Topological Insulator in the Core of the Superconducting Vortex in Graphene

Igor F. Herbut

Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6(Received 29 September 2009; published 10 February 2010)

The core of the vortex in a general superconducting order parameter in graphene is argued to be

ordered, with the possible local order parameters forming the algebra Uð1Þ � Clð3Þ. A sufficiently strong

Zeeman coupling of the magnetic field of the vortex to the electron spin breaks the degeneracy in the core

in favor of the anomalous quantum Hall state. I consider a variety of superconducting condensates on the

honeycomb lattice and demonstrate the surprising universality of this result. A way to experimentally

determine the outcome of the possible competition between different types of orders in the core is

proposed.

DOI: 10.1103/PhysRevLett.104.066404 PACS numbers: 71.10.Pm, 71.10.Li, 72.80.Vp, 74.20.Rp

In addition to the usual metallic state [1], Dirac quasi-particles in graphene may, at least in principle, form amultitude of insulating phases [2]. The insulators shouldbe understood as different types of dynamically or exter-nally generated masses of Dirac fermions, which preservethe quasirelativistic invariance of the low-energy theory,but break some of the space-time symmetries of theHamiltonian [3]. Yet another type of ordered phase ofelectrons on honeycomb lattice is the superconductor,which breaks the Uð1Þ symmetry associated with the con-servation of the particle number [3–6]. The superconduct-ing state should be inducible in graphene by the proximityeffect for example [7]. At the level of the continuum DiracHamiltonian, superconducting states in graphene may beclassified by their symmetry with respect to the inversionof the two inequivalent Dirac points, as the s and p waves,and with respect to translational invariance, as uniform andnot. The variety of such superconducting states notwith-standing, I show here that the core of the superconductingvortex in undoped graphene is always ordered, and that, atleast in the simplest model, always in the same way: thecore of the vortex is the Haldane-Kane-Melle (HKM)topological insulator [8,9].

The above result is obtained within two complementaryDirac representations of the Bogoliubov–de Gennes (BdG)quasiparticle Hamiltonian, in which it arises in related, butdistinct ways. The problem of the internal structure of thesuperconducting vortex in graphene may be mapped ontothe general Dirac Hamiltonian in the presence of two-component mass term with a twist. It is well known thatthe single zero-energy state in this situation would imply aquantum number fractionalization [10]. In graphene, how-ever, the spin 1=2 of the electrons dictates two copies ofDirac fermions, and the concomitant doubling of the num-ber of zero-energy states restores the usual quantizationrule. The same zero-energy states, nevertheless, are re-sponsible for another nontrivial property of the vortexcore: it always harbors a finite local order parameter

(OP) of some kind. An early example of this effect wasprovided in Ref. [11], where the vortex configuration of thein-plane Neel OP on honeycomb lattice was shown toinduce an out-of-plane component of the same OP. Morerecently, the superconducting vortex was argued to exhibitcharge-density waves (CDW) and bond-density waves(BDW) in the core [12]. Here it will be shown that ingeneral the vortex of unit vorticity in the mass of thetwo-copy four-component Dirac Hamiltonian allows fourorder parameters, which close Uð1Þ � Clð3Þ, where Clð3Þstands for the three-dimensional Clifford algebra. Themembers of the algebra depend on the type of the under-lying order; nevertheless, in a superconductor of arbitrarysymmetry, if the magnetic field in the vortex core is suffi-ciently strong the HKM anomalous quantum Hall state isuniversally the ground state. The effect is due to theZeeman coupling of the magnetic field to the electronspin, which, quite unexpectedly, always selects the abovestate out of a wide variety of competing possibilities.The HKM insulator is interesting in its own right, due to

its quantized spin Hall effect, and the accompanying struc-ture of the edge states [9]. Such a topologically nontrivialground state is favored by the spin-orbit interactions ingraphene, which, however, are well known to be way tooweak to lead to an observable effect. It was recentlyproposed that this ground state may also result from theelectron-electron Coulomb repulsion in the presence of abulge in the graphene sheet [13]. The present considera-tions suggest that placing graphene on top of a type-IIsuperconductor in a mixed state may provide an alterna-tive, albeit local, realization of this elusive state of matter.The announced result will be first derived within the

rotationally invariant representation of the paringHamiltonian. Graphene exhibits gapless excitations near

two inequivalent Dirac points at � ~K, with ~K ¼ð1; 1= ffiffiffi

3p Þð2�=a ffiffiffi

3p Þ, for example, where a is the lattice

spacing [2]. One may form an eight-component Dirac-

Nambu fermion as �y ¼ ð�yþ;�y�Þ, where

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�y�ð ~k; !Þ ¼ ðuy�ð ~k; !Þ; vy

�ð ~k; !Þ; �u��ð� ~k;�!Þ; �v��ð� ~k;�!ÞÞ; (1)

where � ¼ � labels the projection of the electron spinalong the z axis, and ~k ¼ ~K þ ~p, with j ~pj � K. u� and v�

are the Grassmann variables corresponding to two triangu-lar sublattices of the honeycomb lattice. The imaginary-time Lagrangian for the excitations near the Dirac points inthis representation becomes

L ¼ �yð ~x; �Þð@� þH0Þ�ð ~x; �Þ; (2)

where the single-particle Hamiltonian is H0 ¼I2 � i�0�ipi, with �0 ¼ �3 � �3, �1 ¼ I2 � �2, and �2 ¼I2 � �1, and i ¼ 1, 2. We may then choose �3 ¼ �1 � �3

and �5 ¼ �2 � �3. fI2; ~�g is the Pauli basis of two-component matrices.

The Lagrangian is invariant under the global Uð4Þ sym-metry generated by the 16 matrices fI2; ~�g �fI4; �3; �5; �35g, where �35 ¼ i�3�5 ¼ �3 � I2, and I4 isthe four-dimensional unit matrix. The SUð2Þ subgroup

generated by the ~S ¼ ~� � I4 is the group of rotations ofthe electron spin, and is, of course, exact. This is due to thefactor ‘‘�’’ in the right half of the Dirac-Nambu fermion inEq. (1), which ensures that the rotation may be accom-plished by the matrix multiplication from the left. Theabove representation is in this sense manifestly rotationallyinvariant. It will also prove helpful to recognize N ¼ I2 ��35 as the number operator.

There are also 16 linearly independent Hermitianmatrices that anticommute with H0: fI2; ~�g �f�0; i�0�3; i�0�5; i�1�2g. The addition of any of thesematrices to the Hamiltonian H0 would gap the Diracspectrum. The matrices that correspond to superconductingorders are those that do not commute with the particlenumber. It is easy to see that h�yðI2 � i�0�3Þ�i ¼Re�s, and h�yðI2 � i�0�5Þ�i ¼ Im�s, where �s repre-sents the spatially uniform, complex s-wave superconduct-ing OP. Consider then the BdG pairing Hamiltonian in thepresence of the vortex in the underlying s-wave super-conducting ground state:

HBdG ¼ H0 þ j�sðrÞjI2 � ½i�0ð�3 sin�þ �5 cos�Þ�; (3)

where (r, �) are the polar coordinates in the grapheneplane. j�sðr ! 1Þj ¼ const, and otherwise arbitrary. Themagnetic field in the vortex will be included shortly. Anindex theorem [14] guarantees that the spectrum of theblock-diagonal Hamiltonian HBdG contains two states with

zero energy:�y0;1 ¼ ðc 0; 0Þ and�y

0;2 ¼ ð0; c 0Þ, where c 0

is the rotationally invariant, four-component bound state[10]. The Hilbert space H 0 spanned by these two zero-energy states is invariant under all operators that commuteor anticommute with HBdG [13], with four falling into thelatter category: fI2; ~�g � �0. These are at the center of ourstudy.

On the other hand, the expectation value of any tracelesssingle-particle operator M that anticommutes with theDirac Hamiltonian is easily seen to derive entirely from

the zero-energy states [11]. At T ¼ 0:

h�yM�i ¼ 1

2

�Xi;oc

� Xi;unoc

�c y

0;ið ~xÞMc 0;ið ~xÞ; (4)

where fc 0;ið ~xÞg is a basis in H 0. SinceH 0 is two dimen-

sional, the four anticommuting operators from above inH 0 reduce to the familiar Pauli matrices fI2; ~�g. In fullanalogy to the spin-1=2 problem [15], any two orthogonalstates in H 0 are then the þ1 and �1 eigenstates ofðn � ~�Þ � �0 for some unit vector n, and theþ1 eigenstatesof I2 � �0. Equation (4) then implies that (a) when one

state is occupied and the second state empty, h�yðn � ~� ��0Þ�i ¼ c y

0 c 0 and h�yðI2 � �0Þ�i ¼ 0, and (b) when

both states are occupied or both empty h�yðI2 � �0Þ�i ¼�c y

0 c 0, with h�yðn � ~� � �0Þ�i ¼ 0 for all n.It is straightforward to rewrite these fermion bilinears in

terms of the original electrons. For example

�yðI2 � �0Þ� ¼ uy�ð ~x; �Þu�ð ~x; �Þ � vy�ð ~x; �Þv�ð ~x; �Þ:

(5)

The corresponding average represents therefore the CDW.Similarly,

�yð ~k;!Þð�3 ��0Þ�ð ~k;!Þ ¼�½uy�ð ~k;!Þu�ð ~k;!Þ�uy�ð� ~k;�!Þu�ð� ~k;�!Þ�� ½u! v�; (6)

and may be recognized as the z component of the vector OPfor the HKM topological insulator [8,9]. The ground statewith a finite expectation value of this operator would breakthe rotational invariance, the sublattice exchange symme-try, and, most importantly, the time reversal for each spinprojection separately.Let us take the effect of the magnetic field of the vortex

into account next. Assume the field to be along the zdirection, orthogonal to the graphene plane. First, thesole orbital effect of the localized magnetic field is thechange of the form of the zero-energy states [16]. Thisfollows from the observation that in the Dirac-Namburepresentation the magnetic field enters the Hamiltonianin Eq. (3) by modifying only the H0 term as

H0 ! H0½A� ¼ I2 � i�0�iðpi � Ai�35Þ¼ e��ð ~xÞI2��0H0e

��ð ~xÞI2��0 ; (7)

where the magnetic field is B ¼ �ij@iAj ¼ @2�. (In our

units @ ¼ e ¼ c ¼ 1.) Since the matrix I2 � �0 anticom-mutes with the mass term in Eq. (3), and acts as the unitoperator within H 0, the zero-energy states with and with-out the magnetic field differ only in the factor exp½�ð ~xÞ�.The Zeeman effect on the electron spin turns out to be moreimportant. With the Zeeman term the Hamiltonian be-comes HBdG þHZ, where

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HZ ¼ gBð ~xÞð�3 � I4Þ (8)

and g � 2 for the electrons in graphene. One may considerHZ as a weak perturbation to the spectrum of HBdG [17].The commutation relations with the four operators thatanticommute with HBdG imply that, within H 0, HZ isproportional to �3 � �0, and therefore it splits the þ1and �1 eigenstates of this particular OP. Equation (4)then implies that

h�yð�3 � �0Þ�i /��������exp

��ð ~xÞ �

Z r

0�ðr0Þdr0

���������2

; (9)

whereas the averages of the other three OPs vanish. As thetotal flux of the magnetic field is hc=2e, we have �ðxÞ �ð1=2Þ lnj ~xj at distances beyond the magnetic field’s pene-tration depth, and the OP is exponentially localized withinthe coherence length � ¼ 1=�sðr ¼ 1Þ.

Interestingly, the same conclusion follows even for thesuperconductor with a different symmetry. Consider theBdG Hamiltonian for a vortex in the p-wave state:

HBdG¼H0þj�pðrÞj�3�½i�0ð�3 sin�þ�5 cos�Þ�; (10)

which breaks the spin-rotational symmetry. This super-conducting OP is odd under the exchange of the Diracpoints, and it is energetically preferable for strong next-nearest neighbor attraction between electrons [5]. TheUð1Þ � Clð3Þ algebra of OPs in the core is now different:Uð1Þ ¼ fI2 � �0g and Clð3Þ ¼ f�3 � �0; �1 �i�1�2; �2 � i�1�2g. The last two matrices correspond tothe x and y components of the Neel OP. The orbital effectof the magnetic field in resolving the degeneracy of thezero-energy states is still null. HZ, on the other hand,within H 0 is again proportional to �3 � �0, as dictatedby the commutation relation of theHZ with the above set ofOPs. This implies the same result in Eq. (9) follows for thep-wave vortex as well.Let us now construct a different representation of the

problem in which the manifest rotational invariance of theLagrangian in Eq. (2) is sacrificed [18]. Doing so willenable us to include the nonuniform insulating andsuperconducting states, and make contact with the recentwork of Ghaemi et al. [12]. Consider a different eight-

component Dirac fermion �y ¼ ð�yþ; ðTð�1 � I2Þ�þÞyÞ,

where the Dirac field �þ is now rather standard [2]:

�yþð ~q;!Þ ¼ ðuyþð ~K þ ~q;!Þ; vy

þð ~K þ ~q;!Þ; uyþð� ~K þ ~q;!Þ; vyþð� ~K þ ~q; !ÞÞ: (11)

T is the time-reversal operator, consisting of spin, momen-tum, and frequency reversal, and particle-hole operatorexchange. The Lagrangian can still be written in the formin Eq. (2), but with the Hamiltonian H0 replaced with~H0 ¼ ðI2 � i�0�1Þp1 þ ð�3 � i�0�2Þp2, and with �0 ¼I2 � �3, �1 ¼ �3 � �2, �2 ¼ I2 � �1, �3 ¼ �1 � �2,and �5 ¼ �2 � �2 [2].

Once the number operator in this representation isrecognized as ~N ¼ �3 � I4, the set of 16 matrices thatanticommute with ~H0 may be divided into those cor-responding to the superconducting (f�1; �2g �f�1; i�0�2; i�1�5; i�1�3g) and the insulating OPs(f�3; I2g � f�0; i�0�3; i�0�5; i�1�2g). In revealing the or-ders corresponding to these fermion bilinears it is useful todiscern P ¼ �3 � �35 as the generator of translations, and

Is ¼ I2 � i�1�5 as the ~K $ � ~K inversion operator [2].This immediately implies that the averages h�yM�i rep-resent: (a) uniform s wave, for M ¼ f�1; �2g � i�1�5,(b) nonuniform s wave, for M ¼ f�1; �2g � i�0�2,(c) uniform p wave, for M ¼ f�1; �2g � i�1�3, and (d)nonuniform p-wave superconductor, for M ¼ f�1; �2g ��1. Explicit rewriting of the above bilinears shows that the

nonuniform superconducting states have the 2 ~K periodic-ity of the OP, and thus represent the LOFF phases [19] withthe Kekule texture. Similarly, the insulating states are(a) CDW, for M ¼ �3 � �0, (b) spin-scalar Kekule BDW[20], for M ¼ f�3 � i�0�3; I2 � i�0�5g, (c) z componentof the HKM, forM ¼ I2 � i�1�2. Exchanging�3 and I2 inthe above further yields (a) Neel, (b) spin-vector BDW, and(c) spin-scalar Haldane’s OP, respectively.

Let us reconsider the Hamiltonian with the vortex in thes-wave superconducting OP. In this representation it takesthe form of

~H BdG ¼ ~H0 þ j�sðrÞjð�1 sin�þ �2 cos�Þ � i�1�5:

(12)

The Uð1Þ � Clð3Þ OP algebra consists now of Clð3Þ ¼f�3 � �0; �3 � i�0�3; I2 � i�0�5g and Uð1Þ ¼ fI2 �i�1�2g. The CDW and BDW belong to the Clð3Þ part, inagreement with the observation in Ref. [12]. In this repre-sentation the Zeeman term becomes proportional to theunit matrix I2 � I4. The Zeeman effect of the magneticfield is thus to shift both zero-energy states equally, so thatboth are above, or below, the Fermi level [21]. From Eq. (4)we see that it is the OP belonging to the Uð1Þ factor thatdevelops an expectation value in this case, which is pre-cisely the HKM state again.Within the � representation it is evident that the above

result holds irrespectively of the symmetry of the super-conducting OP. For each one of the four possible choicesthe Uð1Þ part of the Uð1Þ � Clð3Þ algebra of the core stateswill always be the HKM, since the matrix i�1�2 anticom-mutes with all the superconducting states, while at thesame time it commutes with all the insulating states, whichare the exclusive members of the Clð3Þ. Since the Zeemanterm is proportional to the unity operator only the universalUð1Þ part of the algebra develops the expectation value forany symmetry of the OP.There is always the Uð1Þ � Clð3Þ algebra of possible

OPs in the core, irrespectively of the type of order support-

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ing the vortex. For example, in the � representation avortex in the Neel OP in the x-y plane, ð�1 cos�þ�2 sin�Þ � i�1�2, in the core may exhibit the p-wavesuperconductor, the CDW, and the z component of thesame Neel OP (�3 � i�1�2) [11]. The most generalDirac Hamiltonian with a vortex can be written as

Hgen ¼ M1p1 þM2p2 þ o1M3 þ o2M4; (13)

where ðo1; o2Þ ¼ joðrÞjðcos�; sin�Þ is the OP, and fMig arefour arbitrary eight-dimensional Hermitian matrices satis-fying ½Mi;Mj�þ ¼ 2ij. Since fMig form a representation

of the Clifford algebra Clð4Þ, by a unitary transformation itcan be reduced to a block-diagonal form Mi ¼ i �i,where i and �i are four dimensional [22]. Since fig andf�ig then provide two (in general, different) Hermitianfour-dimensional representations of Clð4Þ, they are equiva-lent [22]. Any Hgen may therefore be transformed into the

block-diagonal form of HBdG as given in Eq. (3), with thesame four-dimensional matrix operator in both blocks. Theonly four operators that anticommute with such aHamiltonian are obviously I2 � �0 and ~� � �0, whichform the algebra Uð1Þ � Clð3Þ.

If the superconducting OP in graphene is induced by alayer of a type-II superconductor laid on top, the Zeemanenergy would roughly be �z � H � kelvin, if the magneticfield H is expressed in tesla. The leading effect of the

graphene lattice, for example, comes from the 2 ~KFourier component of the vortex configuration, which in� representations is proportional to the matrix �1 � I4. Itis straightforward to check using our formalism that thisperturbation favors the scalar BDW in the core, in agree-ment with the result of direct lattice calculation [12]. Themagnitude of this splitting should crudely be �dw j�jða=�Þ � ð102a=�Þ2 kelvin, where a � 1 �A is the latticespacing. In the Nb3Gewith Tc � 23 K for example, �dw �1 K. The upper critical field, however, is 40 T, so at themagnetic fields of few tesla the order in the core shouldpredominantly be the HKM. It is also possible to increasethe HKM component of the OP by adding a component tothe magnetic field parallel to the superconducting layer sonot to suppress the condensate. A distinct sign of the HKMOP in the core would then be the increase of the gap in thelocal density of states with the total magnetic field at afinite temperature. This follows from the generalization ofEq. (4) to finite temperatures:

hOPi ¼ c y0 c 0

2

��tanh

��z þ �dw2kBT

�þ s tanh

��z � �dw2kBT

��;

(14)

with s ¼ 1 for the HKM, and s ¼ �1 for a linear combi-nation of the CDW and BDW. Assuming �z and �dw todiffer by a factor of 5 or more one OP is by an order ofmagnitude larger than the other over a wide range oftemperatures. If the dominant order is any density wave,

for �dw � �z, the OP would in contrast decrease with themagnetic field.It may also be useful to note that the competing orders

behave differently under the changes of the signs of vor-ticity and/or magnetic field. For example, if the sign ofboth the vorticity and the magnetic field is reversed theBDW changes into its complementary pattern [12] whereasthe HKM state remains invariant. A probe sensitive to thesign of the gap would thus help distinguish between thedifferent core states.This work was supported by NSERC of Canada. The

author wishes to thank V. Juricic, B. Roy, B. Seradjeh, andP. Nikolic for discussions, and to the Aspen Center forPhysics where a part of this work was carried out.

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[21] Since the zero-energy states are the Majorana fermions[17] such a shift does not change the particle number.

[22] S. Schweber, An Introduction to Relativistic QuantumField Theory (Harper and Row, New York, 1961).

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