (mod 2π)) given by the eigenvalues of iB̃(q). It has the symmetry
ǫ(−q) = −ǫ(q). The energy gap assumption implies that ǫ(q) never passes
0. It follows that there are n positive and n negative eigenvalues for any q.
Indeed, this is the case for q = 0 due to the ǫ 7→ −ǫ symmetry, hence it is
true for any q by continuity.

11
 It follows from eqs. (22) and (21) that
  
Y Y
Pf B =  Pf B̃(q)  det B̃(q) . (24)
q=−q q6=−q

Remember that q is considered (mod 2π), so q = −q when q = 0 or q = π.

In the q 6= −q case, each {q, −q} pair is counted once. Note that det B̃(q)
is a positive number since iB̃(q) has n positive and n negative eigenvalues.
Hence

Y  sgn(Pf B̃(0)) sgn(Pf B̃(π)) if L is even,
sgn Pf B = sgn(Pf B̃(q)) =
q=−q
 sgn(Pf B̃(0)) if L is odd.
(25)
Finally, we get

M(H) = sgn(Pf B̃(0)) sgn(Pf B̃(π)). (26)

This very general equation can be simplified if superconductivity is a

weak effect, i. e. |∆| ≪ |ǫ(0)|, |ǫ(π)|. Indeed, the right hand side of (26)
makes perfect sense for a U(1)-symmetric Hamiltonian
1 XX  
H0 = Cαβ (m − l) a†lα amβ Cαβ (j)∗ = Cβα (−j) , (27)
2 l,m α,β

where α, β = 1, . . . n refer to fermionic sites. The eigenvalues of C̃(q) (the

Fourier transform of C) form a “single spectrum” ǫ0 (q). The “double spec-
trum” defined above is ǫ(q) = ±ǫ0 (q). It is easy to show that Pf B̃(q) =
det C̃(q) for q = 0, π. Hence

M(H0 ) = (−1)ν(π)−ν(0) , (28)

where ν(q) is the number of negative eigenvalues of C̃(q). Note that ν(π) −
ν(0) equals (mod 2) the number of Fermi points on the interval [0, π]. (A
Fermi point is a point where ǫ0 (q) passes 0). In the most interesting case
ν(π) − ν(0) = 1 (mod 2), the Hamiltonian H0 has a gapless spectrum. So
eq. (28) is only relevant in the presence of superconductivity, i. e. a small
symmetry-breaking perturbation which opens an energy gap.

12
3 Speculations about physical realization
Physical realization of an M = −1 quantum wire is a difficult task because
electron spectra are usually degenerate with respect to spin, so ν(0) and
ν(π) are even. The degeneracy at q = 0 and q = π can be lifted only
if the time reversal symmetry is broken. Thus spin-orbit interaction does
not help. External magnetic field could help, but the Zeeman energy gµB H
is usually small compared to other spectrum parameters, so ν(0) and ν(π)
do not change. The situation may be different for charge and spin density
waves which add fine features to the electron spectrum. Charge density waves
(CDW) tend to occur at the wave vector q∗ = 2qF so that a gap opens at the
Fermi level. In the presence of magnetic field, qF is slightly different for the
↑ and ↓ spin components, so it is possible that q∗ matches only one of them.
The resulting spectrum is shown in fig. 4 in the q∗ /(2π) units. This scenario
can be realized if |∆| < <
∼ ECDW ∼ gµB H.
Another speculative possibility is to use midgap states at the edge of a
two-dimensional p-wave superconductor [12].
ε0(q)

-π π q

spin up
spin down

Figure 4: An electron spectrum in the presence of magnetic field and CDW.

A quantum wire bridge between two superconducting leads (see fig. 5a)
could be used as an experimental test for Majorana fermions. When the phase
parameter θ2 in the right piece of superconductor changes by 2π (relative to
θ1 ), a fermionic quasiparticle is effectively transported to the junction region.

13
At the same time, the Majorana fermions at the ends of the wire switch from
|ψ0 i to |ψ1 i or vice versa. If the quasiparticle stays localized, the junction
parameters change. They change back when θ2 changes by another 2π. Thus
the Josephson current is 4π-periodic as a function of θ = θ2 − θ1 . In fact,
it is more accurate to say that the Josephson energy EJ is 2π-periodic but
2-valued, as shown in fig. 5b. The two levels may not quite cross at θ = π due
to a non-vanishing tunneling amplitude t ∝ e−L/l0 , where L is the distance
between the junction and the closest end of the wire.

EJ(θ)

θ1
θ2
-2π -π π 2π θ

t=0
θ = θ2 − θ1 t≠0

a) b)

Figure 5: A Josephson junction made of quantum wire.

Interesting phenomena can also take place in the simple layout shown
in fig. 1. Suppose that the superconducting island supporting the quantum
wire is connected to a larger piece of superconductor through an ordinary
Josephson junction. If the Coulomb energy is comparable to the Josephson
energy, spontaneous phase slips can occur. Each 2π phase slip is accompanied
by the operator V (see eq. 16). The phase slips occur by tunneling, so the
effective Hamiltonian is
i
Heff.1 = −λV − λ∗ V † = s(L) t b′ b′′ , t = 4 Re λ, (29)
2
where λ is the amplitude of the θ 7→ θ + 2π process while λ∗ corresponds
to the reverse process. Similarly, if the superconducting island supports two
quantum wires, the effective Hamiltonian becomes
1
Heff.2 = −λV1 V2 − λ∗ V1† V2† = s(L1 )s(L2 ) t b′1 b′′1 b′2 b′′2 . (30)
2
Turning λ on and off can be possibly used for quantum gates implementation.

14
Acknowledgements. I am grateful to J. Preskill, M. Feigelman, P. Vigman
and V. Yakovenko for interesting discussions.

References
[1] P. W. Shor, “Fault tolerant quantum computation”, Proceedings of the
37th Symposium on the Foundations of Computer Science (FOCS), 56–
65 (1996); quant-ph/9605011.

[2] D. Aharonov, M. Ben-Or, “Fault-Tolerant Quantum Computation with

Constant Error”, Proc. of the 29th Annual ACM Symposium on Theory
of Computing (STOC) (1997); quant-ph/9611025.

[3] D. Aharonov, M. Ben-Or, “Fault-Tolerant Quantum Computation with

Constant Error Rate”, quant-ph/9906129.

[4] E. Knill, R. Laflamme, W. H. Zurek, “Threshold Accuracy for Quantum

Computation” quant-ph/9610011.

[5] A. Yu. Kitaev, “Quantum Computations: Algorithms and Error Correc-

tion”, Russian Math. Surveys 52:6, 1191–1249 [Russian version: Uspekhi
Mat. Nauk 52:3, 53–112].

[6] A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”,

quant-ph/9707021.

[7] M. H. Freedman, “P/NP, and the quantum field computer”, Proc. Natl.
Acad. Sci. USA, 95, 98–101 (1998).

[8] M. Freedman, M. Larsen, Z. Wang, “A modular functor which is univer-

sal for quantum computation”, quant-ph/0001108.

[9] S. Bravyi, A. Kitaev, “Fermionic quantum computation”,

quant-ph/0003137.

[10] N. Read, D. Green, “Paired states of fermions in two dimensions with

breaking of parity and time-reversal symmetry”, Phys. Rev. B 61, 10267
(2000); cond-mat/9906453.

15
[11] D. A. Ivanov, “Non-abelian statistics of half-quantum vortices in p-wave
superconductors, cond-mat/0005069.

[12] K. Sengupta, I. Zutic, H. Kwon, V. M. Yakovenko, S. Das Sarma,

“Midgap edge states and pairing symmetry of quasi-one-dimensional
organic superconductors”, cond-mat/0010206.

16