Constrained Jackiw-Rebbi model gives McGreevy-Swingle
model
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Ho, S.-H. “Constrained Jackiw-Rebbi Model Gives McGreevySwingle Model.” Physical Review D 84.12 (2011): n. pag. Web.
17 Feb. 2012. © 2011 American Physical Society
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http://dx.doi.org/10.1103/PhysRevD.84.127701
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PHYSICAL REVIEW D 84, 127701 (2011)
Constrained Jackiw-Rebbi model gives McGreevy-Swingle model
S.-H. Ho
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 25 August 2011; published 14 December 2011)
We show that the recently considered McGreevy-Swingle model for Majorana fermions in the presence
of a ’t Hooft-Polyakov magnetic monopole arises when the Jackiw-Rebbi model is constrained to be
conjugation self dual.
DOI: 10.1103/PhysRevD.84.127701
PACS numbers: 03.65.Pm, 03.65.w
The Dirac equation in a topological background has
been studied in various dimensions, such as the background of a kink in one spatial dimension, a vortex in
two spatial dimensions, a ’t Hooft-Polyakov magnetic
monopole and a dyon in three dimensions [1,2], and there
exist normalizable Dirac zero modes in all cases. The zero
modes of Majorana fermions, however, are only found in
the cases of a kink in one dimension [1] and a vortex
background in two dimensions [2]. A well separated pair
of Majorana zero modes can define a quibit since it is a
degenerate two-state system, whose state is stored nonlocally [3]. Because of this feature and obeying nonAbelian statistics, Majorana zero modes caught a lot of
attention in physics because of its potential application on
quantum computing [4].
Recently, McGreevy and Swingle considered a three
spatial dimension model for Weyl fermions coupled to a
’t Hooft-Polyakov monopole and a scalar field in the SUð2Þ
adjoint representation [5,6]. In [5], they solved the zero
mode Dirac equation explicitly and found the exact solutions for the Majorana zero modes. In this brief report, we
show that the Jackiw-Rebbi model with a Dirac fermion in
the fundamental representation of SUð2Þ gauge group,
once the conjugation condition is imposed on the Dirac
field, reproduces the single Wyel fermion case of
McGreevy-Swingle model. This indicates that the
Majorana feature of the model is not only shown in the
zero mode, but in the whole field. However, the quantum
version of the theory is problematic because of the Witten
anomaly [5,7].
Let us start from the Lagrangian density (3.1) in [1]:
A
L ¼ c a i ðD Þab c b gG c a Tab
c b A ;
A
ðD Þab ¼ @ ab igAA Tab
(1a)
(1b)
where c a is a four-component Dirac spinor and a twocomponent SUð2Þ isospinor, AA is the vector potential, T A
is the SUð2Þ generator, g ¼ diagðþ1; 1; 1; 1Þ and
G is a positive dimensionless coupling constant. Here
ða; bÞ are isospin indices while spin indices are suppressed.
(We work in the chiral representation for the gamma
matrices and the fundamental representation for the
1550-7998= 2011=84(12)=127701(2)
SUð2Þ matrices. Thus we use gamma matrix conventions
of [5] rather than [1].)
¼
TA ¼
A
;
2
0
;
0
(1c)
A ¼ 1; 2; 3
(1d)
Here ¼ ð1; Þ
~ and ¼ ð1; Þ.
~
From (1a) we can derive the Dirac equation
g A A
gG A A
cb ¼ 0
i @ ab i A ab 2
2 ab
(2)
or equivalently
g
gG A A
~ ab þ ~ A~ A Aab þ
ab c b
Hab c b ~ p
2
2
¼ i@t c a ¼ E c a ;
(3a)
~
p~ ¼ ir;
(3b)
!
!
~ 0
0 1
~ ¼ 0 ~ ¼
;
¼ 0 ¼
: (3c)
0
~
1 0
The conjugated field
0 i2
c
i2ab c b Cab c b
ca i2
0
(4)
satisfies the equation
Hab c cb ¼ E c ca
(5)
ðCHC1 Þab ¼ ðH Þab :
(6)
owing to
Now we impose
the conjugation constraint on the Dirac
a
spinor a ¼
,
a
127701-1
ca ¼ a ;
a ¼ i2 i2ab b ;
(7a)
a ¼ i2 i2ab b :
(7b)
Ó 2011 American Physical Society
BRIEF REPORTS
PHYSICAL REVIEW D 84, 127701 (2011)
Replacing the unconstrained Dirac spinor c a by the
constrained a we can rewrite (1a) in terms of the twocomponent field a :
1
0 gG A A
@ ig AA A
i
ab
ab
2 ab
2
C
B
C
L¼ ya B
Ab
@
A A
@ ig AA A
gG
i
ab
ab
2
2 ab
(8)
¼
2ya i ig A A
@ ab A ab b
2
gG 2 A
ði Þab A i2 b
2 a
gG y A 2
a ði Þab A i2 b
2
a conjugation constraint is imposed, the equation
reduces to two components and describes Majorana
fermions [8].
We have shown that the McGS model emerges when the
energy reflection conjugation is imposed on JR model with
a Dirac fermion in SUð2Þ fundamental representation. Here
we also note that the Majorana fermion in SUð2Þ adjoint
representation in JR model cannot be achieved since it
seems to be no way to impose the conjugation constraint
in isovector fermion case.1
We thank R. Jackiw for suggesting this calculation and
for discussion with J. McGreevy and B. Swingle. This
work is supported by the National Science Council of
R.O.C. under Grant No. NSC98-2917-I-564-122.
(9)
This is Equation (2.1) in [5]. The single zero mode c 0a
is present both in the unconstrained JR model and the
constrained McGS model since its mode function satisfies
c 0a ¼ Cab c 0
b .
A similar story has been told in two spatial dimensions:
an unconstrained Dirac equation with conjugation
properties like (4) and (5) describes graphene; when
In the case of isovector fermion, the corresponding conjugated field is defined by
0 i2
ca a :
2
i
0
[1] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398
(1976).
[2] R. Jackiw and P. Rossi, Nucl. Phys. B190, 681
(1981).
[3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
(2010).
[4] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das
Sarma, Rev. Mod. Phys. 80, 1083 (2008).
[5] J. McGreevy and B. Swingle, Phys. Rev. D 84, 065019
(2011).
[6] For most recent studies of Majorana zero mode in three
spatial dimension which motivated the study in [5], see:
J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. 104, 046401
(2010); M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi,
K. Walker, and Z. Wang, Phys. Rev. B 83, 115132 (2011);
M. Freedman, M. B. Hastings, C. Nayak, and X.-L. Qi,
arXiv:1107.2731.
[7] E. Witten, Phys. Lett. B 117, 324 (1982).
[8] C. Chamon, R. Jackiw, Y. Nishida, S. Y. Pi, and L. Santos,
Phys. Rev. B 81, 224515 (2010).
1
Once we impose the conjugation constraint c ¼ we only
have the trivial solution ¼ 0.
127701-2