Gapped energy spectra around the Dirac node at the surface of a 3D topological insulator in the presence of the time-reversal symmetry

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by  Tetsuro Habe

Gapped energy spectra around the Dirac node at the surface of a 3D topological
insulator in the presence of the time-reversal symmetry
Tetsuro Habe1 and Yasuhiro Asano1,2
1Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan and
2Center for Topological Science & Technology, Hokkaido University, Sapporo 060-8628, Japan
(Dated: January 7, 2013)
We discuss the excitation spectra around the Dirac node on a surface of a three-dimensional
topological insulator. By using the diagrammatic expansion, we show that the coupling of an
3 electron with the gauge field in the presence of impurity scatterings opens a gap around the Dirac
1
node. The results are consistent with a recent experimental finding by T. Sato, et al. [Nature
0
Phys. 7, 840 (2011)]. We also discuss the consistency between the present results and the bulk-
2
boundary correspondence of the topological insulator in the presence of time reversal symmetry.

n The conclusion can beapplied any two dimensional massless fermions.
a
J PACSnumbers: 73.20.At,73.20.Hb

4

] I. INTRODUCTION makeclearamechanismwhichgeneratesthe gaparound
l
l the Dirac node in the presence of the TRS. We con-
a
h Properties of metallic states at the surface of topolog- sider the massless fermion in two-dimension which cou-
- ically non-trivial materials are a hot issue in the con- ples with the gauge field in the presence of normal im-
s densed matter physics. A three-dimensional(3D) topo- purity scatterings. The self-energy due to the impurity
e
m logical insulator (TI) hosts a metallic state with linear scatteringsandthecouplingwiththegaugefieldiscalcu-
dispersion at its surface as a result of the topologically lated within the lowest order of perturbation expansion.
t. non-trivial nature of the wave function in bulk insulat- The results show that the realpart of the self-energy for
a
m ing region1–4. According to the shape of the dispersion, the upper Dirac cone has the opposite sign to that for
such metallic state is called Dirac cone. A point in the the lowercone. As aresult,two-dimensionalfermionhas
d- momentum space atwhich the upper cone andthe lower gapped energy spectra around the Dirac node. We con-
n one touch with each other is called Dirac node. The cludethattheinterplaybetweentheimpurityscatterings
o bulk-boundary correspondence suggests the presence of and the dynamical electron-electron interactions via the
c a single Dirac node at the surface, which reflects a topo- gauge field generates the gap. The results suggest the
2 [ DloigriaccalnnoudmebisercZon2s=ide1reindtthoebbeulfkraignisleulaagtianignsrtegpieornt.urTbhae- aTbRseSn.cWeeowf igllasphleosws tshuarftaocuersctoantecluevsieonniinsntohteipnrceosnesniscteenotf
v tionswhichbreakthe time reversalsymmetry(TRS)be- with the bulk-boundarycorrespondenceof a TI.We also
3 cause Z is well defined only in the presence of TRS. In concludethatthe translationalsymmetry isnecessaryto
2
6 fact, attaching a ferromagnetic insulator onto a TI5,6 or preserve the gapless surface state in real TIs.
6
introducing magnetic impurities into a TI7,8 remove the This paper is organized as follows. In Sec. II, we ex-
4
. Dirac node from the surface state. On the other hand, plain our theoretical model. The self-energy due to the
1 theDiracnodeisbelievedtoberobustagainstperturba- coupling with the gauge field in the presence of the im-
1 tions which preserve TRS. The Dirac node remains even purity scatterings is calculated in Sec. III. We discuss a
2
in the presence of normal impurity scatterings, which is relation between theoretical results and an experimental
1
: animportantfeatureofthesurfacestatefromtheoretical, one in Sec. IV. The conclusion is given in Sec. V.
v
experimental and applicational view points.
i
X A recent experiment9, however, has reported the
II. THEORETICAL MODEL
r gappedexcitationspectrainthe single Diraccone atthe
a surface of TlBi (Se S ) . Starting with a topological
2 x 1−x 3
insulator TlBi Se , the gradualsubstitution of Selenium LetusconsidertheLagrangiandensitywhichdescribes
2 3
by Sulfur enables to make a series of insulators ended the two dimensional massless Dirac fermions coupling
with a topologically trivial insulator TlBi S . The insu- with the gauge field,
2 3
lators should be topological and should have a gapless
1
single Dirac cone at their surface in the doping range of L = ψ†(p˜ eA )σjψ F Fµν, (1)
j j µν
− − 4
0.5< x 1.0. The experimental results of the angle re-
j=0,1,2
≤ X
solvedphotoemission,however,clearlyshowsthegapped
excitationspectrainthecorrespondingdopingrange. At where σ0 is the 2 2 unit matrix, σj for j = 1 3
× −
present, we do not have any reasonable argument which are the Pauli matrices, ψ and A are the field operators
explains the discrepancy between the theoretical predic- of a surface massless fermion and a U(1) gauge boson.
tion and the experimental results. In Eq. (1), p = p˜ represents energy measured from
0 0
Motivated by the experiment9, we theoretically try to the Dirac node and p = p˜ /v for j = 1 2 represents
j j
−
2
the momenta multiplied by the speed of light c where
v = v /c is a small constant describing dimensionless
F
velocity with the Fermi velocity v . Throughout this
F
paper, we use the units of ~ = c = 1. The field tensor
Fµν is defined by
Fµν =∂µAν ∂νAµ, (2)
−
with ∂µ = ∂/∂x . We consider the Dirac propagator
µ
defined by
p σ0+vp σ
0
G (p)= · , (3)
0 p 2 v2p2+iε
0
−
wherewetakea short-handnotationp=(p0,p). Eq.(3) FIG. 1. The Feynman diagrams under consideration. Figure
represents the particle propagator for positive energy (a) represents the fermion self-energy coupled to the gauge
p > 0 and the hole propagator for negative energy boson. The wavy line denotes the propagator of the gauge
0
p < 0. We employ the Feynman gauge in which the boson. Figure(b)isDiracfermionpropagatorinthepresence
0
propagatorconvergesatthespacialinfinityx with of impurity scatterings within theBorn approximation.
j
→∞
j = 1 2. The propagator of the gauge field at energy
−
q and momentum q is given by
0
The details of calculation are shown in Appendix B,
g
µν
Dµν(q)=q 2 −q2+iε, (4) where we estimate the self-energy in Fig. 1(a). Namely
0 − we calculate Σ(g)(p) in Eq. (8) with using the full Green
g =diag 1, 1, 1 , (5)
µν functionG(p q)inEq.(B1)insteadofthebareG (p q)
{ − − } 0
− −
in Eq. (3). In the limit of small momenta, the real part
where µ and ν are 0, 1, and 2. As shown in Eq. (1),
of the self-energy has an asymptotic form as
the vertex of the coupling between the electron and the
gauge field is described by eσµ in Heaviside units. We
−
assume that the impurity potential is spin-independent p σ0 p σ
and is represented by the delta-function, Re Σ(g)(p) =πe2 0 · . (9)
p02 − p2!
h i
Ui(r)= uσ0δ(r rj), (6) p p
−
j Wenotethattheresultdoesnotcontainγ. Thereforethe
X
impurity scatterings add only negligible corrections to
where r is the position of an impurity and u is the
j Eq.(9) as shownin Appendix B. Frompole ofthe scalar
strength of a single impurity potential. We assume a denominator [G (p)−1 Σ(p)], this self energy cannot
0
constant number density of impurities n . −
i open gap for massless Dirac fermion,
III. SELF-ENERGY p0 = (v p +πe2) πe2sgn(p0)
± | | −
= v p. (10)
± | |
At first, we estimate the self-energy by the impu-
rity scatterings alone. The impurity potential mixes the
Thetwotermsstemmingfromtheself-energycanceleach
stateswithdifferentmomenta. It,however,doesnotmix
otherout,whichreflectsthecovarianceofthegaugefield.
the states with different energies. Therefore impurity
As aconsequence,the coupling withthe gaugefield does
scatterings itself cannot open the gapat the Dirac node.
notchangethe energyspectraofmasslessDiracfermion.
The self-energy within the Born approximation is repre-
Neither the coupling with gauge field alone nor the im-
sented by
purity scatterings alone open the gap at the Dirac node.
Σ(i) = iγp σ0, (7) Theformerchangestheenergyofafermionbutpreserves
0
− the translational symmetry. On the other hand, the lat-
where γ = πu2n /v2 > 0 is the dimensionless expan- ter breaks the translational symmetry and changes the
i
sion parameter smaller than unity (See also Appendix momenta of a fermion. But it preserves the energy of
A).TheresulthasthegeneralformofthemasslessDirac fermion. To have a gap at the Dirac node, we need scat-
fermion10. tering processes which change the momenta and the en-
Secondly, we calculate the self-energy of a Dirac ergy of a fermion at the same time.
fermion coupled with the gauge field which is given by, Finally, we consider a self-energy which represents the
interplay between the coupling with the gauge field and
Σ(g)(p)= ( eσµ)G0(p q)( eσν)Dµν(q). (8) the impurity scatterings. The scattering processes are
− − −
q µν diagrammaticallydescribedinFig.2(a). Suchself-energy
XX
3
The result shows the gapped energy spectra around the
Dirac node. We note that both the impurity scatterings
and the coupling to the gauge field preserve the TRS.
IV. DISCUSSION
According to the features of the spectra in Eq. (17),
the magnitude of the gapshould be independent of tem-
perature. Our results are consistent with the gapped
spectra found in the experiment by T. Sato et al.9. An-
other experiment13 has reported the gapless spectra in
FIG. 2. The Feynman diagrams we consider in the text.
Fig.(a)isthefermionself-energyconsistingofthegaugeboson the similar situation. The energy resolution of experi-
and the impurity scatterings. Fig.(b) represents the vertex ment in Ref. 9 seems to be better than that in Ref. 13.
function including theimpurity scatterings. We discuss effects of electron-electron interactions via
thegaugefieldontheexcitationspectraaroundtheDirac
node. Becauseofthedynamicsofthe gaugefield,the in-
is given by
teraction potential depends on frequency. As a result,
Σ(p)= Γµ(p′,q)G(p q)( eσν)D (q), the interplay between the impurity scatterings and the
0 − − µ,ν interactionviathegaugefieldopensthegapattheDirac
p′0X,q,µ,ν node. On the other hand, in condensed matter physics,
(11)
the electron-electron interactions are usually taken into
Γµ(p ,q)=γ G(p)( eσµ)G(p q), (12) account within the static approximation. Namely, the
0
− −
p Coulombinteractionacts two fermions at an equal time.
X
Therefore such static interaction potential does not de-
whereΓ (p ,q)isthevertexfunctionshowninFig.2(b).
µ 0 pend on the frequency. The two different types of inter-
The vertex function can be derived from the Ward-
actionsinthepresenceofimpurityscatteringsmayaffect
Takahashi’sidentity11,12 withtheself-energyofimpurity
the excitation spectra in two different ways. In one-
scatterings(Eq. (7)),
dimension, sufficiently strong static interactions cause
the gap14. In two-dimension, however, the excitation
(p′ p)µ(σ +Γ (p,p′))=p′ Σ(i)(p′) p Σ(i)(p) .
− µ µ − − − spectra havebeen believedto remaingapless evenin the
(cid:16) (1(cid:17)3) strong static interactions. It is necessary to consider the
electron-electron interactions via the dynamical boson
The vertex function results in
fieldtohavethegapattheDiracnodeintwo-dimension.
Γµ(p,p′)= ieγδ0µ, (14) This argumentis alsoimportant when we consider topo-
−
logicalprotectionofthegaplesssurfacestatebythebulk-
where δµν is the Kronecker’s delta. The correction at
boundary correspondence,which we discuss next.
another vertex of gauge field (Fig. 2 (a)) give the same
Finally, it is necessary to discuss a relationship be-
contribution. Thus the self-energy in Eq. (11) becomes
tweenourresultsandthebulk-boundarycorrespondence
1 of three-dimensional TI in the presence of the TRS. The
Σ(p)=2iγe2 G(p q) . (15)
− q2 q2+iε Z2 invariantiswelldefinedin(1+3)dimensionalfermion
Xq 0 − space of TI1. The bulk-boundary correspondence works
The summation of q can be done in the same method well to understand the existence of the two-dimensional
for deriving Eq. (8). By introducing a parameter x = gapless state on its surface. In this paper, however, we
0
v2p2/p 2, we obtain the expression of the self-energy in considera situationinwhich the fermionin TI naturally
0
the limit of x 0 as coupleswiththe electromagneticfield. Insuchsituation,
0
→
thewholephysicalspaceconsistsofthefermionspaceand
lim Σ(p ,p)=2πe2γ2 (sgn(p )σ0+pˆ σ , (16)
0 0 the gauge boson one. Namely, the (1+3) dimensional
p→0 ·
fermion space is no longer closed independent physical
(cid:0) (cid:1)
where pˆ is the unit vector in the direction of p. This
space. At present, it is unclear if it is possible to apply
is the central result of this paper. The result does not
thetopologicalclassificationdefinedsolelyinthefermion
depend on either the magnitude of the energy or that of
spacetotheintegratedphysicalspaceofthe fermionand
themomentum. Fromthepoleofthescalardenominator
the gauge boson. Even in the integrated space, it may
of [G (p)−1 Σ(p)], we obtain the energy dispersion of
0 be still possible to consider the topological characteriza-
−
perturbed fermions at p 0 as
tion in the fermion space only. In such case, the theory
| |→
p = (v p +2πe2γ2)+2πe2γ2sgn(p ) wouldbepossibletodefinesometopologicalinvariantsin
0 ± | | 0 asimilarwayastheweaktopologicalinvariants15. There-
= (v p +4πe2γ2). (17) fore the predictions by the topological theory1–3,16,17 in
± | |
4
(1+3) dimensional fermion space are not always valid. (MEXT) of Japan.
This makes the background of the gapped energy spec-
traappearingatthetopologicallyprotectedsurfacestate.
The coupling of fermion with another degree of freedom
Appendix A: Self-energy due to impurity scatterings
sometimes changes a topologically non-trivial phase to
a topologically trivial one. Our result demonstrates an
example of the story. We derivethe self-energyof impurity scattering alone.
Within the Born approximation, the self-energy is given
by,
V. CONCLUSION
p σ0+vp σ
Σ(i) =u2n d2p 0 · . (A1)
In conclusion, we have studied effects of the gauge i p 2 v2p2+iε
Z 0 −
field and the impurity scatterings on the energy spec-
tra of massless Dirac fermion at the surface of a three- The term proportional to momentum p vanishes after
dimensional topological insulator. By using the per- integrating of p. The self-energy is calculated as
turbation expansion, we have calculated the self-energy
around the Dirac node. The interplay between the im- d2p p σ0
Σ(i) =u2n 0
purity scatterings and the interaction with gauge field i (√2π)2p02 v2p2+iε
leads to the gapped energy dispersion around the Dirac Z −
=u2n p σ0
node. Our results suggest that the translational invari- i 0
ance is also necessary for the surface state to be gapless d2p ∞ dα
exp[iα(p 2 v2p2+iε)]
in three-dimensional topological insulators. × (√2π)2 ( i) 0 −
Z Z0 −
= iγp σ0. (A2)
0
−
ACKNOWLEDGMENTS γ =πu2n /v2 (A3)
i
This work was supported by the ”Topological Quan- The result is pure imaginary and gives the inverse of the
tum Phenomena” (No. 22103002) Grant-in Aid for Sci- lifetimeinagreementwithapreviousresult10. Weintro-
entific Research on Innovative Areas from the Ministry duce a dimensionless constantγ which contains material
of Education, Culture, Sports, Science and Technology parameters.
Appendix B: Self-energy due to coupling with gauge field
Let us calculate the self-energy due to coupling with the gauge field represented by Fig. 1(a). The propagator for
a Dirac fermion in the presence of impurity scattering is represented by
(1+iγ)p σ0+vp σ
0
G(p)= · . (B1)
(1+iγ)2p 2 v2p2
0
−
By using this propagator,the self-energy in Fig. 1(a) is given by
Σ(g)(p)= ( eσµ)G(p q)( eσ )D (q) (B2)
ν µν
− − −
q µν
XX
(1+iγ)(p q )σ0+v(p q) σ g
= ( eσ ) 0− 0 − · ( eσν) − µν . (B3)
− µ (1+iγ)2(p q )2 v2(p q)2 − q2 q2+iε
Xq0,qXµν 0− 0 − − 0 −
The self-energy is expressed in the parametric integral representation18 as,
∞ dq d2q ∞ ∞ 1
Σ(g)(p)=g lim 0 dα dα ( eσµ) (σ ∂ σ ∂ )( eσν)exp[iS], (B4)
µνx→0Z−∞ (√2π)3 Z0 Z0 1 2 − i 0 x0 − · x −
S =α ((1+iγ)2(p q )2 v 2(p q)2+iε)+α (q 2 q2+iε)+x (1+iγ)(p q ) vx (p q), (B5)
1 0 0 F 2 0 0 0 0
− − − − − − · −
5
wherex andα aredummyvariables. TheintegralwithrespecttothemomentacanbedonebytheFresnelintegral.
µ j
The results become
∞ ∞ dα dα 1
Σ(g)(p)= e2 lim 1 2 σ0∂ +σ ∂
− x→0Z0 Z0 (α1v2+α2) i(α1(1+iγ)2+α2) i x0 · x
(cid:0) (cid:1)
α α (1+iγ)p 2 α α p2 α (1+iγ)p x vx p
1 2p 0 1 2 2 0 0
exp i i(α +α )ε+ + · .
× 1 2 α (1+iγ)+α − α v2+α − α (1+iγ)2+α α v2+α
(cid:20) (cid:26) 1 2 1 2 1 2 1 2(cid:27)(cid:21)
Next we introduce a integration variable ρ=α +α and change the integration variables as α ρα . In this way,
1 2 i i
→
we obtain
1 1 dα dα δ(1 α α ) α (1+iγ) α
Σ(g)(p)= e2 1 2 − 1− 2 2 p σ0 2 vp σ
− Z0 Z0 α1v2+α2 i(α1(1+iγ)2+α2)(cid:18)α1(1+iγ)2+α2 0 − α1v2+α2 · (cid:19)
∞ dρ α α (1+iγ)2 α α
exp ρ εp+ 1 2 p 2 1 2 v2p2 . (B6)
× √iρ − i(α (1+iγ)2+α ) 0 − i(α v2+α )
Z0 (cid:20) (cid:18) 1 2 1 2 (cid:19)(cid:21)
By applying the condition v2 1 and introducing a variable x =v2p2/p 2, the self-energy becomes
0 0
≪
1 dα 1+iγ vp σ
Σ(g)(p) e2 p σ0 · (B7)
0
≃− α((1+iγ)2p 2(1 α) (1+(2iγ+γ2)α)v2p2 1+2iγα − 1 α
Z0 0 − − (cid:18) − (cid:19)
e2 (p1 iγ) 1 dα 1+ γ(2i+γ) 1−α x 1+iγ p σ0 vp·σ . (B8)
≃− p02 − Z0 α((1−x0)−α)(cid:18) 2(1+iγ)21−α−x0 0(cid:19)(cid:18)1+2iγα 0 − 1−α(cid:19)
p p
We neglect terms proportionalto x because we focus on the limit of x 0 in the followings. The results are
0 0
≃
e2 1 dα 1−x0 dα vp σ
Σ(g)(p) + (1+iγ 2iγα)p σ0 · (B9)
0
≃− p02 Z1−x0 i α(α−(1−x0)) Z0 α((1−x0)−α)!(cid:18) − − 1−α(cid:19)
= pe2 (1 iγ) pi(1+iγ)p σ0 log(2α (1px )+2 α(α (1 x ))) 1
0 0 0
− p02 − (cid:18)− h − − p − − i1−x0
(p1+iγ)p σ0 arcsin−2α+(1−x0) 1−x0
0
− 1 x
(cid:20) − 0 (cid:21)0
1
α (1 x ) √α
2γp σ0 α(α (1 x )) (1 x )log − − 0 −
0 0 0
− "p − − − − (cid:12)(cid:12)pα−(1−x0)+√α(cid:12)(cid:12)#1−x0
(cid:12)(cid:12)p 1−(cid:12)(cid:12)x0
(cid:12) (1 x ) α (cid:12)
2iγp σ0 α((1 x ) α) (1 x )arcsin − 0 −
0 0 0
−  − − − − s 1 x0 
−
p 0
  1
vp σ (2α (1 x ))(α 1)+2x 2 x α(α (1 x ))
0 0 0 0
+i · log − − − − − −
√x0 " α−1 p #1−x0
vp σ ( 1 x )(1 α)+2x 1−x0
0 0
· arcsin − − − . (B10)
− √x0 (cid:20) (1−x0)(1−α) (cid:21)0 !
We reach the self-energy as,
e2 1+x +2√x 1 √x
Σ(g)(p) (1 iγ) i(1+iγ)p σ0log 0 0 +πp σ0 2γp σ0√x +2γp σ0(1 x )log − 0
0 0 0 0 0 0
≃− p02 − (cid:18)− 1−x0 − − 1+√x0
vp σ vp σ
p
+iπγp σ x i · log(1+x ) π · . (B11)
0 0 0 0
− √x − √x
0 0 (cid:19)
6
Using x =v2p2/p 2 1, it becomes
0 0
≪
p σ0 p σ p σ0 1 √x p σ0
Σ(g)(p) e2(1 iγ) π 0 π · 2γ√x 0 +2γ(1 x )log − 0 0
0 0
≃− − p02 − p2 − p02 − (cid:18)1+√x0(cid:19) p02
p0σ0p 1+x0p+2√x0 p p0σ0 p σ p
i(1+iγ) log +iπγx ilog(1+x ) · . (B12)
0 0
− p02 1−x0 p02 − p2!
p p p
Finally by putting γ 0, it is possible to obtain the self-energy due to the gauge field alone.
→
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