Identifying the pairing symmetry in sodium cobalt oxide by Andreev edge states

0.93 MB·English

by  Wen-Min Huang

Identifying the pairing symmetry in sodium cobalt oxide by Andreev edge states
Wen-Min Huang and Hsiu-Hau Lin
Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan and
Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
(Dated: January 28, 2009)
We study the Andreev edge states with different pairing symmetries and boundary topologies
on semi-infinite triangular lattice of Na CoO ·yH O. A general mapping from the two dimensional
x 2 2
latticetotheonedimensionaltight-bindingmodelisdeveloped. Itisshownthatthephasediagram
of the Andreev edge states depends on the pairing symmetry and also the boundary topology.
Surprisingly, the structure of the phase diagram crucially relies on the nodal points on the Fermi
9 surface and can be explained by an elegant gauge argument. We compute the momentum-resolved
0 local density of states near the edge and predict the hot spots which are measurable in Fourier-
0 transformed scanning tunneling spectroscopy.
2
n PACSnumbers: 74.78.-w,74.20.-z,74.45.+c
a
J
I. INTRODUCTION ing symmetry of the gap function in Na CoO 19. The
8 x 2
underlaying triangular lattice is proposed to host the
2
The recent discovery of superconductivity in sodium resonating-valence-bond (RVB) state for an unconven-
] tional superconductor20. Base on the RVB picture, the-
cobalt oxide compound intercalated water molecules,
n
o NaxCoO2·yH2O,1 trigged intense attentions and stimu- oretical investigations on the t-J model21,22 favor the
c latedlotsofdiscussions2. Thesuperconductivityinduced dx2−y2 + idxy symmetry. However, within the third-
- intheplanerstructureofCoO issimilarwiththatinthe order perturbative expansions, a stable f-wave pairing
pr CuO planeofcuprates3,4. Ho2wever,theunderlayingtri- is found in the Hubbard model23 with repulsive on-
2
u angular lattice of the Co atoms is fundamentally differ- site interaction. The same conclusion is reached from
s the theoretical study on the single-band extended Hub-
ent from the square lattice of the Cu atoms in cuprates
t. because the antiferromagnetic interactions on the trian- bardmodelwithinrandomphaseapproximations24. Fur-
a
thermore, recent discovery of the Hubbard-Heisenberg
m gular lattice are frustrated. The carrier density in the
model on the half-filled anisotropic triangular lattice
sodium cobalt oxide can be tuned by the Na concentra-
- showthatvaryingthefrustrationt(cid:48)/tchangesthespatial
d tion. By changing the sodium doping, a rich phase dia-
n gram appears and the superconductivity occurs5,6 in the anisotropy of the spin correlations and leads to transi-
o doping regime 1/4 < x < 1/3. Furthermore, the study tions of the pairing symmetries of the superconducting
c oder parameter25. Taking different routes for theoretical
in Co-NMR and Co-NQR found that the spin-lattice re-
[ investigations, other groups demonstrate the possibility
laxation rate at the critical temperature (T ) shows no
1 coherent peak and follows a power below T ,chiniting an of the px +ipy pairing26,27. In addition, starting from
c
v unconventional superconducting phase7,8,9. The node of the fluctuation-exchange approximations, the triplet f-
0 wave and p-wave pairings are favored on the triangular
thesuperconductinggapisconfirmedbythespecific-heat
7 measurements10 and also by the muon spin relaxation lattice28. Withthesameapproximations,solvingthelin-
45 experiments11. earized E´liashberg equation29 leads to dominant pairing
in the spin-triplet f-wave sector. Therefore, the pairing
. However, the symmetry of the Cooper pairs remains
1 symmetry also posts a challenging task for theoretical
unknown at present. In order to identify the pair-
0 understanding from the microscopic perspective.
9 ing symmetry, the measurement of spin susceptibility in
0 the superconducting state through the Knight shift is While it is important to determine the pairing sym-
: helpful12,13,14. The measurements of the powder sam- metry from microscopic approaches, it is equally cru-
v
i ples show that the Knight shifts along the c-axis do not cial to develop phenomenological theories so that one
X decrease below Tc, raising the possibility of spin-triplet can extract the pairing symmetry from the experimen-
r superconducting state13,14,15. On the other hand, recent tal data30,31 such as the Andreev bound states32,33 near
a
measurements on the single-crystal samples16 show that theedgesofthesuperconductors. NotethattheAndreev
the Knight shift decreases below Tc along the a- and c- edgestate34 inasuperconductoristiedupwiththepair-
axes, which suggests for the spin-singlet pairing instead. ing symmetry in the bulk. In addition, recent break-
From the study of the normal-state Fermi surface topol- throughs in the Fourier-transformed scanning tunneling
ogy by the angle-resolved photoemission spectroscopy17 spectroscopy(FT-STS)experiments35,36allowfurtherin-
andtheMndopingeffects18,italsoseemstosupportthe sightintotheedgestateswithmomentumresolutions. In
singletsuperconductingstate. Thus, thepairingsymme- these experiments, not only the spatial profile of the lo-
try of superconductivity in NaxCoO2·yH2O compounds caldensityofstates(LDOS)canbemeasured, thepeaks
remains controversial at the point of writing. of the LDOS in the momentum space can also be deter-
Therearealsotheoreticaleffortstopindownthepair- mined by appropriate Fourier analysis of the experimen-
2
Zigzag edge Flat edge Pairing symmetry II. BOGOLIUBOV-DE GENNES
HAMILTONIAN AT ZIGZAG EDGE
yes no p or f
x
no yes p
y
To accommodate different pairing symmetries within
yes yes d
xy one theoretical framework, it is convenient to start from
no no dx2−y2 or s the Bogoliubov-de Gennes (BdG) Hamiltonian40,
(cid:88) (cid:88)
TABLEI:ExistenceofAndreevedgestateatzigzagandflat H = t c†(r)c (r(cid:48))−µ c†(r)c (r)
BdG σ σ σ σ
edges and its implication for pairing symmetry. (cid:104)r,r(cid:48)(cid:105),σ r,σ
(cid:88) (cid:104) (cid:105)
+ ∆∗(r,r(cid:48))c (r)c (r(cid:48))+∆(r,r(cid:48))c†(r(cid:48))c†(r) ,(1)
↑ ↓ ↓ ↑
(cid:104)r,r(cid:48)(cid:105)
tal data. In a letter published by one of the authors33,
whereonlythenearest-neighborhoppingandpairingare
a theoretical approach was developed to compute the
included. Because the particle-hole symmetry is absent
momentum-resolved LDOS for the Andreev edge state
in the triangular lattice, the sign of the hopping ampli-
in sodium cobalt oxide with f-wave pairing symmetry. tude t is crucial. Recent experiments41,42 suggest that
The exponential decay away from the boundary can be
the maximum of the band occurs at the Γ point, which
comparedwiththeexperimentsdirectly,whilethedepen-
implies t > 0. The pairing amplitudes are either sym-
dence upon the transverse momentum (along the edge metric∆(r,r(cid:48))=∆(r(cid:48),r)orantisymmetric−∆(r(cid:48),r)de-
wherethesystemistranslationalinvariant)canbeseem
pending on the Cooper pairs are spin singlets or triplets.
in Fourier space through scattering processes. Here, we
In this paper, we will discuss two natural boundary
elaborate and extend the previous work by considering
topologies of a triangular lattice – zigzag and flat edges,
gap functions of p-, d- and f-pairing at both zigzag and
as showed in Fig. 1 and Fig. 13 respectively. The con-
flat edges and predict the position of the sharp peaks
ventions for the spatial coordinates and also the pairing
that can be observed in FT-STS experiments.
symmetries can be found in the figures as well. For in-
Westartwiththetwodimensional(2D)Bogoliubov-de stance, the zigzag edge is chosen to lie in the y-axis and
GennesHamiltonianandmapthesemi-infinitetriangular the flat edge along the x-axis in our convention.
lattice to a collection of one-dimensional (1D) chains, la- We start with the zigzag edge first, by cutting the in-
beled by the transverse momentum along the boundary. finite triangular lattice along the y-axis. Note that the
Duetothehiddenstructureoftheseeffective1Dmodels, semi-infinite lattice is still translational invariant along
the AES can be categorized into the positive and neg- the boundary and thus can be mapped onto a collection
ative Witten parity states37 in supersymmetric (SUSY) of semi-inifinite 1D chains, carrying definite transverse
algebra. For readers no familiar with the Witten parity momentum ky after partial Fourier transformation. One
and the SUSY algebra, we have included a brief intro- important subtlety about partial Fourier transformation
duction in Appendix A. By computing the Witten par- is the folding of Brillouin zone. The conventional hexag-
ity states constrained by the boundary conditions, the onal shape must be reshaped into appropriate rectan-
LDOS with specific transverse momentum is obtained. gular one so that the summations over kx and ky are
Furthermore,wecanpredictthehotspotsinFT-STSby decoupled43. For the zigzag edge, the reconstruct rect-
spotting all momentum differences between sharp peaks angle Brillouin zone is shown in the bottom of Figs. 1,
intheLDOS.OurresultsshowthattheexistenceofAES 8 and 10. After the partial Fourier transformation, the
sensitivelydependsonthepairingsymmetryandtheedge Hamiltonian for the collection of the effective 1D chains
topologyandcanthusbeusedasagoodindicatorofthe along x-direction is
underlying pairing symmetry. The existence of the AES (cid:32) (cid:33)
(cid:88) H P
for different pairing symmetries and edge topologies are H = Φ†(k ) ν Φ(k ), (2)
summarized in Table. I. Finally, following an elegant ky y Pν† −H y
gauge argument devised by Oshikawa38,39, we also find
with ν = f,d and p, which are denoted as f-, d- and p-
thatthephasediagramfortheAEScruciallydependson
wave pairing respectively. Here we introduce the Nambu
the nodal points on the Fermi surface where the pairing (cid:104) (cid:105)
amplitude vanishes. basis Φ†(ky) = c†↓(x,ky), c↑(x,−ky) and the semi-
infinite matrix for the hopping term of semi-infinite 1D
Therestofpaperisorganizedasthefollowings. InSec.
chains,
II, we introduce the 2D Bogoliubov-de Gennes Hamilto-
nian for a triangular lattice with the zigzag boundary −µ t t 0 0 ···
1 2
topology. By transforming the Hamiltonian into Super-
 t −µ t t 0 ···
symmetric form and use the generalized Bloch state, the  1 1 2 
 t t −µ t t ···
LDOS of AES is obtained. In Sec. III, in the same spirit H= 2 1 1 2 , (3)
 
and method, we will compute the LDOS of AES for the  0 t2 t1 −µ t1 ···
 
flat edge. We will discuss about the gauge argument of  · · · · · ···
phase diagram and draw a conclusion in Sec. IV. · · · · · ···
3
√
withtheeffectivehoppingamplitudet =2tcos( 3k /2)
1 y
and t = t. The momentum dependence of the matrix
2
elements is a consequence of the partial Fourier trans-
formation. The pairing potential P with different sym-
ν
metries will be studied in details in the following subsec-
tions.
A. f-wave paring
We start with the AES of f-wave pairing symmetry
at zigzag edge. The f-wave symmetry carries angular
momentum l = 3 and thus corresponds to spin-triplet
pairing required by Fermi statistics. It implies that the
pairing potential is antisymmetric, ∆(r,r(cid:48)) = −∆(r(cid:48),r).
Taking the tight-binding approximation, the pairing po-
FIG. 1: (Color Online) Gap function with the f-wave sym-
tential is rather simple ∆(r,r(cid:48))=∆(θ)=∆cos(3θ) with
metry at the zigzag edge of a triangular lattice. The sign
the relative angle θ =2nπ/6 where n is an integer. The
convention of the pairing potential is shown in the shaded
sign convention for different bond orientations is fixed in
hexagon. The bottom figure represents the Fermi surface in
Fig. 1. Wecansolveforthenodallinesbysettingthegap
√ the reshaped Brillouin zone for the zigzag edge. The nodal
(cid:2) (cid:0) (cid:1)(cid:3)
functiontozero, cos(kx/2)−cos 3ky/2 sin(kx/2)= lines of the f-wave gap function are shown in blue lines and
0. These nodal line are drawn in the reshaped Brillouin the nodal points are the intersections of the Fermi surface
zone in Fig. 1. At different fillings (chemical potentials), contour and the nodal lines.
thenodalpointsaretheintersectionsoftheFermisurface
contourandthenodallines. Thesenodalpointsturnout
to be the key for determining the structure of the phase
completelygeneralandworkfordifferentpairingsymme-
diagrams for the AES. The presence of the open bound-
tries.
arycomplicatesthestoryandweneedtowritedownthe
For current case, P† = −P for f-wave pairing, the
pairing potential in the coordinate space. After some al- f f
new basis for the SUSY form is
gebra, the semi-infinite matrix P of Eq. (2) takes the
f
form, (cid:34) (cid:35)
c (x,k )−c†(x,−k )
Ψ (k )= ↓ y ↑ y , (7)
 0 −∆ ∆ 0 0 ··· f y c (x,k )+c†(x,−k )
1 ↓ y ↑ y
 ∆ 0 −∆ ∆ 0 ···
 1 1 
−∆ ∆ 0 −∆ ∆ ··· and the matrix elements of Af are
P = 1 1 , (4)
f  0 −∆ ∆1 0 −∆1 ··· (cid:32)√ (cid:33)
  3
 · · · · · ··· Tf = 2(t∓∆)cos k ,
1,¯1 2 y
· · · · · ···
with ∆ = 2∆cos(cid:0)√3k /2(cid:1). A simple unitary transfor- T2f,¯2 = t±∆. (8)
1 y
mationbringstheHamiltonianintoSUSYformdescribed
in Appendix A. The effective Hamiltonian37,44 in canon- ItwillbecomeclearlaterthatT1f (cid:54)=T¯1f andT2f (cid:54)=T¯2f are
the crucial for the existence of the edge states.
ical SUSY notation is
For the Hamiltonian in Eq. (5), the zero-energy states
(cid:32) (cid:33)
(cid:88) 0 A are “nodal”, i.e. half of the components in the spinor
H = Ψ†(k ) ν Ψ (k ), (5)
ν y A† 0 ν y vanish,andcanbeclassifiedbytheso-calledSUSYparity
ky ν (see Appendix A),
where the matrix A takes the general form
ν (cid:32) (cid:33) (cid:32) (cid:33)
0 ψ (x)
−µ T1ν T2ν 0 0 ··· |Ψ−(cid:105)= ψ (x) , |Ψ+(cid:105)= +0 . (9)
 Tν −µ Tν Tν 0 ··· −
 ¯1 1 2 
A = T¯2ν T¯1ν −µ T1ν T¯2ν ···. (6) It is straightforward to show that the Harper equations
ν  0· T·¯2ν T·¯1ν −·µ T·1ν ······ baebciotm. Tehdeecsoouluptleiodnfowritthhepzoesriot-iveneeWrgiyttsetnatpeasraitnydψsi+m(xp)lifiys
annihilatedbyA†, i.e. itbelongstothenullspaceofthe
· · · · · ··· ν
operator. Similarly, the solution with negative Witten
Althoughweconcentrateonthef-wavesymmetryinthis parity ψ (x) spans the null space of the operator A .
− ν
section, the derivations of the matrix elements of A are It is worth emphasizing that bring the Hamiltonian into
ν
4
theSUSYformsimplifiesthealgebraandallowsanalytic zΓ zΓ
2
calculations for AES as derived here.
Toincludetheopenboundarycondition,theedgestate " " " "
2
can be constructed by the generalized Bloch theorem33.
Taking states with negative Witten parity as a work-
ing example, one can construct an edge state from ap- 1
propriate linear combinations of the zero-energy modes, 1
ψ (x) = (cid:80) a (z )x. Since the zero energy modes sat-
− γ γ γ
isfy A ψ (x) = 0, z is a solution of the following char-
ν −
acteristic equation,
0 0
1 1 ! Π 0 Π ! Π 0 Π
Tν +Tν +Tνz+Tνz2 =µ. (10) 3 3 3 3
¯2 z2 ¯1 z 1 2 zΓ zΓ
! ! ! !
Itisclearthatthealgebraicequationgivesfoursolutions
" " " "
of z for the given chemical potential and the transverse
momentum. However, not all solutions are allowed. For 1 1
the infinite lattice, the wave function must remain finite
at infinities, |ψ(∞)| < ∞ and |ψ(−∞)| < ∞. It implies
that only |z| = 1 solutions are allowed. These are the
plane-wavesolutionswithrealmomentumdefinedasz =
eik. However, for an open boundary with zigzag shape,
the boundary conditions change to
0 0
! Π 0 Π ! Π 0 Π
3 3 3 3
ψ(−1)=0, ψ(0)=0, |ψ(∞)|<∞. (11)
FIG.!2: (Color Online) Th!e ky mo!mentum dependence!of |z|
Thus, |z|≤1 is required to keep the wave function finite
forthegeneralizedBlochstateswiththef-wavesymmetryat
which is less strict than the |z| = 1 criterion for trans- thezigzagedge. Thelineswithdifferentcolorsrepresentfour
lational invariant systems. However, we have additional solutions of the Bloch state with the parameters, t = 1 and
two boundaries conditions at x = 0,−1, the edge state ∆=0.4. Thechemicalpotentialsinthetop-leftandtop-right
does not always exist, unless we have enough |z | ≤ 1 figuresareµ/t=4andµ/t=1respectively. Thebottom-left
γ
zero-modes to construct the edge states. and bottome-right figures are for µ/t=−1 and µ/t=−2.5.
Herecomesthesimplecounting. Ifallofthefoursolu-
tions satisfy |z | ≤ 1, we can construct two edge states.
γ
If three solutions are found, one edge state can be con-
structed. Otherwise, there will be no edge state. In the lines)andthusoneedgestatestartstoemerge. TheAES
case of the f-wave pairing symmetry, we plot the magni- with negative Witten parity is marked by yellow color in
tude of the solutions |z | as a function of the transverse the phase diagram. Moving toward to zone center, the
γ
momentum k in Fig. 2. The z-plot sensitively depends numberofdesiredsolutionreducestoone(greenline)af-
y
on the chemical potential µ. terpassingthetwo-folddegeneratenodalpoint. Thus,no
Now we would like to explain how to obtain the phase edge state in presence in this regime. Due to the parity
diagram for AES from the z-plots. We start with the symmetryiny-direction,thephasediagramissymmetric
first z-plot (upper left) in Fig. 2 where the chemical po- when k →−k .
y y
tential is µ/t=4 and the pairing potential is ∆/t=0.4. What about the edge state |Ψ (cid:105) with positive Wit-
+
There are four intersections with |z| = 1 dashed line. ten parity? One should repeat the derivation for the
These are the nodal points. At larger momentum, the characteristic equation and look for |z| ≤ 1 solutions to
|z| = 1 dashed line intersects with one solution (orange construct the edge states again. However, there is some
line) and gives rise to the nodal point. At small mo- symmetry hidden in the algebraic equation and the rep-
mentum, the dashed line intersects with two degenerate etitionisnotnecessary. SincethematricesAandA† are
solutions (orange and blue lines) at the same time and hermitianconjugatetoeachother,thealgebraicequation
corresponds to a pair of degenerate nodal points. These forthepositiveWittienparitymodescanbeobtainedby
nodal points correspond to the single and double circles replacing z → 1/z in Eq. 10. That is to say, the decay-
in the phase diagram. Now we can proceed to determine ingsolutionsforpositiveWittenparitycanbecalculated
how many edge states |Ψ−(cid:105) with negative Witt√en parity fromthe|z|≥1solutionsinEq.10. Thisrelationisvery
canbefound. Nearthezoneboundaryk =π/ 3,there helpful in constructing the remaining part of the phase
y
are two solutions (green and blue lines) with |z| ≤ 1. diagram. Near the zone center in the first z-plot (upper
Since there are two constraints from the zigzag bound- left) in Fig. 2, there are three |z| ≥ 1 solutions and cor-
ary, no edge state can be constructed. Passing the nodal respond to one edge state with positive Witten parity.
point, there are three solutions (green, blue and orange In other regimes, no such edge state exists. Combining
5
0
Μt"1
!"#$t"%
’ + ’
!
&"#$t"!
’ ’ + ’ ’ 0.6
Π
LDOS0.4
’!"#$t"& 0.2 3
’ + ’ 0
0 ky "
’("#$t"’! 1100
2200
xx 3300 Π
FIG. 3: (Color Online) Phase diagram for AES with the #
40
3
f-wave pairing at different chemical potentials. The single
point circle mark the nodal points without degeneracy while LDOS
thedoublecircledenotethetwo-folddegeneratenodalpoints. 0.30 "
The green/yellow colors denote the SUSY parity ±1 and the
0.25
single/doublelinesmeanone/two-folddegenerateedgestates.
0.20
0.15
Μt"4 0.10
0.05
! x
0 5 10 15 20
0.15 FIG.5: (ColorOnline)momentumresolved(top)andthein-
LDOS0.1 Π tegrated(bottom)LDOSforthef-wavepairingatthechem-
3 ical potential µ/t=1.
0.05
0
0 ky "
1100
2200
xx 3300 Π the edge state at specific transverse momentum ky. We
40 # can also integrate over the Brillouin zone to obtain the
3
spatial profile for LDOS that can be measured directly
LDOS
in the STM experiments. Furthermore, the momentum-
0.08 "
resolvedLDOSprovidesadditionalinformationaboutthe
enhancedspectralweightofthequasi-particlesatspecific
0.06
transverse momenta. Thus, we can predict the evolution
of the so-called “hot spots” in the FT-STS experiments.
0.04
Let us elaborate on the physical properties of the
0.02 AES now. In order to visualize these edge states bet-
ter, we calculate the local density of states D(x,k ) =
y
x (cid:80) |ψ (x)|2δ(E) versus transverse momentum k , as
0 10 20 30 40 j j y
shown in the top panels of Figs. 4, 5 and 6 at different
chemical potentials. Noted that the edge states merge
FIG.4: (ColorOnline)LDOSforthef-wavepairing(topfig-
into the bulk at the nodal points and the weighting of
ure) along the x-direction at different transverse momentum
k atthechemicalpotentialµ/t=4. Thebottomfigureshows theLDOSissuppressedtozero. Furthermore,thelattice
y
the spatial trend of the integrated LDOS over the Brillouin approach reveals a much richer spatial structure in com-
zone that can be measured directly from STM experiments. parison with the conventional Andreev equations in the
continuous limit. For instance, the LDOS has a strong
dependence on the transverse momentum with transpar-
ent peak structures. For 2 < µ/t < 6, there are three
the results for both Witten parities, the first part of the peaks separated by the nodal points and the peak posi-
phase diagram in Fig. 3 is obtained. tions change with the chemical potential. At µ/t = 2,
Since we compute the value of z for each transverse the outer peaks move to the boundary of Brillouin zone
momentum k , the quasiparticle wave function of Bloch andmergeintoone. Therefore,for0<µ/t<2,thereare
y
statescanbeobtainstraightforwardly. Thus,inaddition only two peaks located at the center and the boundary
to the phase diagram, we can also compute the LDOS of of the Brillouin zone and the locations of the peaks do
6
Μt"#1 !ky
"
3
!
0.6
Π "
LDOS0.4
0.2 3 2 3
0
0 ky " peak 1
1100 peak 2
2200
xx 3300 Π 0 µ/t
# !3 !2 0 2 4 6
40
3
LDOS
" FIG. 7: (Color Online) Predicted hot spots in FT-STS ex-
0.5 perimentforthef-wavesymmetrynearthezigzagedge. The
hot spots are obtained by locating the momentum difference
0.4
betweenlargepeaksinLDOS.Afterµ/t<2,thesecondpeak
0.3 disappearssincetheouterpeaksmergeintooneatthebound-
ary of Brillouin zone.
0.2
0.1
√
x zone. Thus,themomentumtransferisalwaysπ/ 3that
0 5 10 15 20
is half of the Brillouin zone.
FIG.6: (ColorOnline)momentumresolved(top)andthein-
tegrated(bottom)LDOSforthef-wavepairingatthechem-
ical potential µ/t=−1. B. d -wave pairing
xy
For easier experimental comparisons, we would also
workoutsomeotherpairingsymmetriesexplicitly. Since
not change with the chemical potential. Further reduc- the derivations are rather similar, we would skip the re-
ing the chemical potential to the regime −2 < µ/t < 0, peatedpartsandconcentratedonthedifferentoutcomes.
therelativeweightsofthepeakschangebutthelocations Now we turn to the d pairing symmetry at the zigzag
xy
remain fixed. edge. For d pairing symmetry, Cooper pairs form spin
xy
By integrating over the Brillouin zone, we can com- singlets and the gap function in the coordinate space
pute the spatial profile of the LDOS in coordinate space is thus symmetric, ∆(r,r(cid:48)) = ∆(r(cid:48),r). Again, within
as shown in the bottom panels of Figs. 4, 5 and 6 at dif- the tight-binding approximations, the pairing potential
ferentchemicalpotentials. Ontopofthedecayingtrend, is rather simple, ∆(r,r(cid:48)) = ∆sin2θ, with relative angle
theLDOSalsoshowsnon-trivialoscillationduetoquan- θ =2nπ/6wherenisaninteger. Thenodallines,satisfy-
√
tum interferences due to different zero modes. These ing the constraint sin(k /2)sin( 3k /2) = 0, are shown
x y
oscillations can only be captured faithfully in the lattice inthereconstructedBrillouinzoneinFig. 8. TheHamil-
approach. For instance, at µ/t = 4, the LDOS at the tonian for the hopping is identically the same so that we
outmost edge site is not the largest as one would naively donotputitdownagain. Ontheotherhand,thepairing
expectsointhecontinuoustheory. Furthermore, thede- potential consists of another semi-infinite matrix P ,
d
cay length is smaller as the chemical potential decreases.
The momentum-resolved LDOS can also help us to  0 −i∆ 0 0 0 ···
2
determine the hot spots due to quasi-particle scatter-
i∆ 0 −i∆ 0 0 ···
ing/interferences in FT-STS experiments. By Fourier  2 2 
 0 i∆ 0 −i∆ 0 ···
analysis of the STM data, the momentum transfer be- P = 2 2 , (12)
d  
tween quasi-particle scattering is revealed. The momen-  0 0 i∆2 0 −i∆2 ···
 
tum transfer associated with the scattering process be-  · · · · · ···
tween peaks in LDOS will emerge after Fourier trans- · · · · · ···
formation. In Fig. 7, the momentum transfers between
peaksinLDOSareplottedversusthechemicalpotential. with the effective 1D pairing potential ∆ =
√ √ 2
Forµ/t>2,therearethreepeaksgivingtwospecificmo- 3∆sin( 3k /2). Note that the next nearest-neighbor
y
mentumtransfers. Forµ/t<2, thereareonlytwopeaks pairing potentials are absent due to the nodal structure
located at the center and the boundary of the Brillouin of d -wave pairing symmetry along the x-direction (see
xy
7
FIG. 9: (Color Online) Phase diagram for AES with the d
xy
pairing in the presence of the zigzag edge. The meanings of
the labels are the same as in the f-wave case.
nodal points move backward to the center again. For
FIG. 8: (Color Online) Gap function with the d symmetry −3<µ/t<−2, no edge state can be found. It is worth
xy
at the zigzag edge of a triangular lattice. The bottom figure mentioning that the nodal point connecting edge states
representsthereshapedBrillouinzone,nodallinesandnodal with opposite Witten parities must be two-fold degener-
pointsforthed symmetricgapfunctionwiththesamecon- ate by simple counting. Finally, we also calculated the
xy
vention as explained in the f-wave case. hot spots at different chemical potentials, as shown in
Fig. 12, which can be measured in FT-STS experiment.
Fig. 8). Making use of P = P†, a unitary transforma-
d d C. px-wave paring
tion is devised,
(cid:34) (cid:35) We come to the last case at the zigzag edge – the
c (x,k )−ic†(x,−k )
Ψd(x,ky)= c↓↓(x,kyy)+ic†↑↑(x,−kyy) , (13) ppxroxpiamiraitnigonss,ymthmeetgrayp. fuWncitthioinn itshe∆(trig,hrt(cid:48))-bi=ndi∆ngsinapθ-,
with relative angle θ = 2nπ/6 where n is an inte-
to bring the BdG Hamiltonian into the SUSY form in ger. The nodal lines in the momentum space, as
Eq.(5). Although the pairing symmetry is different, the shown in Fig. 10, is determined by the constraint,
√
(cid:2) (cid:3)
structure of the SUSY Hamiltonian remain the same 2cos(k /2)+cos( 2k /2) sin(k /2) = 0. Following
x y x
form. After some algebra, the off-diagonal components the same steps, the semi-infinite matrix P is
p
of the semi-infinite matrix A are,
d
 
0 ∆ ∆ 0 0 ···
(cid:32)√ (cid:33) (cid:32)√ (cid:33) 3
T1d,¯1 = 2tcos 23ky ∓√3∆sin 23ky , −−∆∆3 −∆0 ∆03 ∆∆ ∆0 ······
P = 3 3  (15)
p  
T2d,¯2 = t. (14)  0 −∆ −∆3 0 ∆3 ···
 · · · · · ···
As mentioned before, the hidden SUSY in the BdG · · · · · ···
Hamiltonian makes the zero-energy modes nodal for all
√
pairingsymmetries. Repeatingthesamecalculations,the with the effective gap potential ∆ =∆sin(cid:0) 3k /2(cid:1). It
3 y
z-plotsareobtainedatdifferentchemicalpotentials. The is clear that P = −P†. Since the semi-infinite matrix
only differences are the matrix elements Td and Td p p
1,¯1 2,¯2 Pp sharethesamepropertyasPf forthef-wavepairing,
due to different pairing symmetry. By counting the de- the same basis, Eq.(7), can utilized to bring the BdG
caying modes with |z| ≤ 1, we can construct the phase Hamiltonian into the SUSY form.
diagram for AES with dxy pairing symmetry as shown After some algebra, the matrix elements of the semi-
in Fig. 9. The phase diagram for dxy symmetry ap- infinite matrix Ap in Eq.(5) can be computed,
pears to be much simpler since the number of nodes are
reduced and the only double node lie in k = 0. Start- (cid:32)√ (cid:33)
y 3
ing from the regime 2 < µ/t < 6, there exists an edge T1p,¯1 = (2t±∆)cos 2 ky ,
statewiththepositive/negativeWittenparitydepending
on the sign of the transverse momentum. When reach- Tp = t±∆. (16)
2,¯2
ing µ/t = 2, the nodal points move to the boundary of
the reshaped Brillouin zone so that the edge state ex- Following the same steps to obtain the z-plot, we can
ists for every transverse momentum. Further reducing count the number of decaying modes with |z| ≤ 1. The
the chemical potential to the regime −2 < µ/t < 2, the sameconstructionleadstothephasediagramofAESfor
8
!k
y p
x
" d
xy
3
"
2 3
0 µ/t
!3 !2 0 2 4 6
FIG. 12: (Color Online) Predicted hot spots in FT-STS ex-
perimentforthed orp pairingsymmetriesnearthezigzag
xy x
edge. Thehotspotsareobtainedbylocatingthemomentum
FIG. 10: (Color Online) Gap function with the px symme- difference between large peaks in LDOS.
try at the zigzag edge of a triangular lattice. The bottom
figurerepresentsthereshapedBrillouinzone,nodallinesand
nodalpointsforthep symmetricgapfunctionwiththesame
x
convention as explained in the f-wave case.
instead. Later, we will find that it also happens for the
p pairing symmetry at the flat edge. Again, the under-
y
lying reason is that the gap function only changes signs
across the open boundary of the system.
III. BOGOLIUBOV-DE GENNES
HAMILTONIAN AT FLAT EDGE
By cutting the triangular lattice in another direction
(along the x-axis), we end up with a semi-infinite lat-
tice with a flat edge as shown in Figs. 13 and 15.
Since the semi-infinite lattice is still translational invari-
FIG. 11: (Color Online) Phase diagram for AES with the p
x antalongthex-direction,thesemi-infintecanbebrought
pairing in the presence of the zigzag edge. The meanings of
into the sum of the 1D chains by partial Fourier trans-
the labels are the same as in the f-wave case.
formationalongtheedgedirection. Notedthat,tomain-
tain the Fermi statistics between the lattice operators,
the Brillouin zone must be reshaped in a different way
thep pairingsymmetryasplottedinFig. 11. Although as shown in Figs. 13 and 15. In the Nambu basis,
x (cid:104) (cid:105)
we do not show the momentum-resolved LDOS for the Φ(cid:101)†(kx,y) = c†↓(kx,y), c↑(−kx,y) , The BdG Hamilto-
present case, it can be computed in a similar way as
nian of the ν-wave pairing symmetry can be represented
for the f-wave pairing. Fig. 12 shows the evolution of
as,
the momentum transfer between the peaks in LDOS at
different chemical potentials and can be compared with (cid:32) (cid:33)
theAhsoFtisgp.ot1s1inshtohwesF,Tfo-SrTtShempexaspuareirminegntssy.mmetry, all H(cid:101) =(cid:88)kx Φ(cid:101)†(kx,y) PH(cid:101)(cid:101)ν† −P(cid:101)Hν(cid:101) Φ(cid:101)(kx,y). (17)
edgestatesliveinthenullspaceofthesemi-inifinitema-
trix A†p, which are rather different from the f- and d- HereH(cid:101) isasemi-infinitematrixfortheeffectivehopping
wavesymmetries. Thatistosay, onlyAESwithpositive in the 1D chains labeled by different momentum k ,
x
Witten parity (according to the our convention here) ex-
ists! Thequalitativedifferencearisesfromthesignofthe −µ(cid:101) (cid:101)t1 0 0 0 ···
gap function across the open boundary. For the p pair-
ing symmetry, the pairing potentials at the edge sxites all  (cid:101)t1 −µ(cid:101) (cid:101)t1 0 0 ···
share the same sign. The pairing potential only changes H(cid:101) = 0 (cid:101)t1 −µ(cid:101) (cid:101)t1 0 ···, (18)
 
signs when crossing the edge along the y-direction. As a  0 0 (cid:101)t1 −µ(cid:101) (cid:101)t1 ···
 
result, the null space of the semi-infinite matrix A van-  · · · · · ···
p
ishes and all edge states belong to the null space of A† · · · · · ···
p
9
FIG. 14: Color Online) Phase diagram for AES with the d
xy
pairing in the presence of the flat edge. The meanings of the
labels are the same as in the f-wave case.
FIG. 13: (Color Online) Gap function with the d symme-
xy rows instead of four,
tryattheflatedgeofatriangularlattice. Thebottomfigure
rpeopinretssefnotrstthheedreshsaypmemdeBtrriiclloguaipnfzuonncet,ionnodwailthlintheseasanmdenocodna-l −µ(cid:101) T(cid:101)1ν 0 0 0 ···
xy
vention as explained in the f-wave case.  T(cid:101)¯1ν −µ(cid:101) T(cid:101)1ν 0 0 ···
A(cid:101)ν = 00 T(cid:101)0¯1ν −T(cid:101)¯1µν(cid:101) −T(cid:101)1µν(cid:101) T(cid:101)01ν ······. (21)
 · · · · · ···
with the momentum-dependent hopping amplitude (cid:101)t1 = · · · · · ···
2tcos(k /2). Note that the chemical potential is renor-
x
malized,µ=µ−2tcos(k )afterthepartialFouriertrans-
(cid:101) x The matrix elements can be worked out explicitly from
formation. Notonlythehoppingmatrixisdifferentfrom
thesemi-infinitematrixP(cid:101)ν inEq.(17)whichdependson
thatforthezigzagedge,theothersemi-infinitematrixP(cid:101)ν thepairingsymmetry. Forthed pairingsymmetry,the
xy
for the pairing potentials with the ν-wave pairing sym-
metrywouldbedifferentaswell. Inthefollowing,wewill semi-infinite matrix satisfies P(cid:101)d =P(cid:101)d†. Thus, the unitary
transformation to the SUSY form is
study the phase diagrams of AES with different pairing
symmetries near the flat edge in details. (cid:34)c (k ,y)−ic†(−k ,y)(cid:35)
Ψ(cid:101)d(kx,y)= c↓(kx,y)+ic↑†(−kx,y) . (22)
↓ x ↑ x
A. d -wave pairing It is straightforward to work out the matrix elements of
xy
the semi-infinie matrix A(cid:101)d,
Forthedxy pairingsymmetry, theAESexistsforboth (cid:18)k (cid:19) √ (cid:18)k (cid:19)
thezigzagandtheflatedges. AfterpartialFouriertrans- T(cid:101)1d,¯1 = 2tcos 2x ∓ 3∆sin 2x . (23)
formation in the x-direction, the pairing potential P(cid:101)d in
Eq.(17) can be explicitly worked out,
Again, the zero-energy modes exhibit the nodal struc-
ture and can be classified into two categories with oppo-
 
0 −i∆(cid:101)1 0 0 0 ··· site Witten parities,
i∆(cid:101)1 0 −i∆(cid:101)1 0 0 ··· (cid:32) (cid:33) (cid:32) (cid:33)
P(cid:101)d = 00 i∆(cid:101)01 i∆(cid:101)01 −i0∆(cid:101)1 −i0∆(cid:101)1 ······ (19) |Ψ(cid:101)−(cid:105)= ψ(cid:101)−0(y) , |Ψ(cid:101)+(cid:105)= ψ(cid:101)+0(y) . (24)
 
 · · · · · ···
· · · · · ··· Here ψ(cid:101)−(y) and ψ(cid:101)+(y) belong to the null space of the
semi-infinite matrices A(cid:101)ν and A(cid:101)†ν respectively. The edge
√ state is constructed from the generalized Bloch theorem.
with ∆(cid:101)1 = 3∆sin(kx/2). To obtain the zero-energy For instance, the edge state with negative Witten parity
states, itisconvenienttobringtheeffectiveHamiltonian is ψ (y)=(cid:80) a (z )y, where z satisfies,
into the SUSY form as in the zigzag case, − λ λ λ
1
H(cid:101) =(cid:88)kx Ψ(cid:101)†ν(kx,y)(cid:32)A(cid:101)0†ν A(cid:101)0ν (cid:33)Ψ(cid:101)ν(kx,y), (20) The above algebraicT(cid:101)¯1eνqzua+tiTo(cid:101)1nνzg=iveµ(cid:101)s.two solutions fo(r25z).
In the presence of the flat edge, the boundary conditions
where ν denotes the pairing symmetry considered. For are slightly different,
the flat edge, the semi-infinite matrix A(cid:101)ν is simpler than (cid:12) (cid:12)
thatforthezigzagedgesinceitonlyhastwooff-diagonal ψ(cid:101)(0)=0, (cid:12)(cid:12)ψ(cid:101)(∞)(cid:12)(cid:12)<∞. (26)
10
FIG. 16: Color Online) Phase diagram for AES with the p
y
pairing in the presence of the flat edge. The meanings of the
labels are the same as in the f-wave case.
FIG. 15: (Color Online) Gap function with the py symmetry !kx p
at the flat edge of a triangular lattice. The bottom figure " y
representsthereshapedBrillouinzone,nodallinesandnodal d
xy
points for the p symmetric gap function with the same con-
y
vention as explained in the f-wave case.
"/2
As before, only decaying modes with |z|≤1 are allowed.
But, only one constraint is required at the flat edge in
contrast to the two constraints for the zigzag edge. The
0 µ/t
simplification is due to the missing matrix elements T(cid:101)¯2ν !3 !2 0 2 4 6
and T(cid:101)ν at the flat edge which makes searching for the
2
AES much easier here. The phase diagram for the AES FIG. 17: (Color Online) Predicted hot spots in FT-STS ex-
with dxy pairing symmetry is shown in Fig. 14. Using periment for the dxy or py pairing symmetries near the flat
the Bloch wave function of those edge states, we obtain edge. Thehotspotsareobtainedbylocatingthemomentum
the LDOS for all transverse momenta k . Then, we can difference between large peaks in LDOS.
x
proceed to predict the sharp peaks in STM data after
Fourier analysis by finding out the momentum transfer
between peaks in the LDOS. The results are plotted in √
Fig. 17 versus the chemical potential µ/t. with ∆(cid:101)2 = 3∆cos(kx/2). Because of P(cid:101)p = P(cid:101)p†, the
HamiltoniancanbebroughtintoSUSYforminthebasis,
(cid:34) (cid:35)
c (k ,y)−c†(−k ,y)
Ψ(cid:101)p(kx,y)= c↓(kx,y)+c†↑(−kx,y) . (28)
↓ x ↑ x
B. p -wave pairing
y Afterstraightforwardalgebra,thecomponentsofthema-
trix A(cid:101)p for the py-wave pairing are
We now continue to study the AES with the p -wave
y (cid:18) (cid:19)
pairing symmetry at the flat edge as shown in Fig. 15. T(cid:101)p = (2t±∆)cos kx . (29)
The nearest-neighbor gap amplitude of p -wave pairing 1,¯1 2
y
takes the form, ∆(r,r(cid:48)) = ∆cosθ, with relative angle
The effective 1D chains for the flat edge are univer-
θ = 0,π/3,2π/3,...,5π/3. The nodal lines in reshaped
sal. Thus, the whole discussions and calculations for the
Brilliouin zone, shown in Fig. 15, are determined by the
√
d-wave pairing with the flat edge can be applied here.
equation sin( 3k /2)cos(k /2) = 0. Again, applying
partial Fourier tryansformatxion to the gap function, we SubstitutingthematrixelementsT(cid:101)1p andT(cid:101)¯1p intoEq.(25)
obtain the semi-infinite matrix for the pairing potential, and combining the boundary conditions of the flat edge,
Eq.(26), we obtain the phase diagram of AES for the
p pairing symmetry as shown in Fig. 16. As we men-
y
 0 ∆(cid:101)2 0 0 0 ··· tioned in previous section, the p-wave pairing potential
changes sign across the edge boundary and lead to edge
−∆(cid:101)2 0 ∆(cid:101)2 0 0 ··· states with positive Witten parity only. Finally, using
P(cid:101)p = 0 −∆(cid:101)2 0 ∆(cid:101)2 0 ···, (27) the Bloch wave function of those edge states, we obtain
 0 0 −∆(cid:101)2 0 ∆(cid:101)2 ··· the LDOS for all transverse momenta k . Then, we can
  x
 · · · · · ··· proceed to predict the sharp peaks in STM data after
· · · · · ··· Fourier analysis by finding out the momentum transfer