Quantizing Majorana fermions in a superconductor
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Chamon, C. et al. “Quantizing Majorana fermions in a
superconductor.” Physical Review B 81.22 (2010): 224515. ©
2010 The American Physical Society.
As Published
http://dx.doi.org/10.1103/PhysRevB.81.224515
Publisher
American Physical Society
Version
Final published version
Accessed
Thu May 26 04:48:23 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/60849
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PHYSICAL REVIEW B 81, 224515 共2010兲
Quantizing Majorana fermions in a superconductor
C. Chamon,1 R. Jackiw,2 Y. Nishida,2 S.-Y. Pi,1 and L. Santos3
1Physics
Department, Boston University, Boston, Massachusetts 02215, USA
Department, MIT, Cambridge, Massachusetts 02139, USA
3Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
共Received 19 January 2010; revised manuscript received 7 May 2010; published 17 June 2010兲
2Physics
A Dirac-type matrix equation governs surface excitations in a topological insulator in contact with an s-wave
superconductor. The order parameter can be homogenous or vortex valued. In the homogenous case a winding
number can be defined whose nonvanishing value signals topological effects. A vortex leads to a static,
isolated, zero-energy solution. Its mode function is real and has been called “Majorana.” Here we demonstrate
that the reality/Majorana feature is not confined to the zero-energy mode but characterizes the full quantum
field. In a four-component description a change in basis for the relevant matrices renders the Hamiltonian
imaginary and the full, space-time-dependent field is real, as is the case for the relativistic Majorana equation
in the Majorana matrix representation. More broadly, we show that the Majorana quantization procedure is
generic to superconductors, with or without the Dirac structure, and follows from the constraints of fermionic
statistics on the symmetries of Bogoliubov-de Gennes Hamiltonians. The Hamiltonian can always be brought
to an imaginary form, leading to equations of motion that are real with quantized real-field solutions. Also we
examine the Fock space realization of the zero-mode algebra for the Dirac-type systems. We show that a
two-dimensional representation is natural, in which fermion parity is preserved.
DOI: 10.1103/PhysRevB.81.224515
PACS number共s兲: 74.20.⫺z
I. INTRODUCTION
Majorana bound states arise as zero-energy states in twodimensional 共2D兲 systems involving superconductors in the
presence of vortices.1–4 These zero modes have attracted
much attention recently, in part, because of the possibility
that they can realize “half-qubits” within topological quantum computing schemes.3 The basic idea is that two far away
Majorana bound states, real fermions, can be put together
into a complex fermion acting on a two-dimensional Hilbert
space spanned by the states 兩0典 and 兩1典. Hence, two Majorana
fermions comprise one qubit, which is protected against the
environment if the vortices binding the Majorana fermions
are kept far away from each other.
The first example of a zero mode in a two-dimensional
superconductor was presented in Ref. 1. More recently it has
been stated that the proximity effect at the interface between
an s-wave superconductor and the surface of a topological
insulator can be described by a planar Dirac equation,4 providing a physical realization of the mathematical structure of
Ref. 1. Other examples of Majorana bound states arise in
systems with a nonrelativistic kinetic term and a p⫾
⬅ px ⫾ ipy interaction with a vortex order parameter 共i.e.,
p-wave superconductors兲.2,3 These types of bound states
have been the subject of much recent interest.5–8 The focus
of the discussions of Majorana fermions in superconductors
have focused thus far on the zero modes.
However, Majorana’s original work9 was actually quite
more general and did not address a single mode but instead a
whole field. What he showed was that it was possible to
construct a representation of the Dirac equation that admits
purely real solutions. The particles that follow from his construction are their own antiparticles and thus necessarily neutral. What was striking about Majorana’s proposal was that
these particles were fermions—bosonic neutral particles rep1098-0121/2010/81共22兲/224515共10兲
resented by real fields are common, pions, and vector
bosons, such as photons, being simple examples 共see Ref. 10
for a perspective on Majorana fermions兲.
In this paper we look at three issues regarding the quantization of Majorana fermions, beyond simply the zero
modes, in superconductors. First, we look specifically at the
case of Dirac-type systems describing s-wave-induced superconductivity on the surface of topological insulators. There,
we find that the entire ␺ field of the superconductor model
共and not merely particular modes兲 obeys equations that are
analogous to the Majorana equations of particle physics. The
equations of motion for the fields can be brought to a real
form, and the fermionic solutions are real and therefore their
own antiparticles. Indeed, other than the fact that surface
states are 2D, the topological insulator-superconductor system can be brought to the exactly same form that was discussed in Majorana’s original formulation of real relativistic
fermions.
Second, we note various topological features of the Diractype model. We compute the Pontryagin index associated
with the k-space dispersion, and find it to be ⫾1 / 2, which is
an indication of the existence of zero modes in the presence
of vortices. We then present the Fock space-level structures
that accommodate an isolated, zero-energy state, which
arises in the presence of a vortex. In particular, we show that
fermion parity can be preserved, even with a single zeroenergy state. As we discussed above the Majorana zero
modes are usually thought of “half” qubits, as two of them
make up a complex fermion with a two-dimensional Hilbert
space. Here we ruffle this simple view by quantizing the
theory in the infinite plane in the presence of a single vortex.
A sole Majorana zero mode exists but a two-dimensional
Hilbert space remains. In a finite system, another zero mode
would appear at the edge, which is however absent in the
infinite plane.
224515-1
©2010 The American Physical Society
PHYSICAL REVIEW B 81, 224515 共2010兲
CHAMON et al.
Third, we show that the Majorana quantization procedure
that we discuss for the Dirac-type equations describing
s-wave-induced superconductivity on the surface of topological insulators does extend, more broadly, to any superconductor. A description of Bogoliubov-de Gennes 共BdG兲
Hamiltonians using Majorana modes has been noted by Senthil and Fisher11 for systems where spin-rotational symmetry
is broken 共classes D and DIII of Ref. 12兲. Here we show
rather generically that lack of spin-rotation symmetry is not a
necessity and thus classes C and CI of Ref. 12 also realize
Majorana fermions. All one actually needs is to have fermions and hence these results hold for any superconducting
system made of half-integer spin particles, regardless of the
size of the spin. What we show is that the constraints imposed by fermionic statistics on the symmetries of
Bogoliubov-de Gennes Hamiltonians always allow one to
bring the Hamiltonian in the Nambu representation to an
imaginary form. In turn, Schrödinger’s equation with this
imaginary Hamiltonian leads to a real equation of motion for
the fields, as in Majorana’s construction. The real-field solutions in the constrained doubled Nambu space can then be
quantized as Majorana fields.
II. QUANTUM STRUCTURE OF THE
SUPERCONDUCTING MODEL
Let us start by analyzing the planar Dirac-type systems
realized on the surface of a topological insulator, placed in
proximity to an s-wave superconductor. The Hamiltonian
density for the model under discussion acts on two spatial
dimensions.1,4
spin 1/2 fermions with “Majorana mass” 兩⌬兩.13
A static solution to Eq. 共3兲, equivalently Eq. 共4兲, with a
vortex profile for ⌬, can be readily found. It corresponds to a
zero-energy mode. With f and g real in the Ansatz
␺↑ = f共r兲exp兵− i␲/4 − V共r兲其,
␺↓ = g共r兲exp兵i共␪ + ␲/4兲 − V共r兲其,
V⬘共r兲 ⬅ v共r兲.
Equation 共3兲 reduces to
共rg兲⬘ = ␮rf ,
f ⬘ = − ␮g.
f共r兲 = NJ0共␮r兲,
g共r兲 = NJ1共␮r兲
H=
1
1
· p − ␮兲ij␺ j + ⌬␺ⴱi i␴2ij␺ⴱj − ⌬ⴱ␺ii␴2ij␺ j . 共2兲
2
2
␺
␺ = 共 ␺↑↓ 兲
2
and ␴ comprises the two Pauli matrices
Now
共␴1 , ␴ 兲. The 共2 + 1兲-dimensional equations of motion for
Eqs. 共1兲 and 共2兲
共3兲
can be presented in two-component matrix notation.
i⳵t␺ = 共␴ · p − ␮兲␺ + ⌬i␴2␺ⴱ .
⌿=
冢冣
␺↑
␺↓
␺ⴱ↓
=
冉 冊
␺
.
i ␴ 2␺ ⴱ
共8兲
− ␺ⴱ↑
An extended Hamiltonian density H leads to equations for
⌿, which are just two copies of Eq. 共3兲 or 共4兲.
冉
冊
␴·p−␮
⌬
1
1
⌿ ⬅ ⌿ⴱTh⌿. 共9兲
H = ⌿ⴱT
ⴱ
2
2
−␴·p+␮
⌬
Here T denotes transposition. Because the last two components of ⌿ are constrained by their relation to the first two,
⌿ satisfies the 共pseudo-兲 reality constraint
C⌿ⴱ = ⌿
共10兲
−i␴2
i⳵t␺↑ = p−␺↓ − ␮␺↑ + ⌬␺ⴱ↓ ,
i⳵t␺↓ = p+␺↑ − ␮␺↓ − ⌬␺ⴱ↑
共7兲
with N as a real normalization constant.14
While the static, zero-energy mode is readily obtained
from Eq. 共3兲, for the finite-energy modes, we must take account of the fact that ␺↑ , ␺↓ mix with their complex conjugates. Therefore, one cannot separate the time dependence
with an energy phase. Correspondingly one cannot construct
a Hamiltonian energy eigenvalue problem, which is the usual
first step in the quantization procedure. Progress is achieved
by doublings the system with a four-component spinor.
共1兲
␺ⴱi 共␴
共6兲
共Dash signifies r—differentiation兲. Regular solutions are
Bessel functions
H = ␺ⴱ↑ p−␺↓ + ␺ⴱ↓ p+␺↑ − ␮共␺ⴱ↑␺↑ + ␺ⴱ↓␺↓兲 + ⌬␺ⴱ↑␺ⴱ↓ + ⌬ⴱ␺↓␺↑ .
Here ␺↑,↓ are electron field amplitudes, p⫾ ⬅ −i⳵x ⫾ ⳵y , ␮ is
the chemical potential 共which was omitted in the Ref. 1兲 and
⌬共r兲 is the order parameter that is constant in the homogenous case or takes a vortex profile in the topologically interesting case: ⌬共r兲 = v共r兲ei␪, in circular coordinates. Equivalently, in a two-component notation
共5兲
with C = C−1 = Cⴱ = CT = C† ⬅ 共 0i␴2 0 兲.
To proceed, one ignores the constraint in Eq. 共10兲 on ⌿
and works with an unconstrained four-spinor ⌽ = 共 ␸␺ 兲. Time
can now be separated with the usual phase Ansatz and the
energy eigenvalue spectrum can be found.
h⌽ = i⳵t⌽,
共4兲
When the chemical potential is absent, and ⌬ is constant, the
above system is a 共2 + 1兲-dimensional version of the
共3 + 1兲-dimensional, two component Majorana equation,
which in 共3 + 1兲-dimensional space-time describes chargeless
⌽ = e−iEt⌽E ,
h⌽E = E⌽E .
共11兲
These are the Bogoliubov-de Gennes equations for the superconductor problem. In the particle physics application, the
224515-2
PHYSICAL REVIEW B 81, 224515 共2010兲
QUANTIZING MAJORANA FERMIONS IN A SUPERCONDUCTOR
unconstrained four-component equation is just the Dirac
equation describing charged spin 1/2 fermions. When the
共pseudo-兲 reality constraint is imposed, one is dealing with
the four-component version of the Majorana equation.13
Observe that h in Eq. 共9兲 possesses the conjugation symmetry
C−1hC = − hⴱ ,
共12兲
which has the consequence that to each positive-energy
eigenmode there corresponds a negative-energy mode.
ⴱ
C⌽+E
= ⌽−E .
冉
共␳,J兲 = ⌽ⴱi ⌽i,⌽ⴱi
共13兲
A quantum field may now be constructed by superposing
the energy eigenmodes ⌽E with appropriate creation and annihilation quantum operators. It is here that we again encounter the Majorana construction: the unconstrained fermion four-spinor ⌽ is like a “Dirac” fermion spinor,
governed by a Hamiltonian, which satisfies a conjugation
symmetry in Eq. 共12兲 that leads to Eq. 共13兲. Then the spinor
⌿, which satisfies the 共pseudo-兲 reality constraint in Eq. 共10兲,
is like a Majorana spinor, viz., a Dirac spinor obeying a
共pseudo-兲 reality condition.
With the eigenmodes one can construct a quantum field
⌽̂. It can be an unconstrained Dirac field operator.
⌽̂ =
† −iEt
aEe−iEt⌽E + 兺 b−E
e ⌽E
兺
E⬎0
E⬍0
=
兺 aEe−iEt⌽E + E⬎0
兺 bE† eiEtC⌽Eⴱ .
E⬎0
共15a兲
兵⌽̂i共r兲,⌽̂†j 共r⬘兲其 = ␦ij␦共r − r⬘兲.
共15b兲
For the superconductor/topological insulator system under
consideration ⌽ → ⌿ and the quantum field ⌿̂ satisfies the
constraint
共16兲
This is achieved by setting b = a in Eq. 共14兲.
兺 共aEe−iEt⌽E + aE† eiEtC⌽Eⴱ 兲.
E⬎0
冉 冋 册
共␳5,J5兲 = ⌽ⴱi
册 冊
⌽j .
共19兲
ij
0
I
0 −I
⌽ j,⌽ⴱi
ij
冋 册 冊
␴ 0
0 ␴
⌽j .
共20兲
ij
But with nonvanishing ⌬ this is not conserved.
冉
0 ⌬
⳵
␳5 + ⵱ · J5 = − 2i⌽ⴱi
− ⌬ⴱ 0
⳵t
冊
⌽j .
共21兲
ij
共␳5,J5兲 ⇒ 2共␺ⴱT␺, ␺ⴱT␴␺兲,
共22兲
⳵
␳5 + ⵱ · J5 ⇒ 2⌬␺ⴱT␴2␺ⴱ + 2⌬ⴱ␺T␴2␺ .
⳵t
共23兲
Thus no conserved current is present in the superconductor
model in Eq. 共1兲.
The Majorana/reality properties are obscured by the representation of the Dirac matrices employed in presenting the
4 ⫻ 4 Hamiltonian h in Eq. 共9兲. As written, the matrices in h
are given in the Weyl representation.
␣=
兵⌿̂i共r兲,⌿̂ j共r⬘兲其 = Cij␦共r − r⬘兲,
共18a兲
兵⌿̂i共r兲,⌿̂†j 共r⬘兲其 = ␦ij␦共r − r⬘兲.
共18b兲
obeying conThese also follow from Eq. 共17兲 with aE ,
ventional anticommutators. We have ignored possible zero-
冉
␴ 0
0 −␴
冊 冉 冊
␤=
0 I
␥5 =
I 0
冉 冊
I
0
0 −I
h = ␣ · p − ␮␥5 + ␤⌬R − i␤␥5⌬I .
,
共24兲
共25兲
⌬R,I are the real and imaginary parts of the order parameter.
One may pass to the Majorana representation by conjugating
with the unitary matrix
共17兲
Owing to the constraint in Eq. 共16兲 the anticommutators take
a Majorana form.
aE†
␴ 0
0 −␴
When the above is evaluated on the constrained field ⌿, all
terms vanish. This is to be expected for a Majorana field
which carries no charge.
One may also consider a chiral current constructed with
the Dirac field ⌽.
共14兲
兵⌽̂i共r兲,⌽̂ j共r⬘兲其 = 0,
Cij⌿̂†j = ⌿̂i .
冋
These results persist when the constraint in Eq. 共16兲 is imposed on ⌽ → ⌿.
Here the aE operator annihilates positive-energy excitations
共conduction band兲 and the bE† operator creates negativeenergy excitations 共valence band兲. Since ⌽̂ is unconstrained,
a and b are independent operators. Their conventional anticommutators ensure that the unconstrained fields satisfy
Dirac anticommutation relations.
⌿̂ =
energy states; they will be discussed at length below.
In the final result in Eq. 共17兲, ⌿̂ retains the Majorana
feature of describing excitations that carry no charge. This is
true for the entire quantum field ⌿̂, not only for its zeroenergy modes 共if any兲, which are emphasized in the
condensed-matter literature. Explicitly we see this by examining the conserved current that is constructed with the unconstrained Dirac field ⌽.
V=
冉
Q−
Q+
冊
Q+
ei␲/4,
− Q−
1
Q⫾ ⬅ 共1 ⫾ ␴2兲.
2
Then h becomes
V−1hV =
冉
− py
px␴1 + i⌬I
px␴1 − i⌬I
py
冊冉
+
共26兲
冊
␮ ␴ 2 − ⌬ R␴ 2
.
− ⌬ R␴ 2 − ␮ ␴ 2
共27兲
This is manifestly imaginary and the conjugation matrix C in
Eq. 共10兲 becomes the identity so that the 共pseudo-兲 reality
constraint on ⌿̂ becomes a reality condition.17
224515-3
PHYSICAL REVIEW B 81, 224515 共2010兲
CHAMON et al.
III. HOMOGENOUS ORDER PARAMETER
operator ⌿̂ is constructed as in Eq. 共17兲, which with notational changes 关aE → an共k兲 ; ⌽E → ⌽n共k兲 ; n = 共+ , −兲兴 reads explicitly
i␻
For constant ⌬ = me , we pass to momentum space with
an eik·r Ansatz in Eq. 共11兲. The energy eigenvalue is
E = ⫾ 冑共k ⫾ ␮兲2 + m2
⌿̂共t,r兲 = 兺
共28兲
n
with no correlation among the signs. For fixed k there are
two 共⫾␮兲 positive-energy solutions and two negative-energy
solutions. They become doubly degenerate at ␮ = 0. The degeneracy occurs because at ␮ = 0, h commutes with S
i␻ 3
= 共 e0−i␻e␴3␴ 0 兲 when the phase ␻ of ⌬ is constant. The energy in
Eq. 共28兲 is nonvanishing for all values of the parameters;
there is no zero-energy state.
The operators aE , aE† and the eigenmodes ⌽E, which are
explicitly presented in Appendix A, are labeled by the momentum k and a further 共+ , −兲 variety describing the twofold
dependence on ␮ of E⫾ ⬅ 冑共k ⫿ ␮兲2 + m2 ⱖ 兩m兩. The quantum
冉冊冕
冉
␺ˆ ↑
␺ˆ =
␺ˆ ↓
+
=
1
冑2E−
d 2k
共2␲兲2
再 冉
1
冑2E+
a−共k兲e−i共E−t−k·r兲
a+共k兲e−i共E+t−k·r兲
⌬
†
†
i共E−t−k·r兲
h ⬘ = ⌺ an a
共a = 1,2,3兲.
⌺i =
冉
Here ni = ki共i = 1 , 2兲 and n3 is ⌬e−i␻ ⬅ m, i.e., the constant
phase of ⌬ is removed so m is a real constant but of indefinite sign. The matrices ⌺a
冑E+ − k + ␮e−i␸兩− 典
␧ ␴
冊
共30兲
.
− iei␻␧ij␴ j
0
ie
冊冎
−i␻ ij j
0
冊
,
⌺3 =
冉
␴3 0
0 ␴3
冊
共32兲
satisfy the SU共2兲 algebra, as is explicitly recognized after a
further unitary transformation.
U−1⌺aU =
U⬅
冉
P+
e
−i␻
冉 冊
冊
− e i␻ P −
P−
P+
␴a 0
,
0 ␴a
,
1
P⫾ ⬅ 共I ⫾ ␴3兲.
2
共33兲
共34兲
When the Hamiltonian is of the form 共31兲, we can consider the topological current in momentum space.20
K␮ =
共31兲
i共E+t−k·r兲
冑E− + k + ␮e−i␸兩+ 典
IV. TOPOLOGICAL NUMBERS
When ␮ is absent and S commutes with h, we may
equivalently work with h⬘ ⬅ Sh, which possesses the same
eigenvectors as h, common with the eigenvectors of S. However, h⬘ has the appealing form
共29兲
with positive-energy eigenfunction ⌽n共k兲 carrying energy eigenvalue En. The conjugation condition in Eq. 共13兲 now
states that C⌽ⴱn共k兲 is a negative energy solution at 共−k兲.
Actually we can suppress the lower two components of
⌽n共k兲 in Eq. 共29兲, because they repeat the information contained in the upper two components, owing to the subsidiary
condition in Eq. 共16兲. In this way from the four-component
spinors recorded in Appendix A, we arrive at a mode expansion for the electron field operators ␺ˆ ↑,↓.
冑E− + k + ␮ 兩− 典 + a−共k兲e
Here 兩 ⫾ 典 are the two-component eigenvectors of ␴ · k̂ : 兩+典
where
␸
arises
as
k̂
⬅ 冑12 共 e1i␸ 兲 , 兩−典 ⬅ 冑12 共 −e1i␸ 兲,
= 共cos ␸ , sin ␸兲. One verifies that Eq. 共30兲 satisfies Eq. 共3兲.
The Majorana character of this expression manifests itself
in that the particle annihilation operators a⫾, associated with
the positive energy eigenvalues E⫾, are partnered with their
†
, which are associHermitian adjoint creation operators a⫾
ated with the negative-energy −E⫾ modes. By contrast, for a
Dirac field the negative-energy modes are associated with the
†
, which anticommute with
antiparticle creation operators b⫾
†
a⫾ , a⫾. In other words, in the Majorana field operator in
Eq. 共30兲 the antiparticle 共hole兲 states are identified with the
particle states.
d 2k
兵an共k兲e−i共Ent−k·r兲⌽n共k兲
共2␲兲2
+ a†n共k兲ei共Ent−k·r兲C⌽ⴱn共k兲其
冑E+ − k + ␮ 兩+ 典 − a+共k兲e
⌬
冕
1 ␮␣␤
␧ ␧abcn̂a⳵␣n̂b⳵␤n̂c共n̂ ⬅ n/兩n兩兲
8␲
共35兲
and evaluate the topological number by integrating over twodimensional k space.
N=
冕
d2kK0共k兲 =
1
8␲
冕
d 2k
m
m
=
. 共36兲
共k2 + m2兲3/2 2兩m兩
The nonvanishing answer ⫾1 / 2, depending only on the sign
of m, is evidence that the model belongs to a topologically
nontrivial class. It is also a hint that topologically protected
zero modes exist in the presence of a vortex. 共Although
vortex-based zero modes are also present for ␮ ⫽ 0, we do
224515-4
PHYSICAL REVIEW B 81, 224515 共2010兲
QUANTIZING MAJORANA FERMIONS IN A SUPERCONDUCTOR
Again there are two towers of states
not know how to define a winding number in that case.兲
We can understand the fractional value for N. The unit
vector
n̂a = 共k cos ␸,k sin ␸,m兲/冑k2 + m2
共37兲
maps R共2兲共⫽S共2兲兲 to S共2兲. When k begins at k = 0, n̂a is at the
north or south pole 共0 , 0 , ⫾ 1兲. As k ranges to ⬁, n̂a covers a
hemisphere 共upper or lower兲 and ends at the equator of S共2兲.
Thus only one half of S共2兲 is covered.
†
†
aE† aE⬘aE⬙ . . . 兩f典,
兩0+典 =
When ⌬ takes the vortex form, ⌬共r兲 = v共r兲ei␪, Eq. 共11兲
possesses an isolated zero-energy mode
冉 冊
␺0v
i␴2␺0vⴱ
共38兲
with ␺0v determined by Eqs. 共5兲 and 共7兲. Note that
C⌿0vⴱ = ⌿0v .
共39兲
There are also continuum modes.
The operator field ⌿̂ is now given by an expansion such
as Eq. 共17兲, except there is an additional contribution due to
the zero mode controlled by the operator A.
⌿̂ ⬅
共aEe−iEt⌽E + aE† eiEtC⌽Eⴱ 兲 + A冑2⌿0v .
兺
E⬎0
共40兲
共The 冑2 factor will be explained later.兲 Due to Eqs. 共16兲 and
共39兲, A is Hermitian A = A†, anticommutes with 共aE , aE† 兲 and
obeys
兵A,A其 = 2A2 = 1.
共41兲
The question arises: how is A realized on states? Two
possibilities present themselves: two disconnected onedimensional representations or one two-dimensional representation. In the first instance, we take the ground state to be
an eigenstate of A. The possible eigenvalues are ⫾ 冑12 , so
there are two ground states, 兩0+典 with eigenvalue + 冑12 , and
兩0−典 with eigenvalue − 冑12 . No local operator connects the two
and the two towers of states built upon them
†
†
aE† aE⬘aE⬙ . . . 兩0⫾典
define two disconnected spaces of states. Moreover, one observes that A has a nonvanishing expectation value
具0⫾兩A兩0⫾典 = ⫾ 冑12 . Since A is a fermionic operator, fermion
parity is lost.21
In the second possibility, with a two-dimensional realization, we suppose that the vacuum is doubly degenerate: call
one “bosonic” 兩b典, the other “fermionic” 兩f典, and A connects
the two
1
共42兲
1
共43兲
A兩f典 =
冑2 兩b典,
A兩b典 =
冑2 兩f典.
共Phase choice does not loose generality.兲
†
but now A connects them. With this realization, fermion parity is preserved when 兩b典 and 兩f典 are taken with opposite
fermion parity. Of course since A is Hermitian, it can be
diagonalized by the eigenstates.
V. SINGLE-VORTEX ORDER PARAMETER
⌿0v =
†
aE† aE⬘aE⬙ . . . 兩b典
兩0−典 =
1
冑2 共兩b典 + 兩f典兲,
1
冑2 共兩b典 − 兩f典兲.
共44兲
This regains the two states of the two one-dimensional realizations. But the combination 兩b典 ⫾ 兩f典 violates fermion parity
as it superposes states with opposite fermion parity.
There does not seem to be a mathematical way to choose
between the two possibilities. But physical arguments favor
the fermion parity preserving realization. First of all, there is
no reason to abandon fermion parity; if possible it should be
preserved since it is a feature of the action. Also arguments
against combining states of opposite fermion parity may be
given: Since bosons and fermions transform differently under 2␲-spatial rotations; the 兩0⫾典 states in Eq. 共44兲 are not
rotationally covariant but transform into each other. 关This
argument is completely convincing in a 共3 + 1兲-dimensional
theory. In 共2 + 1兲 dimensions the anyon possibility clouds the
picture, and in 共1 + 1兲 dimensions the argument cannot be
made, because spatial rotations do not occur.兴 Furthermore,
time inversion transformations work differently on bosons
and fermions: T2 is I for bosons and −I for spinning fermions. The superposed states in Eq. 共44兲 are not invariant under
T2, rather they transform into each other. 关This argument can
be made for 共2 + 1兲 dimensional models, but in 共1 + 1兲 dimensions spin is absent so the fermion parity violating option
cannot be ruled out. Furthermore, the fermion-boson equivalence of 共1 + 1兲-dimensional models obscures the status of
fermion-boson mixing. Indeed it is argued within supersymmetry that fermion parity is lost in the presence of solitons in
共1 + 1兲 dimensions “due to boundary effects.”22兴
关Any argument based on time inversion transformations
requires viewing the complex valued vortex configuration as
arising from the degrees of freedom of an enlarged model, in
which the vortex emerges from the dynamics of the extended
model 共Abrikosov, Ginzburg, and Landau兲. Otherwise, a vortex background is not T invariant.兴
In the next section we examine the vortex/antivortex
background and argue that the two-state, two-dimensional,
fermion parity preserving realization can be established. The
physical picture that emerges is that there are two towers of
states, one built on an “empty” zero-energy state 兩b典, the
other on the “filled” zero-energy state 兩f典, and the A operator,
which connects the two “vacua,” fills or empties the zeroenergy state.
VI. VORTEX/ANTIVORTEX ORDER PARAMETER
Insight on physical states in the presence of a vortex in a
superconductor adjoined to a topological insulator can be
224515-5
PHYSICAL REVIEW B 81, 224515 共2010兲
CHAMON et al.
gotten by considering a vortex/antivortex background. The
zero-energy mode for an antivortex at the origin, ⌬共r兲
= v共r兲e−i␪, is given by
␺↓ = NJ0共␮r兲exp兵i␲/4 − V共r兲其,
␺↑ = NJ1共␮r兲exp兵− i共␪ + ␲/4兲 − V共r兲其.
共45兲
To simplify the discussion, we omit the chemical potential
and evaluate V共r兲 ⬅ 兰rdr⬘v共r⬘兲 with the asymptotic form of
v共r兲 → m. Thus the zero-energy mode for the vortex be-
a⑀†兩f典 = 0.
can be built either on the
The remaining states, created by
vacuum 兩⍀典 : aE† aE† aE† . . . 兩⍀典 or on the low-lying state
⬘ ⬙
兩f典 : a⑀†兩⍀典 : aE† aE† ⬘aE† ⬙ . . . 兩f典.
Now let us remove the antivortex by passing R to infinity.
vv̄
vv̄
v
Both ⌿⫾
⑀ collapse to their zero-mode limit, ⌿⫾⑀ → ⌿0, and
the expansion in Eq. 共42兲 becomes
⌿̂ = ⌿̂cont +
r→⬁
comes, approximately
␺0v
⬇ Ne
−i␲/4 −mr
e
冉冊
1
共46a兲
0
= Ne
i␲/4 −m兩r−R兩
e
冉冊
0
1
The corresponding four spinors that solve Eq. 共11兲 at zero
energy are
冢 冣
冢 冣
⌿0v =
Ne−i␲/4e−mr
0
0
共47a兲
,
− Nei␲/4e−mr
0
⌿0v̄ =
Ne
Ne
i␲/4 −m兩r−R兩
e
−i␲/4 −m兩r−R兩
e
0
.
共47b兲
Consider now a configuration with a vortex at the origin
and an antivortex at R. No zero mode is present in the spectrum of h; rather there are two bound states, one with positive, exponentially small energy ␧ ⬇ e−mR and the other with
equal magnitude, but opposite sign.
The former, called ⌽⑀vv̄, consists of portions localized at
the origin 共vortex兲 and at r = R 共antivortex兲. The latter is
given by ⌽−vv̄⑀ = C⌽⑀vv̄ⴱ and has similar structure. Both contribute unambiguously to the expansion of the quantum field
operator ⌿̂, the former with an annihilation operator, the
latter with a creation operator.
⌿̂ ⬅ ⌿̂cont + a⑀e−i␧t⌽⑀vv̄ + a⑀†ei␧tC⌽⑀vv̄ⴱ .
共48兲
The first term on the right is the continuum contribution, as
in Eq. 共40兲. The Fock space spectrum is clear. There is a
vacuum state 兩⍀典 annihilated by a⑀
a⑀兩⍀典 = 0.
a⑀†兩⍀典 = 兩f典,
共49b兲
a⑀兩f典 = 兩⍀典,
共49c兲
a⑀ + a⑀†
2
2⌿0v = ⌿̂cont + A冑2⌿0v .
共50兲
1
A兩b典 =
冑2 兩f典,
A兩f典 =
冑2 兩b典
1
共51兲
and two towers of states are built upon 兩b典 and 兩f典.
In this way we justify the two-dimensional, fermion parity
preserving realization of the zero-mode algebra in a
superconducting/topological insulator system. 关Note the occurrence of the factor 冑2 modifying ⌿0v. This explains its
first appearance in Eq. 共40兲. This factor compensates in the
completeness sum for the loss of the antivortex wave function.兴
Because no explicit solutions in a vortex/antivortex background are available, the argument in this section is qualitative, without explicit formulas. However, one may consider a
one-dimensional example with Majorana fermions in the
presence of a kink and/or a kink antikink pair.21 In that
model one can solve equations explicitly and verify the behavior described here for the two-dimensional vortex case. In
this way one also establishes that even in one spatial dimension 共in the absence of rotation and spin to enforce fermion
parity兲 the two-dimensional realization of the zero-mode algebra is appropriate.
In Appendix B we present an approximate determination
of the low-energy eigenvalues in the presence of a vortex/
antivortex pair. The result supports the above qualitative argument: an exponentially small splitting of the zero-energy
mode is established. Also in the appendix, we study the two
vortex background, and find, within the same approximation
that no energy splitting occurs; rather two zero modes persist
as anticipated by index theorems.
共49a兲
A low-lying state is gotten by operating on 兩⍀典 with a⑀†.
冉 冑 冊冑
兵A,A其 = 1,
共46b兲
.
⑀→0
Moreover the action of A ⬅ 冑12 共a␧ + a†␧兲 = A† may be read off
Eq. 共49兲. Renaming 兩⍀典 as 兩b典, we find
while the antivortex at r = R, in the same approximation
leads to
␺0v̄
共49d兲
aE†
VII. QUANTIZING MAJORANAS FERMIONS IN
GENERIC SUPERCONDUCTORS
In the preceding sections we showed that Majorana’s
quantization prescription of the Dirac equation directly applies to the full quantum field describing the proximity effect
of an s-wave superconductor to surface states of a topologi-
224515-6
PHYSICAL REVIEW B 81, 224515 共2010兲
QUANTIZING MAJORANA FERMIONS IN A SUPERCONDUCTOR
cal insulator. Below we shall show that Majorana’s quantization prescription of real neutral fermions is rather generic in
superconductors with or without Dirac-type dispersions. The
construction below is possible for any half-integer spin 共fermionic兲 particle. The reality conditions on the fermionic
fields follow from symmetries of the BdG Hamiltonian for
superconductors constructed in the Nambu basis.
Let us consider a system with fermionic degrees of free†
, where r labels position, n the possible
dom ␺r,n,s and ␺r,n,s
flavors 共bands, for instance兲, and s the spin 共half-integer兲
along a chosen quantization axis. For simplicity, we shall
define an index ␣ ⬅ 共r , n , s兲 that encodes all these degrees of
freedom. The Hamiltonian describing superconductivity in
such a system can be written as
− hⴱ = CⴱhC.
We stress that fermionic statistics underlies this result, as it is
the reason for the minus signs and the complex conjugation
in both terms in the second row of Eq. 共55兲.
It follows from this symmetry that positive and negative
eigenmodes of h are paired
h⌽E = E⌽E ⇒ h共C⌽Eⴱ 兲 = − E共C⌽Eⴱ 兲
ⴱ
= ⌽−E .
C⌽+E
B. Generic Majorana basis and its real equation of
motion
we can write
It follows that
− h̃ⴱ = Vⴱ共− hⴱ兲Vⴱ† = Vⴱ共CⴱhC兲Vⴱ† = VⴱCⴱV†VhV†VCVⴱ†
= 共VCVT兲ⴱh̃共VCVT兲 = C̃ⴱh̃C̃
共52兲
共53兲
冊
共54兲
␺
␺†
H
⌬
1
⌿ ⬅ ⌿†h⌿.
H = ⌿† †
2
⌬ − Hⴱ
That HT = Hⴱ follows from H = H†. Notice that ⌬ = −⌬T is
enforced because of fermionic statistics and consequently we
can also write
h=
冉
冊
⌬
1 H
.
ⴱ
2 − ⌬ − Hⴱ
共55兲
冉 冊
共56兲
Let us define
C=
0 I
I 0
so that C = CT = Cⴱ = C† = C−1. The operators ⌿ must satisfy
the constraint
Cab⌿†b = ⌿a ,
so the transformation law of C is
共57兲
and
兵⌿a,⌿†b其 = ␦ab .
共⳵t + ih̃兲⌿̃ = 0
One can easily check that any BdG-type h as in Eq. 共55兲
possesses the following conjugation symmetry
共65兲
so the equation of motion for the field is purely real and thus
admits purely real solutions. Notice that this path mirrors
Majorana’s formulation of the Dirac equation for spin 1/2
particles 共he constructed a purely imaginary representation of
the Dirac matrices, obtaining an equation of motion that was
real兲.
Notice that in this basis the commutation relations become
兵⌿̃a,⌿̃b其 = C̃ab = ␦ab
and
兵⌿̃a,⌿̃†b其 = ␦ab
共66兲
corresponding to real fermions
⌿̃a = ⌿̃†a .
共58兲
A. Conjugation symmetry
共64兲
共Notice that C̃C̃ⴱ = VCVTVⴱCⴱV† = I so C̃−1 = C̃ⴱ still.兲
We will construct below a unitary matrix V such that C̃
= I. This basis is the appropriate Majorana representation for
the generic superconducting system of half-integer spin particles 共for any number of flavors兲. In this basis, one has h̃ =
−h̃ⴱ so that h̃ is imaginary or equivalently ih̃ is real. It follows from Schrödinger’s equation that
where the index a ⬅ 共␣ , p兲 with p = ⫾ the Nambu grading
共⌿␣,+ = ␺␣ and ⌿␣,− = ␺␣† 兲. The fermionic commutation relations of the fields ␺ , ␺† translate into
兵⌿a,⌿b其 = Cab
共63兲
C → C̃ = VCVT .
冉 冊
冉
共62兲
h → h̃ = VhV† .
1
1
1
= 兺 ␺␣† H␣␤␺␤ + ␺␣共− HT兲␣␤␺␤† + ␺␣† ⌬␣␤␺␤†
2
2
␣,␤ 2
⌿=
共61兲
Consider a unitary transformation V, under which
1
1
1
1
ⴱ
= 兺 ␺␣† H␣␤␺␤ − ␺␤H␣␤␺␣† + ␺␣† ⌬␣␤␺␤† + ␺␤⌬␣␤
␺␣
2
2
2
2
␣,␤
Defining
共60兲
or equivalently
1
1
ⴱ
H = 兺 ␺␣† H␣␤␺␤ + ␺␣† ⌬␣␤␺␤† + ␺␤⌬␣␤
␺␣
2
2
␣,␤
1
†
+ ␺␣⌬␣␤
␺␤ .
2
共59兲
共67兲
C. Construction of V
The appropriate unitary transformation V which makes
C̃ = I is constructed as follows. Because of fermionic statistics, the time-reversal operator ⌰ squares to −1. One can
y
write ⌰ = TK, where K is complex conjugation and T = ei␲S ,
224515-7
PHYSICAL REVIEW B 81, 224515 共2010兲
CHAMON et al.
with Sy the y component of the angular-momentum operator
共in a representation such that Sy is a purely imaginary matrix兲. T is a real antisymmetric matrix 共T = Tⴱ and TT = −T兲
y
with T2 = ei2␲S = −Ispin⫻flavor when spin is half-integer. For instance, for spin 1/2 particles T = i␴2.
Consider the following transformation:
V=
冉
Q− − iQ+
iQ+ − Q−
冊
ei␲/4,
1
Q⫾ ⬅ 共1 ⫿ iT兲
2
APPENDIX A
We present the four-component, positive energy solutions
to Eq. 共11兲. The eigenvalues
E⫾ = 冑共k ⫿ ␮兲2 + 兩⌬兩2
are associated with the eigenvectors
共68兲
关compare with Eq. 共26兲.兴 Notice that Q⫾ are projectors
2
†
= Q⫾兲, that Q⫾
= Q⫾, and that Q+2 + Q−2 = Ispin⫻flavor and
共Q⫾
ⴱ
T
= Q⫾
= Q⫿. One can
Q+Q− = Q−Q+ = 0. Also notice that Q⫾
then easily check that the above defined V is such that
C̃ = VCVT = I.
共69兲
1
⌽+共k兲 =
冑E +
⌽−共k兲 =
1
冑E −
冢
冢
⌬
冑E+ − k + ␮ 兩+ 典
冑E+ − k + ␮兩+ 典
⌬
冑E− + k + ␮ 兩− 典
冑E− + k + ␮兩− 典
冣
冣
,
.
ⴱ
共−k兲. 兩 ⫾ 典 are
The negative-energy spinors are given by C⌽⫾
defined in the text.
VIII. SUMMARY
In this paper we studied mainly three issues regarding the
quantization of Majorana fermions in superconductors, following closely Majorana’s original definitions, and looked
beyond just the Majorana zero-energy modes that are bound
to topological defects such as vortices. We started by analyzing the specific case of Dirac-type systems describing
s-wave-induced superconductivity on the surface of topological insulators. We showed that the entire ␺ field of the
superconductor model 共and not merely particular modes兲
obeys equations that are analogous to the Majorana equations of particle physics.
We then analyzed the quantization of the theory in the
presence of vortices. We showed that fermion parity can be
preserved even with a single zero-energy state. This quantization scheme shows that one can obtain a two-dimensional
Hilbert in the presence of a single vortex in an infinite plane,
presenting a case where each Majorana fermions can be,
when present in odd numbers, more than half a qubit.
Finally, we showed that the Majorana quantization procedure that we discussed for the Dirac-type equations describing s-wave-induced superconductivity on the surface of topological insulators does extend, more broadly, to any
superconductor. The constraints imposed by fermionic statistics on the symmetries of Bogoliubov-de Gennes Hamiltonians are sufficient to allow real-field solutions in the constrained doubled Nambu space that can then be quantized as
Majorana fields. This results follows simply from fermionic
statistics plus superconductivity, irrespectively of the presence or absence of any other symmetries in the problem,
such as spin-rotation invariance or time-reversal symmetry.
APPENDIX B
We study the low-lying energy levels of the Dirac-type
Hamiltonian h in Eq. 共9兲 with ␮ set to zero and ⌬ chosen
first in an approximate vortex/antivortex profile
⌬vv̄ = mei⍀共r−R/2兲e−i⍀共r+R/2兲
共B1兲
and then similarly with two vortices.
共B2兲
⌬vv = mei⍀共r−R/2兲ei⍀共r+R/2兲
Here ⍀ is the argument of the appropriate vector
x + iy
= e i␪ ,
r
ei⍀共r兲 ⬅
ei⍀共⫾R兲 ⬅ ⫾
共X + iY兲
= ⫾ ei⌰
R
with x = r cos ␪ , y = r sin ␪ , X = R cos ⌰ , Y = R sin ⌰. One
vertex is located at r ⬇ R / 2, the antivortex or the second
vortex at r ⬇ −R / 2.23
1. Vortex/antivortex
Near the vortex at r ⬇ R / 2 the order parameter ⌬vv̄ is
approximated by
⌬vv̄ → ⌬v = mei⍀共r−R/2兲e−i⍀共R兲 = mei⍀共r−R/2兲e−i⌰ . 共B3兲
The zero mode in the presence of ⌬v differs from Eq. 共47a兲
by a phase due to the additional phase ei⌰ in ⌬v. Also the
location is shifted by R / 2.
ACKNOWLEDGMENTS
冢冣
v
We thank V. Sanz, who participated in an early stage of
this investigation, for useful discussions. N. Iqbal and D.
Park evaluated numerically the integral in Eqs. 共B11兲 and
共B12兲. This research is supported by DOE under Grants No.
DEF-06ER46316 共C.C.兲, No. 05ER41360 共R.J.兲, and No.
91ER40676 共S.-Y. P.兲.
␺0v =
0
0
− vⴱ
,
v⬅
m
冑␲ e
−i共␲/4+⌰/2兲 −m兩r−R/2兩
e
.
共B4兲
Similarly, with the order parameter near the antivortex at r
⬇ −R / 2 taken as
224515-8
PHYSICAL REVIEW B 81, 224515 共2010兲
QUANTIZING MAJORANA FERMIONS IN A SUPERCONDUCTOR
⌬v̄v → ⌬v̄ = mei⍀共−R兲e−i⍀共r+R/2兲 = − mei⌰e−i⍀共r+R/2兲
⑀ = 4m2
共B5兲
冕
⬁
dre−mr
0
1
␲
冕
␲
d␪关1 − rm共1 + cos ␪兲兴e−mD
0
共B11兲
the zero mode solution replacing Eq. 共47b兲 reads
冢冣
0
␺0v̄ =
v̄
v̄
ⴱ
m
v̄ ⬅
,
冑␲ e
i共3␲/4+⌰/2兲 −m兩r+R/2兩
e
共B6兲
.
and the energy is shifted from zero by ⫾⑀.
Numerical integration of Eq. 共B11兲 at large R yields a
result consistent with
⑀→
0
␺0v
R→⬁
␺0v̄
and
of h
Next we evaluate the matrix element between
in Eq. 共9兲, with ␮ = 0 and order parameters as in Eq. 共B1兲. We
find that the diagonal matrix elements vanish 具v兩h兩v典
= 具v̄兩h兩v̄典 = 0. The energy shift ⌬E, determined by the offdiagonal elements of h, is
⌬E = ⫾ 兩具v̄兩h兩v典兩,
ⴱ
具v̄兩h兩v典 = 具v兩h兩v̄典 =
冕
共B7兲
d2rv̄ⴱ p+v = − i
m3
␲
冕
ⴱ
d r共v̄ p+v +
⬁
rdre−mr
冕
␲
−␲
0
A further shift of ␪ by ⌰ leaves
d2rv̄ⴱ p+v = − i
2m3
␲
ⴱ
v̄⌬vv̄v兲
− H.c. 共B8兲
The order parameter in Eq. 共B2兲 describing two vortices
located at r ⬇ ⫾ R / 2 reduces at r ⬇ R / 2 to
冕
⬁
rdre−mr
0
冕
␲
⌬vv → ⌬v+ = mei⍀共r−R/2兲ei⍀共R兲 = mei⍀共r−R/2兲ei⌰
共B13a兲
while the one at r ⬇ −R / 2 becomes
d␪ei共␪−⌰兲 exp − m关r2
⌬vv → ⌬v− = mei⍀共−R兲ei⍀共r+R/2兲 = − mei⌰ei⍀共r+R/2兲 .
共B13b兲
共B9a兲
The corresponding zero modes differ by phases from the
vortex solution in Eq. 共47a兲 or 共B4兲 but they retain their
spinor structure.
d␪ cos ␪e−mD ,
0
共B9b兲
␺0v+ =
For the second integrand, a similar shift, first by R / 2 and
then by ⌰ gives
d2rv̄⌬ⴱvv̄v = i
2m3
␲
冕
⬁
rdre−mr
0
冕
␲
0
d␪
r + R cos ␪ −mD
e
.
D
冕
冕
⬁
dr共1 − rm兲e−mr
0
冕
␲
0
,
v−
We notice that the ␪ integrand may also be presented as
⳵
− m1 ⳵r e−mD, thereby transforming Eq. 共B10a兲 after an integration by parts into
2m2
␲
0
− v+ⴱ
␺0v− =
共B10a兲
d2rv̄⌬ⴱvv̄v = i
冢冣
冢冣
v+
D ⬅ 冑r2 + R2 + 2Rr cos ␪
冕
共B12兲
2. Two vortices
2
+ R2 + 2rR cos 共␪ − ⌰兲兴1/2 .
冕
8
m共mR兲1/2e−mR .
␲
This may be derived analytically with the following argument. We replace the upper limit 共⬁兲 of the r integral by R
and approximate D by R + r cos ␪. The ␪ integral now leads
to modified Bessel functions I0 and I1, and we keep only
their large argument, exponential asymptote. The remaining r
integral yields Eq. 共B12兲.
The evaluation of the first integrand proceeds by recalling
that p+v = ⌬v共r − R / 2兲vⴱ and yields, after a shift of integration variable by R / 2
冕
冑
d␪e−mD .
0
共B10b兲
0
0
.
− v−ⴱ
The explicit expressions for v+ and v− are not needed because the above form of the spinors guarantees that all matrix
elements of h vanish. Thus, within our approximation, the
two-vortex background retains its two zero modes. This is to
be expected because asymptotically such a configuration is
indistinguishable from a double vortex
⌬vv → mei⍀共r兲ei⍀共r兲 = me2i␪
Thus we find that
r+⬁
具v̄兩h兩v典 = i⑀ ,
共B14兲
and a double vortex possesses two zero modes.1
224515-9
共B15兲
PHYSICAL REVIEW B 81, 224515 共2010兲
CHAMON et al.
R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 共1981兲.
Read and D. Green, Phys. Rev. B 61, 10267 共2000兲.
3
D. A. Ivanov, Phys. Rev. Lett. 86, 268 共2001兲.
4 L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 共2008兲.
5 V. Gurarie and L. Radzihovsky, Phys. Rev. B 75, 212509
共2007兲.
6
S. Tewari, S. Das Sarma, and D.-H. Lee, Phys. Rev. Lett. 99,
037001 共2007兲.
7 P. Ghaemi and F. Wilczek, arXiv:0709.2626 共unpublished兲.
8 D. L. Bergman and K. Le Hur, Phys. Rev. B 79, 184520 共2009兲.
9 E. Majorana, Nuovo Cim. 14, 171 共1937兲.
10
F. Wilczek, Nat. Phys. 5, 614 共2009兲.
11
T. Senthil and M. P. A. Fisher, Phys. Rev. B 61, 9690 共2000兲.
12
A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 共1997兲.
13 L. S. Brown, Quantum Field Theory 共Cambridge University
Press, Cambridge, UK, 1995兲, p. 363; M. F. Peskin and D. V.
Schroeder, An Introduction to Quantum Field Theory 共Perseus,
Cambridge, MA, 1995兲, p. 73.
14
The same zero-energy mode with damped Bessel function oscillations occurs also in descriptions of bilayer graphene with the
condensate ⌬ arising from states bound by interlayer Coulomb
forces and the role of the chemical potential ␮ taken by an
1
2 N.
external constant biasing voltage 共Refs. 15 and 16兲.
Seradjeh, H. Weber, and M. Franz, Phys. Rev. Lett. 101,
246404 共2008兲.
16
R. Jackiw and S. Pi, Phys. Rev. B 78, 132104 共2008兲.
17 Note that the Hamiltonian h acting on an unconstrained Dirac
field ⌽ also arises in a description of graphene. There the 共↑ , ↓兲
duality refers to two sub-lattices in the graphene hexagonal lattice and ⌬ results from a hypothetical Kékulé distortion 共Refs.
18 and 19兲.
18 C. Y. Hou, C. Chamon, and C. Mudry, Phys. Rev. Lett. 98,
186809 共2007兲.
19
R. Jackiw and S. Pi, Phys. Rev. Lett. 98, 266402 共2007兲.
20 X. Qi, T. Hughes, and S. Zhang, Phys. Rev. B 78, 195424
共2008兲.
21 G. Semenoff and P. Sodano, Electron. J. Theor. Phys. 3, 157
共2006兲.
22 A. Losev, M. A. Shifman, and A. I. Vainshtein, Phys. Lett. B
522, 327 共2001兲.
23 Similar calculations for the p ⫾ ip model are by M. Cheng, R.
x
y
Lutchyn, V. Galitski, and S. Das Sarma, Phys. Rev. Lett. 103,
107001 共2009兲.
15 B.
224515-10