Topological superconductors as nonrelativistic limits of
Jackiw-Rossi and Jackiw-Rebbi models
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Nishida, Yusuke, Luiz Santos, and Claudio Chamon.
"Topological superconductors as nonrelativistic limits of JackiwRossi and Jackiw-Rebbi models." Physical Review B 82.14
(2010): 144513. © 2010 The American Physical Society
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http://dx.doi.org/10.1103/PhysRevB.82.144513
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Detailed Terms
PHYSICAL REVIEW B 82, 144513 共2010兲
Topological superconductors as nonrelativistic limits of Jackiw-Rossi and Jackiw-Rebbi models
Yusuke Nishida,1 Luiz Santos,2 and Claudio Chamon3
1Center
for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
3Physics Department, Boston University, Boston, Massachusetts 02215, USA
共Received 19 July 2010; published 15 October 2010兲
We argue that the nonrelativistic Hamiltonian of px + ipy superconductor in two dimensions can be derived
from the relativistic Jackiw-Rossi model by taking the limit of large Zeeman magnetic field and chemical
potential. In particular, the existence of a fermion zero mode bound to a vortex in the px + ipy superconductor
can be understood as a remnant of that in the Jackiw-Rossi model. In three dimensions, the nonrelativistic limit
of the Jackiw-Rebbi model leads to a “p + is” superconductor in which spin-triplet p-wave and spin-singlet
s-wave pairings coexist. The resulting Hamiltonian supports a fermion zero mode when the pairing gaps form
a hedgehoglike structure. Our findings provide a unified view of fermion zero modes in relativistic 共Dirac-type兲
and nonrelativistic 共Schrödinger-type兲 superconductors.
DOI: 10.1103/PhysRevB.82.144513
PACS number共s兲: 74.90.⫹n, 74.20.Rp, 74.25.Ha, 03.65.Ge
I. INTRODUCTION
Fermion zero modes bound to topological defects have
been discovered by Jackiw and Rebbi in 1976 共Ref. 1兲 and
recently received renewed interest in condensed matter physics 共see, for example, Ref. 2兲. Vortices in a certain class of
superconductors in two dimensions 共2D兲 support zero-energy
Majorana bound states and obey non-Abelian statistics,3
which can be potentially used for topological quantum
computation.4 Although vortices in the ordinary nonrelativistic s-wave superconductor do not support Majorana zero
modes, the weakly paired phase of the px + ipy superconductor, which is believed to be realized in Sr2RuO4,5 does
support Majorana zero modes bound to vortex cores.3,6,7
It is also known from the pioneering work by Jackiw and
Rossi that the relativistic s-wave superconductor in 2D
共Jackiw-Rossi model兲 has similar properties.8 Remarkably it
has been shown that such a system can be realized on the
surface of the three-dimensional 共3D兲 topological insulator in
contact with the s-wave superconductor.9 Besides these examples, there is a number of proposals to realize Majorana
zero modes using heterostructures of semiconductor and
superconductor,10–12 superconductor and ferromagnet,13 and
quantum 共anomalous兲 Hall state and superconductor.14
Although the nonrelativistic px + ipy superconductor and
the relativistic Jackiw-Rossi model share similar properties,
the existence of a fermion zero mode bound to a vortex has
been discussed separately in the two systems.8,9,15–19 In this
paper 共Sec. II兲, we argue that they are actually linked by
showing that the former Hamiltonian can be derived from the
latter by taking the limit of large Zeeman magnetic field and
chemical potential. In particular, the fermion zero mode
bound to a vortex persists under taking this limit.
Then in Sec. III, we turn to the relativistic JackiwRebbi model in 3D, which is known to exhibit a fermion
zero mode associated with a pointlike topological defect
共hedgehog兲.1,18,20,21 The limit of large mass and chemical potential 共nonrelativistic limit兲 leads to a “p + is” superconductor in which spin-triplet p-wave and spin-singlet s-wave
pairings coexist. We show that the resulting nonrelativistic
1098-0121/2010/82共14兲/144513共7兲
Hamiltonian supports a fermion zero mode when the pairing
gaps form a hedgehoglike structure.
We note that the analysis presented in this paper is largely
motivated by the recent paper by Silaev and Volovik:22 The
nonrelativistic Hamiltonian of the Balian-Werthamer 共BW兲
state of the superfluid 3He was derived from the relativistic
superconductor with the odd parity pairing23 and their topological properties were studied. In this paper, we shall
broadly use “relativistic” to indicate Dirac-type Hamiltonians and “nonrelativistic” to indicate Schrödinger-type
Hamiltonians. For readers’ convenience, references to the
main results are summarized in Table I.
II. JACKIW-ROSSI MODEL IN 2D AND ITS
NONRELATIVISTIC LIMIT
A. Jackiw-Rossi model and fermion zero mode at a vortex
We start with the Hamiltonian describing 2D Dirac fermions coupled with an s-wave pairing gap 共Jackiw-Rossi8 or
Fu-Kane9 model兲
H=
1
2
冕
dx⌿†H⌿
共1兲
with ⌿† = 共␺† , −i␺T␴2兲 and
TABLE I. References to the equations in which the Hamiltonian, zero-energy solution bound to a defect, and its normalizability condition are shown for the relativistic model and its nonrelativistic descendant both in 2D and 3D.
2D
2D
3D
3D
144513-1
relativistic
nonrelativistic
relativistic
nonrelativistic
Hamiltonian
Solution
Normalizability
共2兲
共13兲
共29兲
共46兲
共6兲
共17兲
共36兲 or 共38兲
共50兲 or 共51兲
共7兲
共19兲
共39兲
共53兲
©2010 The American Physical Society
PHYSICAL REVIEW B 82, 144513 共2010兲
NISHIDA, SANTOS, AND CHAMON
H=
冉
冊
␴ · p + ␴ zh − ␮
⌬
.
− ␴ · p + ␴ zh + ␮
⌬ⴱ
This Hamiltonian can be realized on the surface of the 3D
topological insulator in contact with the s-wave
superconductor.9 h is the Zeeman magnetic field and ␮ is the
chemical potential. When the pairing gap ⌬ is spatially dependent, p ⬅ 共px , py兲 has to be regarded as derivative operators 共−i⳵x , −i⳵y兲. The energy eigenvalue problem is
␧
冢冣 冢冣
u1
u2
v2
=H
v1
u1
u2
v2
1
2␲
The low-energy spectrum in such a limit can be obtained by
eliminating small components u2 and v2.26 Substituting the
following two equations from Eq. 共3兲:
共␧ + h + ␮兲u2 = p+u1 + ⌬v1
共␧ − h − ␮兲v2 = − p−v1 + ⌬ⴱu1
共3兲
.
v1
冕
ˆ ⳵⌬
ˆ
dli⑀ab⌬
a i b ⬅ Nw ,
共4兲
where ⌬ˆ a ⬅ ⌬a / 冑⌬21 + ⌬22 and the line integral is taken at spatial infinity. However, in the presence of h and ␮, the index
theorem is no longer valid: h and ␮ terms in the Hamiltonian
can couple zero modes and they become nonzero energy
states so that two states form a pair with opposite energies.
Therefore, in general, only one zero mode survives for odd
Nw while no zero mode survives for even Nw.2,18,24
If we work in polar coordinates 共r , ␪兲 with the gap function given by the vortex form
⌬共x,y兲 = 兩⌬共r兲兩ein␪ with 兩⌬共⬁兲兩 ⬎ 0,
冉冊冋
冑␮ + hJl共冑␮2 − h2r兲e−共␲/4兲i
冑␮ − hJl+1共冑␮2 − h2r兲e共␲/4兲i+i␪
共5兲
册
⫻e
and v1 = −uⴱ1, v2 = uⴱ2 with an integer l ⬅ 共n − 1兲 / 2. We note
that the zero-energy solution, Eq. 共6兲, is normalizable as long
as
共7兲
is satisfied and there is a topological phase transition at ␮2
+ 兩⌬兩2 = h2 关see also Eq. 共24兲 below兴.
B. Derivation of px + ipy superconductor and fermion zero
mode
We now derive a clear connection between the JackiwRossi model and the nonrelativistic px + ipy superconductor.
Suppose we are interested in the low-energy spectrum of
Hamiltonian 共2兲 in the limit where both h ⬎ 0 and ␮ ⬎ 0 are
equally large
共␧ − h + ␮兲u1 =
p2u1 + p−⌬v1 − ⌬p−v1 + 兩⌬兩2u1
+
␧+h+␮
␧−h−␮
共␧ + h − ␮兲v1 =
p2v1 − p+⌬ⴱu1 ⌬ⴱ p+u1 + 兩⌬兩2v1
+
. 共10兲
␧−h−␮
␧+h+␮
␧
冉冊
u1
v1
=
冢
p2
− ␮nr
2m
1
兵p−,⌬nr其
2
2
1
p
ⴱ
+ ␮nr
兵p+,⌬nr
其 −
2
2m
冣
冉冊
u1
v1
,
共11兲
where we defined the nonrelativistic mass, chemical potential, and pairing gap as
m ⬅ h,
␮nr ⬅ 冑␮2 + 兩⌬兩2 − h,
⌬nr ⬅
⌬
.
h
共12兲
The resulting Hamiltonian
Hnr =
il␪−兰rdr⬘兩⌬共r⬘兲兩
共6兲
␮2 + 兩⌬共⬁兲兩2 ⬎ h2
共9兲
Here we introduced p⫾ ⬅ px ⫾ ipy.
In the limit under consideration, Eq. 共8兲, we can neglect ␧
compared to h + ␮ and approximate 冑␮2 + 兩⌬兩2 by h. The remaining components u1 and v1 obey the new energy eigenvalue problem
it is easy to find the explicit zero-energy solution for odd
Nw = n 共Ref. 25兲
u1
=
u2
共8兲
into the remaining two equations, we obtain
When h and ␮ are both zero, the number of fermion zero
modes 共␧ = 0兲 bound to a vortex formed by ⌬共x , y兲 ⬅ ⌬1
+ i⌬2 is determined by the winding number of the two scalar
fields8,15,18
Index H =
␧,兩冑␮2 + 兩⌬兩2 − h兩 Ⰶ h ⬃ ␮ .
共2兲
冢
p2
− ␮nr
2m
1
兵p−,⌬nr其
2
1
p2
ⴱ
+ ␮nr
其 −
兵p+,⌬nr
2
2m
冣
共13兲
describes the nonrelativistic px + ipy superconductor. We note
that when h ⬍ 0, one obtains the Hamiltonian of the px − ipy
superconductor where p+ and p− are exchanged in Eq. 共13兲.
The first nontrivial check of this correspondence is the
comparison of spectrum in a uniform space where ⌬ is constant. The relativistic Hamiltonian 共2兲 has the energy eigenvalues
␧2 = p2 + h2 + ␮2 + 兩⌬兩2 ⫾ 2冑p2␮2 + h2共␮2 + 兩⌬兩2兲. 共14兲
Its low-energy branch 共lower sign兲 at small p is correctly
reproduced by the energy eigenvalue of the nonrelativistic
Hamiltonian 共13兲
2
=
␧nr
冉
p2
− ␮nr
2m
冊
2
+ p2兩⌬nr兩2
共15兲
under the assumptions in Eq. 共8兲.
Because the above “nonrelativistic limit” does not rely on
the spatial independence of ⌬, the fermion zero mode found
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PHYSICAL REVIEW B 82, 144513 共2010兲
TOPOLOGICAL SUPERCONDUCTORS AS…
TABLE II. Properties under the time-reversal operator T. ␶-matrices act on the particle-hole space and ⴰ
共 ⫻ 兲 indicates even 共odd兲 under T. Replacement of ␶0 by ␶3 exchanges the roles of ⌬1 and ⌬2.
T
TT / T
h
␮
⌬1
⌬2
−1
+1
⫻
⫻
ⴰ
⫻
ⴰ
ⴰ
⫻
ⴰ
␴2 丢 ␶0
␴1 丢 ␶1
in Eq. 共6兲 persists into the px + ipy superconductor, Eq. 共13兲.
In order to demonstrate it, we consider the simplified vortex
configuration with a constant 兩⌬nr兩 ⬎ 0
⌬nr共x,y兲 = ein␪兩⌬nr兩.
共16兲
When n is odd, we can find the explicit zero-energy solution
共␧ = 0兲 to Eq. 共11兲 共Ref. 17兲
u1 = Jl关冑2m␮nr − 共m兩⌬nr兩兲2r兴e−共␲/4兲i+il␪−m兩⌬nr兩r
共17兲
and v1 = −uⴱ1. One can see that this zero-energy solution is the
direct consequence of that in Eq. 共6兲 because Eqs. 共8兲 and
共12兲 lead to
冉
␮2 − h2 = 共␮ + h兲共␮ − h兲 ⬇ 2m ␮nr −
冊
m兩⌬nr兩2
.
2
共18兲
Thus we have established that the existence of a fermion
zero mode bound to a vortex in the px + ipy superconductor,
Eq. 共13兲, is a remnant of that in the Jackiw-Rossi model, Eq.
共2兲. In particular, the condition for the normalizability of the
zero-energy solution, Eq. 共7兲, is translated into
␮nr ⬎ 0
共19兲
which coincides with the well-known topological phase transition in the px + ipy superconductor existing at ␮nr = 0 共Refs.
3, 17, and 27兲 关see also Eq. 共25兲 below兴. Our finding also
clarifies why a vortex with winding number Nw in the px
+ ipy superconductor cannot support 兩Nw兩 zero modes in contrast to in the Jackiw-Rossi model with h = ␮ = 0.16,17 In order
to derive the px + ipy superconductor as a nonrelativistic limit
of the Jackiw-Rossi model, one needs to introduce h and ␮
which split an even number of zero modes into positive- and
negative-energy states. Therefore, only one zero mode survives for odd Nw in the px + ipy superconductor.
C. Altland-Zirnbauer symmetry class (Refs. 28 and 29) and
topological invariant
Finally, we note the Altland-Zirnbauer symmetry class of
the Hamiltonians that we have investigated in this section.
Hamiltonian 共2兲 with spatially dependent ⌬1,2 ⫽ 0 has the
charge conjugation symmetry
C−1HC = − Hⴱ with C =
冉
0
− i␴2
i␴2
0
冊
.
␹−1H␹ = − H with ␹ =
冊
␴3 0
,
0 − ␴3
共21兲
is present only if h = ␮ = 0 and essential for the index theorem, Eq. 共4兲. Therefore, the relativistic Hamiltonian 共2兲 with
h , ␮ ⫽ 0 and thus the resulting nonrelativistic Hamiltonian
共13兲 belong to the symmetry class D.
According to Refs. 30 and 31, the class D Hamiltonians
defined in compact 2D momentum spaces can be classified
by an integer-valued topological invariant, which is the first
Chern number32–34
C1 ⬅
冊
共22兲
兺 具␧a,p兩 ⳵ pi 兩␧a,p典.
␧ ⬍0
共23兲
−i
2␲
冕 冉
dp
⳵ a y ⳵ ax
−
⳵ px ⳵ p y
with
ai共p兲 ⬅
⳵
a
We shall use Eqs. 共22兲 and 共23兲 as a definition of the topological invariant C1 even for relativistic 共Dirac-type兲 Hamiltonians while C1 in this case can be a half integer. However,
for superconductors, C1 is always an integer because of the
Nambu-Gor’kov doubling. We find that the topological invariant for the relativistic Hamiltonian 共2兲 is given by
C1 =
再
冎
for ␮2 + 兩⌬兩2 ⬎ h2
0
− sgn共h兲 for ␮2 + 兩⌬兩2 ⬍ h2
共24兲
while the topological invariant for the nonrelativistic Hamiltonian 共13兲 is given by
C1 =
再
冎
1 for ␮nr ⬎ 0
0 for ␮nr ⬍ 0.
共25兲
Therefore, in general, the topological invariant of the momentum space Hamiltonian is not preserved by the nonrelativistic limit.35
Nevertheless, both values of C1 computed for the relativistic and nonrelativistic Hamiltonians are consistent with recent conjectures relating the topological invariant of a momentum space Hamiltonian to the number of fermion zero
modes bound to a vortex.2,36 For class D superconductors
defined in compact momentum spaces 共as is the case for
nonrelativistic Hamiltonians兲, Teo and Kane in Ref. 2 conjecture that the number of fermion zero modes is
␯ = C1Nw mod 2.
共20兲
The properties of each term under the time-reversal operator
T 共T−1HT = Hⴱ at h = ␮ = ⌬1,2 = 0兲 are summarized in Table II.
In particular, the so-called chiral symmetry,
冉
共26兲
This formula gives ␯ = 1 for ␮nr ⬎ 0 and ␯ = 0 for ␮nr ⬍ 0 for
an odd winding number Nw. On the other hand, Santos et al.
in Ref. 36 do not constrain Hamiltonians to be defined in
compact momentum spaces, allowing for relativistic 共Dirac-
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PHYSICAL REVIEW B 82, 144513 共2010兲
NISHIDA, SANTOS, AND CHAMON
type兲 Hamiltonians, and conjecture that the number of fermion zero modes is
␯ = 共C1 + N f 兲Nw mod 2,
共27兲
where N f is the number of Dirac flavors 关N f = 1 for the
Jackiw-Rossi model, Eq. 共2兲, and N f = 0 for the px + ipy superconductor, Eq. 共13兲兴. For an odd winding number Nw,
their formula gives ␯ = 1 for ␮2 + 兩⌬兩2 ⬎ h2 and ␮nr ⬎ 0 and
␯ = 0 for ␮2 + 兩⌬兩2 ⬍ h2 and ␮nr ⬍ 0. Therefore, the conjectured
counting of fermion zero modes in terms of the momentum
space topological invariant works both in the relativistic and
nonrelativistic Hamiltonians, even though the value of C1 is
not preserved by the nonrelativistic limit.
III. JACKIW-REBBI MODEL IN 3D AND ITS
NONRELATIVISTIC LIMIT
A. Jackiw-Rebbi model and fermion zero mode at a hedgehog
In this section, we extend the above developed analysis to
three dimensions. For this purpose, we consider the following Hamiltonian describing 3D Dirac fermions coupled with
three real scalar fields 共⌬ ⬅ ⌬1 + i⌬2 and ⌬3兲:
1
H=
2
冕
is determined by the winding number of the three scalar
fields1,18,20
dx⌿†H⌿
共28兲
Index H =
=
冉
冊
␣ · p + ␤ m − ␮ − i ␥ 5␤ ⌬ 3
⌬
.
ⴱ
− ␣ · p + ␤ m + ␮ + i ␥ 5␤ ⌬ 3
⌬
共29兲
This Hamiltonian with zero mass m and zero chemical potential ␮, after an appropriate unitary transformation and renamings 共␤ ↔ i␥5␤ , ⌬1 → ␾1 , ⌬2 → −␾2 , ⌬3 → −␾3兲, was
studied initially by Jackiw and Rebbi1 and recently by Teo
and Kane21 in the context of ordinary and topological insulators coexisting with superconductivity. When the scalar
fields ⌬1,2,3 are spatially dependent, p ⬅ 共px , py , pz兲 has to be
regarded as derivative operators 共−i⳵x , −i⳵y , −i⳵z兲. The energy eigenvalue problem is
␧
冢冣 冢冣
u1
u2
v2
u1
u2
=H
v2
⌬共x,y兲 = 兩⌬共r兲兩ein␪ with 兩⌬共⬁兲兩 ⬎ 0
共34兲
⌬3共z → ⫾ ⬁兲 → ⫾ 兩⌬3兩,
共35兲
冉冊
u1
u2
␣=
冉 冊
␴ 0
,
␤=
and hence
i ␥ 5␤ =
冉 冊
1
0
0 −1
冉
0 − i1
i1
0
␥5 =
,
冊
冉 冊
0 1
1 0
共31兲
=
冤
冑␮ + mJl共冑␮2 − m2r兲e−共␲/4兲i
0
0
冑␮ − mJl+1共冑␮2 − m2r兲e共␲/4兲i+i␪
⫻ eil␪−兰 dr⬘兩⌬共r⬘兲兩−兰 dz⬘⌬3共z⬘兲
r
z
共32兲
共36兲
⌬3共z → ⫾ ⬁兲 → ⫿ 兩⌬3兩
共37兲
with the same ⌬共x , y兲 in Eq. 共34兲, we have Nw = −n and the
zero-energy solution in Eq. 共36兲 is replaced by
冉冊
u1
=
u2
冤
0
冑␮ + mJl+1共冑␮2 − m2r兲e共␲/4兲i+i␪
冑␮ − mJl共冑␮2 − m2r兲e−共␲/4兲i
0
⫻ eil␪−兰 dr⬘兩⌬共r⬘兲兩+兰 dz⬘⌬3共z⬘兲 .
r
z
冥
共38兲
We note that the zero-energy solution, Eq. 共36兲 or 共38兲, is
normalizable as long as
␮2 + 兩⌬共⬁兲兩2 ⬎ m2
.
冥
and v1 = i␴2u1, v2 = i␴2u2 with an integer l ⬅ 共n − 1兲 / 2. On the
other hand, when
where u1,2 and v1,2 are two-component fields. Here we employ the standard representation of Dirac matrices
0 ␴
共33兲
it is easy to find the explicit zero-energy solution for odd
Nw = n 共Ref. 38兲
v1
v1
dSi⑀ijk⑀abc⌬ˆ a⳵ j⌬ˆ b⳵k⌬ˆ c ⬅ Nw ,
and
共30兲
,
冕
where ⌬ˆ a ⬅ ⌬a / 冑⌬21 + ⌬22 + ⌬23 and the surface integral is taken
at spatial infinity. However, in the presence of m and ␮, the
index theorem is no longer valid: m and ␮ terms in the
Hamiltonian can couple zero modes and they become nonzero energy states so that two states form a pair with opposite energies. Therefore, in general, only one zero mode survives for odd Nw while no zero mode survives for even
Nw.2,18,37
Here, instead of the symmetric hedgehog 共⌬i ⬀ x̂i兲, we assume the hedgehoglike configuration in which ⌬1,2 depend
only on 共x , y兲 and form a vortex and ⌬3 depends only on z
and forms a kink. They have the same winding number but
the latter has the advantage that an analytic solution can be
found even with m , ␮ ⫽ 0. If we work in cylindrical coordinates 共r , ␪ , z兲 with the gap functions given by the forms
with ⌿† = 共␺† , −i␺T␣2兲 and
H
1
8␲
共39兲
is satisfied.
B. Derivation of p + is superconductor and fermion zero mode
When m and ␮ are both zero, the number of fermion zero
modes 共␧ = 0兲 bound to a hedgehog formed by ⌬1,2,3共x , y , z兲
We now study the nonrelativistic limit of the above
Jackiw-Rebbi model. Suppose we are interested in the low-
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PHYSICAL REVIEW B 82, 144513 共2010兲
TOPOLOGICAL SUPERCONDUCTORS AS…
energy spectrum of Hamiltonian 共29兲 in the limit where both
m ⬎ 0 and ␮ ⬎ 0 are equally large
␧,兩冑␮2 + 兩⌬兩2 − m兩 Ⰶ m ⬃ ␮ .
共40兲
The low-energy spectrum in such a limit can be obtained by
eliminating small components u2 and v2.26 Substituting the
following two equations from Eq. 共30兲:
共␧ + m + ␮兲u2 = 共␴ · p − i⌬3兲u1 + ⌬v1
共␧ − m − ␮兲v2 = − 共␴ · p + i⌬3兲v1 + ⌬ⴱu1
共41兲
complex but its phase is locked to the phase of ⌬t 关see Eq.
共45兲兴 and thus there are three independent degrees of freedom. The last term in the diagonal elements resembles the
Zeeman coupling ␴ · B with “magnetic field” Bi =
−⳵i⌬3 / 共2m兲 generated by the gradient of ⌬3 = ⌬s / ⌬t. We note
that the nonrelativistic Hamiltonian 共46兲 in the absence of ⌬s
is the BW state of the superfluid 3He and studied in Ref. 22.
The first nontrivial check of this correspondence is the
comparison of spectrum in a uniform space where ⌬ and ⌬3
are constant. The relativistic Hamiltonian 共29兲 has the energy
eigenvalues
␧2 = p2 + m2 + ␮2 + 兩⌬兩2 + ⌬23
into the remaining two equations, we obtain
关p2 + ⌬23 − ␴ · 共⵲ ⌬3兲兴u1 + 共␴ · p + i⌬3兲⌬v1
␧+m+␮
共␧ − m + ␮兲u1 =
− ⌬共␴ · p + i⌬3兲v1 + 兩⌬兩2u1
+
␧−m−␮
关p +
2
共␧ + m − ␮兲v1 =
⌬23
+ ␴ · 共⵲ ⌬3兲兴v1 − 共␴ · p − i⌬3兲⌬ u1
␧−m−␮
共42兲
Here the derivative operator ⵲ in ␴ · 共⵲⌬3兲 acts only on ⌬3.
In the limit under consideration, Eq. 共40兲, we can neglect
␧ compared to m + ␮ and approximate 冑␮2 + 兩⌬兩2 by m. The
remaining components u1 and v1 obey the new energy eigenvalue problem
冉冊
u1
v1
=
冤
␴ · 共 ⵲ ⌬ 3兲
p2
− ␮nr −
2m
2m
⫻
1
兵␴ · p,⌬ⴱt 其 − i⌬sⴱ
2
冉冊
u1
v1
1
兵␴ · p,⌬t其 + i⌬s
2
␴ · 共 ⵲ ⌬ 3兲
p
+ ␮nr −
2m
2m
2
−
冥
2
=
␧nr
⌬23
2m
⌬
⌬ 3⌬
and ⌬s ⬅
.
m
m
Hnr =
冤
1
兵␴ · p,⌬t其 + i⌬s
2
1
兵␴ · p,⌬ⴱt 其 − i⌬sⴱ
2
␴ · 共 ⵲ ⌬ 3兲
p2
−
+ ␮nr −
2m
2m
2
+ p2兩⌬t兩2 + 兩⌬s兩2
共48兲
When n is odd, we can find the explicit zero-energy solution
共␧ = 0兲 to Eq. 共43兲
u1 =
冦 冋冑
Jl
2m␮nr − 共m兩⌬t兩兲2 +
0
冏 冏 册冧
⌬s
⌬t
2
r
⫻e−共␲/4兲i+il␪−m兩⌬t兩r−兩⌬s/⌬t兩兩z兩
共44兲
共45兲
The resulting Hamiltonian
␴ · 共 ⵲ ⌬ 3兲
p2
− ␮nr −
2m
2m
冊
共49兲
冥
u1 =
冦 冋冑
Jl+1
0
2
r
⫻e共␲/4兲i+i共l+1兲␪−m兩⌬t兩r−兩⌬s/⌬t兩兩z兩
共51兲
corresponding to the lower sign in Eq. 共49兲, and v1 = i␴2uⴱ1.
One can see that this zero-energy solution is the direct consequence of that in Eq. 共36兲 or 共38兲 because Eqs. 共40兲, 共44兲,
and 共45兲 lead to
冋
describes the p + is superconductor in which spin-triplet
p-wave and spin-singlet s-wave pairings coexist. ⌬s can be
冏 冏 册冧
⌬s
2m␮nr − 共m兩⌬t兩兲 +
⌬t
2
␮2 − m2 ⬇ 2m ␮nr −
共46兲
共50兲
corresponding to the upper sign in Eq. 共49兲, or
and the spin-triplet p-wave and spin-singlet s-wave pairing
gaps as
⌬t ⬅
p2
− ␮nr
2m
⌬t共x,y兲 = ein␪兩⌬t兩 and ⌬s共x,y,z兲 = ⫾ ein␪ sgn共z兲兩⌬s兩.
where we defined the nonrelativistic chemical potential as
␮nr ⬅ 冑␮2 + 兩⌬兩2 − m −
冉
under the assumptions in Eq. 共40兲.
Because the above nonrelativistic limit does not rely on
the spatial independence of ⌬ and ⌬3, the fermion zero mode
found in Eq. 共36兲 or 共38兲 persists into the p + is superconductor, Eq. 共46兲. In order to demonstrate it, we consider the
simplified hedgehoglike configuration resulting from Eqs.
共34兲, 共35兲, and 共37兲 with constant 兩⌬t兩 ⬎ 0 and 兩⌬s兩 ⬎ 0:
共43兲
,
共47兲
Its low-energy branch 共lower sign兲 at small p and ⌬3 is correctly reproduced by the energy eigenvalue of the nonrelativistic Hamiltonian 共46兲
ⴱ
⌬ⴱ共␴ · p − i⌬3兲u1 + 兩⌬兩2v1
+
.
␧+m+␮
␧
⫾ 2冑p2␮2 + m2共␮2 + 兩⌬兩2兲 + ␮2⌬23 .
冉 冊册
m兩⌬t兩2
1 ⌬s
+
2
2m ⌬t
2
.
共52兲
Thus we have established that the existence of a fermion
zero mode bound to the hedgehoglike structure, Eq. 共49兲,
formed by ⌬t and ⌬s / ⌬t in the p + is superconductor, Eq.
共46兲, is a remnant of that in the Jackiw-Rebbi model, Eq.
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PHYSICAL REVIEW B 82, 144513 共2010兲
NISHIDA, SANTOS, AND CHAMON
TABLE III. Properties under the time-reversal operator T. ␶-matrices act on the particle-hole space and ⴰ
共 ⫻ 兲 indicates even 共odd兲 under T. Replacement of ␶0 by ␶3 exchanges the roles of ⌬1 and ⌬2.
T
TT / T
␮
m
⌬3
⌬1
⌬2
−1
−1
+1
−1
ⴰ
ⴰ
⫻
⫻
⫻
ⴰ
⫻
ⴰ
ⴰ
⫻
ⴰ
⫻
ⴰ
ⴰ
ⴰ
ⴰ
⫻
⫻
ⴰ
ⴰ
␣2 丢 ␶0
␥ 5␣ 2 丢 ␶ 0
␤␣2 丢 ␶1
␥5␤␣2 丢 ␶1
共29兲. In particular, the condition for the normalizability of the
zero-energy solution, Eq. 共39兲, is translated into
␮nr +
冉 冊
1 ⌬s
2m ⌬t
2
⬎ 0.
共53兲
C. Altland-Zirnbauer symmetry class (Refs. 28 and 29)
Finally, we note the Altland-Zirnbauer symmetry class of
the Hamiltonians that we have investigated in this section.
Hamiltonian 共29兲 with spatially dependent ⌬1,2,3 ⫽ 0 has the
charge conjugation symmetry
C−1HC = − Hⴱ with C =
冉
0
− i␣2
i␣2
0
冊
.
共54兲
The properties of each term under the time-reversal operator
T 共T−1HT = Hⴱ at m = ␮ = ⌬1,2,3 = 0兲 are summarized in Table
III. In particular, the so-called chiral symmetry,
␹−1H␹ = − H with ␹ =
冉
冊
␤ 0
,
0 −␤
共55兲
is present only if m = ␮ = 0 and essential for the index theorem, Eq. 共33兲. Therefore, the relativistic Hamiltonian 共29兲
with m , ␮ ⫽ 0 and thus the resulting nonrelativistic Hamiltonian 共46兲 belong to the symmetry class D. There is no
topological classification of class D Hamiltonians in 3D momentum spaces.30,31
IV. SUMMARY
We have studied the nonrelativistic limit of the JackiwRossi model in 2D and the Jackiw-Rebbi model in 3D, both
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Jackiw-Rossi model leads to the px + ipy superconductor. Because the fermion zero mode persists under taking this limit,
we obtain a clear understanding of the existence of a fermion
zero mode bound to a vortex in the px + ipy superconductor as
a remnant of that in the Jackiw-Rossi model. Similarly, the
nonrelativistic limit of the 3D Jackiw-Rebbi model leads to
the p + is superconductor in which the spin-triplet p-wave
pairing gap ⌬t and the spin-singlet s-wave pairing gap ⌬s
coexist. We showed that the resulting Hamiltonian supports a
fermion zero mode when ⌬t and ⌬s / ⌬t form a hedgehoglike
structure. Fermion zero modes in the superconductors studied in this paper correspond to Majorana fermions and the
associated pointlike defects obey non-Abelian statistics both
in 2D 共Refs. 3 and 7兲 and 3D.21,39
Our findings provide a unified view of Majorana zero
modes in relativistic 共Dirac-type兲 and nonrelativistic
共Schrödinger-type兲 superconductors. It should be possible to
generalize our analysis to other interesting cases and find
new examples of nonrelativistic Hamiltonians, which are
more common in condensed matter systems, with topological
properties that descend from Dirac-type Hamiltonians, which
are generally easier to analyze.
ACKNOWLEDGMENTS
The authors thank M. A. Silaev and G. E. Volovik for
sharing their note regarding Ref. 22 and R. Jackiw and S.-Y.
Pi for valuable discussions. This research was supported by
MIT Department of Physics Pappalardo Program and DOE
under Grants No. DE-FG02-94ER40818 共Y.N.兲 and No.
DEF-06ER46316 共C.C., L.S.兲.
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PHYSICAL REVIEW B 82, 144513 共2010兲
TOPOLOGICAL SUPERCONDUCTORS AS…
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24 Two exceptional cases are h = ⫾ ␮. When N ⬎ 0共⬍0兲, the zerow
energy solutions at h = ␮ = 0 still solve Eq. 共3兲 with h = +共−兲␮
⫽ 0 and thus there are 兩Nw兩 zero modes. This can be easily seen
if one rewrites the Hamiltonian in the basis where the chiral
operator defined in Eq. 共21兲 has the form ␹ = diag共1 , 1 , −1 , −1兲
and recognizes that the 兩Nw兩 zero-energy solutions are eigenstates of ␹ with the eigenvalue +1共−1兲 for Nw ⬎ 0共⬍0兲.
25
The same solution with n = 1 was obtained independently in I. F.
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26 K. Capelle and E. K. U. Gross, Phys. Lett. A 198, 261 共1995兲;
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35 The topological invariant can be matched if we properly regularize the large p behavior of the relativistic Hamiltonian: Replacing h in Eq. 共2兲 by h共1 + 兩␧兩p2兲, the Chern number becomes
sgn共h兲 for ␮2 + 兩⌬兩2 ⬎ h2 and 0 for ␮2 + 兩⌬兩2 ⬍ h2, which coincides
with that of Eq. 共13兲.
36 L. Santos, S. Ryu, C. Chamon, and C. Mudry, Phys. Rev. B 82,
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37
Two exceptional cases are m = ⫾ ␮. When Nw ⬎ 0共⬍0兲, the zeroenergy solutions at m = ␮ = 0 still solve Eq. 共30兲 with m =
+共−兲␮ ⫽ 0 and thus there are 兩Nw兩 zero modes. This can be easily
seen if one rewrites the Hamiltonian in the basis where the chiral
operator defined in Eq. 共55兲 has the form ␹ = diag共1 , 1 , −1 , −1兲
and recognizes that the 兩Nw兩 zero-energy solutions are eigenstates of ␹ with the eigenvalue +1共−1兲 for Nw ⬎ 0共⬍0兲.
38 The same problem with m = 0 was studied in T. Fukui, Phys. Rev.
B 81, 214516 共2010兲.
39 M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi, K. Walker,
and Z. Wang, arXiv:1005.0583 共unpublished兲.
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