Coherence Effects of Caroli-de Gennes-Matricon Modes in Nodal Topological Superconductors

0.25 MB·

by  Yasumasa Tsutsumi

JournalofthePhysicalSocietyofJapan LETTERS
Coherence Effects of Caroli-de Gennes-Matricon Modes in Nodal
Topological Superconductors
Yasumasa Tsutsumi∗ and Yusuke Kato
6
1 Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan
0
2
Coherence effects by the impurity scattering of Caroli–de Gennes–Matricon (CdGM) modes in a vortex

r fornodaltopologicalsuperconductorshavebeenstudied.Thecoherenceeffectsreflectatopologicalnumber
p
defined on a particular momentum space avoiding the superconducting gap nodes. First, we analytically
A
derivedtheeigenvalueandeigenfunction oftheCdGMmodes,includingthezero-energy modes,inanodal

4 topologicalsuperconductingstatewithoutimpurities,wherewefocusedonapossiblesuperconductingstate
1 ofUPt3 asanexample.Then,westudiedimpurityeffectsontheCdGMmodesbyintroducingtheimpurity
self-energy, which are dominated by the coherence factor depending on the eigenfunction of the CdGM

] modes. For the zero-energy CdGM modes, the coherence factor vanishes in a certain momentum range,
n
which is guaranteed bytopological invariance characterized by theone-dimensional winding number.
o
c
-
r
p
u
Topologicalmaterials,suchasquantumHallsystems,1 the E planar state described by the d-vector d(k)
s 1u ∝
. topological insulators,2,3 and topological superconduc- (xk +yk )(5k2 k2)asanexampleofthenodaltopolog-
t y x z−
a tors/superfluids,3–5 havebeenattractingwide attention, ical superconducting state. The gap function has point
m
andinvestigationoftheirtopologicalphaseshasbeende- nodes at the north and south poles on the Fermi surface
- veloping beyond the field of condensed matter physics.6 and two horizontal line nodes at k = k /√5, where
d z ± F
The topological phase is characterized by topological k is the Fermi wave number.
n F
o numbers defined in gapped states, where topological in- Ouraimistoclarifythesignificanceofthetopological
c variance guarantees the presence of gapless edge modes. classification of nodal superconductors by studying im-
[
Alsofornodalsuperconductors/superfluids,thetopolog- purity effects on the CdGM modes. In a vortex along

2 ical numbers can be defined on a particular momentum the z-direction, the E1u planar state has many zero-
v spaceavoidinggapnodes.7,8 However,whenthereismo- energyCdGMmodeswithmomentum k <k between
z F
5 mentum transfer due to the presence of impurities, the the point nodes. In this paper, we ha|ve|demonstrated
1
validity of the procedure of taking a particular momen- that the zero-energy modes show characteristic impu-
8
tum space is not clear. rity effects owing to the coherence factor vanishing in
4
0 Zero-energy vortex modes in a topological phase are a certain momentum range. The coherence factor of the
. also characterized by topological numbers defined on zero-energy modes reflects a topological number, which
1
0 a base space (k,φ), which consists of the momentum doesnotchangeunlesscrossingthehorizontallinenodes.
6 and a parameter φ on a circle surrounding the vortex.9 Thus, the topological classification of nodal supercon-
1 In superconductors/superfluids, the zero-energy vortex ductors is effective even when momentum transfers are
:
v modes are derived from the Caroli–de Gennes–Matricon caused by impurity scattering.
i (CdGM)modes.10 Inparticular,onsuperconductors,the First,wederiveCdGMmodeswithoutimpuritiesfrom
X
robustnessof the CdGM modes,namely their relaxation the Bogoliubov–de Gennes (BdG) equation. The BdG
r
time,canbeobservedbyfluxflowconductivitymeasure- equation for an inhomogeneous order parameter is de-
a
ments.11Therefore,impurityeffectsontheCdGMmodes scribed by20
arepromisingforprovidingacriterionfortheavailability
ˆǫ(r ,r ) ∆ˆ(r ,r )
ofthetopologicalclassificationofnodalsuperconductors. dr 1 2 1 2 ~u (r )=E ~u (r ).
2 ∆ˆ†(r ,r ) ǫˆT(r ,r ) ν 2 ν ν 1
The heavy fermion superconductor UPt312,13 is a Z (cid:18) 2 1 − 2 1 (cid:19)
(1)
highly probable nodal topological superconductor. As a
result of numerous experimental and theoretical studies The normal-state Hamiltonian omitting the vector po-
over three decades, the possible gap functions in UPt3 tential, which can be neglected when considering the
have been narrowed to the E1u planar state,14,15 E1u CdGMmodesinlowfieldswithalargeGinzburg–Landau
chiral state,16 and E2u chiral state.17 All of the pair- parameter κ 1,10 is given by
ing states in the low-temperature and low-field phase, ≫
i.e., the B-phase, show topologically protected gapless ǫˆ(r ,r )
1 2
edge modes.18,19 In this paper, we mainly focus on
∗[email protected]
1
J.Phys. Soc. Jpn. LETTERS
1.5
~2 1 1
=δ(r r ) ∂2 ∂ ∂2 ∂2 k2 σˆ ,
1− 2 2m − ρ − ρ ρ− ρ2 φ− z − F 0 1
(cid:20) (cid:21)
(2)
0.5
wheremistheparticlemass,σˆ isthe 2 2unitmatrix,
0
and (∂ ,∂ ,∂ ) are differential operator×s in cylindrical 0
ρ φ z
coordinates. The pair potential is
-0.5
∆ˆ(r ,r )= dk ∆ˆ(r,k)eik·r′, (3)
1 2 (2π)3 -1
Z
with r = (r1 +r2)/2 and r′ = r1 r2, and the wave -1.5
function is − -1 -0.5 0 0.5 1
u↑(r)
ν
u↓(r)
~uν(r)=v↑ν(r). (4) Fig. 1. (Color online) Eigenvalue of the CdGM modes Eν =
vν↓(r) −lωq for |l| ≤ 5 within the superconducting gap ∆0|(5cos2α−
 ν  1)sinα|,whereq/kF=cosα.
 
For the E planar state,14,15 the gap function is de-
1u
scribed by
∆ˆ(r,k) id(r,k) σˆσˆ
≡ · y Eν calculated for ∆(ρ) = ∆0tanh(ρ/ξ) with kFξ = 5
=i∆(r)(xk +yk )(5k2 k2)/k3 σˆσˆ , (5) in Fig. 1, where the coherence length is defined by
y x z − F F· y ξ ~v /∆ .
F 0
with the Pauli matrix σˆ. When we consider the singly ≡
Next, we consider impurity effects on the CdGM
quantizedvortexstatedescribedby∆(r)=∆(ρ)eiφ,the
modes. The Dyson equation obeyed by the Mat-
spin-degenerate eigenvalue of the CdGM modes is given
subara Green’s function with impurity self-energy
by21 Σ (r ,r ,ω ) is described by
imp 1 2 n
E = lω , (6)
ν q
− b G(r,r′,ω )=G(0)(r,r′,ω )
n n
where
ω 5cos2α 1 0∞ |∆kF(ρρ′′)|e−2χq(ρ′)dρ′, (7) + bdr1 dr2G(0b)(r,r1,ωn)Σimp(r1,r2,ωn)G(r2,r′,ωn).
q ≡| − |R 0∞e−2χq(ρ′)dρ′ Z Z b b b (11)
with R The impurity self-energy is given by
χq(ρ)= |5cos~2vαF−1|Z0ρ|∆(ρ′)|dρ′, (8) Σimp(r1,r2,ωn)=πΓNnFF(r1−r2)G(r1,r2,ωn)
by using the Fermi velocity v . The quantum number
F b γF(r r )G(rb,r ,ω ), (12)
1 2 1 2 n
ν = (l,q) consists of the angular momentum l and the ≡ −
ewiagvenefunnucmtiboenr oaflotnhgetChdeGvMortmexodliensefqor≡thkeFucpo-sα(d.oTwhne- wthheerneorNmFalisstathtee adtenthsietyFeorfmsitaetneesbrg(yD,OΓSn)ispetrhespiminpuin-
)spin state is given by21 rity scattering rate in the normal state, and F(r1 r2)
−
describes the spatial dependence of the squared impu-
~uν↑(↓)(r)=Ul(φ)~uν↑(↓)(ρ)eiqz/√2π, (9) rity potential. Here, F(r1 r2) = δ(r1 r2) gives the
− −
impurity self-energy by the self-consistent Born approx-
where U (φ) diag(ei(l+1)φ,eilφ,ei(l−1)φ,eilφ)/√2π and
l ≡ b imation.22–24 Instead, we introduce the spatial depen-
J (k ρ) dence of the impurity potential in the z-direction as
b l+1 q
0 F(r r )=δ(ρ ρ )f(z z ),whereρ indicatesthe
~u↑(ρ) ↑ e−χq(ρ), 1− 2 1− 2 1− 2
ν ≡Nν sqJl−1(kqρ) two-dimensionalcoordinatesperpendiculartothevortex
 0  line. The Green’s function without impurities is derived
  (10) from the BdG wave function in Eq. (4) as
0
J (k ρ) τ ~u (r)~u†(r′)
~u↓ν(ρ)≡Nν↓ l 0q e−χq(ρ), G(0)(r,r′,ωn)= zEν iων , (13)
ν n
ν −
 sqJl(kqρ) X b
−  where τ isbthe Pauli matrix in the Nambu space. By
 
with the Bessel function J , k k sinα, s =
l q F q deriving G(r,r,ω ) from Eq. (11), we can obtain the
≡ n
sgn(5cos2α 1), and the normalizationfactor ↑(↓) for
ν b
− N
the up- (down-)spin state. Here, we show the eigenvalue b
2
J.Phys. Soc. Jpn. LETTERS
30
DOS as
1
N(ω)= drIm Trτ G(r,r,ω )
Z (cid:20)π z n (cid:12)iωn→ω+i0+(cid:21) 20
1 b b 1 (cid:12)(cid:12)
= Im
− π ω E +σ (ω)+i0+
ν (cid:20) − ν ν (cid:21) 10
X
N (ω). (14)
ν
≡
ν
X 0
Here, we consider the DOS of the CdGM modes 40
N (ω)=N↑ (ω)+N↓ (ω) by the approximationof ne-
l,q l,q l,q
glecting the contributions from the continuum state to
30
theimpurityself-energyoftheCdGMmodes.Withinthe
approximation,the relaxationtime of the CdGM modes
depends on the modified self-energy σ↑(↓)(ω) for the up- 20
ν
(down-)spin mode:23
10
γ
σ↑(↓)(ω)= dq′f˜(q q′)
ν −(2π)2 −
l′ Z 0
X -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
M↑(↓)
ν,ν′
, (15)
× ω−Eν′ +σν↑(′↓)(ω)+i0+
wheref˜(q)istheFouriertransformoff(z).Themodified 0Fidge.riv2e.du(sCinoglotrhoenmlinoed)ifiDeOdSseolff-CendeGrgMyimnoEdqe.s(N15l,)q=w0itfhorf˜(−q5−≤q′)l=≤
self-energyreflectsthewavefunctionoftheCdGMmodes 1(a)andf˜(q−q′)=θ(kF/10−|q−q′|)(b).
in Eq. (10) through the coherence factor
2
M↑(↓) = ρdρ ~u↑(↓)†(ρ)τ ~u↑(↓)(ρ)
ν,ν′ ν z ν′
Z (cid:12) (cid:12) the eigenvalue, the CdGM modes are dominantly trans-
(cid:12) 2 (cid:12)
=Z ρdρ(cid:12)(cid:12)mν↑,(ν↓′)(ρ)(cid:12)b e−2χq(ρ)(cid:12)−2χq′(ρ), (16) fFeorrresdqbsqe′tw=een1,stsaintecsewmit↑(h0,qt)h,(e0,sqa′)me=an0g,utlhaermzeormo-eennteurmgy.
where (cid:12) (cid:12) CdGM modes in the up-spin state are hardly scattered,
(cid:12) (cid:12)
m↑ (ρ)= ↑ ↑ which is similar to the impurity effects on the CdGM
ν,ν′ NνNν′ modes in a parallel vortex for the chiral p-wave state.23
[Jl+1(kqρ)Jl′+1(kq′ρ) sqsq′Jl−1(kqρ)Jl′−1(kq′ρ)], The different contributions from the up- and down-spin
× −
statestothecoherencefactoraresimilartothecaseofan
m↓ (ρ)= ↓ ↓
ν,ν′ NνNν′ s-wave topological superconductor by Rashba-type spin
orbit coupling.24
[J−l(kqρ)J−l′(kq′ρ) sqsq′J−l(kqρ)J−l′(kq′ρ)].
× − Thus, impurity scattering with the transitionbetween
(17)
the zero-energy CdGM modes with sqsq′ =1 hardly oc-
Altogether, the coherence factor dominates the relax- curs; however, the scattering between the CdGM modes
ation time of the CdGM modes. If Mν,ν′ = 0, the ν′th with sqsq′ = 1 generally occurs, regardless of whether
−
CdGM mode does not contribute to the modified self- the excitation energy is zero or not, because Mν,ν′ =0.
energy on the νth CdGM mode. When νth and ν′th InFig.2,weshowtheDOSofeachCdGMmodeN6
l,q=0
CdGMmodeshavemomentuminsideoroutsidethehor- for 5 l 0, where the impurity scattering rate Γ of
izontal line nodes, namely sqsq′ = 1, m↓ν,ν′(ρ) = 0 irre- the−CdG≤M≤mode is reflected by the width of the DOS
spective of l and l′. Thus, impurity scattering with the spectrum. In the calculation of the modified self-energy
transitionbetween these CdGM modes inthe down-spin in Eq. (15), we sum the CdGM modes with l′ 10
| | ≤
state does not occur, which is similar to the impurity and set the impurity scattering rate in the normal state
effects on the CdGM modes in an antiparallel vortex to Γ = 0.05π∆ . We again take ∆(ρ) = ∆ tanh(ρ/ξ)
n 0 0
for the chiral p-wave state.23,25 Here, “antiparallel” in- with k ξ = 5, which fixes χ (ρ) in the coherence factor
F q
dicates that the vorticity is antiparallel to the orbital and the eigenvalue of the CdGM modes Eν′ as shown
chirality. in Fig. 1. The difference between Figs. 2(a) and 2(b) is
On the other hand, m↑ν,ν′(ρ) 6= 0 even for sqsq′ = 1 the spatial dependence of the impurity potential in the
except when l′ = l. If the impurity scattering rate of z-direction.
−
the CdGM modes Γ is smaller than the level spacing of In Fig. 2(a), we put f˜(q q′) = 1 in Eq. (15), which
−
3
J.Phys. Soc. Jpn. LETTERS
implies f(z) = δ(z). Since the modified self-energy is of the winding number is related to the eigenvalue of Γ
given by the integral over all q′ < k , there are many for the zero-energy mode, namely its chirality. Impurity
F
| |
pairsofzero-energyCdGMmodeswithsqsq′ = 1across effectsoftheshort-rangepotentialinthekz-spacedonobt
−
thelinenodes;thus,themanyrelaxationprocessesgivea affect the zero-energy CdGM modes around q = 0 with
largeΓforthezero-energymodes.ForN withl <0, the same chirality. This is because perturbations couple
l,q=0
the lower-energymode has the broader spectrum, which the zero-energy modes with opposite chirality.27
is similar to the CdGM modes in a vortex for the s- Since we have already demonstrated that the topo-
wave state and in a parallel vortex for the chiral p-wave logical argument of nodal superconductors is effective
state.23,25 even when momentum transfer is caused by impurity
Next, we consider the step function f˜(q q′) = scattering, we can discuss the robustness of the CdGM
−
θ(k q q′ ), which is the Fourier transform of f(z)= modes against impurities for the other possible super-
−| − |
sin(kz)/(πz).InthecalculationoftheDOSoftheCdGM conducting states of UPt without complicated calcula-
3
modes inFig.2(b),we takek =k /10.Inthis situation, tions. The E and E chiral states have the gap func-
F 1u 2u
the down-spin CdGM modes have the DOS Nl↓,q=0(ω)= tions ∆ˆ(k,φ) = ∆0eiφ(kx + iky)(5kz2 − kF2)/kF3σˆx and
δ(ω−El,0)owingtothecoherencefactorM(↓l,0),(l′,q′) =0 t∆ˆh(ekg,aφp)f=un∆ct0ioeniφs(kgxiv+e tihkey)w2iknzd/iknF3gσˆnxu,mrebseprecwtiv=elyw. B=oth0
within q′ < k /√5. In Fig. 2(b), we show the DOS 0 π
N↑ f|or|the uFp-spin CdGM modes, in which only the for any kz30 owing to the difference in the spin state
l,q=0 from the E planar state. The trivial topological num-
1u
zero-energy mode has a sharp DOS, which is due to the
ber reflects the absence of zero-energy CdGM modes.
fact that M↑ = 0. Except for the zero-energy
(0,0),(0,q′) For the E1u chiral state, the lowest CdGM modes in-
CdGM mode, the lower-energy mode has the broader
deed have a gap owing to the magnetic field in the
spectrum with an asymmetric tail toward the low en-
z-direction. Since the magnetic field deforms the wave
ergy,wherethe asymmetricspectrumreflectstheconvex
function of the CdGM modes with the energy shift,
dispersion of the CdGM modes for l<0 around q =0.
M =0.Therefore,thelowestCdGMmodeshave
(0,q),(0,q)
The robust zero-energy modes demonstrated in 6
abroadenedDOSevenbycolumnardefects withoutmo-
Fig. 2(b) can be understood from the one-dimensional
mentum transfer on k .
z
winding number, which is a topological number defined
In summary, we have studied impurity effects on
on a closed path with particular values of momentum.
CdGMmodesinavortexfornodaltopologicalsupercon-
For the calculation of the winding number, we consider
ductors by focusing on the E planar state, which is a
1u
the semiclassical BdG Hamiltonian parametrized by the
possiblesuperconductingstateinUPt .Wehavedemon-
3
angle φ on a circle surrounding the vortex as
strated that the coherence factor of the zero-energy
ǫˆ(k) ∆ˆ(k,φ) CdGM modes vanishes for quasiparticle pairs with mo-
(k,φ)= . (18)
H ∆ˆ†(k,φ) ǫˆT( k) mentaonthesamesideofahorizontallinenode.Theco-
(cid:18) − − (cid:19) herencefactorreflectstheone-dimensionalwindingnum-
Here, ǫˆ(k)bis the Hamiltonian in the normal state of berdefinedonaparticularmomentumspace,whichdoes
UPt3, which has the D6h hexagonal symmetry. For the notchangeunlesscrossingthehorizontallinenode.Thus,
E1u planar state with a vortex, whose pair potential is theclassificationofnodaltopologicalsuperconductorsby
described by ∆ˆ(k,φ) = ∆0eiφ(−kyσˆz + ikxσˆ0)(5kz2 − thetopologicalnumberonaparticularmomentumspace
kF2)/kF3,thesemiclassicalBdGHamiltonianhasmagnetic is effective even when there is momentum transfer. The
reflection symmetry on the xz-plane.18 From the combi- characteristic coherence effects of nodal topological su-
nationofthemagneticreflectionsymmetryandparticle- perconductorsshowninFig.2canbeconfirmedbycom-
hole symmetry, we obtain chiral symmetry on ky = 0 paring flux flow conductivities, which depend on the re-
and φ = 0 or π as Γ, = 0, where Γ τ and laxationtimeτ =~/ΓoftheCdGMmodes,11 beforeand
φ=0,π x
{ H } ≡
(k ,k = 0,k ,φ = 0,π). The chiral sym- after spreading artificial columnar defects.31,32 We may
φ=0,π x y z
H ≡ H
metry enables us to dbefibne the one-dimensionbal wbinding directly observe the q-dependent coherence factor of the
nbumber as26b,27 zero-energyCdGMmodesbyinelasticneutronscattering
1 experiments or quasiparticle interference imaging.
wφ=0,π(kz)=−4πi dkxTr ΓHφ−=10,π∂kxHφ=0,π , Acknowledgments We thank T. Hanaguri, S. Hoshino,
ZBZ h (i19) Y.Masaki,andS.Kobayashiforhelpfuldiscussion.Y.T.acknowl-
bb b edges financial supportfromtheJapanSocietyforthePromotion
of Science (JSPS). This work was supported by JSPS KAKENHI
wherethelineintegralshouldbeperformedovertheBril-
GrantNumbers15K17715and15J05698.
louin zone. Since the winding number depends on the
sign of the gap function,27 we evaluate w = w = 2
0 π
− −
for k <k /√5,w = w =2fork /√5< k <k ,
z F 0 π F z F
| | − | |
and w0 = wπ = 0 for kz > kF.18 The difference in the 1) R. E. Prange and S. M. Girvin, The Quantum Hall Effect
| |
winding number (w w )/2 = N guarantees the ex- (Springer-Verlag,Berlin,1987).
0 π
istence of at least N−zero-energy modes.28,29 The sign 2) M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010).
| |
4
J.Phys. Soc. Jpn. LETTERS
3) X.-L.QiandS.-C.Zhang,Rev.Mod.Phys.83,1057(2011).
4) T. Mizushima, Y. Tsutsumi, M. Sato, and K. Machida, J.
Phys.:Condens.Matter 27,113203(2015).
5) T. Mizushima, Y. Tsutsumi, T. Kawakami, M. Sato,
M. Ichioka, and K. Machida, J. Phys. Soc. Jpn. 85, 022001
(2016).
6) G.E.Volovik,TheUniverseinaHeliumDroplet(Clarendon,
Oxford,2003).
7) B.B´eri,Phys.Rev.B81,134515(2010).
8) M.Sato,Phys.Rev.B81,220504 (2010).
9) J.C.Y.TeoandC.L.Kane,Phys.Rev.B82,115120(2010).
10) C. Caroli,P. de Gennes, and J. Matricon, Phys.Lett. 9,307
(1964).
11) N. B. Kopnin, Theory of Nonequilibrium Superconductivity
(Clarendon,Oxford,2001).
12) G. R. Stewart, Z. Fisk, J. O. Willis, and J. L. Smith, Phys.
Rev.Lett.52,679(1984).
13) R.JoyntandL.Taillefer,Rev.Mod.Phys.74,235(2002).
14) Y.Machida,A.Itoh,Y.So,K.Izawa,Y.Haga,E.Yamamoto,
N. Kimura, Y. O¯nuki, Y. Tsutsumi, and K. Machida, Phys.
Rev.Lett.108,157002 (2012).
15) Y. Tsutsumi, K. Machida, T. Ohmi, and M. Ozaki, J. Phys.
Soc.Jpn.81,074717 (2012).
16) K.Izawa,Y.Machida,A.Itoh,Y.So,K.Ota,Y.Haga,E.Ya-
mamoto,N.Kimura,Y.Onuki,Y.Tsutsumi,andK.Machida,
J.Phys.Soc.Jpn.83,061013(2014).
17) C.H.ChoiandJ.A.Sauls,Phys.Rev.Lett.66,484(1991).
18) Y. Tsutsumi, M. Ishikawa, T. Kawakami, T. Mizushima,
M.Sato, M.Ichioka, andK.Machida,J.Phys.Soc.Jpn.82,
113707(2013).
19) P.GoswamiandA.H.Nevidomskyy,Phys.Rev.B92,214504
(2015).
20) T. Kawakami, T. Mizushima,andK. Machida, J.Phys. Soc.
Jpn.80,044603 (2011).
21) The derivation of the eigenvalue and eigenfunction of the
CdGMmodesisshownintheSupplemental Material.
22) N.B.KopninandV.E.Kravtsov,JETPLett.23,578(1976).
23) Y.MasakiandY.Kato,J.Phys.Soc.Jpn.85,014705(2016).
24) Y.MasakiandY.Kato,J.Phys.Soc.Jpn.84,094701(2015).
25) Y.KatoandN.Hayashi,J.Phys.Soc.Jpn.71,1721(2002).
26) M.SatoandS.Fujimoto,Phys.Rev.B79,094504(2009).
27) M.Sato, Y.Tanaka, K.Yada, and T.Yokoyama, Phys.Rev.
B83,224511(2011).
28) K.ShiozakiandM.Sato, Phys.Rev.B90,165114(2014).
29) Y. Tsutsumi, T. Kawakami, K. Shiozaki, M. Sato, and
K.Machida,Phys.Rev.B91,144504(2015).
30) S. Kobayashi, Y. Tanaka, and M. Sato, Phys. Rev. B 92,
214514(2015).
31) L. Civale, A. D. Marwick, T. K. Worthington, M. A. Kirk,
J.R.Thompson,L.Krusin-Elbaum,Y.Sun,J.R.Clem,and
F.Holtzberg,Phys.Rev.Lett. 67,648(1991).
32) M. Toulemonde, S. Bouffard, and F. Studer, Nucl. Instrum.
Methods Phys.Res.,Sect.B91,108(1994).
5
J. Phys. Soc. Jpn. LETTERS
Supplementary Material
Caroli-de Gennes-Matricon modes for the E planar state
1u
In the Supplemental Material, we derive the Caroli-de Gennes-Matricon (CdGM) modes1 in a vortex for the E
1u
planarstate,2,3 whichis a strongcandidate for the superconducting state of UPt .We startoutthe derivationfrom
3
the Bogoliubov-de Gennes (BdG) equation for the inhomogeneous order parameter:
ˆǫ(r ,r ) ∆ˆ(r ,r )
dr 1 2 1 2 ~u (r )=E ~u (r ). (S.1)
2 ∆ˆ†(r ,r ) ǫˆT(r ,r ) ν 2 ν ν 1
Z (cid:18) 2 1 − 2 1 (cid:19)
The normal state Hamiltonian omitting the vector potential is given by
~2 1 1
ˆǫ(r ,r )=δ(r r ) ∂2 ∂ ∂2 ∂2 k2 σˆ , (S.2)
1 2 1− 2 2m − ρ − ρ ρ− ρ2 φ− z − F 0
(cid:18) (cid:19)
where m is the particle mass, k is the Fermi wave number, σˆ is the unit matrix in the spin space, and (∂ ,∂ ,∂ )
F 0 ρ φ z
are differential operators in cylindrical coordinates. The pair potential is
∆ˆ(r ,r )= dk ∆ˆ(r,k)eik·r′, (S.3)
1 2 (2π)3
Z
with r =(r +r )/2 and r′ =r r , and the wave function is
1 2 1 2
−
u↑(r)
ν
u↓(r)
~uν(r)=v↑ν(r). (S.4)
ν
v↓(r)
 ν 
 
The gap function for the E planar state is described by
1u
∆(r)
∆ˆ(r,k)=iσˆσˆ (xk +yk )(5k2 k2), (S.5)
y · k3 y x z − F
F
with the Pauli matrix σˆ. By substituting Eq. (S.5) into Eq. (S.3) and following the procedure in Ref.,4 the pair
potential is rewritten as
i
∆ˆ(r ,r )= ∆(r)ˆ(r )δ(r r ), (S.6)
1 2 2 1 2
−k D −
F
where
5∂2+k2
ˆ(r) iσˆσˆ (x∂ +y∂ ) z F. (S.7)
D ≡ y · y x k2
F
When we consider a vortex along the z-direction, the order parameter ∆(r) is uniform to the z-direction. By the
integral in Eq. (S.1), the BdG equation is reduced to
ǫˆ(r) i ∆(r)ˆ(r)+ 1 ˆ(r)∆(r)
kF D 2D ~u (r)=E ~u (r), (S.8)
 i ∆∗(r)ˆ†(r)+ 1 ˆ†(r)∆∗(r) h ǫˆT(r) i ν ν ν
kF D 2D −
 h i 
where
~2 1 1
ǫˆ(r) ǫ(r)σˆ ∂2 ∂ ∂2 ∂2 k2 σˆ . (S.9)
≡ 0 ≡ 2m − ρ − ρ ρ− ρ2 φ− z − F 0
(cid:18) (cid:19)
Here, the BdG equation can be separated into the up-spin state and the down-spin state, where u↑ (u↑,v↑)T
under ∆(r) is equivalent to u↓ ∗ (u↓,v↓)† under ∆∗(r). ν ≡ ν ν
When we consider the axisymν me≡tricνvorνtex state described by ∆(r) = ∆(ρ)exp(iκφ) with a vorticity κ Z, we
(cid:0) (cid:1) ∈
can derive the BdG wave function as
u↑ν(ρ)eiκ+21φ
~u (r)= 1 u↓ν(ρ)eiκ−21φ eilφeiqz, (S.10)
ν 2π vν↑(ρ)e−iκ+21φ
vν↓(ρ)e−iκ−21φ
 
with ν = (n,l,q) consisting of n Z related to the phase of the wave function, integer l for odd κ or half-integer l
∈
6
J. Phys. Soc. Jpn. LETTERS
for even κ, and q =k cosα. The BdG equation for the up-spin state is described by
F
ǫ(r) kiF ∆(r)D(r)+ 12D(r)∆(r) u↑ν(ρ)eiκ+21φ eilφeiqz
(cid:18)kiF ∆∗(r)D∗(r)+ 12D∗(r)∆∗(r) (cid:2) −ǫ(r) (cid:3)(cid:19) vν↑(ρ)e−iκ+21φ!
(cid:2) (cid:3) =E u↑ν(ρ)eiκ+21φ eilφeiqz, (S.11)
ν vν↑(ρ)e−iκ+21φ!
where
1 5∂2+k2
(r) ieiφ ∂ +i ∂ z F. (S.12)
D ≡ ρ ρ φ k2
(cid:18) (cid:19) F
By the symmetry betweenthe up-spinstate andthe down-spinstate, eigenvalueandeigenfunctionofthe down-spin
∗
statearegivenbyE and u↑ ,respectively,underthe orderparameterwiththeoppositevorticity κ.
n,−l,−q n,−l,−q −
From now on, we only consider(cid:16)the up-sp(cid:17)in wave function u (ρ) u↑(ρ) which has similar features to the chiral
ν ≡ ν
p-wave state5–7 except for the q-dependence. The following derivation of the CdGM modes obeys Ref..7 By taking
real ∆(ρ) 0, Eq. (S.11) is rewritten as
≥
ǫ 1 d ∆(ρ) 5κ 1
F (α)τˆ ∆(ρ) + (5cos2α 1)+ ∆(ρ)cos2α ∆′(ρ) τˆ +E τˆ u (ρ)
k2Lm 0− k dρ 2ρ − 2ρ − 2 x ν z ν
(cid:20) F F (cid:26)(cid:18) (cid:19) (cid:27) (cid:21)
(κ+1)l l
= ǫ τˆ i∆(ρ) (5cos2α 1)τˆ u (ρ), (S.13)
F (k ρ)2 z − k ρ − y ν
(cid:20) F F (cid:21)
where ǫ is the Fermi energy, τˆ and τˆ are the unit matrix and the Pauli matrix, respectively, in the Nambu space,
F 0
and
d2 1 d m2
(α) + +k2sin2α, (S.14)
Lm ≡ d2ρ ρdρ − ρ2 F
with m l2+ (κ+1)2.
≡ 4
Here, weqintroduce a radius ρc that ∆(ρ) = 0 for ρ < ρc, where l kFρc kFξ on the coherence length
ξ ~v /∆( ) defined by using the Fermi velocity v . The solution of|t|h≪e wave fu≪nction in Eq. (S.13) is obtained
F F
≡ ∞
in the range of ρ<ρ as
c
uJ (k+ρ)
uν(ρ)= NνvJl+κ+21(kν−ρ) , (S.15)
Nν l−κ+21 ν !
where u and v are the normalization factors and we define
Nν Nν
E
k± k sinα ν . (S.16)
ν ≡ F ± ~v sinα
F
(1)
For ρ > ρ , the wave function in Eq. (S.13) is assumed to consist of the Hankel function H and the slow
c m
functions varying over the order of ξ, ϕ (ρ), as
ν
u (ρ)=H(1)(k sinαρ)ϕ (ρ)+c.c. (S.17)
ν m F ν
By using the ordinary condition k ξ 1, Eq. (S.13) is reduced to
F
≫
d ∆(ρ) 5κcos2α
τˆ (5cos2α 1) i τˆ ϕ (ρ)
dρ 0− ~v − − 2k sinαρ x ν
(cid:20) F (cid:26) F (cid:27) (cid:21)
E (κ+1)l ∆(ρ) l
= i ν τˆ (5cos2α 1)τˆ ϕ (ρ). (S.18)
~v sinα − 2k sinαρ2 z − ~v k sinαρ − y ν
(cid:26) (cid:20) F F (cid:21) F F (cid:27)
By the assumptions E ∆( ) and k ρ l , the right-hand-side of Eq. (S.18) can be regarded as a small
ν F c
| | ≪ ∞ ≫ | |
perturbation. The solution within the first order on ψ 1 is given by
ν
| |≪
ϕ (ρ)=A e−χq(ρ)+iκχ˜q(ρ) 1+iψν(ρ) A e−χq(ρ)+iκχ˜q(ρ) eiψν(ρ) , (S.19)
ν ν (cid:18)−sq[1−iψν(ρ)](cid:19)≈ ν (cid:18)−sqe−iψν(ρ)(cid:19)
7
J. Phys. Soc. Jpn. LETTERS
where s sgn(5cos2α 1),
q
≡ −
5cos2α 1 ρ 5cos2α ρ ∆(ρ′)
χ (ρ) | − | ∆(ρ′)dρ′, χ˜ (ρ) s dρ′, (S.20)
q ≡ ~v q ≡ q2~v sinα k ρ′
F Z0 F Z0 F
and
ψ (ρ) ∞ Eν (κ+1)l ∆(ρ′) l 5cos2α 1 e−2[χq(ρ′)−χq(ρ)]dρ′. (S.21)
ν ≡− ~v sinα − 2k sinαρ′2 − ~v k sinαρ′| − |
Zρ (cid:20) F F F F (cid:21)
In order to obtain the solution of Eq. (S.13), the wave functions in Eq. (S.15) for ρ < ρ and in Eq. (S.17) for
c
ρ >ρ are matched at ρ= ρ . By using the asymptotic forms of J (x) and H(1)(x) in x m, the wave function
c c m m
≫| |
in Eq. (S.15) becomes
u (ρ )= 2 Nνucos kν+ρc+ (l+2kκF+2s1in)2α−ρ1c/4 − 2(l+κ4+21)+1π . (S.22)
ν c rπkFsinαρc Nνvcoshkν−ρc+ (l−2kκF+2s1in)2α−ρ1c/4 − 2(l−κ4+21)+1πi
 h i
at ρ=ρ and the wave function in Eq. (S.17) is rewritten as
c
uν(ρ)=rπkFs2inαρe−χq(ρ) −sAq[νAeνiηeν+iη(ν−ρ)(ρ+)+A∗νAe∗ν−ei−ην+iη(ν−ρ)(ρ)]!, (S.23)
for ρ ρ , where
c
≥
m2 1/4 2m+1
η±(ρ) k sinαρ+ − π+κχ˜ (ρ) ψ (ρ). (S.24)
ν ≡ F 2k sinαρ − 4 q ± ν
F
Since χ (ρ ) = χ˜ (ρ ) = 0, in order to match two expressions u (ρ ) in Eq. (S.22) and in Eq. (S.23) at ρ = ρ ,
q c q c ν c c
ψ (ρ ) in Eq. (S.24) should be satisfied both conditions
ν c
E (κ+1)l π κ+1
ν
ψ (ρ )= ρ + + m l γ , (S.25)
ν c ~v sinα c 2k sinαρ 2 − − 2 − ν
F F c (cid:18) (cid:19)
and
E (κ+1)l π κ+1
ν
ψ (ρ )= ρ + m l+ +γ +nπ, (S.26)
ν c ~v sinα c 2k sinαρ − 2 − 2 ν
F F c (cid:18) (cid:19)
with the relation among the coefficients, that is v = s u for even n or v = s u for odd n and A =
Nν − qNν Nν qNν ν
(Nνu/2)eiγν. When γν =(m−l−n)π/2, Eqs. (S.25) and (S.26) become the identical expression:
E (κ+1)l π κ+1
ν
ψ (ρ )= ρ + + n . (S.27)
ν c ~v sinα c 2k sinαρ 2 − 2
F F c (cid:18) (cid:19)
The generalexpressionofψ (ρ)inEq.(S.21)forρ>ρ shouldcorrespondto Eq.(S.27)atρ=ρ .This condition
ν c c
provides the eigenvalue of the BdG equation (S.11) as
κ+1
E = κlω n ω′, (S.28)
ν − q− − 2 q
(cid:18) (cid:19)
where
ω 5cos2α 1 0∞ ∆kF(ρρ′′)e−2χq(ρ′)dρ′ ∆(∞)2, (S.29)
q ≡| − |R 0∞e−2χq(ρ′)dρ′ ∼ ǫF
and R
π~v
ω′ sinα F ∆( ). (S.30)
q ≡ 2 ∞e−2χq(ρ′)dρ′ ∼ ∞
0
Since ωq ≪ ωq′, the eigenvalue of the CdGM modRes for singly quantized vortex states with κ = ±1 is given by
n=(κ+1)/2.8
By using the eigenvalue E = κlω , the phase ψ (ρ) of the eigenfunction of the CdGM modes for κ = 1 is
ν q ν
− ±
given by Eq. (S.21) as
E (κ+1)l
ψ (ρ)= ν ρ+ +κlψ˜ (ρ), (S.31)
ν ~v sinα 2k sinαρ q
F F
8
J. Phys. Soc. Jpn. LETTERS
where
ψ˜ (ρ) |5cos2α−1| ρ 1 2ωqρ′ ∆(ρ′)e−2[χq(ρ′)−χq(ρ)]dρ′. (S.32)
q ≡ ~v sinα k ρ′ − ~v
F Z0 (cid:20) F F (cid:21)
From Eq. (S.23), the eigenfunction for the up-spin state is given by
J [k+ρ+κlψ˜ (ρ)+κχ˜ (ρ)]
u↑(ρ)= ↑ l+κ+21 ν q q e−χq(ρ), (S.33)
ν Nν κsqJl−κ+21[kν−ρ−κlψ˜q(ρ)+κχ˜q(ρ)]!
whichismatchedtoEq.(S.15)atρ=ρ becauseχ (ρ )=χ˜ (ρ )=ψ˜ (ρ )=0duetotheassumption∆(ρ<ρ )=0.
c q c q c q c c
The evaluation of the argument in the Bessel function can be advanced to
k±ρ κlψ˜ (ρ)+κχ˜ (ρ)=k sinαρ Eν ρ 1 |5cos2α−1|∆(ρ′) e−2[χq(ρ′)−χq(ρ)]dρ′+κχ˜ (ρ)
ν ± q q F ± ~v sinα − ω k ρ′ q
F Z0 (cid:18) q F (cid:19)
E
ν
k sinαρ f (ρ)+κχ˜ (ρ). (S.34)
≡ F ± ~v sinα q q
F
Since f (ρ)/ξ . 1, the second term in Eq. (S.34) can be neglected by E ∆( ). Moreover, the third term in
q ν
| | | | ≪ ∞
Eq. (S.34) is much less than the first term for any ρ owing to k ξ 1. Finally, from the symmetry between u↑ and
F ≫ ν
u↓, the BdG wave function in Eq. (S.10) is given by
ν
u↑ν(ρ) = ↑ Jl+κ+21(kqρ) e−χq(ρ), u↓ν(ρ) = ↓ Jl+κ−21(kqρ) e−χq(ρ), (S.35)
(cid:18)vν↑(ρ)(cid:19) Nν κsqJl−κ+21(kqρ)! (cid:18)vν↓(ρ)(cid:19) Nν −κsqJl−κ−21(kqρ)!
with k k sinα.
q F
≡
9
J. Phys. Soc. Jpn. LETTERS
References
1) C. Caroli, P. de Gennes, and J. Matricon, Phys. Lett. 9, 307 (1964).
2) Y. Machida, A. Itoh, Y. So, K. Izawa, Y. Haga, E. Yamamoto, N. Kimura, Y. O¯nuki, Y. Tsut-
sumi, and K. Machida, Phys. Rev. Lett. 108, 157002 (2012).
3) Y. Tsutsumi, K. Machida, T. Ohmi, and M. Ozaki, J. Phys. Soc. Jpn. 81, 074717 (2012).
4) M. Matsumoto and R. Heeb, Phys. Rev. B 65, 014504 (2001).
5) N. B. Kopnin and M. M. Salomaa, Phys. Rev. B 44, 9667 (1991).
6) M. Stone and S.-B. Chung, Phys. Rev. B 73, 014505 (2006).
7) T. Mizushima and K. Machida, Phys. Rev. A 81, 053605 (2010).
8) K. M. Suzuki, T. Mizushima, M. Ichioka, and K. Machida, Phys. Rev. A 77, 063617 (2008).
10