Effective response theory for zero energy Majorana bound states in three spatial dimensions

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by  Pedro L. S. Lopes

Effective response theory for zero energy Majorana bound states in three spatial
dimensions
Pedro L. e S. Lopes,1,2,∗ Jeffrey C. Y. Teo,3 and Shinsei Ryu2
1Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Campinas, SP 13083-970, Brazil
2Department of Physics and Institute for Condensed Matter Theory,
University of Illinois, 1110 W. Green St., Urbana IL 61801-3080, U.S.A.
3Department of Physics, University of Virginia, Charlottesville, VA 22904, U.S.A.
We propose a gravitational response theory for point defects (hedgehogs) binding Majorana zero
modesin(3+1)-dimensionalsuperconductors. Startingin4+1dimensions,wherethepointdefectis
extendedintoaline,acouplingofthebulkdefecttexturewiththegravitationalfieldisintroduced.
5 Diffeomorphism invariance then leads to an SU(2)2 Kac-Moody current running along the defect
1 line. The SU(2)2 Kac-Moody algebra accounts for the non-Abelian nature of the zero modes in
0 3+1 dimensions. It is then shown to also encode the angular momentum density which permeates
2 throughout the bulk between hedgehog-anti-hedgehog pairs.

n
a
J I. INTRODUCTION It is our objective here to propose and analyze a
topological field theory that describes point defects in
6
1 Topologicalphasesaregappedquantumphasesofmat- (3+1)D superconductors with broken time-reversal sym-
metry, belonging to the symmetry class D in Altland-
ter, which are impossible to be characterized via sponta-
] Zirnbauer classification.15
r neous symmetry breaking. The list of known topological
e states in condensed matter has been greatly expanded
h
recently; in addition to quantum Hall states which fea-
t
o ture a genuine, intrinsic topological order which do not
t. require any symmetry, topological states with (or pro-
a Developing an effective (response) theory of (topologi-
tected by) symmetries, such as symmetry-protected and
m cal) superconductors is beset with difficulties, at the ut-
symmetry-enriched topological phases, have been dis-
most, connected to the chargeless nature of their low-
- covered and extensively discussed recently.1–3 For non-
d energyquasi-particles. Aknownapproachbasedontopo-
interactingfermionsystemswithacertainsetofdiscrete
n logical field theories is to introduce new Majorana fields
o symmetries, the classification of all topological phases is for these low energy degrees of freedom.16 Other options
c possible and summarized in terms of a periodic table.4–6
involve cleverly constructing the topological supercon-
[
Topological defects introduce further complexity and ducting phase by dimensional reduction17 or using grav-
1 possibilities in topological phases of matter.7 For in- itational fields to infer about thermal13 and viscous18–20
v stance, while the periodic table does not list topologi-
responses. Thegravitationalapproach,whichisthemain
9 cal superconductors with broken time-reversal symme-
focus of this manuscript, has a strong appeal due to
0 try in three spatial dimensions,5,6 one can endow trivial
recent advances in relating the geometrical and entan-
1
4 superconductors with non-trivial topological properties glement properties of these systems.21,22 The latter, in
by introducing point defects. These defects may be re-
0 particular, has been shown to encode subtle topological
. alized, for example, as superconducting vortices on the characteristics of these phases.23
1
surface of topological insulators with proximity-induced
0
superconductivity.8 These topological defects host zero
5
1 energy Majorana bound states (MBS) at their cores
: which are robust against any perturbation weaker than
v
thebulkenergygap, andareshowntoobeynon-Abelian The paper is organized as follows. In Sec. II we in-
i
X statistics in (3+1) dimensions.9–11 troduce the physics of defects in superconductors with
r WhileRef.9providesthedescriptionsofsuchtopologi- brokentime-reversalsymmetry. Weexplainhowtocalcu-
a
caldefectsintermsofsingle-particleHamiltonians,char- latetopologicalinvariantsfromthesingle-particleHamil-
acterization of the defects beyond the non-interacting tonian and relate the (3+1)D point defect case with a
limit remains a challenge. Often, a good way to tackle higher dimensional (4+1)D situation with a line-defect,
this problem comes from appealing to topological field the latter case being the starting point to define our ef-
theories.12,13 They are desirable as they directly sug- fective field theory. In Sec. III, our main section, we
gest the presence of boundary excitations for topologi- concretelly describe our gravitational action. We show
cal insulators12 and the quasiparticle braiding behavior how frame-rotation invariance of our action leads to the
in(2+1)Dtopologicallyorderedphases14. Theyareusu- introduction of chiral modes living along the line defect
ally related to bulk-boundary anomalies, surviving the which, upon dimensional reduction, back to (3+1)D, de-
effects of interactions and giving the phenomenological scribe the hedgehog-bound MBS. Finally in Sec. IV we
responses expected from the low-energy excitations. present our concluding remarks.
2
II. TOPOLOGICAL DEFECTS AND
DIMENSIONAL AUGUMENTATION
A. Bogoliubov de Gennes Hamiltonian and
dimensional argumentation
Figure 1. Dimensionally reducing a line defect in (4+1) di-
We begin by reviewing the hedgehog defects discussed
mensions to a point defect in (3+1) dimensions. (Left) The
in Ref. 9 in terms of a non-interacting fermionic Hamil- point defect (red dot) is spatially surrounded by a sphere S2
tonians. They are classical fields of certain order pa-
(yellowshell). x istheradialparameter,x istheazimuthal
1 2
rameters, varyingadiabaticallyinspace-time, coupledto one and x , the polar parameter, is not shown. (Right) The
3
the Bloch Hamiltonian to be studied. Such topological line defect (red line) parallel to the x direction sits inside a
4
defects carry extensive textures around them. In this cylindrical three-dimensional hypersurface, where each cross
case, the electronic band Hamiltonian can be written section is a sphere S2 not intersecting the line defect.
as H(k,x), where the momentum is k = (cid:0)k1, k2, k3(cid:1)T
and the space-time coordinates x = (x,t) characterize
line-defect(seeFig.1). Thedimensionofthespherethat
the defect. Concretely, a representative for the class D
wraps the defect is still D = 2 such that now the defect
Bogoliubov-de Gennes (BdG) Hamiltonian is
dimension is δ = d−D = 4−2 = 2. According to Ref.
7, such an object is classified by an integer topological
H(k,x)=Γ·k+Λ·n(x), (1)
invariant
ˆ ˆ
Dwhirearce nba(nxd) =ga(pm(mx),wRiteh∆(txhe),sIump∆er(cxo)n)dcuocmtinbgineosrdtheer ν = 1 (cid:18) i (cid:19)3 Tr(cid:0)F3(cid:1)∈Z (3)
3! 2π
parameter ∆, and the Γ = (Γ ,Γ ,Γ ) and Λ = S2 BZ4
1 2 3
(Λ1,Λ2,Λ3)matricesobeytheCliffordalgebra{Γi,Γj}= for F =dA+A2 is the Berry curvature, and BZ4 is the
{Λ ,Λ } = 2δ and {Γ ,Λ } = 0. In the case of point
i j ij i j 4D Brillouin zone. This counts the number of Majorana
defects in (3+1)D, we have a d = 3-dimensional Bril-
chiral modes along the defect. In the following, we will
louinzone BZ3 anddefinesa D =d−1=2dimensional
consider a field theory describes topological excitations
(spherical)surfaceS2aroundapointdefect(seefigure1.)
along the defect line. Through a subsequent compactifi-
ThisleadstoaZ topologicalclassification, accordingto
2 cation procedure, we will infer the effective field theory
the periodic table of defects,7 signaling the presence or
description of the (3+1)D theory with the point defect.
absence of protected Majorana bound states. The ap-
pearance of the MBS is guaranteed by the non-trivial
bulk invariant (−1)ν, where
B. Topological defects
ˆ ˆ
2 (cid:18) i (cid:19)3
ν = Q ∈Z (2) ConsidernowaclassDsystemin(3+1)Dwithapoint-
3! 2π S2 BZ3 5 2 defectstructure. ThisisdescribedbyEq.(1). Thevector
nˆ(x) = n(x)/|n(x)| defines a hedgehog looking vector
and Q = Tr(cid:2)AdA+(3/2)A3dA+(3/5)A5(cid:3) is the
5 field around the point defect. The winding of this vec-
Chern-Simons 5-form and A =(cid:104)u |du (cid:105) is the Berry
mn m n tor field determines the presence or absence of particles
connection constructed from the occupied BdG-states
bound to the defect. A unit winding corresponds to a
|u (k,x)(cid:105) of H(k,x).
m quantum vortex across a superconducting interface be-
In the context of bulk topological insulators and su-
tweenatopologicalandtrivialinsulator,anobjectknown
perconductors (i.e., those without topological defects),
to bind a Majorana zero-mode.
topological states characterized by a Z topological in-
2 Extending to (4+1)D, time reversal symmetric topo-
variant are closely related to (in fact, “descend from”) logical insulators are Z classified5,6 and the 3D interface
their higher-dimensional parent state characterized by a
between a 4D bulk primitive topological insulator (with
Z topological invariant. Here in our context, we also
index±1)andatrivialinsulatorhostsasingleWeylnode.
foundittobeconvenienttoconsidera“dimensionalaug-
In the presence of superconductivity, a quantum vortex
mentation”. In this case the original point defect is ex-
line through the 3D hyper-interface binds a chiral Majo-
tendedintoalinein(4+1)D,which,beinga(1+1)Dsys-
rana mode. Unlike the anti-periodic boundary condition
tem, should support a simple chiral theory as its bound
ofafermionringinreal3-spacewherefermionsphysically
state.
rotate by 2π going around a cycle enclosing the vortex
We thus augment the space with a further direction,
line, compactifying the fourth dimension simply closes
in which case the Brillouin zone is extended to 4D. The
thevortexloopwithaperiodicboundaryconditiononits
BdG Hamiltonian may be written as in (1) but with
chiralMajoranamode. Thezero-energyzero-momentum
k=(cid:0)k1, k2, k3, k4(cid:1)T. This dimensional “augmentation” Majorana mode is associated with the protected Majo-
is done such that the direction of k4 defines now a line ranaboundstateatthepoint-defectin(3+1)D,surviving
crossing the former point-defect, which now becomes a as the lowest energy mode after compactification.
3
We introduce a differential 2-form θ = 1θ dxµ∧dxν III. EFFECTIVE GRAVITATIONAL THEORY
2 µν
withθ = 1nˆ·∂ nˆ×∂ nˆ. Here,µ=0,...,4inthespace-
µν 2 µ ν
time coordinates. Again, its winding counts the number A. Coupling between defects and gravity
of zero-modes along the line defect and the factor of 1/2
was introduced such that this count matches the chiral
At this point, we are ready to introduce an ansatz for
central charge of the modes along the line as
the gravitational response of this system. We fix the
ˆ ansatz by a twofold argument. The first requirement is
1 thattheactionistopologicalinnature. Thisimpliesthat
c ≡c−c¯= θ (4)
− 4π the(4+1)Dactioninvolvesadifferential5-formasitsin-
S2
tegrand which is independent of the metric (or the vol-
where c,c¯ are the central charges of the left and right ume form). The second is that the action should reflect
moving modes, and S2 is the spherical surface that sur- the chiral central charge along the line defect as this is
rounds the vortex line (see Fig. 1). The chiral cen- a direct consequence of the topological field theory. The
tral charge relates to thermal transport behavior24,25 following is the unique gravitational action that obeys
by equating the energy current at temperature T to theserequirementsinthepresenceofthestaticorderpa-
I =(π/12)c T2. rameter field θ,
−
ˆ ˆ
Equation (4) may be related to the bulk’s Berry con-
1
nection,whichalsocontainstheinformationonthequasi- S = θ Qω (7)
4π 3
particles living at the defect.9 S2 Σˆ2+1
1
ThechiralcentralchargeofthegaplessMajoranamode = (c−c¯) Qω,
4π 3
along the vortex line relates to the winding of the differ- Σ2+1
ential 1-form as well as the integral invariant (3) by the
where Qω is the gravitational Chern-Simons 3-form such
topological index theorem 3
that dQω = Tr[R∧R] is the second Chern form, with
3
ˆ ˆ ˆ R the Riemann curvature tensor.
1 θ = 1 1 (cid:18) i (cid:19)3 Tr(cid:2)F3(cid:3), (5) In order to exploit the similarities between tangent
4π 23! 2π bundles and internal bundles, we choose to parametrize
S2 S2 BZ4
the manifold in non-coordinate basis in terms of local
frame fields (also known as vielbeins) ea. These are vec-
The (4+1)D space can be foliated into Σ4+1 = S2 ×
torvaluedone-formswhosevectorcomponentsrunalong
Σ2+1(seeFig.2). S2isthesphericalsurfacethatencloses
the local bases as a=0,...,4 and which satisfy the local
the point defect in (3+1)D and wraps the vortex line
orthogonality condition on the manifold eaebgµν = ηab,
in (4+1)D. Σ2+1 is an open semi-infinite surface that µ ν
where g is the manifold metric and η is the local flat
terminatesalongthelinedefect,orthogonaltoS2atevery
(Minkowski)metric. Intermsofthelocalbasis,themain
point. It may be decomposed as Σ2+1 = ΣR+ ×Σ1+1. geometric quantity of our interest, the Riemann curva-
HereΣ1+1encompassesthetime(x0)andline-defect(x4)
ture, is written as a tensor-valued two form as
directionswhereour(1+1)DchiralMajoranamodeslive.
Thisdefinesa(1+1)Dconformalfieldtheory(CFT).ΣR+ Rab =dωab+ωac∧ωcb (8)
is the positive radial direction orthogonal to the defect
sphere and ends at the sphere’s origin. In summary one where ω = ωµdxµ is the spin connection, generated by
may write the affine (Christoffel, in coordinate basis) connection
ωab =ea∂ eνb+eaeρbΓν (9)
Σ4+1 =S2×ΣR+ ×Σ1+1. (6) µ ν µ ν ρµ
forΓν =gνκ(∂ g +∂ g −∂ g )/2. Itarisesasthe
ρµ ρ µκ µ ρκ κ ρµ
connection which corrects derivatives and parallel trans-
ports of frame fields in the manifold. It is intimately re-
lated to spin angular momentum and also allows for the
covariant differentiation of spinors. In terms of the spin-
connection1-formω,theRiemanncurvatureissimilarto
a non-Abelian gauge field strength R = dω+ω∧ω and
Qω = Tr(cid:2)ω∧(cid:0)dω+ 2ω∧ω(cid:1)(cid:3) is just the usual Chern-
3 3
Simons form.
Topological field theories with SU(N) gauge groups
Figure 2. Foliation of the (4+1)D space into S2×Σ2+1. The whichhavesomesimilaritieswithEq.(7)werepreviously
spherical surface S2 (blue) surrounds the vortex line (red).
studied, e.g., in Refs. 26 and 27. In the following, we
ThesurfaceΣ2+1(green)orthogonallyintersectsS2atapoint
willderive, fromtheaction(7), thepropertiesofdefects,
and terminates along the vortex line.
whichweshowareconsistentwiththelocalizedMajorana
zero modes at the defects.
4
B. Gauge Invariance Thesecorrespondtorotationswherebothpairsofplanes
rotate by the same angle (for A) or opposite angles
We aim at describing the physics of the modes living (for B). These isoclinic rotations are equivalent to unit
atthelinedefect. Fromourpreviousdiscussions,theline quaternion elements, which themselves are elements of
defect acts as the edge of the open manifold and consists the SU(2), denoted by lower case letters a and b respec-
of the radial, time and defect directions (which we call tively. This gives the well known double cover,
x1, x0 and x4 for concreteness.) It is a known fact that SU(2)×SU(2)
the Chern-Simons action is not gauge invariant in open SO(4)≡ Z ; (14)
2
manifolds. The restoration of gauge invariance demands
the introduction of extra chiral fields living at the edge whereZ2isthegroupofinversionsgeneratedby(−1,−1)
of the manifold28 giving rise to the boundary physics. in SU(2)×SU(2). Thus, we may separate the SO(4)
Sowestartaddressingthegaugetransformationprop- WZW action in a pair of SU(2) WZW ones as
erties of our action. Since the spin-connection is defined Tr (cid:2)(dO)O−1(cid:3)3
in a coordinate independent basis, Qω is reparametriza- SO(4)
3
tioninvariant. Itisnotinvariant, ontheotherhand, un- =Tr (cid:2)(dA)A−1(cid:3)3+Tr (cid:2)(dB)B−1(cid:3)3 (15)
SO(4) SO(4)
droetratLioornenotfztthreanfrsafomrmeafiteioldnssionf othuer fsryasmteemfiewldorsk29s.aTshains =2(cid:110)TrSU(2)(cid:2)(da)a−1(cid:3)3+TrSU(2)(cid:2)(db)b−1(cid:3)3(cid:111),
SO(4,1) gauge transformations of the spin-connection
wherewenoticedthattheSO(4)traceistwiceofthatof
ω →O−1ωO+O−1dO. (10) a SU(2) one. For completeness and to address the unfa-
miliarized reader, we present in the appendix a detailed
Under such a transformation we have29 description of this mapping
Afurthercaveatmustbetakenintoaccount. Thecom-
S →S+δSˆ pactification of the line defect shrinks one of the rota-
δS = − 1 θ∧Tr(cid:2)(dO)O−1(cid:3)3 tion directions and reduces the SO(4) group to SO(3)=
12πˆ Σ4+1 SU(2)/Z2. It contains the rotations in the dimension
1 reduced 3+1D physical space. This is just the diagonal
− θ∧α , (11)
4π 2 subgroup of SO(4) in Eq. (14), i.e. a = b. This means
∂Σ4+1
that the compactification of the defect line confines the
where α2 = Tr(cid:2)(dO)O−1∧ω(cid:3). To maintain gauge in- twoSU(2)theoriesintoasingleone. Thusthefirstterm
variance, the factor δS has to be compensated by the of the action (12) becomes
introduction of extra degrees of freedom. ˆ
The general form of δS dictates which type of degrees − 4c− Tr (cid:2)(da)a−1(cid:3)3
of freedom needs to be introduced. A detailed study of 12π ˆΣ2+1 SU(2)
the most general behavior of δS is thus imperative. We =− 2 Tr (cid:2)(da)a−1(cid:3)3, (16)
startbyfoliatingthe(4+1)Dmanifoldinspheresaround 12π SU(2)
Σ2+1
the point defect. Integrating over the spherical surface
whereweidentifiedc =1/2forasinglechiralMajorana
around the defect gives −
fermion at the line defect. The SO(4) WZW theory is
ˆ ˆ
δS =− c− Tr(cid:2)(dO)O−1(cid:3)3− c− α , (12) nowreducedtoasingleSU(2)2, andtheoverallfactorof
12π 4π 2 2fixestheleveloftheaffineKac-Moodycurrentrunning
Σ2+1 Σ1+1 along the boundary of Σ2+1, which is the (1+1)D vortex
wherec =c−c¯forthechiralcentralchargeseenin(4). line.
−
The first term corresponds to a Wess-Zumino-Witten Finally we see,
ˆ
(tWermZWco)upalcetsiothneisnpina-ncoonpneencti3o-nmtaontihfoeldcurwrehnilte(dtOhe)Ola−s1t δS =− 2 Tr(cid:2)(da)a−1(cid:3)3
12π
along the line-defect. The topological nature of the ˆ Σ2+1
WZW action allows us to restrict our consideration to − 2 Tr(cid:2)(da)a−1∧ω(cid:3), (17)
4π
the SO(4) subgroup in SO(4,1), which deformation re- Σ1+1
tracttoSO(4)sincetheboostdirectionsarecontractible. wherethetraceisunderstoodtobetakeoverSU(2)ma-
As a consequence, we assume without loss of generality trices. The original action must be modified to compen-
that all transformations O are SO(4) rotations. sate for this. We address this issue in the next section,
Rotations in four dimensional manifolds may be de- defining the edge theory.
composedasrotationsofpairsofplanes. Inparticular, a
general SO(4) rotation may be separated into the prod-
uctof,uptoapairofglobalinversions,so-calledleft-and C. Edge Theory
right-isoclinic rotations.
As discussed, the line-defect acts as a boundary to the
O =AB =(−A)(−B) (13)
manifold, rendering the action defined in Eq. (7) not
5
gauge invariant. Typically one fixes then a gauge and, by x4 and x0 =t (see figure 1). With these coordinates,
solving the equations of motion for the resulting con- the coupling becomes
straint, obtains the action for the edge theory. One says ˆ ˆ
2 1
thatthelossofgaugeinvariancesetsfreeextradegreesof Tr(J ∧ω)= dx4dtTr(J ω −J ω ).
freedom, whichthenareallowedtobecomedynamical.28 4π Σ1+1 2π 0 4 4 0
Inthepresentcontext,gaugeinvarianceamountstoin- (21)
variance under Lorentz transformations. This invariance
Inaalmost-flatgeometry,theframefieldcomponentsare
is closely connected to energy-momentum conservation.
We would like to be able to do Lorentz transformations
1
at the boundaries as well as in the bulk and gauge fixing ea ≈δa + ha (22)
µ µ 2 µ
is therefore not desirable.
We simply notice that introducing the proper set of and,forsymmetricperturbations,thespin-connectionre-
degrees of freedom we may recover the gauge invariance duces to
desired. Thisisachieved,inourcase,bysubstitutingthe
1
original action (7) by ωa = ηaρ(∂ h −∂ h ). (23)
µb 2 b ρµ ρ µb
ˆ
1
S = θ∧Tr(Qω) Aftercompactification,x4 =x4+2πR,forR→0,andall
4π 3
Σ4+ˆ1 x4 derivatives should vanish and the metric to decouple
+ 4 θ∧Tr(cid:0)J3(cid:1) from the other directions, such that ω4ab = 0. Then the
12π coupling (21) simplifies into
ˆ Σ4+1
4 ˆ
+ θ∧Tr(J ∧ω), (18) 1
4π − dx4dtTr(J ω ). (24)
∂Σ4+1 2π 4 0
where
The dynamical spin-density then reads
J =(ds)s−1, s∈SU(2) (19)
2 δS 1
lb ≡ √ =− Jb +... (25)
is the SU(2) current operator of the edge theory for 0a gδωa π 4a
0b
s ∈ SU(2) along the vortex line Σ1+1. Under a gauge
where the unspecified terms are bulk contributions and
transformation by an SO(3) rotation of the frame-field,
are suppressed by its gap.
thegravitationalparttransformsasfollowsfromEq.(10)
while s→as. The changes in the two counterterms can-
cel the changes in the first one, giving the desired invari-
ance.
The boundary condition (as specified by the sign of
the winding number) of the defect field θ determines the
chirality of the current operators. This theory still lacks
dynamics,asalldefectsintopologicalphases,intheform
of a non-linear sigma model. In the presence of a kinetic
term, however the sign of the WZW action (the sign in
frontofthelevel)fixestheconservationequationsforthe
corresponding chirality.
We are not going to consider further the dynamics
along the line defect Σ1+1, which is treated to be static
in this manuscript. After integrating with static defect
field θ over the spherical surface around it, we have an
Figure3. Non-locallycorrelatedangularmomentumbetween
SU(2) WZW term on the (2+1)D space Σ2+1 orthogo-
2 zero energy Majorana bound states (red rings) at 0 and ∞
nal to the surface around the line defect together with a
through the SU(2) field s in the bulk.
couplingbetweenthebulkgeometryandtheedgemodes.
Explicitly, we obtain
This angular momentum density may be rewritten in
ˆ ˆ
S = 2 Tr(cid:0)J3(cid:1)+ 2 Tr(J ∧ω) (20) terms of the SU(2) fields (we omit the matrix indices)
12π 4π
Σ2+1 ∂Σ2+1
1 1
wheretheintegrationofθ overS2 givesthechiralcentral l0 =−πJ4 =−π∂4logs. (26)
charge c =1/2 and is absorbed in the coefficient.
−
The last term in (20) acts as a coupling between the Weconjecturethatthisexpressionrepresentsanon-local
SU(2) current J and the bulk geometry through the storage of angular momentum between two defects, one
2
spin connection ω. The vortex line Σ1+1 is parametrized placed at the origin and the other at infinity (see Fig.
6
3). The defect at infinity is not seen because in Eq. (20) the WZW action scales linearly with the winding num-
as only the origin boundary at x1 = 0 was taken into ber. On the other hand, one would expect the vortex
account. The more general expression would include the line to consist of n copies of SU(2) theories and hold n
2
other Majorana mode chiral Majorana modes. We suspect this could be recov-
ered by first relating SU(2) and SU(2n) through the
2n 2
l0 ∼∂4logs|∞−∂4logs|0, (27) level rank duality, then regard (SU(2)2)n as a conformal
embeddinginSU(2n) . Inthiscase, thetheorycaptures
2
which reduces to the previous result if s| is a constant.
∞ only the spin part while other flavor degrees of freedom
The total angular momentum L is obtained by integrat-
are either uncoupled to gravity or are confined by the
ing over x4, and we may write it suggestively as
locality of electrons.
ˆ ˆ
1 2πR ∞
L= dx dx ∂ ∂ logs. (28)
π 0 4 0 1 1 4 ACKNOWLEDGEMENTS
The dimensional augumentation resulted in a “fatten-
P.L.S.L. acknowledges support from FAPESP under
ing” of the string connecting the two defects allowing us
grants 2009/18336-0 and 2012/03210-3. S. R. is sup-
tofollowthetheangularmomentumstoredinthesystem.
ported by Alfred P. Sloan Foundation.
The semi-extensive nature of the point defects through
the bulk in these systems (which wascrucialto the iden-
tification of their non-Abelian properties in Ref. 9 is im-
Appendix: The SO(4) double cover
pliedandencodedbythebulkSU(2)texturefields. This
issimilartoabulk-boundarycorrespondence,wherespa-
tiallyseparatedlowenergydegreesoffreedomconnectto Herewepresentamoredetaileddiscussionoftheiden-
each other through the bulk high energy modes. tification SO(4)≡ SU(2)×SU(2). In particular we present
Z
2
Thetotalangularmomentuminaclosedsystemiscon- anexplicitformulamappingtheSO(4)AandB fieldsto
served. This means twisting the Majorana mode at one theSU(2)aandbones. Letusstartthinkingofrotations
end will generate an anti-twist at the other end. We in a four-dimensional space. If we write a general point
attribute this non-local angular momentum transfer to as r = (w, x, y, z), we may decompose a general rota-
the extensive bulk SU(2) texture. This angular momen- tion in rotations of orthogonal planes, say (w, x, 0, 0)
tum pump could be a gravitational version of the Thou- and (0, 0, y, z) for simplicity. These rotations leave the
less charge pump across an insulating chain30,31 and the normal vectors of the planes fixed. If these rotations
fermion parity pump across a topological p-wave super- turn the planes around their normals for the same angu-
conducting chain.7,32 lar displacement they are called isoclinic rotations. For
ourparticularexampletheymaybewritten,foranangle
θ,
IV. CONCLUSION
cosθ −sinθ 
sinθ cosθ
WehavestudiedpointdefectsinclassDtopologicalsu- O =  (A.1)
iso  cosθ −sinθ 
perconductors in (3+1) space-time dimensions from the
sinθ cosθ
point of view of topological field theories. From symme-
tryarguments,wehaveproposedaminimalgravitational
If one exchanges the ordering of the basis vectors y and
Chern-Simons model coupled with a bulk texture in the
z, we arrive at another, equally reasonable, possibility,
extended (4+1)D space.
namely
We have shown that the point defect extends to a line
defect after the ”dimensional augumentation”, and acts cosθ −sinθ 
acosvaenr Lefforeecntitvzeinbvoaurniadnacrey, oonrevoisrttehxenlinfoer.ceIdntoordinetrrotodurcee- Oi(cid:48)so = sinθ cosθ cosθ sinθ . (A.2)
extra boundary degrees of freedom. This is given by a −sinθ cosθ
chiral SU(2) WZW theory coupled to the bulk geom-
2
etry through the spin connection. The chirality of the Rotations with like-signs (θ, θ) are called left-isoclinic
new fields is fixed by the bulk texture. Under periodic while those of opposite signs (θ, −θ) are called right-
boundary condition, the compactified vortex line traps isoclinic. From the shape of the matrices in this par-
a single zero energy Majorana mode, and through bulk ticular case, the SU(2) nature of the isoclinic rotations
coupling, we have shown non-local angular momentum already becomes apparent. One may be more general,
correlation between spatially separated defects. callingAandBtheleft-andright-isoclinicmatrices,they
As a final concluding remark, we consider the possi- may be written
bility of higher winding of the defect field. In this case,
integrating over this field yields an SU(2) WZW theory A=α1−iσyα2−iτyσzβ1−iτyσxβ2,
at level 2n, for a n-fold winding, since the coefficient of B =γ −iτ σ γ −iτ δ −iτ σ δ , (A.3)
1 z y 2 y 1 x y 2
7
where σ and τ are Pauli matrices and we omitted the where a and b are SU(2) transformations. Naturally,
identitymatrices. Theeasiestwaytoseethatindeedone multiplying both a and b by −1 gives the same result,
may decompose a general rotation in four dimensions in which accounts for the double-cover. Now simply taking
such rotations, namely O = AB, is to complexify the generalaandbmatricesandseparatelyequatingeachone
coordinates. In this case, we have of them to the identity and expanding, one may relate
these to the SO(4) isoclinic rotations A and B. At the
(cid:18) (cid:19) end of a short calculation, we find an explicit relation
y z
r=(x , x , x , x )→X = , (A.4) between them, which may be used in the fields of the
1 2 3 4 −z¯ y¯
main text, as follows
(cid:18) (cid:19)
α β
wherey =x +ix andz =x +ix andthebarsrepresent A→a= , (A.6)
1 2 3 4 −β¯ α¯
complex conjugation. Now the most general transforma-
(cid:18) (cid:19)
tion on the matrix X which preserves its determinant γ δ
B →b= (A.7)
(and as such, the modulus of r) reads −δ¯ γ¯
where α = α +iα (with α as defined above for A)
1 2 1,2
(cid:48)
X =aXb, (A.5) and the notation follows similarly for β, δ and γ.
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