Non-Abelian toplogical superconductors from topological semimetals and related systems under superconducting proximity effect

0.18 MB·English

by  Jian-Hua Jiang

epl draft
Non-Abelian states from topological semimetals and related sys-
tems under superconducting proximity effects
Jian-Hua Jiang1,3 and Si Wu2
1 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2
13 Corresponding author; email: [email protected]
0
2

b PACS 03.65.Vf–Topological phases
e
F PACS 71.10.Pm–Anyons: electronic structure
PACS 03.67.Lx–Quantumcomputation
6
Abstract–Non-Abelianstatesarecharacterizedbytheexistenceofzero-modeMajoranafermions

] bound in the quantized vortices, which in the bulk is characterized by an odd Chern number. In
l
e this Letter, we find that in topological semimetals with a single two-band-touching node all the
- gapped pairing states are non-Abelian states, regardless of the pairing details. The property is
r
t furthergeneralizedtosituationswheresuchtwo-band-touchingaregappedand/ordeformed. The
s
. nonzeroBerryphaseontheFermisurfaceplaysanimportantroleinthetopological propertiesof
t
a thepairing states.
m
-
d
n
o
c Introduction. – Since the discovery of the quantum Semiconductornanostructureswithstrongspin-orbitcou-
[
Hall effects [1], efforts devoted to understanding various pling proximity to superconductors are thought to be an-

7topological states of matter and their phase transitions othergoodcandidatetorealizethelongpersuedMajorana
vgreatly enriched the basic concepts of condensed mat- fermions and topological quantum computations [16,18].
4ter physics [2–4]. One of the significant aspect is that Inspired by those studies in this paper we search for non-
7
two topologically distinct states, which can not be adi- Abelian states in topological semimetals and related sys-
2
abatically connected to each other, can have the same tems due to superconducting proximity effect.
4
.symmetry and hence breaks down the Landau-Ginzburg Topological semimetals are systems in which there are
7
paradigm of phase transitions. Moreover, exotic excita- band-touching nodes with nonzero winding number. In
0
1tions obeying non-Abelian statistics have been found in particular,here we focus on the situation with singletwo-
1genuine and model condensed matter systems [5,6]. In band-touching node. The winding number is related to
:one known paradigm, the non-Abelian states are closely theBerryphaseenclosedbyanypathsenclosingthenode,
v
Xirelated to superconducting/superfluid pairing states with Nw = π1 dk·hΨ(k)|i∇k|Ψ(k)i where C is a contour en-
odd Chern number [7], in which there is a topologically closing thCe two-band touching point K in k-space and
rprotected Majorana fermion in each quantized vortex. Ψ(k) is Hthe wavefunction (single-valued and continuous
a
The braiding of Majorana fermions gives rise to the non- function of k) in the band above (or below) the node.
Abelian statistics [8]. The non-Abelian statistics is pro- Such nodes can be viewed as k-space vortices [19]. Ex-
posed to be exploited to realize the topological quantum amples are Dirac cones and quadratic band touching [see
computation [9–11]. Fig. 1], where the winding number is N = 1 and 2
w
± ±
Besides the known non-Abelian states in fractional respectively. When the Fermi level is away from the two-
quantum Hall system [5], spin liquids [6], 3He [12] and band touching node, the Fermi surface encloses the node
Sr RuO [13], recently there are theoretical proposals of and gains a nonzero Berry phase. The Berry phase on
2 4
realizingnon-Abelianstatesinspin-orbitcoupledsemicon- the Fermi surfaceplays anessentialrole as itmodifies the
ductor heterostructures [15–18] and in cold-atom systems effectivepairingbetweenpairedstatesandhencethetopo-
[14]. Especially, the proposal of achieving non-Abelian logical property of the superconductor.
states on the surface of topological insulators proximity Remarkably, we find that, in topological semimetals
to s-wave superconductors invoked a lot of studies [15]. with a single k-space vortex, all gapped pairing states
p-1
Jian-Hua Jiang and Si Wu
are non-Abelian states, regardless of the specific details EEEkkk EEEkkk
of pairing. We further generalize such property to situ-
ations where such two-band-touching are gapped and/or
kkk
deformed. The paper is organizedas follows: we first dis- kkk
cusstopologicalsemimetalswithak-spacevortexcarrying
even winding number. We then study the situation when
b
thewindingnumberisodd,whichcanbethoughtasgener- a
ky hz ky hy
alizationsofsingleDiracconesystems. As weaimatvery
generalpropertiesofthesystem,inallthestudies,wecon-
siderageneralformofpairing,i.e.,weassumeallthes, p, kx hx kx hx
d ... -wave pairing interaction coexist, in contrast that in
the literature mainly s-wave pairing interaction is consid-
ered. We then discuss the situations when i) the k-space
c d
vortexisgappedandii)bothgappedanddeformedsothat
the two bands go in the same direction. The situation ii)
Fig. 1: (Color online) Illustration of the (a) quadratic band
for systems with odd winding number can be viewed as
touching(b) Diraccone systems with Fermi surface above the
generalizationsof spin-orbit coupledtwo-dimensionalsys- touchingpoint. Illustrationofthedirectionofthefield(hx,hz)
tems [16]. or(hx,hy)(alsorepresentsthepseudo-spinorspinpolarization
direction)atFermisurfacefor(c)quadraticbandtouchingand
Topologicalsemimetalswithasinglek-spacevor-
(d) Dirac cone system.
tex carrying even winding number. – In spinless
(or spin-polarized) many-fermion systems in 2D lattices
withmultipleorbitsineachunitcell,theremayexisttwo- Oneexampleofsuchtwo-bandtouchingisthequadratic
band-touching nodes (k-space vortices) in the spectrum. bandtouchinginthecheckerboardlatticesystems[20,21].
In systems with a single k-space vortex as well as time- It is shown [21,22] that in such systems in the vicinity
reversal and space inversion symmetry, the general form of K = (π,π), h (k) = t k2, h (k) = 2t k k , h (k) =
0 0 x x x y z
of the Hamiltonian is t (k2 k2), and h (k) = 0 with t , t , and t (t <
z x − y y 0 x z | 0|
t , t ) being the band parameters. When the Fermi
x z
H0(k)=h0(k)σˆ0+hx(k)σˆx+hz(k)σˆz. (1) l|ev|el|is| away from the band-touching point K, there is
only one band crosses the Fermi level. In this situation,
HerethePaulimatricesactontheWannierorbits(pseudo-
the wavefunction(or the pseudo-spin polarization)on the
spins), σ is the 2 2 identity matrix. Due to time-
0 FermisurfacewindsN times,aswhichenclosestheband-
× w
reversal symmetry, the two-band touching node can only
touching node K.
be at a time-reversal invariant momentum K with K=
− The general form of the Bogoliubov-de Gennes (BdG)
Ktiona,s ktheirsemiseaosnulryeda fsrionmgleKn.ode.h (Inkt)he=abhove(ke)qufoar- Hamiltonian is H = 12 kΨ†(k)HkΨ(k), where Ψ(k) =
ν = 0,x,z and h (k) = 0 due νto−the timeν-reversal ψ (k),ψ (k),ψ†T( k)P,ψ†T( k) , and
y ↑ ↓ ↑ − ↓ −
and space inversion symmetry. The spectrum is εk = (cid:16) (cid:17)
h0(k) ± h2x(k)+h2y(k). For semimetals, |h0(k)±| < Hk = H∆ˆ0(k()−k)µ H−(∆ˆ(kk))+µ . (4)
h2(k)+qh2(k). The eigenstates of H (k) are (cid:20) ∗ − − 0∗ − (cid:21)
x y 0
q Here ∆ˆ(k)=i∆0(k)σˆy+∆z(k)σˆx+i∆y(k)σˆ0 ∆x(k)σˆz
−
u (k) = 1 e−iφk +1 , with ∆0 and ∆ν (ν =x,y,z)being the singlet and triplet
| + i 2 ie−iφk i pairing respectively. The above Hamiltonian represents
(cid:18) − (cid:19)
a general form of pairing, i.e., includes all the s, p, d ...
1 ie iφk i
u (k) = − − , (2) -wave pairing interaction.
| − i 2 e−iφk 1
(cid:18) − − (cid:19) In the weak pairing regime where ∆ µ, one
ν
| | ≪ | |
with φk = Arg[hz(k)+ihx(k)] being the direction of the can ignore pairing between states separated ≥ |µ|. As
pseudo-spin polarization in z-x plane. The winding num- there is only one band crosses the Fermi level and an-
ber of the k-space vortex can also be written as other is well below, one can project into the subspace
with only the band crossing the Fermi level. To the lead-
1 ing order, the projected BdG Hamiltonian is H =
Nw = 2π ICdφk, (3) 12 kΨ†P(k)HkPΨP(k) with ΨP(k)=(ck±,c†−k±)TPBadnGd
dowfihrteihcchetioifisneplodhfyhtsh)iceoanpllsytehuterdaocn-ossnpptiaonrueprnotl:a.NrAizwsathiisotn(h(ekowr)it=nhdehind(girkeo)cf,tttiohhnee P HkP =(cid:20) ε∆k±∗eff−(kµ) −∆εke±ff(+k)µ (cid:21). (5)
ν ν
C −
winding number can only be an even integer. Here the + and indices are applied to the µ > 0 and
−
p-2
Non-Abelian superconducting states from topological semimetals ...
<0 cases respectively. The effective pairing is oddinteger. ThisisreasonableastheZ topologyguaran-
2
teesoddnumberofDiracconesonthesurface[4],whichif
1 merged by adiabatic tuning of the Hamiltonian can form
∆ (k)=eiφk i∆ sgn(µ) (∆ i∆ )e iφk .
eff " y− 2 x± z ± # a single k-space vortex carrying any odd integer winding
X± number. The general Hamiltonian of such k-space vortex
(6)
is
The eigenstates can be obtained by directly diagonalizing
H (k)=h (k)σ +h(k) σˆ, (9)
the above Hamiltonian. The Chern number is given by 0 0 0 ·
[23] where the Pauli matrices now denote true-spin and k
measured from K. It is required that h < h so
0
1 | | | |
NC = dk ez [∇k Ψn i∇k Ψn ], (7) that the k-space vortex is well-defined. Without chang-
2π · ×h | | i ing the winding property of the Dirac cone, one can set
n Z
X
h (k) = 0 (This can also be achieved by adiabatic tun-
z
where Ψn are the wavefunctions of the occupied bands. ing of the Hamiltonian of the system.). The spectrum is
Direct calculation yields [for details, see Appendix]
ε±k =h0(k)± h2x+h2y andtheeigenstatesare|u±(k)i=
N = sgn(µ) 2π dθk∂ θ (k) , (8) √12(e−iψk|↑i±q|↓i) with ψk =Arg[hx(k)+ihy(k)] is the
C 2π θk ∆ direction of spin polarization.
Z0 (cid:12)FS
(cid:12) The winding numberofsuchk-spacevortexcanalsobe
(cid:12)
where θ (k) = Arg[e iφk∆ (k)]. The (cid:12)Chern number written as
∆ − eff 1
is nothing but the winding number of e−iφk∆eff(k) at the Nw = dψk. (10)
2π
Fermisurface(denotedas’FS’above). TheChernnumber IC
is thus fully determined by the effective pairing at Fermi Again it is transparent that N is the winding number
w
surface,which is due to the factthatthe superconducting of the direction of the spin polarization (or the direction
gapisopenedattheFermisurface[24]. FromEqs.(6)and of the spin-orbit field). The winding number can only be
(8), the Berry phase contribution to the Chern number, an odd integer as h( k) = h(k) due to time-reversal
− −
through e iφk, is clearly seen. symmetry in spinful system. In particular, N = 1 for
± w
±
Theerroroftheeigenstatesobtainedfromtheprojected Dirac cone.
Hamiltonian is on the order of (D /µ). However, this Followingtheargumentinprevioussection,intheweak
O | | | |
inducesnoerrorinthecalculatedChernnumberofthesu- pairing regime one can study the topological property of
perconductor. This is because one can adiabatically tune thesystemwiththeprojectedBdGHamiltonian[Eq.(5)].
all the pairing ∆ (ν = 0,x,y,z) to infinitesimally small But here
ν
by a universal scaling, which does not close the gap and
1
hence keeps the Chern number unchanged. In such a way ∆ (k)=eiψk sgn(µ)∆ + (∆ i∆ )e iψk . (11)
eff 0 x y ±
2 ∓
the error can be tuned to infinitesimally small. Therefore
h X± i
the above approach faithfully deduces the Chern number
The Chern number N is the winding number of
in the weak pairing regime. C
e iψk∆ (k)attheFermisurfaceasinEq.(8). Itisnoted
A crucial observation is that e iφk∆ (k) is always an − eff
− eff that e iψk∆ (k) is alwaysan even function of k. There-
odd function of k as ∆ (ν = x,y,z) are odd functions − eff
ν
fore the Chern number of each gapped pairing state is
ande iφk are evenfunctions ofk. Hence the Chern num-
±
even.
ber N is odd for all the gapped pairing states (for nodal
C
As the concerned system is on the boundary of two
states the Chern number is ill-defined). Therefore all the
topologically distinct three-dimensional systems, it does
gappedpairingstatesarenon-Abelianstates,regardlessof
not have well defined edges [4]. One way to circumvent
the specific pairing interactions.
thisproblemistocirculatethesuperconductingstatewith
Topologicalsemimetalswithasinglek-spacevor- a ferromagnetic insulating state with the same H (k) as
0
tex carrying odd winding number. – In spinful 2D wellasamagnetizationalongz-directionMσ [15]. When
z
many-fermionsystemsthere mayexisttwo-bandtouching M > µ [26], the quasi-particles can not propagate into
| | | |
points in the spectrum as k-space vortex. The fermion the ferromagnetic region. On the boundary between the
doubling theorem states that in 2D lattice systems there superconducting region and the ferromagnetic one, there
can only be even number of such vortices [25]. How- aregaplessMajoranaedgestates. Theferromagneticinsu-
ever, in the boundaries of two three-dimensional systems lating state is topologically equivalent to a superconduct-
with different Z topology, there can be odd number of ing massive Dirac fermion system with M > µ. It has
2
| | | |
such k-space vortices. Such as a single Dirac cone on a Chern number of sgn(M)N . For instance, there may
w
the surface of strong topological insulator. The systems be n clockwise moving edge states and n anti-clockwise
c a
concerned here have a single such k-space vortex at K moving edge states. According to bulk-edge correspon-
(with K=K)andtime-reversalsymmetry. Ratherthan dence, n n =N sgn(M)N . Thedifferencen n
c a C w c a
− − − −
N = 1,hereweassumethewindingnumbercanbeany is fixed by topology and is always odd as N is even and
w C
±
p-3
Jian-Hua Jiang and Si Wu
Nw is odd. Therefore the total number of edge states where ηk =Arg[hy+i h2x+h2z] and φk =Arg[hz+ihx].
=n +n is definitely odd. Atk =0, h =h =0whereash =0. Atlargek, h
Nedge c a x z p y 6 | y|≪
The above analysis can also be applied to the Majo- h2x+h2z andηk →π/2. Werestrictthediscussiontothe
ranaboundstatesinthecoreofaquantizedvortex,which situations where h ( k)=h (k). For the k-space vortex
p y − y
can be viewed as edge states live in the small circular carrying odd winding number, the eigenstates are
edge of the vortex with vacuum at the center [7]. As the
boundary condition at the center does not affect the ex- u (k) = cosζ2ke−iψk ,
istence of the zero-energy Majorana bound state, it can | + i sinζk
(cid:18) 2 (cid:19)
be tuned that the vacuum at the center is a supercon- sinζke iψk
ducting massive Dirac fermion with M > µ. Therefore u (k) = 2 − . (14)
| | | | | − i cosζk
there are edge number of Majorana states in core of a (cid:18) − 2 (cid:19)
N
quantized vortex. In reality, there are inevitable mixing
between those states (due to, e.g., disorder) or interac- where ζk =Arg[hz+i h2x+h2y] andψk =Arg[hx+ihy].
tion between them, which lift the degeneracy. However, Here at k = 0, hx =qhy = 0 and hz = 0, whereas at
6
tzhereop-eanrteircgley-Mhoalejosryamnambeotruyngdusatraatnetiefestheeisxiosdtedn.cAecocfoornde- large k, |hz|≪ h2x+h2y and ζk →π/2. We also assume
ingly, all the gapped pairing states aNreedngoen-Abelian states hz(−k)=hz(k)q.
Let’s first consider the case where the single k-space
as is definitely odd. This argument (essentially the
Nedge vortex carrying even winding number is gapped. When
same as that in Ref. [7]), verifies the existence of the Ma-
h (0) < µ and h < h2 +h2 +h2 [see Fig. 2a] one
jorana zero modes in the vortex without explicitly solving | y | | | | 0| x y z
the Schro¨dinger equations as such property is dictated by cansimilarly reduce the pqroblemto that described by the
the bulk topology [7]. Recent theories also present dif- projected BdG Hamiltonian in Eq. (5) for each band in
ferent proofs of the relation of the number of Majorana the weak pairing regime. The effective pairings are
zeromodestothe Chernnumber(difference)[27]whichis
consistent with the discussion here. ∆±eff(k) = eiφk ±i∆ysinηk
Ek Ek −∆nx[cosφk±isinφkcosηk]
∆z[ icosφkcosηk sinφk] , (15)
− ± −
k o
where+and areforthe higherandlowerbandsrespec-
k
−
tively. Direct calculation yields that the Chern number is
a b stillgivenbyEq.(8),whichisbecauseagapopenedbelow
Ek Ek or above the Fermi level can be closed by adiabatic tun-
ing. A nontrivial situation is when the k-space vortex is
both gapped and deformed so that h > h2 +h2+h2.
0 x y z
k
In this situation the two bands go in theqsame direction
k
at large k [see Fig. 2b]. In this case, when µ < h (0),
y
| | | |
c d following the derivation in Appendix, one can show that
the Chern number is
Fig. 2: (Color online) Illustration of the dispersions in (a) 2π dθk
gapped(b)gappedanddeformedquadraticbandtouchingsys- NC = 2π ∂θkθ∆−(k) +sgn[hy(0)]Nw, (16)
temsaswellas(c)gappedand(d)gappedanddeformedDirac Z0 (cid:12)FS
cone systems. (cid:12)
(cid:12)
where θ∆± = Arg[e−iφk∆±eff(k(cid:12))]. It is seen that as Nw is
even, the Chern number is again odd for all the gapped
Systems with asinglegapped/deformedk-space states. Therefore in this situation all the gapped pair-
vortex. – The general Hamiltonian is ing statesare non-Abelianstates regardlessof the specific
pairing details. When the Fermi level is so high such that
H (k)=h (k)σˆ +h (k)σˆ +h (k)σˆ +h (k)σˆ . (12)
0 0 0 x x z z y y
both bands cross it, the total Chern number consists of
The spectrum is εk± =h0(k)± h2x(k)+h2y(k)+h2z(k). the contributions from each band, which is
When k-space vortex carrying eqven winding number, the 2π dθk
eigenstates are NC = 2π ∂θkθ∆±(k) . (17)
|u+(k)i= √12(cid:18) iccoossηη22kkee−−iiφφkk +−siisninη2kη2k (cid:19), Hence the total CXh±ernZ0number become(cid:12)(cid:12)(cid:12)(cid:12)sFSeven (Abelian
1 isinηke iφk icosηk topologicalsuperconductor),when the Fermi levelcrosses
|u−(k)i= √2(cid:18) −sin2η2ke−−iφk−−cosη22k (cid:19). (13) two bands.
p-4
Non-Abelian superconducting states from topological semimetals ...
Now we turn to the situation where the single k-space Conclusion and discussions. – In this paper
vortexcarryingoddwindingnumberisgapped. The effec- we discussed the topological properties of two classes of
tive pairing in the two bands are semimetal systems under superconducting proximity ef-
fect in general situations. Those two classes are semimet-
∆±eff(k) = eiψk cos2(ζ2k)(∆x±i∆y)e∓iψk ∓∆0sin(ζk) ai)lsevweinthii)a osidndglweitnwdoin-bgannudm-tobuecrshinregspneocdtievewlyh.ichRecmararrikes-
+sihn2(ζk)(∆ i∆ )e iψk . (18) ably, we found that in those systems all the gapped pair-
x y ±
2 ∓ ing states are non-Abelian states regardless of the pairing
i
details. This property is further generalized to systems
Again the Chern number is the same as that at h =
z
whensuchtwo-bandtouchingisgappedand/ordeformed.
0 when µ > h (0) and h (k) < h2 +h2+h2 [see
| | | z | | 0 | x y z Those findings supply useful information and a route for
Fig. 2c] as the gap is opened below qor above the Fermi the search for non-Abelian states.
level. When the k-space vortex is gapped and deformed, Although some of special systems, such as single Dirac
sothath (k)> h2 +h2+h2 wherethetwobandsgoin conesystem[15]andtwo-dimensionalelectronsystemwith
0 x y z
Rashba spin-orbit coupling [16], have been discussed in
thesamedirectioqnatlargek [seeFig.2d],at µ < hz(0),
| | | | the literature (results here are consistent with those), the
one can show that
results obtained in this paper reveals a more general pic-
2π dθk ture of the topological property in a general topological
NC = 2π ∂θkθ∆−(k) +sgn[hz(0)]Nw, (19) semimetalsandrelatedsystemswithgeneralpairinginter-
Z0 (cid:12)FS
(cid:12) actions. The theory presented above can be directly gen-
(cid:12)
withθ∆± =Arg[e−iψk∆±eff(k)].(cid:12)It is seenthate−iψk∆eff(k) eralizedtosystemswithmultipletwo-bandtouchingnodes
is always an even function of k as e iψk, ∆ and ∆ are whenthepairingbetweendifferentnodescanbeneglected.
± x y
odd functions while ζk and ∆0 are even functions of k. It is noted from Eq. (8) that the Chern number changes
Thus the Chern number is odd for all the gapped pairing sign when the chemical potential goes through the band-
states. When the Fermilevelis higherso thatboth bands touching node, which indicates that there is a topological
crossit,thetotalChernnumberisgivenbyEq.(17),which phase transitioninthe strongpairing regime. Fromspon-
is always even. taneous symmetry breaking under attractive interaction,
the gapped pairing states usually gain more condensation
Relevant physical systems. – Besidesthesystems
energy compared to the nodal ones and thus become en-
already discussed in the literature, such as systems with
ergetically favored [29]. Hence the above special property
single Dirac cone and semiconductor quantum wells with
also implies that the non-Abelian states may be energeti-
Rashba spin-orbit coupling, there are many unexplored
cally favored to be the ground states in the superconduct-
candidate systems which fit into the above discussions.
ing/superfluid phase [22].
Below we list some examples of physical systems that re-
alize the models discussed above:
∗∗∗
Semiconductor nanostructures with Zeeman (ex-
• Work at the Weizmann Institute was supported by
change) splitting and arbitrary spin-orbit coupling.
the German Federal Ministry of Education and Research
Given that the winding number at Fermi surface
(BMBF) within the framework of the German-Israeli
is nonzero, two-dimensional systems arbitrary spin-
projectcooperation(DIP)andbytheIsraelScienceFoun-
orbitcoupling andZeemansplitting canachievenon-
dation(ISF).JHJthankZoharRingelforilluminatingdis-
Abelian states when there is only one band crossing
cussions and comments.
the Fermi level. This is a direct generalization of the
studies in the literature. Appendix: DetailsofthederivationoftheChern
number. – Consider, e.g., systems with a single k-
Thin films of topological Weyl semimetals. In space vortex carrying even winding number with µ > 0.
•
Ref. [28], it is found that in the thin film of topo- TherearetwooccupiedbandsafterdiagonalizingtheBdG
logical Weyl semimetal HgCr2Se4 the Chern number Hamiltonian: onecomesfromthebandcrossingtheFermi
depends on the thickness of the film. At the critical level, the other from the band below the chemical poten-
thickness, there will be a single two-band touching tial. Intheweakpairingregimewhere ∆ µ,onecan
ν
with even winding number Nw = 2. Around the ignore pairing between states separate|d |≪µ|. |By doing
critical thickness, there are gapped±quadratic band so, one can obtain the approximate wave≥fu|nc|tions of the
touching which can also give rise to triplet pairing former and latter occupied bands, which are:
and non-Abelian states.
sinξk cosφkeiθ∆
2 2
Optical lattices with a single quadratic band touch- sinξk sinφkeiθ∆
• ing. Such as, checkerboard lattices and kagome lat- Ψo =eiφk/2 co2sξk c2osφk ,
tices [20,22].  −cosξ2k sinφ2k 
 − 2 2 
 
p-5
Jian-Hua Jiang and Si Wu
sinφk [10] M. Freedman et al., Bull. Am.Math. Soc. 40, 31 (2002);
2
Ψv(k)=e−iφk/2 −co0sφ2k , (20) [11] SA..TYe.wKairtia,eSv., ADnans.SPahrmysa.,(NC..YN.)a3y0ak3,, 2C.(2Z0h0a3n).g, and P.
 0  Zoller, Phys. Rev.Lett. 98, 010506 (2007).
  [12] G.E.Volovik,TheUniverseinaHeliumDroplet,(Claren-
 
respectively. Here ξk = Arg[εk+ µ + i∆eff(k)] and don, Oxford, 2003); arXiv:1111.4627
θ = Arg[e iφk∆ (k)]. An impor−tant pro|perty i|s that [13] G.M.Luke,etal.,Nature394,558(1998);A.P.Macken-
∆ − eff zie and Y.Maeno, Rev.Mod. Phys.75, 657 (2003).
the Chern number N does not change without closing
C [14] C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma,
the gap. One can then simplify the calculation of N
C Phys. Rev. Lett. 101, 160401 (2008); M. Sato, Y. Taka-
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