Time-Reversal Phase Transition at the Edge of 3d Topological Band Insulator

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by  Cenke Xu

Time-Reversal Phase Transition at the Edge of 3d Topological Band Insulator
Cenke Xu1
1Department of Physics, Harvard University, Cambridge, MA 02138
(Dated: January 19, 2010)
We study the time reversal (T) symmetry breaking of 2d helical fermi liquid, with application
totheedge states of 3d topological band insulators with only one two-component Dirac fermion at
finite chemical potential, as well as other systems with spin-orbit coupling. The T-breaking Ising
orderparameterisnotover-dampedandthetheoryisdifferentfromtheordinaryHertz-Millistheory
for order parameters at zero momentum. We argue that the T-breaking phase transition is an 3d
0 Ising transition, and thequasiparticles are well-defined in thequantum critical regime.
1
0 PACSnumbers:
2

n
Time reversal (T) symmetry is the key to guarantee couples to the Dirac fermions as
a
J the stability of both 2d and 3d topological insulators
(TBI) [1–4], therefore it is meaningful to study the T- L = Lf +Lb+Lbf,
9
1 symmetry breaking in these systems. Because the bulk
L = (∂ φ)2− v2(∂ φ)2−rφ2−uφ4,
of TBI is always an insulator, the T-breaking transition b t b i

el] osynmlyminetvroilcvpeshatshee,aedngdethsteastpese,ctwruhmichopaernesguapplaesgsaipnwahTen- Lbf = gφψ¯ψ. iX=x,y (2)
-
r T is broken. The simplest version of 3d TBI has only ψ¯ψ order breaks T, and drives the edge to a quantum
t
s one two-component Dirac fermion at the edge, which Hall phase. Identifying the leading spin order instability
t. can be perfectly realized in materials based on Bi2Se3 requires detailed knowledge of the fermion interaction,
a
m and Bi2Te3 [5–8]. The time reversal symmetry can ei- hence we focus on the universal physics at the quantum
ther be brokenexplicitly by magnetic impurities, or bro-
critical point, assuming the existence of the phase tran-
-
d ken spontaneously by strong enough interactions. The sition. In the current work we only discuss the discrete
n effects ofmagnetic impurities andquencheddisorderson symmetry breaking,the transition with continuous sym-
o the edge states of 2d and 3d TBI has been discussed in
metrybreakingwillbestudiedinanotherpaper[15]. The
c Ref. [9, 10] and Ref. [11] respectively. Spontaneous
Lagrangian Eq. 2 can also describe the phase transition
[
T-breakingphase transitionis mostrelevantto the tran- of magnetic impurities doped into the system, and the
5 sitionmetalversionofthe3dTBIwithinterplaybetween
order parameter φ stands for the global magnetization
v
spin-orbit coupling and strong interaction [12], and it is of the magnetic impurities. The uφ4 term represents ei-
7
the goal of the current paper.
4 thertheself-interactionbetweenthemagneticimpurities,
1 Without loss of generality, the edge state of 3d TBI or the higher order spin-spin interactions between heli-
2 is described by the following time-reversal invariant La- cal fermions. In this paper we assume u > 0 and large
. grangian [3, 13]:
8 enough to ensure a second order transition.
90 Lf =ψ¯(γ0(i∂t−µ)+vfiγj∂j)ψ. (1) beLcoemt eusstfihrestHitgagkse-Yµuk=aw0a minodEeql,. w2h,icnhowis btheilsievmedodteol
0 γ0 = σz, γ1 = iσx, γ2 = iσy, ψ¯ = ψ†γ0. v is the be equivalent to the Gross-Neveu model [16, 17] L =
: f
v fermi velocity at the Dirac point, µ is the chemical po- iψ¯γ ∂ ψ −γ(ψ¯ψ)2, at least when v = v . The tran-
µ µ f b
i
X tential. The Pauli matrices in Eq. 1 represent the pseu- sition of φ is not 3d Ising transition because the cou-
dospin, which is a combination between real spin space pling g is relevant at the 3d Ising fixed point, based
r
a and orbital space. For conciseness we will call σa the on the well-known scaling dimensions [ψ] = 1/2, and
spin hereafter. The spin σa of the electrons are per- [φ]=(d−1)/2+η/2=0.518 at the 3d Ising fixed point
pendicular with their momenta. This helical spin align- [18]. IfthereareN flavorsofDiracfermions,Thecritical
ment has been successfully observed in a recent ARPES exponentsofthistransitionwithlargeN havebeencalcu-
measurement [14]. The T-symmetry guarantees that in latedby means of 1/N and ǫ=4−d expansions[19–22],
the Lagrangian the Dirac mass gap ψ¯ψ does not appear andasecondordertransitionwithnon-Isinguniversality
explicitly, although a mass generation can occur when class was found. In our current case with N = 1, there
the T-symmetry is spontaneously broken. The Dirac is no obvious small parameter to expand, we conjecture
gap is simply the z−spin magnetization, hence the gap that the transition is still second order, with different
can be spontaneously generated with strong enough fer- universality class from the 3d Ising transition.
romagnetic interaction between z−component of spins: Let us now turn on a finite chemical potential µ, but
−(ψ¯ψ)r′V~r,~r′(ψ¯ψ)~r′. To describe this T-breaking transi- stillmakeµmuchsmallerthantheband-width2Λofthe
tion, we can define an Ising order parameter φ, which edge states. Now the edge states become a helical fermi
2
S
q
q
(−k , 0) (+k , 0)
S f f a b
0
a b
FIG.1: ThefermisurfaceofDiracfermions,withfinitechem-
ical potential. a, when we translate the fermi surface with a
small momentum ~q, at the intersection the spins are almost c d
parallel; b,the two patchesof fermi surface Eq. 6 describes.
liquid, with spins aligned parallel with its fermi surface.
Thetuningparameterr inEq. 2willberenormalizedby
the static and uniform susceptibility of σz of the helical
e f
fermi liquid
∆rφ2 =Re[χ(0,0)]φ2 ∼g2(µ−Λ)φ2. (3) FIG. 2: The one-loop Feynman diagrams for boson, fermion
self-energy, vertex correction, and φn term generated with
Therefore the phase transition of φ can be driven by fermion loop. The dashed line and solid line represent the φ
propagator and fermion propagator respectively.
tuning the chemical potential µ. Also, it is straightfor-
ward though a little tedious to check that the momen-
tum andfrequency dependence ofRe[χ] arenonsingular: parallel the matrix element of σz vanishes. Mathemati-
Re[χ(ω,q)]∼c0−c1ω2−c2q2+···. cally this intuition is manifested as |hk|ψ¯kψk+q|k+qi|2
As the ordinary Hert-Millis theory [23] of quantum vanishesasq2inthelimitofq →0. Thereforeinthiscase
phase transition inside fermi liquid, the singular correc-
φ is not overdamped at low momentum and frequency.
tiontotheeffectiveLagrangianoforderparametercomes
If we ignore the self-interaction between φ, and take
fromtheimaginarypartofthesusceptibility. Atthecrit-
the GaussianpartofL , we cancalculate the self-energy
b
ical point, the critical mode of Ising order parameter φ
correction of fermion ψ through Feynman diagram Fig.
can be damped through particle-hole excitations. The
2b. Evaluated close to |ν| ∼ ǫ , the imaginary part of
q
damping rate can be calculated from the Feynman dia-
fermion self-energy scales as
gram Fig. 2a, or through the Fermi-Golden rule
1
d2k Σ(ν)′′ ∼ d2kω [θ(ǫk+q)δ(ν−ǫk+q−ωk)
Im[Σφ(ω,q)] ∼ (2π)2[f(ǫk+q)−f(ǫk)] Z k
Z − θ(−ǫk+q)δ(ν −ǫk+q+ωk)]
× δ(|ω|−ǫk+q+ǫk)|hk|gψ¯kψk+q|k+qi|2
× |hq|gψ¯qψk+q|k+qi|2 ∼g2ν2Sign[ν]+···(5)
|ω|q ω2
∼ g2v k2 1− v2q2. (4) Unlike the Hertz-Millis theory, the scaling of Σ(ν)′′ is
f fs f similar to fermi liquid, which means that the quasiparti-
cles are well-defined even at the quantum critical point.
Thisresultisobtainedinthelimitq ≪k ,k isthefermi
f f
The above calculations are only one-loop level. To
wave-vector. When|ω|>v qthescatteringratevanishes
f
evaluate higher loop diagrams, we had better simplify
for kinematic reasons,therefore when v >v this decay
b f
the problemby consideringtwopatches of the fermisur-
rate is unimportant because the Green’s function of φ
face around two opposite points (±k ,0), and label the
will peak when ω ∼ v q. From now on we will assume f
b
fermionsintermsofitsmomentump =k −k ,p =k .
thatv <v . Thedecayrateobtainedabovediffersfrom x x f y y
b f
Now the action becomes
the Hertz-Millis theory[23]whichusually takesthe form
|ω|/q for order parameters at zero momentum. This re- L = ψ†(p~)(ω−v p τz −v p2)ψ(p~),
f f x y y
sult can be physically understood as following: φ(~q) can
transfer momentum ~q to the fermi surface, and if we de- L = ηω2|φ(p~)|2−(v2 p2 +v2 p2)|φ(p~)|2+···
b bx x by y
notethefermisurfaceasS0,anddenotethefermisurface
translatedbyasmallmomentum~q asSq~,thenaslongas L = igq φ(~q)ψ†(p~)τzψ(p~+~q)+··· (6)
bf y
~q is small enough Sq~ and S0 will have almost the same
spin directions at their intersection. Because σz always Here both |p | and |p | are much smaller than k , and
x y f
flips spin direction in the XY plane, when two spins are v = v /(2k ). τz is the Pauli matrix operating on the
y f f
3
space of two fermi patches (±k ,0). This isolated patch avoid naive divergence one has to choose the first set of
f
approximation is based on the observation that φ most scalingdimensions,[g2]=−1impliesthattheself-energy
q~
strongly couples to the patch with ~q ⊥ K~ , where the shouldhavedimension2,whichisconsistentwiththere-
f
particle-hole excitation with momentum ~q is soft. Also, sultΣ(ν)′′ ∼g2ν2 weobtainedbefore. Theone loopver-
at low energy limit, none of the scattering process will tex correctioncanbe calculatedusing the secondscaling
mix these fermions with those from other patches. For and Fig. 2f, the result is Vq ∼qy2/(|qy|+cg2|ω|).
instance,ifweintegrateoutthebosonφ ,interactionbe- Now let us discuss the nature of the T-breaking tran-
q~
tweendifferentpatcheswillbeinduced,butthestandard sition. The pure bosonLagrangianLb in Eq. 2 describes
scaling argument for ordinary fermi liquid suggests that a 3d Ising transition. At the g2 order the perturbation
the only important interaction at low energy has ~q = 0 at the 3d Ising transition is included in the self-energy
i.e. the δnθδnθ′ interaction. However, when ~q = 0 the correction to φ, whose singular contribution is in the
interactionvertexvanishes. Therefore the isolatedpatch imaginary part. The imaginary part of the self-energy
approximation is reasonable. is given by both Eq. 4 and Eq. 9, evaluated with the
Under discrete symmetry transformations T, P and the full fermi surface and isolated patch approximation
x
P , the physical quantities in Eq. 6 transform as respectively. In both cases this self-energy mix φ at dis-
y
tinct points in space-time, their actual scaling dimen-
T : t→−t, ψ →τxψ, k →−k , φ→−φ, i→−i,
i i sions at the 3D Ising critical point can be estimated as
P : x→−x, ψ →τxψ, k →−k , φ→−φ,
x x x D−(2+D−2+η) =−η, η ∼0.037 [18]. Therefore at
Py : y →−y, ψ →ψ, ky →−ky, φ→−φ. (7) the g2 orderthere is no relevantperturbationinducedat
the 3d Ising fixed point.
and the action is invariant. Had we only kept one single
The higher loop diagrams are more complicated, al-
fermipatchat(+k ,0)likeRef. [24,25],theactionwould
f
thoughinthepreviousparagraphweshowedthatinboth
not be invariant under these discrete transformations.
choices of scalings g is irrelevant, it does not immedi-
The fermion-boson vertex is proportional to q of φ,
y
ately imply none of the higher order loops can generate
therefore for any loop diagram with φ external line, the
important terms at the 3d Ising fixed point. This is be-
loop diagram will vanish as q → 0 for each φ exter-
y
cause when we evaluate the fermi loop, in order to avoid
nal line. There are two different ways to assign scaling
naivedivergencewehavetotakethesecondscalinginEq.
dimensions to operators in Eq. 6:
8, which is different from the 3d Ising fixed point with
Scaling 1, [ω]=1, [px]=1, [py]=1, [vy]=−1, isotropic scaling dimensions in space-time. For instance
5 the leading φn term generated at gn order perturbation
[φ] = − , [ψ]=−2, [η]=[v ]=[v ]=0,
bx by
2 is given by diagram Fig. 2c, which should take the form
1
[g] = − ,
2 n
gn [q φ(~q )]×f (ω ,~q ). (10)
Scaling 2, [ω]=2, [p ]=2, [p ]=1, [v ]=0, i,y i n j j
x y y i=1
7 Y
[φ] = [ψ]=−2, [η]=[vb2x]=−2, [vb2y]=0, Noticethatallthe φn termswithnoddareforbiddenby
1 symmetry. This term is irrelevant based on the second
[g] = − . (8)
2 scaling of Eq. 8, but in order to know its scaling dimen-
sion at the 3d Ising fixed point, we need to evaluate its
For both scaling choices, [g] < 0, i.e. according to the
formmoreexplicitly. Thefunctionf(ω ,~q )isintegralof
naive scaling the coupling between fermions and bosons j j
the following fermion loop:
are irrelevant, and the loop diagrams are suppressed.
When we evaluate loop integrals, irrelevant terms can
f (ω ,~q )∼ dωdp dp ×δ( ω )δ( ~q )
in general be ignored, but in order to avoid divergence n j j x y j j
fromintegratingaconstant,wehaveto makeadiagram- Z n j X Xj
dependent choice of scaling from the two options in Eq. ×Tr[ G(ω+ ω ,p~+ ~q )]. (11)
i i
8, otherwise some irrelevant terms have to be kept in j=1 i=1 i=1
Y X X
the integral. For instance, we can reproduce the results
obtainedpreviouslyfromscalingargument: atthe g2 or- After the integral, this term has a very complicated de-
pendenceoftheexternalfrequencyω andmomentump~ ,
der,choosingthe secondscalinginEq. 8,the self-energy j j
but since we are only interested in its scaling dimension,
correctionofφ should havedimension3, whichis consis-
the following schematic form will be good enough:
tent with the direct calculation with action Eq. 6 and
Feynman diagram Fig. 2a: |Ω|
f ∼ . (12)
Im[Σφ]∼g2|ω||qy|, (9) n |Qy| jn=−12(Ωj +vfQjx)+···
X
which due to energy conservation is valid when |ω − Ω and Q represPent lineQar combination between external
v q | < v |q |. For the fermion self-energy, in order to frequency andmomentumq respectively. In the denomi-
f x f y
4
nator,theellipsesincludetermswithhigherpowerofmo- the Rashba model [28, 29] with inner and outer fermi
mentum compared with the leading term. We can easily surfaceswithopposite inplane helicalspindirection,and
verify that when n = 2 Eq. 12 reproduces the well- the results are very similar to our paper. Another sys-
known result |ω|/|q |. At the 3d Gaussian fixed point, tem is graphene with N = 4 flavors of Dirac fermion,
y
thecoefficientofthe φn termwillhavescalingdimension our analysis applies to order parameters ψ¯ψ and ψ¯Taψ
1−n/2,whichshouldbeirrelevantforanyn≥4. Eq. 12 (Ta ∈ SU(N)). For instance the phase transition of
is applicable to the kinematic regime with all the exter- Quantum Spin Hall order ψ¯S~ψ belongs to the 3d O(3)
nal momenta nearly parallel to yˆ, when φ couples most universality class, when the fermi energy is tuned away
strongly with particle-hole excitations. For more general from the Dirac point. In future we shall try to make
kinematic regime the φn term generated is expected to connectionbetweenourresultsandrealisticphysicalsys-
be no more singular than Eq. 12. tem, after a suitable physical system with both topolog-
So far we have only considered the leading φn term, ical band structure and strong interaction is discovered,
whichisgeneratedatgnorder. Higherordercontribution like the one studied theoretically in Ref. [12].
to φn always involve one or more internal boson lines
Theauthorappreciatetheveryhelpfuldiscussionwith
like Fig. 2e, and because of the suppression of p at the
y Max Metlitski and Xiaoliang Qi. This work is sponsored
internalvertices,we expect these higher orderterms will
by the Society of Fellows, Harvard University.
notbemorerelevantthantheleadingorder. Forinstance
the result of diagram Fig. 2e with one internal boson
line has the same scaling dimension as Fig. 2c. Based
ontheseobservations,theT-breakingphasetransitionin
thehelicalfermiliquidwithfiniteµisexpectedtobea3d [1] C. L. Kane and E. J. Mele, Phys. Rev. Lett 95, 226801
Isingtransition. Ifwetakeintoaccountoftheinteraction (2005).
between φ at the 3d Ising universality class i.e. using [2] C. L. Kane and E. J. Mele, Phys. Rev. Lett 95, 146802
the fully dressed boson propagator in Fig. 2d, the self- (2005).
[3] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,
energy of the fermion will be even more suppressed due
106803 (2007).
to self-screening between bosons. One reasonable result [4] L. Fu and C. L. Kane, Phys.Rev. B 76, 045302 (2007).
could be Σ(ν)′′ ∼ g2|ν|2+ηSign[ν], η ∼ 0.037 [18] is the
[5] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin,
anomalousdimensionofφatthe3dIsingtransition,since A.Bansil,D.Grauer,Y.S.Hor,R.J.Cava,etal.,Nature
η > 0, the quasiparticle is always well-defined at the Physics 5, 398 (2009).
quantum critical regime. [6] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier,
J.Osterwalder, L.Patthey,J.G.Checkelsky,N.P.Ong,
TheLagrangianEq. 1isinvariantwhenspinandspace
et al., Nature460, 1101 (2009).
are rotated by the same and arbitrary angle, which is
[7] H.Zhang,C.-X.Liu,X.-L.Qi,X.Dai,Z.Fang,andS.-C.
generically larger than the symmetry of the microscopic Zhang, Nature Phys.5, 438 (2009).
system. For instance in material Bi2−xSnxTe3 the fermi [8] Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K.
surface of edge states is not circular when the chemi- Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang,
cal potential is large, instead it is a hexagonal star with et al., arXiv:0904.1829 (2009).
six sharp corners [8]. Therefore with large chemical po- [9] C.XuandJ.E.Moore,Phys.Rev.B73,045322(2006).
[10] C.Wu,B.A.Bernevig,andS.-C.Zhang,Phys.Rev.Lett.
tential, terms with higher order momentum should be
96, 106401 (2006).
considered in the free electron Lagrangian L of Eq. 1.
f [11] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang,
These higher order terms can lead to many new effects, Phys. Rev.Lett. 102, 156603 (2009).
for instance it may align the spins slightly along zˆdirec- [12] D. A. Pesin and L. Balents, arXiv:0907.2962 (2009).
tion instead of completely within the XY plane [26, 27], [13] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
althoughtheintegralofσz vanishesalongthewholefermi (2008).
surface. Ifthespinshavezˆcomponent,thenφ∼ψ¯ψ will [14] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Di, F. Meier,
L. Patthey, J. Osterwalder, A. Fedorov, H. Lin, et al.,
cause a deformation of the fermi surface, and is over-
arXiv:0904.1260 (2009).
damped for small momentum, in this case the ordinary
[15] C. Xu,arXiv:0909.2647 (2009).
z =3 Hertz-Millis theory becomes applicable. [16] D. Gross and A. Neveu,Phys.Rev.D 10, 3235 (1974).
In summary, we studied the time-reversal symmetry [17] K. Wilson, Phys.Rev. D 7, 2911 (1973).
breakingforsingleDiracfermionwithfinitechemicalpo- [18] M.Hasenbusch,K.Pinn,andS.Vinti,Phys.Rev.B59,
tential. UnliketheordinaryHertz-Millistheory,theIsing 11471 (1999).
order parameter is not overdamped, and we argue that [19] J. Zinn-Zustin,Nucl.Phys. B 367, 105 (1991).
[20] L. Karkkainen, R. Lacaze, P. Lacock, and B. Petersson,
thecouplingbetweenIsingorderparameterandfermions
Nucl. Phys. B 415, 781 (1994).
is weak in the infrared limit. The transition most likely
[21] J. A.Gracey, Int.J. Mod. Phys. A 6, 395 (1991).
belongs to the 3d Ising universality class. The analysis [22] J. A.Gracey, Phys. Lett. B 297, 293 (1992).
in our paper can be generalized to many other systems. [23] J. A.Hertz, Phys. Rev.B 14, 1165 (1976).
For instance we can consider the spin order ψ†σzψ in [24] J. Polchinski, Nucl. Phys. B 422, 617 (1994).
5
[25] S.-S.Lee, arXiv: 0905.4532 (2009). [28] E. I.Rashba, Sov.Phys. Solid State2, 1106 (1960).
[26] H.-J. Zhang, C.-X. Liu, X.-L. Qi, X.-Y. Deng, X. Dai, [29] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039
S.-C. Zhang, and Z. Fang, arXiv:0901.2762 (2009). (1984).
[27] L. Fu,arXiv:0908.1418 (2009).