Table Of ContentTopological Fermi arcs in superfluid 3He
M.A. Silaev1 and G.E. Volovik2,3
1 Institute for Physics of Microstructures RAS, 603950 Nizhny Novgorod, Russia
2 Low Temperature Laboratory, Aalto University, P.O. Box 15100, 00076 Aalto, Finland
3 L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia
(Dated: December 5, 2012)
WeconsiderfermionicstatesboundondomainwallsinaWeylsuperfluid3He-Aandoninterfaces
2 between 3He-A and a fully gapped topological superfluid 3He-B. We demonstrate that in both
1 cases fermionic spectrum contains Fermi arcs which are continuous nodal lines of energy spectrum
0 terminatingattheprojectionsoftwoWeylpointstotheplaneofsurfacestatesinmomentumspace.
2 ThenumberofFermiarcsisdeterminedbytheindextheoremwhichrelatesbulkvaluesoftopological
invarianttothenumberofzeroenergysurfacestates. Theindextheoremisconsistentwithanexact
c
e spectrum of Bogolubov- de Gennes equation obtained numerically meanwhile the quasiclassical
D approximation fails to reproduce the correct number of zero modes. Thus we demonstrate that
topology describes theproperties of exact spectrum beyond quasiclassical approximation.
4
PACSnumbers:
]
n
o I. INTRODUCTION count). Bound states emerging at one of the representa-
c tives of such interface – the continuousˆl-soliton – have
-
r Chiral Weyl fermions represent the fermionic sector been calculated in Ref. 17. Here we consider bound
p
u in the Standard Model of particle physics. The point states emerging on the singular domain wall in 3He-A
s nodes in the spectrum of chiral quarks and leptons are discussedinRefs. 18,19. Zeroenergyedgestatesonsuch
t. topologically protected, as well as their condensed mat- domain wall in a thin film of 3He-A and the topological
a ter counterparts, which are called Dirac or Weyl points bulk-edge correspondence for this 2+1 system were con-
m
(on topology of Weyl points in particle physics and con- sidered in Ref. 20. The Fermi arc emerging in the 3+1
- densed matter see e.g. Refs. 1–5). systemformsaspecialconfigurationinmomentumspace,
d
n The nodal topological systems with Weyl fermions see Fig. 2b. We also consider the Fermi arc emerging at
o demonstrate different types of the bulk-surface and the theinterfacebetweentwotopologicallydifferentquantum
c bulk-vortexcorrespondence. Due to the bulk-vortexcor- vacua: the 3He-Awith Weylpoints andthe fully gapped
[ respondence, the cores of some vortices in the Weyl su- 3He-B, which also has nontrivialtopology in momentum
4 perfluid 3He-A contain a dispersionless branch of bound space12,21–23 (AB interface).
v states with zero energy which is one-dimensional flat In the present paper we consider the Fermi arcs in su-
8 band, discussed first by Kopnin and Salomaa in 19916. perfluid 3He solitons which differ qualitatively from that
6 The end points of this flat band are determined by the existinginothersystemssuchasthesurfaceof3He-Aand
3
projectionsofWeyl points to the directionofvortexaxis Weyl semimetals. The reason is that they can be ob-
3
. (see Refs. 7,8 for the topological originof this flat band, tained ultimately beyond the quasiclassical approxima-
9 andRefs. 9–12fordiscussionofthetopologicalflatbands tion. It was found that the quasiclassical approximation
0
in general). Due to the bulk-surface correspondence,the yields a large number of spectrum branches which inter-
2
surfaceofasystemwithWeylpointscontainsanotherex- sect the Fermi level.17,25 The number of such branches
1
: oticobject–theFermiarc–the 1DFermiline inthe2D depends on the parameters of the solition in contradic-
v momentumspace,whichterminatesontheprojectionsof tion with the topological index theorem. We show that
i
X two Weyl points to the plane of the surface. The Fermi thereasonforthiscontradictionisthatthequasiclassical
r arc on the surface of 3He-A shown in Fig.2a has been Bogoluibov-deGennes(BdG)equationsinherentlymiss
a consideredinRef. 13andthatonthe surfaceoftopolog- the normal reflection of quasiparticles from the spatially
ical semi-metals with Weyl points – in Refs. 14–16. The inhomogeneoussuperfluidorderparameterandtakeinto
flatbandinthevortexcoreandtheFermiarconthesur- accountonlyAndreevreflection. Thusthesystemofqua-
facearemomentum-spaceanalogsofthereal-spaceDirac siclassical BdG equations is often called the system of
string terminating on two magnetic monopoles9. Andreev equations. As we will discuss below the nor-
Fermiarcsappearalsoattheinterface,whichseparates mal reflection is crucial to describe the Fermi arcs in
two degenerate states of 3He-A with the opposite direc- superfluid 3He solitons. In this case fermionic zero en-
tions of the orbital anisotropy vectorˆl (in chiral super- ergy states are realized on the quasiparticle trajectories
fluid3He-Avectorˆldeterminestheorientationofaspon- passing at the small sliding angle to the soliton plane.
taneous orbital angular momentum of this anisotropic Forsuchtrajectoriesthenormalreflectionisstronglyen-
liquid and also determines the positions p = p ˆl of hancedandchangequalitatively the behaviorof spectral
F
±
two Weyl points on the Fermi surface with topological branchesnearthezeroenergypointsonFermiarcs. Here
charges N = 2, if spin degeneracy is taken into ac- we find numerically exact solutions of the spectral prob-
3
±
2
lems for BdG equations which confirms the topological
indextheorempredictions. Thusourresultsdemonstrate
that topological bulk-edge correspondence describe the
properties of the exact spectrum of localized boundary
states which in some cases are missed in the quasiclassi-
cal approximation.
The structure of the paper is as follows. In section
II we introduce the model which is the BdG equations
and the order parameter structures of domain wall in A
phaseandABinterface. Theresultsarepresentedinsec-
tion III where we consider at first topological nature of
Fermi arcs and then the calculations of fermionic spec-
trum. Wediscussthefailureofquasiclassicalapproxima-
FIG. 1: (Color online) Model spatial dependencies of the or-
tion to describe Fermi arcs in the fermionic spectrum of der parameter components corresponding to the most sym-
domain wall in A phase and AB interface in section IV. metric and energetically preferable 1D topological defects in
We summarize the results in section V. superfluids 3He according to Refs.(18,19,31,32): (a) domain
wall of the A phase and (b) AB interface. The order param-
eter is normalized tothe bulkvalue.
II. THE MODEL
is neglected, and we put dˆ = xˆ. Then the gap function
WecalculatethespectrumofeigenstatesofBogolubov- is given by
de Gennes (BdG) Hamiltonian describing spin-triplet p-
wave superfluids/superconductors ∆ˆ =c σˆ [pˆ +if(x)p ] , (3)
AA 0 x x y
Hˆ =[ε(pˆ) µ]τˆ +τˆ Re∆ˆ τˆ Im∆ˆ, (1) where c0 = ∆0/pF is the parameter which plays the
− 3 1 − 2 role of the longitudinal speed of light in bulk 3He-A2,
where ε(pˆ) = (pˆ2 +pˆ2 +pˆ2)/2m, µ is chemical poten- f(x) is arbitrary monotonic function with asymptotic
x y z
tial, pˆ = i and τˆ are Pauli matrices of Bogolubov– f( )= 1. Thetopologyofboundstatesdoesnotde-
Nambiusp−in∇acitingonithewavefunctionψ =(u,v)T with pe±nd∞on th±e details of the function f(x) in Eq.(3) if it is
particleandholecomponentsuandv. Thegapoperator monotonic. Notethatthesamemodel(modulounimpor-
is ∆ˆ =A σˆ pˆ/p , where σˆ are Pauli matrices of spin tantdˆ vectororientation)describesdomainwallbetween
αi α i F α
(in 3He it is nuclear spin), pF =√2mµ is the Fermi mo- chiraldomainsinp+ipsuperconductorslikeSr2RuO424
mentum. We will consider the spectral problem of BdG thus the fermionic spectra are identical in these cases.
Hamiltonian(1)forthefermionicstatesboundonthe1D AB interface. Let us fix the order parameter on the
solitons in superfluid 3He namely the domain wallof the B phase side of the interface, i.e. at x = + , in the
∞
A phase and the AB interface. following form
The order parameter in superfluid 3He is 3 3 ma-
× A (x=+ )=∆ (xˆ xˆ +yˆ yˆ +zˆ zˆ). (4)
trix Aαi where the Greek and Latin indices correspond αi ∞ B α i α i α i
to the spin and orbital variables. We will consider the
The configuration of the A phase at x can be
fermionic modes localized on different 1D order parame- → −∞
differentfordifferentrealizationsofthe domainwall(see
ter solitons, namely the domain wall of a 3He-A and the
e.g. Ref. 31). They are described by the relative ori-
AB interface with order parameter inhomogeneity along
entations of the vectorsˆl, dˆ and nˆ (the normal to the
the vector nˆ = xˆ so that the momentum projections p
y AB interface which in our case is nˆ = xˆ). First we shall
and p are conserved. At first we introduce the order
z consider the case dˆ = xˆ, ˆl = ˆz which has the lowest
parameter structure corresponding to the discussed soli-
energy32
tons.
Domain wall of A phase. We consider the configu-
A (x= )=∆ xˆ (xˆ iyˆ ). (5)
ration of one of the domain walls possible in 3He-A ac- αj −∞ A α j − j
cording to the symmetry classification19– the one which WemodelthedomainwallattheABinterfacewith∆ =
A
is shown in Fig. 1a. The orbital vectorˆl of the order ∆ = ∆ and the switching between bulk phases (4,5)
B 0
parameter points down (ˆl ˆz) as x and points as follows
k − → −∞
up (ˆl ˆz) as x + . This domain wall configuration
can bekapproxim→ated∞by the following Ansatz: Axx =∆0 =const (6)
A =A =∆ f (x) (7)
yy zz 0 1
A (x)=∆ dˆ [xˆ +if(x)yˆ], (2)
αi 0 α i i Axy = i∆0f2(x) (8)
−
wheretheunitvectordˆ representsthespinpartoftheor- where f (x) are arbitrary monotonic functions with
1,2
derparameter,whichisfixedifthespin-orbitinteraction asymptotics f ( )=0, f (+ )=1 and f ( )=1,
1 1 2
−∞ ∞ −∞
3
f2(+ ) = 0. The order parameter components (6) are pz
∞ ~ 2D trivial ~
shown in Fig.(1b). Then the gap operator is given by N3(pz) = 0 insulator N3(pz) = 0
∆ˆAB =c0σˆx[pˆx−if2(x)py]+c0f1(x)[σˆypy +σˆzpz]. Weyl pF
(9) point Weyl
point
p
y
III. RESULTS N~3(pz) = 1 px2inDsu tloaptoorlogical Fermi arcE (pp)=0y,zFermi arc N~ 3(pz) = - 1
- p
F
Weyl
A. Topology of Fermi arcs in superfluid 3He point
Weyl
point
~ 2D trivial x=0 ~
N3(pz) = 0 insulator N3(pz) = 0
FIG.3: Topologyofthefermionicboundstatesonthedomain
wall in 3He-A. Momentum space topology of Weyl points in
bulk 3He-A on two sides of the wall prescribes existence of
the Fermi arcs in the spectrum of the fermionic states in the
soliton or at the interface between the bulk states with dif-
ferent positions of Weylpoints. In theconsidered case of the
domain wall the Weyl points on two sides of the interface
have the same positions in momentum space, but opposite
topological invariants N3. This leads to two Fermi arcs ter-
minating on the projections of the Weyl points to the plane
of domain wall (Fig. 2b) according to the index theorem
N˜3(left) −N˜3(right) = 2. This is distinct from the single
Fermi arc on the surface of the3He-Ain Fig. 2a.
FIG. 2: (Color online) The manifold of zero energy states in
py,pz plane in the spectrum of bound fermions ε(py,pz)=0
forms (a) the Fermi arc (solid blue line) on the boundary
of 3He-A according to Ref. 13; (b) two Fermi arcs on the
domain wall in Eq.(2). Only single spin projection is con- 2a. Topological origin of two Fermi arcs on the domain
sidered. Thick arrows show directions of Fermi velocity at wall is demonstrated in Fig. 3. The Weyl points on two
the Fermi arcs. In (a) the Fermi arc has topological charge sides of the interface have the same positions in momen-
N =+1,whichsatisfiestheindextheoremfollowingfromthe tum space, but opposite values of topological invariant
bulk-surfacecorrespondenceandthemomentumspacetopol- N .
ogy of Weyl points in bulk 3He-A. Fermi arc terminates on 3
the projections of Weyl points to the surface of 3He-A. The The bulk spectrum in the plane with fixed pz in mo-
spectrumofboundstatesterminatesatthedashedlinewhere mentum space has no nodes if pz =pF and thus corre-
| |6
the spectrum merges with the bulk spectrum. The region of spondstothespectrumof2Dinsulator. Thisinsulatoris
continuous spectrum is shown by yellow shading. In (b) at topologicalfor p <p ,sinceitisdescribedby nonzero
z F
pz = 0 the Fermi velocity at the Fermi arcs is in the same topological inva|ria|nt introduced in Refs. 27–30 for 2+1
direction, vF = −vFy which demonstrates that both Fermi systems:
arcs have the same topological charge N = +1, which to-
gethersatisfy theindex theorem N˜3(left)−N˜3(right)=+2,
iinn abgurlekem3Henet-AwiothnmtwoomesindteusmofsptahceedtoopmoaloingywoafllWineyFlpigo.int3s. N˜3(pz)
FermiarcsterminateontheprojectionsofWeylpointstothe = 1 tr dp dp dω G∂ G−1G∂ G−1G∂ G−1 .
interface. The spectrum of bound states has discontinuity at 4π2 (cid:20)Z x y px py ω (cid:21)
py = 0 where the spectrum merges with the bulk spectrum (10)
(theedge of continuumis shown by dashed line).
As we have mentioned in the Introduction the spec- Here G is the Green’s function matrix, which in our
trum of fermions bound to the surface of superfluid 3He non-interacting models is G = iω Hˆ −1. One has
A is known to contain the Fermi arc13. Let us now com- (cid:16) − (cid:17)
pare the Fermi arcs on a domain wall in 3He-A with the N˜3(pz < pF) = +1 on one side of the wall and
| |
Fermi arc on a surface of 3He-A shown in Figs.(2a) and N˜ (p < p ) = 1 on the other side. According to
3 z F
(2b) correspondingly. In both cases Fermi arcs originate the|ind|ex theorem2−,20, the difference between these two
from the Weyl points in the bulk spectrum. But in case values determines the number of the zero modes at the
of the domain wall the Weyl points exist in the bulk liq- interfacebetweenthe 2+1topologicalinsulatorsforeach
uids on both sides of the wall, as a result there are two p < p . As a result one has two Fermi arcs at the
z F
| |
FermiarcsinFig. 2binsteadofasingleFermiarcinFig. soliton wall.
4
B. Calculation of the spectrum pointscomeclosetogetherformingtheboundstatewhich
correspondto the so called ’zerothbranch’ of the energy
At first we note that the spectral problem for the do- spectrum17,25. This case can be treated with the help
main wall can be significantly simplified since the BdG of another approach - the quasiclassical approximation
Hamiltonian (1) with the gap operator given by Eq.(3) when one can use reduced order Andreev equations for
commutes with fermionic spin σˆ . Thus we transform the envelope wave functions discussed in Section IIIB2.
x
the quasiparticle wave function ψ = (u,v)T to remove ThequasiclassicalapproximationinBdGequationsisap-
the spin dependence of the order parameter u˜ = σˆ u, plicableaslongasthenormalreflectioncanbeneglected.
x
v˜=v which yields the gap operator in the form For the structures considered in the present paper the
normal reflection is crucial to describe properly the zero
∆ˆ =c [pˆ +if(x)p ]. (11) modes of the fermionic spectrum. This canbe done only
0 x y
with the help of numerical solution of the exact BdG
FurthermorethespectralproblemfortheABinterface system of equations discussed in SectionIIIB3.
with the gap operator given by Eq.(9) can be mapped Hereafter we will choose the model form of Eq.(11)
on the case of domain wall of the A phase. Let us with f(x) = tanh(x/r) where r ξ is width of the
transform the quasiparticle wave function components domain wall and ξ = ~v /∆ is t∼he coherence length
F 0
as follows u˜ = σˆxu, v˜ = v. We choose the spin basis where vF = pF/m is the Fermi velocity. The fermionic
χσ = (ipz,σ⊥p⊥ py) of the eigenstates of the opera- spectrum in this configurationwas consideredin Ref. 25
−
tor σˆ⊥ = σˆzpy σˆypz corresponding to the eigenvalues in quasiclassicalapproximation. Here we will implement
−
σ⊥ =±1 where p⊥ =qp2y+p2z. Then the order param- aclnasasnicaslyinsisoordfetrhetofesrtmudioynitchesptehcetrFuemrmbieayrocnsdstuhpepoqrutainsig-
eterisdiagonalinspinspaceandhastheformofEq.(11)
Majorana states.
where
f(x)=σ⊥p⊥f1(x)−pyf2(x). (12) 1. Semiclassical spectrum.
In order to obtain the bound states the function (12)
has to satisfy the condition f(+ )f( ) < 0 which We begin the analysis of the spectrum of localized
yields σ⊥py > 0. Thus in contras∞t to t−he∞case of the A states from the semiclassical approximation. Provided
phase domainwallconsideredabovethe spindegeneracy the condition px ξ 1 isvalidwe usethe expressionfor
| | ≫
is removed since the proper spin state is determined by the energy (13) substituting the local value of the order
the condition of bound state existence. parameter. In this case we obtain
Thecontinuouspartofthespectrumε (p)offermionic
c
excitations is determined by the eigen states of Hamilto- ε= [p2/2m µ ]2+c2[p2 +f2(x)p2]. (14)
nian (1) at the bulk regions x r ±q x − x 0 x y
| |≫
whereµ =µ (p2+p2)/2m. FromEq.(14)thefunction
x − y z
εc(p)=±q[ε(p)−µ]2+c20(p2x+p2y). (13) oprxbi=tspinx(x(p) c,axn) pbheafsoeusnpdacweh.icIhn dgeesncerriablesthseemoribcliatssscicaanl
x
Thestateslocalizedatthedomainwallarecharacterized have two types of stationary points determined by the
by a single discrete quantum number which enumerates nature of quasiparticle reflection.
the energy branches and by the two continuous quan- The normal reflection occurs at the points x = xn1,2
tum numbers which are the projections of the quasipar- defined by px(xn1,2)=0 or equivalently
ticle momentum p onto the domain wall plane. The
y,z
energy of localized states is confined within the region c p f(xn )= ε2 ε2 (15)
ε <min ε (p). 0 y 1,2 ±q − n2
| | px c
In general the eigenvalue problem of BdG Hamilto- and exists at the energy interval ε > ε > ε where
n1 n2
nian (1) yields a system of differential equations which ε = (c p )2+µ2 and ε = µ . T|h|e Andreev re-
cparonxinmotatbeemseotlhvoeddsawnahliychticcaalnly.heHlpowtoevsetrudtyheqrueaalirteataivpe- flne1ctionpocc0urys at x x= xa1,2 wn2here|Ixm|(px) becomes non-
zero
features of the spectrum. First of all we will employ the
semiclassical approach26 when the momentum operator c p f(xa )= ε2 ε2 (16)
is approximated by a number pˆx = px(x) thus turning 0 y 1,2 ±q − a2
thedifferentialBdGequationsintothealgebraicones. In
and exists if ε > ε > ε where ε =
a1 a2 a1
this case the spectrum of bound states is determined by | |
c p2+2mµ (mc )2 and ε =c 2mµ (mc )2.
Bohr-Sommerfeld quantization of classical periodic mo- 0 y x− 0 a2 0 x− 0
tion between reflection points discussed in the Section qOne can see that ε > ε and εp> ε therefore
n1 a1 n2 a2
IIIB1. The semiclassical approximation fails when the Eq.(14) determines two qualitatively different types of
distance between reflection points is too small. Particu- the enclosed quasiparticle orbits in (p ,x) space. That
x
larly important is the case when two Andreev reflection is for the energies (i) ε > ε > ε the orbits have
n1 a1
| |
5
only normal reflection points and for (ii) ε > ε >ε
n2 a2
| |
onlyAndreevreflectionpoints. Theorbitsoftype(i)and
(ii) are shown in the Fig.(4a) by red dashed and green
solidlinescorrespondingly. Provided(iii)ε >ε there
a1 n2
is also the third regime when the orbit has both normal
andAndreevreflectionpointswhichisshownbytheblue
dash-dotted line in the Fig.(4). The energy spectrum is
determined by the Bohr-Sommerfeld quantization
p dx=2π(n +γ) (17)
x s
I
wheren isinteger, γ =1/2whenthe numberofturning
pointsisseven26. Semiclassicalspectralbranchesforn = FIG. 4: (Color online) (a) Closed orbits px = px(x) deter-
s mined by the semiclassical Eq.(14) at the domain wall for
1,2 are shown in Fig.(4b) by solid lines. pz = 0, py = 0.5pF and c0/vF = 2.5 10−3. Shown by
From the above semiclassical consideration we obtain red dashed, green solid and blue dash-dotted curves are the
thattheenergyoflocalizedstatesisnotboundedasfunc- regimes (i), (ii) and (iii) discussed in the text. The posi-
tionofpy andpz. Indeedletus considerthecase(i)and tions of Andreev xa1,2 and normal xn1,2 reflection points on
assume ε2 µ2 c p ε. In this case we can thesemiclassicalorbitsisindicated. (b)Comparisonofenergy
put f(x)p= x−/rxin≪the0|Eqy.| ≪(14|)|which yields then the spectrumgivenbyquasiclassicalandsemiclassicalapproxima-
tions. Thequasiclassical spectrum given byEq.(24)is shown
harmonic oscillator spectrum
for nq =0,1 bydashedred and bluelines. Quasiclassical ap-
proximationtakesintoaccountonlyAndreevreflectionhence
ε (p ,p )= µ +ω(n +1/2) (18)
ns y z | x| s thespectrumisdoublydegeneratebysign(px). Thesemiclas-
sical approach describes both Andreev and normal reflection
where ω = c (p /p r) 2µ/µ . One can see that the
0 | y| F | x| hence the degeneracy by sign(px) is removed. The branches
Eq.(18) yields ε > µx p. Hence the energy branches at given by Bohr-Sommerfeld quantization (17) of semiclassical
| |
µx <0 can have arbitrary high energy when µx →−∞. orbits with ns = 1,2 are shown by solid magenta lines. The
This tendency can be seen in the Fig.(4b). number of semiclassical branches is doubled due to the nor-
Besides the energy branches determined by Bohr- mal reflection when the Andreev orbits shown in the panel
Sommerfeld quantization with n 1 there exists a so- (a) by green solid lines merge into the single one shown by
s
called’zerothbranch’ofthespectru≥mwhichcontainsthe blue dash-dotted line with both normal and Andreev reflec-
tion points.
zeromodesε=0. Thisbranchcannotbe obtainedfrom
the semiclassical approach therefore the different treat-
ment is needed17.
we will see below the normal reflection is crucial to de-
scribetheFermiarcsandMajoranastatesatthe domain
wall considered. However away from the Fermi arcs the
2. Quasiclassical approximation.
quasicalssicalapproximationmatchestheexactspectrum
obtained numerically.
To find the zeroth branch analytically we assume the To obtain analytical solutions of Andreev equations
restriction (px/pF)pxξ 1 where px =√2mµx is a con- first let us introduce the transformationof the quasipar-
≫
stant number. Qualitatively this restriction corresponds ticle wave function ψ =(u,v)T
to the regime (iii) in Fig.4 when the deviation p˜ (x) of
x
momentum p (x) = √2mµ +p˜ (x) along the orbit is g+ =v+u (19)
x x x
much smaller than its average value p˜x √2mµx. In g− =v u (20)
| | ≪ −
thiscasewecanneglectthesecondordertermsinp˜ still
x
so that the system of Andreev equations acquires the
treating it as an operator so that [ε(pˆ) µ] iv ∂
x x
− ≈ − form
where the projection of the Fermi velocity on the x axis
icsonvsxid=er tphxe/mreg=im±esqwph2Fen−tph2ze−clpo2ys/emd.orbTithsisbaeclloomwse stoo (cid:18)ε−hc+0px ε+hc−0px(cid:19)(cid:18)gg+−(cid:19)=0 (21)
smallthatsemiclassicalapproximationdoesnotapply. In where h± = ivx∂x ic0f(x)py.
particular it allows to calculate the zeroth branch which The system−(21) c±an be transformed to the decoupled
appears when the distance between Andreev reflection second-order equations
points is much smaller than the coherence length ξ. In
this way one obtains a system of abridged equations for ∂x2+U0+cosh−2x g+ =Eg+ (22)
the wave function envelopes which is called the system (cid:2)∂x2+U0−cosh−2x(cid:3)g− =Eg− (23)
of quasiclassical Andreev equations. The quasicalssical
(cid:2) (cid:3)
approximationinherently misses the normalreflectionof where E = (r/ξ)2 +α2 (p ε/p ∆ )2, U = α+α2,
F x 0 0+
−
particles from the order parameter inhomogeneities. As α = rc0py/vx and U0−(py) = U0+( py). The Eq.(22)
−
6
has eigenvalues The normal reflection is particularly important for the
quasiparticle trajectories passing almost parallel to the
Enq =(1+2nq−p1+4U0+)2/4=(nq−α)2 mdoamlareinflewctailolnplleaandeswtohetnhepdxr→asti0c.chInantgheisincatsheetshpeecntorar-l
wheren 0isinteger,whichresultsinthequasiclassical
spectralqb≥ranches17,25 branches behaviour near the zero energy point shown in
Fig.5(comparethedashedandsolidlines). Theinfluence
p of normal reflection on the spectrum of bond fermions
ε = ∆ | x| 2αn n2+(r/ξ)2. (24)
nq ± 0 pF q q− q wasanalyzedaboveintheframeworkofsemiclassicalap-
proach in the section (IIIB1). Comparing the Figs.(4b)
Note that the energy branches (24) are twofold degen- and(5)onecanseethatthesemiclassicalapproachqual-
erate with respect to the sign change of px projection. itatively describes the doubling of spectral branches due
The zeroth spectral branch is given by the Eq.(24) with to the normal reflection but gives rather large quantita-
nq =0 tive discrepancy with the exact spectrum.
In accordance with general topological argument in
ε (p ,p )=c sign(p )p . (25)
0 y z 0 y | x| Sec. (IIIA) the only one zeroth branch remains at each
fixedvalueofmomentumprojectionp (andspinprojec-
The dependencies of zeroth branch (25) ε = ε (µ ) at y
0 0 x
tionσ forthedomainwalloftheAphase). Exactzeroth
fixed p and ε = ε (p ) at fixed p are shown in Fig.5 x
y 0 0 y z
branchesforp = 0.5p areshownintheFig.5bythick
by red dashed lines and in Fig.6 by blue dashed lines y F
±
red solid lines. The spectrum in Fig.5 is invariant with
correspondingly. At p = 0 this spectral branch is dis-
y
continuousandmergestheedgeofcontinuumε = c p∗ respect to the transformation ε(py,σ) = ε( py, σ)
at p = 0. c ± 0 y where σ = σx for the case of domain wall−and−σ =−σ⊥
y
± for that of the AB interfere. Note that in contrast to
Substituting the energy of zeroth branch into the
the quasiclassical zeroth branch shown by red dashed
Eq.(16) for the coordinates of Andreev reflection points
we obtain x = p−1(r/ξ)(p /p ) p−1 provided line the exact zerothbranchintersects at the Fermi level
| a1,2| F F | y| ∼ F at finite value of µ = µ > 0. This crossing point
p is not too small. For such a small distance between x x0
y
| | determines the positions of Fermi arcs shown in the
reflection points the semiclassical approximation is not
Fig.(2b) by blue and green solid lines. For example the
validandthereforeitcannotdescribethezerothbranch.
Fermi arcs cross the line p = 0 at p = p where
The spectral branches with n > 0 given by Eq.(24) z y y0
q ±
p = 2m(µ µ )<p .
exist only in the limited range of parameters p /ξ y0 x0 F
F −
|px| < p∗x where p∗x = |p∗y|nq/[n2q −(r/ξ)2] provpided th≪at theNleiamrpitthpe pro0jeactnidonpof Wepyl pthoeinbteohnavpioyuprzopflaznereotinh
n > r/ξ. At p = p the branch merges continuum. y z F
q x x → | | →
branch can be found analytically using the approach of
Thus behaviour us demonstrated in Figs.4b where the
Ref.(20). We will treatthe firstterm inthe Hamiltonian
energybranches(24)withn =0,1areshownbydashed
q
red and blue lines as function of µ =µ (p2 +p2)/2m (1)asperturbation. Therestofthetermsformzeroorder
x − y z Hamiltonian
for the parameters r = ξ/2 (width of the domain wall),
c /v = 2.5 10−2 and p = 0.5p . The zeroth branch
0 F y F Hˆ =τˆ pˆ τˆ p f(x) (26)
mergescontinuumatpy =0wherethespectrumEq.(25) 0 1 x− 2 y
isdiscontinuous. Thisbehaviourisshownbydashedblue
where pˆ = i∂ . We assume the model form of the do-
x x
lines in Fig.(6). −
main wall f(x) = tanh(x/r). The Hamiltonian (26) has
zero energy eigen state with the wave function compo-
nents ψ =(u ,v )
3. Exact spectrum near the Fermi arcs. 0 0 0
u =0 (27)
0
As pointed out by Nakahara25 all quasiclassical v =N−1/2cosh−α(x/r) (28)
branches (25) formally contain zero energy states at 0
tphxe=lim0ibtupt Eq.(025a)nids aarpbpiltircaarbylevaiflu(epsx/opfFp)pxaξnd≫p1.thIne where α= rpy and N = −∞∞cosh−2α(x/r)dx. The per-
x → y z turbation of the zero enerRgy level is given by
spectrum canbe found numericallyby solvingthe eigen-
value problem for the Eq. (21). The resulting several p¯2
lowestenergy spectrum branchesε=ε(µx) are shownin ε0(px,py)= 2mx −µx (29)
the Fig.5 by solid red and blue lines as the function of
µ = µ (p2 +p2)/2m. Remarkably the twofold degen- where p¯2 = ψ pˆ2 ψ . The Fermi arc is then given by
x − y z x h 0| x| 0i
eracy p p of the quasiclassical spectrum is broken
x x
at0<µ →−µandatµ <0. Thereforeeachofthequa- p2 =p2 p¯2 p2 (30)
x ≪ x z F − x− y
siclassicalenergybranches(25)foundbyNakaharasplits
bTyhetwspolimttiondgesisacsauµsxed→by+n0or(manadl rpexfle=ctio√n2omfµqxua→sipa0r)-. Ianrcthisegliimveint pbyy→p20=wepo2btai2np2p¯2xa=ndp2yensdosthaattpthe=Ferpmi
z F − y z ± F
ticles from the order parameter spatial inhomogeneity. in accordance with general topological arguments. The
7
Eq.(29) describes analytically the zeroth branch of the
spectrum shown by thick red lines in Fig.(5) near the
projections of Weyl points when p 0.
y
→
The splitting of quasiclassical zeroth branches ε =
ε (p ) at fixed value of p = 0 is shown in Fig.6. Red
0 y z
solidlinesistheexactspectrumobtainednumericallyand
blue dashed lines are the quasiclassical zeroth branches
(25). Away from the level ε = 0 the correspondence of
quasiclassical and exact spectra is of a very good accu-
racy. The splitting takes place in the small vicinity of
p = p∗ where the quasiclassical branch intersect the
y ± y
Fermi level. The behaviour of spectral branches near
this point is shown in the zoom inset in the Fig.(6). All
branches are discontinuous at p = 0 where they merge
y
the continuum. Shown in Fig.(6) the spectrum of bound
fermionic states at the domain wall in 3He-A at p = 0
z
coincides with the spectrum of electronic states at do-
main walls in p+ip superconductor Sr RuO .
2 4
FIG. 6: (Color online) The lowest energy spectral branches
of the surface states at the domain wall of the A phase and
at the AB interface as functions of py at fixed pz. Shown
by blue dashed line is the quasiclassical zero energy branch
b(2r5a)ncwhhesichsheonwdnsbayt psoyli=d r±edp∗yliwnehserceonpt∗yin=uepatp2F|p−y|p>2z.pE∗y.xaActt
py = 0 the discrete spectrum merges with continuum shown
by the yellow shading. Inset: zoomed area demonstrating in
detail the splitting of zeroth branches and the behavior of
exact spectral branches near the end point of quasiclassical
spectrum branch.
IV. DISCUSSION
In contrast to the Fermi arcs on 3He-A surface13 the
Fermi arcs on the domain wall in 3He-A and on the AB
interface can not be obtained in the framework of qua-
siclassical approximation. Indeed the quasiclassical ap-
proximation when applied to these systems yields that
any of the subgap spectral branches given by Eq.(24)
FIG. 5: (Color online) The spectrum of surface states at the intersects the Fermi level25. Thus the number of zero
A phase domain wall and the AB interface ε = ε(µx) where modes in quasiclassics is model depended and does not
µx =µ−(p2y+p2z)/2m. The parameters are r =ξ/2 (width satisfy the index theorem. The reason behind this dis-
of the domain wall) and c0/vF = 2.5 10−2, |py| = 0.5pF. crepancy is that the zero modes exist for p = 0 (see
x
Shown by solid red and blue lines are the spectral branches
dashed lines in Fig.4b) which corresponds to the quasi-
determined by the exact solution of BdG Eq.(1). Thick red
classical trajectories almost parallel to the plane of do-
solid linesshowthezerothbranchesat py =±0.5pF. Dashed
mainwallwhenoneneedstogetintoaccountthenormal
lines represent quasiclassical energy branches(24) for nq =0
(redline)andnq =1(blueline)degeneratebysign(px). The reflectionofqasiparticlesfromtheorderparameterinho-
degeneracy by sign(px) is removed at the region |µx| ≪ µ mogeneity along the x axis. Since the normal reflection
and µx < 0 where each quasiclassical branch splits by two is inherently missing in quasiclassicalAndreev equations
modes. In accordance with general topological argument the (without diagonal potential) this approximation fails to
onlyonezerothbranchremainsatfixedvalueofpy (andspin describe zero modes and Fermi arcs on the domain wall
projection σx for the domain wall of the A phase) shown by in 3He-A and the AB interface.
theredsolidline. Thespectrumofdelocalizedstatesisshown
The modification of quasiclassical results due to the
by yellow shading and the edge of continuum by thin dotted
normalreflectioncanbeanalyzedbyemployingthesemi-
line.
classicalapproximation(see Section IIIB1) whichyields
threepossibletypesofclosedorbitsin(p ,x)phasespace.
x
8
In case when quasiclassical approximation is valid the establishedonlybeyondthequasiclassicalapproximation
regime (ii) of only Andreev reflection is realized with by solving the exact BdG system numerically. The do-
two separate orbits at p > 0 and p < 0 shown by the main wall in A-phase contains 4 Fermi arcs, if one takes
x x
greensolidlines inFig. 4a. The Bohr-Sommerfeldquan- into account spin degrees of freedom, while the AB in-
tization (17) yields the same energy of bound states for terface contains 2 Fermi arcs, i.e. the same number as
the two orbits whichis degenerateby sign(p ) andcoin- thesurfaceoftheA-phase. Thereasonforthisdifference
x
cides well with quasiclassical result for n >0. However is that the Fermi arc is determined by the Weyl points
q
for smaller µ two Andreev orbits merge into a single in the bulk states on two sides of the wall. The 3He-B,
x
one shownin Fig.4a by blue dash-dottedline. This orbit though being a topological superfluid, does not contain
hasbothAndreevandnormalreflectionpointsandthere- Weyl points and thus does not contribute to the number
foreisnotdescribedbythequasiclassicalapproximation. of the Fermi arcs. Note that the spectrum of localized
Moreover merging of Andreev orbits doubles the orbit fermions at the domain wall of the A phase is identical
area. Therefore the number of subgap states accord- tothespectrumofelectronicstatesboundatthedomain
ingtoBohr-Sommerfeldquantizationisdoubledwhichis wall in p+ip superconductor Sr RuO .
2 4
manifested by the appearance of the second semiclassi- Majorana fermions living on the Fermi arc at the AB
cal branchin Fig.4b simultaneously with merging of two interface may give an additional contribution to the cal-
Andreev orbits in Fig.4a. In exact spectrum obtained culatedfriction forceacting on the movinginterface33 at
by solving numerically the BdG system the branch dou- very low temperatures. This is important for the devel-
bling is manifested in splitting of quasiclassicalbranches opmentoftheKelvin-Helmholtzinstabilityofthemoving
in Figs.5,6. Thus the branch splitting is the direct re- AB interface observedin Ref. 34 (see also Chapter 27 of
sult of normal reflection of quasiparticles moving almost Ref. 2). The latter instability has an analogue with the
parallel to the domain wall. ergoregioninstabilityofblackholesdiscussedinChapter
Affected by the enhanced normal reflection the be- 32 of Ref. 2.
haviour of the energy branches changes in px 0 limit We considered the simplest most symmetric realiza-
→
qualitatively compared to the quasiclassical result. At tions of the A-A and A-B interfaces. The problem for
fixedvalueofmomentumprojectionpy (andspinprojec- future investigations is weather the Fermi arc survives
tionσx incaseofAphasedomainwall)thereis onlyone or not, if the symmetry of the interface is violated. For
branchleft whichintersectstheFermilevelandsupports that the relative homotopy group formalism applied to
Majorana states in accordance with the topological in- Green’s function35 is probably required.
dex theorem. The rest of spectral branches become non-
monotonic at small values of p = √2mµ at µ > 0
x x x
| |
and turn upwards at µ <0 as can be seen in the Fig.5.
x
VI. ACKNOWLEDGEMENTS
V. CONCLUSION This work was supported, in part by the Academy of
Finland (Centers of Excellence Programme 2012-2017),
We demonstrated that Fermi arcs on the domain wall the EU 7th Framework Programme (FP7/2007-2013,
in the A-phase andat the interface betweenthe A-phase Grant No. 228464 Microkelvin), by Russian Foundation
and the B-phase of superfluid 3He obey the index theo- forBasicResearchGrantNo. 11-02-00891-a,Presidential
rem,whichconnectstopologicalpropertiesofbulkstates RSSCouncil(GrantNo. MK-4211.2011.2),byPrograms
on two sides of the walls with topology of zero energy of RAS “Quantum Physics of Condensed Matter” and
bound states. In contrast to the other systems support- “Stronglycorrelatedelectronsinsemiconductors,metals,
ing Fermiarcsthe bulk-edge correspondenceherecanbe superconductors and magnetic materials”.
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