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illustrated in Fig. 1 (a).
B.
Origins of phase of pair potential
Let us comment on the correspondence of our bo-
golon model with the original electron degrees of free-
dom, on which bogolons are based. While the origins of
the pair potential of bogolon have already been discussed
in Refs. [21, 22, 37], we briefly revisit the discussion with
a particular focus on the origin of the phase of the pair
potential for bogolons, which is crucial in the Josephson | [
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potential for bogolons, which is crucial in the Josephson
junction. Let us consider the impurity scattering as an
origin of the self-energy. The phase of pair potential for
bogolon can be understood by the following two steps.
The first step is to recognize that impurity effects on
anomalous self-energy for bogolons enter through α†α-
type diagonal scattering (U1) and α†α†-type off-diagonal
one (U2) [37]. The second step is to recognize that the lat- | [
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] |
one (U2) [37]. The second step is to recognize that the lat-
ter scattering potential is composed of the product of two
types of wave functions u (electron wave function) and
v (hole wave function). Therefore, the superconducting
phase of the original electron’s pair potential (arg uv) is
reflected in U2, and is then also inherited to bogolon’s
pair potential Γ2. On the other hand, U1 is composed of
products of u∗u and v∗v, both of which do not carry the | [
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] |
products of u∗u and v∗v, both of which do not carry the
phase of superconducting pair potential. The more de-
tailed expressions are given in Appendix A and Ref. [21].
C.
McMillan Green’s function
In order to solve the Gor’kov equations given in
Eqs. (2) and (3), we employ McMillan formalism [30, 94,
97–102]. We consider the following two eigenequations
for a given ωn:
[ ˆH0(x) + ˆΣ(iωn, x)]Ψ(x) = iωnΨ(x),
(8)
˜Ψ(x)T[ ˆH0(x) + ˆΣ(iωn, x)] = iωn ˜Ψ(x)T.
(9) | [
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] |
[ ˆH0(x) + ˆΣ(iωn, x)]Ψ(x) = iωnΨ(x),
(8)
˜Ψ(x)T[ ˆH0(x) + ˆΣ(iωn, x)] = iωn ˜Ψ(x)T.
(9)
Once these eigenequations are solved, the Green’s func-
tion is then expressed as the bilinear form of the eigen-
functions Ψ and ˜Ψ as will be shown later [see Eq. (15)].
In Eqs. (8) and (9), ˆH0(x) + ˆΣ(iωn, x) can be regarded
as an effective Hamiltonian which is not necessarily Her-
mitian. Then, as we will discuss in Sec. II D, we evalu-
ate the Green’s function using a biorthogonal basis em- | [
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-0.0911206379532814
] |
ate the Green’s function using a biorthogonal basis em-
ployed for non-Hermitian Hamiltonian systems [95, 96].
Note that the junctions with the non-Hermitian system
have been discussed in previous studies [103, 104]. On
the other hand, the non-Hermiticity discussed in this
paper is derived from self-energy.
Therefore, the re-
lation ˆΣ∗(−iωn, x) = ˆΣ†(iωn, x) derived from standard
Lehmann representation is always satisfied in our formu-
lation, from which any unphysical results, such as imagi- | [
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] |
lation, from which any unphysical results, such as imagi-
nary physical quantities, are excluded. In this paper, the
non-Hermitian formalism is just a methodology to solve
Eqs. (8) and (9).
In this subsection, we consider the case of ωn > 0.
Since the self-energies does not depend on the spatial
coordinate x once x > 0 or x < 0 is specified [see Eqs. (6)
and (7)], the eigenfunction for Eq. (8) can be expressed
by the plane wave, as in an ordinary scattering problem. | [
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] |
by the plane wave, as in an ordinary scattering problem.
Although the following argument in this subsection is
essentially the same as the case for the conventional SC
junction, whose ˆΣ given by Eq. (C1) is Hermitian [30,
94, 97–102], it can also be applied to our bogolon model
with Eq. (5).
We need to consider the four types of indepen-
dent eigenfunctions [outgoing/incoming (out/in), parti-
cle/hole (+/−)] shown in Fig. 2 [30, 94, 98–102], which
depend on the incident plane wave indicated by dashed | [
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0.06537697464227676,
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0.05829315632581711,
0.016604211181402206,
0.3812260925769806
] |
depend on the incident plane wave indicated by dashed
lines. For example, the outgoing particle wave function
Ψ(+)
out is given by
Ψ(+)
out(x) =
eik+
L x
�
u+
L
v+
L
�
+ a(+)
outeik−
L x
�
u−
L
v−
L
�
+b(+)
oute−ik+
L x
�
u+
L
v+
L
�
(x < 0),
c(+)
outeik+
Rx
�
u+
R
v+
R
�
+ d(+)
oute−ik−
R x
�
u−
R
v−
R
�
(x > 0),
(10)
which is a linear combination of bulk solutions. In the
above expression, the wave number k±
r , kinetic energy
Ωr(iωn), and bulk wave functions u± | [
-0.19482184946537018,
-0.09552610665559769,
0.7428033351898193,
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0.7081224918365479,
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-0.35047295689582825,
1.1036258935928345,
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-0.21382394433021545,
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-0.5518333911895752,
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-0.6677850484848022,
0.1374848186969757,
-0.00270909839309752,
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0.18270492553710938,
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0.08154570311307907,
-0.6548637747764587,
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0.36215895414352417,
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0.341650128364563,
0.7168784737586975,
0.21019421517848969,
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0.19192247092723846,
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1.1416079998016357,
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0.37271809577941895,
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0.03953371196985245,
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0.3767072856426239,
0.384624183177948,
-0.3338462710380554,
0.1892756223678589,
0.27032437920570374,
0.10543566942214966,
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0.32074496150016785,
0.47280481457710266,
-0.13766181468963623,
0.012086647562682629,
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-0.22123336791992188,
-0.24783219397068024,
-0.7675005793571472,
0.31082674860954285,
0.36590084433555603,
0.35241326689720154,
0.7474074959754944,
-0.26846933364868164,
0.11699317395687103,
0.46598201990127563,
0.26538699865341187,
0.6059508919715881,
1.1325793266296387,
-0.07063165307044983,
0.8462768197059631,
-0.354424387216568,
0.3363751769065857,
-0.49285611510276794,
0.2509304881095886,
0.4458414912223816,
-0.31619054079055786,
0.310036838054657,
0.25821641087532043,
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0.14016535878181458,
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0.7363594174385071,
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0.13870814442634583,
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0.5580756664276123,
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1.0593760013580322,
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0.7691839337348938,
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0.5942059755325317,
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0.4104468822479248,
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0.33764851093292236,
0.22881007194519043,
0.12350244075059891,
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0.4429873526096344,
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0.20961177349090576,
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0.0145034771412611,
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0.19019576907157898,
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0.911352813243866,
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0.005103599280118942,
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0.07637721300125122,
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0.40885964035987854,
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0.5854595899581909,
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0.030709443613886833,
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0.06588303297758102,
0.23553813993930817,
0.3603086471557617,
0.47314539551734924,
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0.06133846193552017,
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0.26334038376808167,
0.3997216820716858,
0.03748048096895218,
0.10326157510280609,
0.0946599543094635,
-0.3607337474822998,
0.3417041599750519,
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0.2921242415904999,
0.16258366405963898,
-0.3631070554256439,
0.26980480551719666,
-0.10278106480836868,
-0.33370596170425415,
-0.7345629930496216,
0.0649866983294487,
-0.4007551372051239,
-0.3093447685241699,
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0.27892017364501953,
0.15678080916404724,
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0.37658292055130005,
0.6747259497642517,
0.006189195439219475,
-0.41101181507110596,
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0.4248749911785126,
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0.036733195185661316,
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0.0005502756685018539,
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0.6308286786079407,
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0.3969234526157379,
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0.4861276149749756,
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0.10683288425207138,
0.13121525943279266,
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0.3797699809074402,
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0.38201621174812317,
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0.020453892648220062,
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0.761689305305481,
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0.7419683933258057,
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0.06356782466173172,
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0.7741353511810303,
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0.5943453311920166,
0.6831373572349548,
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0.4421424865722656,
0.7860627174377441,
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0.09674836695194244,
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0.29196712374687195,
0.5670053958892822,
0.2016080617904663,
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0.16425439715385437,
0.04074867442250252,
0.11815745383501053,
0.087728351354599,
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0.5630850791931152,
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0.3972950875759125,
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0.3654698431491852,
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0.11737193167209625,
0.2001226842403412,
0.5097736716270447,
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0.3655649721622467,
0.56320720911026,
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0.8862020373344421,
0.6494595408439636,
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0.3118475377559662,
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0.3911106586456299,
1.018904209136963,
-0.2582091987133026,
0.4176550805568695,
-0.4854434132575989,
0.8395390510559082,
-0.613455057144165,
-0.08637364208698273,
0.6123838424682617,
-0.1250702142715454,
-0.2002079039812088,
-0.026494096964597702,
-0.017551304772496223,
-0.4081806540489197,
0.4092123210430145,
0.10043632239103317,
-0.3056873679161072,
-0.25454795360565186,
-0.356810599565506,
-0.23216839134693146,
0.6080306768417358,
0.26332029700279236,
0.055659834295511246,
-0.12169842422008514,
0.3012830913066864,
0.012392728589475155,
-0.07239838689565659,
0.17405152320861816,
-0.06711790710687637,
-0.19626814126968384,
-0.22151575982570648,
0.7537152767181396,
-0.5689879059791565,
0.3106265068054199,
0.10678759962320328,
0.3895302712917328,
0.49580562114715576,
-0.5104957222938538,
-0.4885309040546417,
-0.051460251212120056,
-0.1865653395652771,
0.5159444808959961,
0.4942895174026489,
0.5322152376174927,
0.07906338572502136,
-0.2121727615594864,
-0.17183467745780945,
0.8451370596885681,
0.1482597440481186,
-0.3832302689552307,
-0.411946177482605,
0.41313883662223816,
-0.1993463635444641,
0.12926435470581055,
-0.33101969957351685,
-0.3380512595176697,
-0.2415883094072342,
0.06368544697761536
] |
above expression, the wave number k±
r , kinetic energy
Ωr(iωn), and bulk wave functions u±
r , v±
r (r = L, R) sat-
4
FIG. 2. Four types of eigenfunctions. The red (+) arrows and
blue (−) arrows indicate the particle- and the hole-like plane
waves, respectively. The dashed lines represent the incident
waves. The directions of arrows indicate the group velocity.
The sign in the superscript of Ψ, a, b, c, and d indicates
particle- (+) or hole-like particle (−) incident.
isfy the following relations:
k±
r = | [
-0.1980050653219223,
0.0109260194003582,
0.7035232186317444,
-0.06478799879550934,
-0.2719748020172119,
0.4392850697040558,
-0.056251417845487595,
-0.7555632591247559,
-0.19666653871536255,
-0.5277140736579895,
0.00479583814740181,
0.42628347873687744,
-0.18105469644069672,
0.43945425748825073,
0.41625475883483887,
0.4486277103424072,
0.8579098582267761,
0.281357079744339,
-0.7022110223770142,
-0.693717360496521,
0.3818577229976654,
-0.01810343563556671,
0.6761797070503235,
-0.3521353006362915,
0.3563023507595062,
0.059161461889743805,
0.237603098154068,
-0.13822762668132782,
0.4320372939109802,
0.25043511390686035,
-0.3461243510246277,
0.727724015712738,
-0.1508905291557312,
0.2521268427371979,
0.342629998922348,
0.13368025422096252,
0.0898604467511177,
0.1471218466758728,
0.21852628886699677,
-0.19367432594299316,
0.02928108721971512,
-0.4951520264148712,
0.16601033508777618,
0.2748555839061737,
-0.37328705191612244,
-0.3684048354625702,
-1.0473332405090332,
0.052222732454538345,
-0.3975332975387573,
-0.05814531445503235,
-0.01519895438104868,
0.20436441898345947,
0.2684638500213623,
0.8208326697349548,
-0.319729208946228,
-0.07082995772361755,
-0.3228568136692047,
0.006579932756721973,
0.5322309136390686,
-0.5685257911682129,
0.28681105375289917,
-0.30671513080596924,
0.12154459208250046,
0.5657281875610352,
0.2453795075416565,
0.7964537739753723,
-0.11902277171611786,
-0.01080262940376997,
-0.49491918087005615,
0.04551476240158081,
0.6171585321426392,
0.003085408126935363,
-0.39993172883987427,
-0.1351364701986313,
0.00945736002177,
0.2418629378080368,
-0.2225782573223114,
0.001520550693385303,
-0.1681036800146103,
0.19843299686908722,
-0.16846655309200287,
-0.14152772724628448,
0.18183620274066925,
-0.5525217652320862,
0.131045401096344,
0.00029414918390102684,
0.06890332698822021,
0.30951401591300964,
-0.6562381982803345,
0.7843984961509705,
-0.1901751458644867,
-0.1805678755044937,
0.010275784879922867,
0.4621858596801758,
0.2960447669029236,
0.5442101955413818,
0.33674561977386475,
-0.7330365777015686,
0.5344103574752808,
0.08618339896202087,
-0.5527581572532654,
-0.23975351452827454,
-0.024620331823825836,
-0.16661742329597473,
-0.4177316129207611,
-0.44164296984672546,
-0.5980856418609619,
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-0.2570444345474243,
-0.03406716510653496,
0.012453676201403141,
-0.14086738228797913,
-0.08070880174636841,
-0.02819431759417057
] |
particle- (+) or hole-like particle (−) incident.
isfy the following relations:
k±
r =
�
(2m/ℏ2)[µ ± Ωr(iωn)],
(11)
�
± Ωr(iωn)ˆτ z + ˆΣr(iωn)
� �
u±
r
v±
r
�
= iωn
�
u±
r
v±
r
�
.
(12)
The solution of Eq. (12) is discussed in the next sub-
section in detail.
Note that Ωr(iωn) can also be rep-
resented in another form as shown later [see Eq. (22)].
In Eq. (12), (u+(−)
r
, v+(−)
r
)T corresponds to the particle
(hole) eigenfunction. The eigenfunctions corresponding
to ˜Ψ in Eq. (9) are denoted as (˜u± | [
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] |
(hole) eigenfunction. The eigenfunctions corresponding
to ˜Ψ in Eq. (9) are denoted as (˜u±
r , ˜v±
r )T.
As seen in Fig. 2, the coefficients a(±)
out and a(±)
in
corre-
spond to the reflection from particle to hole. Then, we
call this type of reflection as Andreev reflection of bo-
golons, which is analogous to the conventional SC junc-
tions. Similarly, we call “b(±)
out, b(±)
in ” and “c(±)
out, c(±)
in ” in
Eq. (10) and Fig. 2 as normal reflection and normal trans- | [
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] |
in ” and “c(±)
out, c(±)
in ” in
Eq. (10) and Fig. 2 as normal reflection and normal trans-
mission, respectively. The transmission from particle to
hole is reflected in “d(±)
out, d(±)
in ”. The specific expressions
of a(±)
out, a(±)
in , b(±)
out, b(±)
in , c(±)
out, c(±)
in , d(±)
out, and d(±)
in
are de-
termined by the boundary conditions at x = 0 [30, 94, 97–
102], which are given by
Ψ(±)
out(x = +0+) = Ψ(±)
out(x = −0+),
(13)
dΨ(±)
out(x)
dx
�����
x=0+
− dΨ(±)
out(x)
dx
�����
x=−0+
= 2mVB
ℏ2
Ψ(±) | [
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] |
out(x = −0+),
(13)
dΨ(±)
out(x)
dx
�����
x=0+
− dΨ(±)
out(x)
dx
�����
x=−0+
= 2mVB
ℏ2
Ψ(±)
out(x)
����
x=0
.
(14)
We also impose the same conditions on Ψ(±)
in . These con-
ditions are similar to those in the scattering problem with
the barrier potential of quantum mechanics. The spe-
cific forms of a(±)
out, a(±)
in , b(±)
out, b(±)
in , c(±)
out, c(±)
in , d(±)
out, and
d(±)
in
are listed in Eqs. (B11)–(B18).
We also evaluate
the eigenfunction ˜Ψout(x), ˜Ψin(x) in Eq.(9) in a manner | [
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] |
We also evaluate
the eigenfunction ˜Ψout(x), ˜Ψin(x) in Eq.(9) in a manner
similar to Ψout(x), Ψin(x).
Using the four types of eigenfunctions described in
Fig. 2, the Green’s function is expressed in the follow-
ing form [30, 94, 98–102]:
ˆG(iωn, x, x′) =
�
α1Ψ(+)
in (x)˜Ψ(+)
out(x′)T + α2Ψ(+)
in (x)˜Ψ(−)
out(x′)T + α3Ψ(−)
in (x)˜Ψ(+)
out(x′)T + α4Ψ(−)
in (x)˜Ψ(−)
out(x′)T
(x < x′),
β1Ψ(+)
out(x)˜Ψ(+)
in (x′)T + β2Ψ(−)
out(x)˜Ψ(+)
in (x′)T + β3Ψ(+)
out(x)˜Ψ(−)
in (x′)T + β4Ψ(−)
out(x)˜Ψ(−)
in (x′)T | [
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0.3121572732925415,
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0.25053033232688904,
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] |
in (x′)T + β2Ψ(−)
out(x)˜Ψ(+)
in (x′)T + β3Ψ(+)
out(x)˜Ψ(−)
in (x′)T + β4Ψ(−)
out(x)˜Ψ(−)
in (x′)T
(x > x′),
(15)
where coefficients α1, · · · , α4 and β1, · · · , β4 are deter-
mined by the boundary conditions at x = 0, which are
given by
ˆG(iωn, x → x′ + 0+, x′) = ˆG(iωn, x → x′ + 0−, x′) (16)
and
∂
∂x
ˆG(iωn, x, x′)
����
x=x′+0+ − ∂
∂x
ˆG(iωn, x, x′)
����
x=x′−0+ = 2m
ℏ2 ˆτ z.
(17)
We note that Eq. (17) is derived from the Gor’kov equa-
tions in Eqs. (2) and (3). The Green’s function for ωn < 0 | [
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] |
tions in Eqs. (2) and (3). The Green’s function for ωn < 0
can be evaluated by the conjugate relation
ˆG(−iωn, x, x′) = ˆG†(iωn, x′, x),
(18)
which is derived from the Lehmann representation of the
Green’s function.
We emphasize again that Eq. (15) can be used for
both bogolon (non-Hermite case) and s-wave SC (Her-
mite case). In the next subsection, we evaluate the spe-
cific form of u±
r , v±
r , ˜u±
r , and ˜v±
r by using a biorthogonal
basis, which can be applied to both the Hermite (s-wave | [
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] |
r , and ˜v±
r by using a biorthogonal
basis, which can be applied to both the Hermite (s-wave
SC junction) and the non-Hermite cases (boglon junc-
tion).
D.
Solution of auxiliary non-Hermitian problem
Now we consider the concrete eigenvalue problem, in
which the characteristic feature of bogolons, i.e. Non-
5
Hermitian nature of effective Hamiltonian, is reflected.
The following discussion can be used for both ωn > 0
and ωn < 0. First we write down Eq. (12) for the r-side | [
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] |
and ωn < 0. First we write down Eq. (12) for the r-side
(r = R for x > 0 and r = L for x < 0) as
�
±Ωr(iωn) − iΓ1rsgn ωn
Sr(iωn)
S+
r (iωn)
∓Ωr(iωn) − iΓ1rsgn ωn
� �
u±
r
v±
r
�
= iωn
�
u±
r
v±
r
�
.
(19)
For the bogolon junction, we set Γ1r > 0, Sr(iωn) =
−iΓ2rsgn ωn, S+
r (iωn) = −iΓ∗
2rsgn ωn [If we set Γ1r = 0
and Sr(iωn) = S+
r (iωn)∗ = ∆r, Eq. (19) applies to the s-
wave SC.]. Since the matrix in Eq. (19) is non-Hermitian,
it is necessary to consider the following Hermite conju-
gate version:
� | [
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] |
it is necessary to consider the following Hermite conju-
gate version:
�
±Ωr(iωn)∗ + iΓ1rsgn ωn
S+
r (iωn)∗
Sr(iωn)∗
∓Ωr(iωn)∗ + iΓ1rsgn ωn
� �
˜u±∗
r
˜v±∗
r
�
= (iωn)∗
�
˜u±∗
r
˜v±∗
r
�
.
(20)
The wave functions u±
r , ˜u±
r , v±
r , and ˜v±
r satisfy the fol-
lowing biorthogonal condition [105] , which is used in the
context of non-Hermitian quantum mechanics [95, 96]:
˜u±
r u±
r + ˜v±
r v±
r = 1.
(21)
From Eqs. (19) and (20), Ωr(iωn) is determined as
Ωr(iωn) =
�
(iωn + iΓ1rsgn ωn)2 − Sr(iωn)S+
r (iωn). | [
-0.5955019593238831,
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0.2590499520301819,
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0.4767538905143738,
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0.3651103079319,
0.4074132740497589,
0.5930148363113403,
0.918774425983429,
0.16323962807655334,
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] |
Ωr(iωn) =
�
(iωn + iΓ1rsgn ωn)2 − Sr(iωn)S+
r (iωn).
(22)
We note that Ωr(iωn) is an even function of ωn in the
bogolon model (this is also true for the s-wave SC).
As discussed in Sec. II C, the coefficients α1, · · · , α4
and β1, · · · , β4 in Eq. (15) are determined by the bound-
ary conditions for the Green’s function in Eqs. (16) and
(17). The specific form of α1, · · · , α4 and β1, · · · , β4 are
listed in Eqs. (B3)–(B10). Using these coefficients, the | [
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] |
listed in Eqs. (B3)–(B10). Using these coefficients, the
Green’s function Eq. (15) for x, x′ < 0 is reduced to
ˆG(iωn, x, x′) = m(iωn + iΓ1Lsgn ωn)
ikFℏ2ΩL(iωn)
��
eik+
L |x−x′| + ˜b(+)
oute−ik+
L (x+x′)� �
u+
L ˜u+
L u+
L ˜v+
L
v+
L ˜u+
L
v+
L ˜v+
L
�
+ ˜a(+)
oute−ik+
L x+ik−
L x′ �
u+
L ˜u−
L u+
L ˜v−
L
v+
L ˜u−
L
v+
L ˜v−
L
�
+
�
e−ik−
L |x−x′| + ˜b(−)
outeik−
L (x+x′)� �
u−
L ˜u−
L u−
L ˜v−
L
v−
L ˜u−
L
v−
L ˜v−
L
�
+ ˜a(−)
outeik−
L x−ik+
L x′ �
u−
L ˜u+
L u−
L ˜v+
L
v−
L ˜u+
L
v−
L ˜v+
L
� �
,
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] |
v−
L ˜v−
L
�
+ ˜a(−)
outeik−
L x−ik+
L x′ �
u−
L ˜u+
L u−
L ˜v+
L
v−
L ˜u+
L
v−
L ˜v+
L
� �
,
(23)
where kF is a Fermi wavevector of bogolon and ˜a(±)
out and ˜b(±)
out are coefficients in ˜Ψ(±)
out(x). This expression, which
includes six terms, provides a clearer physical meaning than Eq. (15) [98]. Namely, the first line describes particle
behavior, while the second line describes hole behavior. The contributions from the bulk are represented by the first | [
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] |
and fourth terms. The second and the fifth terms, which are proportional to ˜b(±)
out, correspond to the normal reflection.
The third and sixth terms proportional to ˜a(±)
out represent the contributions from the Andreev reflection.
E.
Concrete form of Green’s function
In the following of this paper, we focus on the bogolon junction with Γ1L = Γ1R and |Γ2L| = |Γ2R|. The concrete
form of the Green’s function in this subsection is one of the central results of this paper. Using the functional forms
6
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6
of u±
L , v±
L , ˜u±∗
L , and ˜v±∗
L
given in Appendix B, the Green’s function for x, x′ < 0 is obtained as follows:
ˆG(iωn, x, x′) =
m
2ikFℏ2ΩL(iωn)
��
eik+
L |x−x′| + ˜b(+)
oute−ik+
L (x+x′)� �
iωn + iΓ1Lsgn ωn + ΩL(iωn)
−iΓ2Lsgn ωn
−iΓ∗
2Lsgn ωn
iωn + iΓ1Lsgn ωn − ΩL(iωn)
�
+ ˜a(+)
oute−ik+
L x+ik−
L x′ �
i|Γ2L|sgn ωn
−eiθL[iωn + iΓ1Lsgn ωn + ΩL(iωn)]
−e−iθL[iωn + iΓ1Lsgn ωn − ΩL(iωn)]
i|Γ2L|sgn ωn
�
+
�
e−ik−
L |x−x′| + ˜b(−)
outeik−
L (x+x′)� �
iωn + iΓ1Lsgn ωn − ΩL(iωn)
−iΓ2Lsgn ωn
−iΓ∗
2Lsgn ωn | [
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] |
+
�
e−ik−
L |x−x′| + ˜b(−)
outeik−
L (x+x′)� �
iωn + iΓ1Lsgn ωn − ΩL(iωn)
−iΓ2Lsgn ωn
−iΓ∗
2Lsgn ωn
iωn + iΓ1Lsgn ωn + ΩL(iωn)
�
+ ˜a(−)
outeik−
L x−ik+
L x′ �
i|Γ2L|sgn ωn
−eiθL[iωn + iΓ1Lsgn ωn − ΩL(iωn)]
−e−iθL[iωn + iΓ1Lsgn ωn + ΩL(iωn)]
i|Γ2L|sgn ωn
� �
(24)
with
˜a(±)
out(iωn) = −i|Γ2L|sgn ωn[(iωn + iΓ1Lsgn ωn) sin2(θ/2) ± iΩL(iωn) sin(θ/2) cos(θ/2)]
Z2ΩL(iωn)2 + (iωn + iΓ1Lsgn ωn)2 + |Γ2L|2 cos2(θ/2)
,
(25)
˜b(±)
out(iωn) = −
Z(Z ± isgn ωn)ΩL(iωn)2 | [
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0.5398556590080261,
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0.2198341190814972,
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] |
,
(25)
˜b(±)
out(iωn) = −
Z(Z ± isgn ωn)ΩL(iωn)2
Z2ΩL(iωn)2 + (iωn + iΓ1Lsgn ωn)2 + |Γ2L|2 cos2(θ/2).
(26)
We have introduced the relative phase of the pair poten-
tial θ = θR − θL with −π < θ ≤ π and defined
Z = mVB
kFℏ2 ,
(27)
which represents a magnitude of delta-function barrier
potential at x = 0.
F.
Physical quantities
1.
s-wave component
Let us consider the physical quantities derived from
McMillan Green’s function. From the local Green’s func- | [
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] |
McMillan Green’s function. From the local Green’s func-
tion (x = x′), we define the s-wave component (i.e., sym-
metric in terms of the exchange of x and x′) of the pair
amplitude as
Fs(iωn, x) =
�
ˆG(iωn, x, x)
�
12.
(28)
We employ the Matsubara frequency representation for
pair amplitudes, which is useful to recognize the even-
and odd-frequency components.
We can also evaluate the local density of states (LDOS)
of bogolons from the local Green’s function. The LDOS | [
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0.23948973417282104,
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] |
of bogolons from the local Green’s function. The LDOS
of bogolons is defined by using the diagonal compo-
nents of the retarded Green’s function, which is ob-
tained through analytical continuation with respect to
frequency:
D(ω, x) = − 1
π Im Tr ˆG(ω + i0+, x, x),
(29)
with which the probability density of bogolons is identi-
fied.
2.
p-wave component
Next, we define p-wave component of Green’s function
(i.e., antisymmetric in terms of the exchange of x and x′)
as
ˆGp(iωn, x)
= lim
∆x→0
1
∆x
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0.1233391985297203,
-0.8124182224273682,
0.2933214604854584,
0.7277987599372864,
0.19624342024326324,
-0.1939588487148285,
0.00578826479613781,
0.2456413209438324,
-0.28117242455482483,
0.08692958950996399,
0.24929587543010712,
0.05608373135328293,
-0.27138492465019226,
0.28917473554611206,
-0.7059044241905212,
0.6266438364982605,
0.13706746697425842,
0.46643978357315063,
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0.44655442237854004,
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0.1945798546075821,
0.3029786944389343,
0.3489396274089813,
-0.8227646350860596,
0.12346184998750687,
0.12069862335920334,
-0.32386282086372375,
0.18774080276489258,
0.04220997169613838,
-0.1383814960718155,
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-0.37452957034111023,
-0.34443309903144836,
-0.1469733715057373,
-0.03654758259654045
] |
(i.e., antisymmetric in terms of the exchange of x and x′)
as
ˆGp(iωn, x)
= lim
∆x→0
1
∆x
�
ˆG(iωn, x + ∆x, x) − ˆG(iωn, x, x + ∆x)
�
=
� ∂
∂x − ∂
∂x′
�
ˆG(iωn, x, x′)
����
x′→x
.
(30)
Inserting Eq. (24) into Eq. (30) and assuming k+ ≃ k− ≃
kF, one finds that the bulk parts in Eq. (24) vanishes.
This fact implies that ˆGp(iωn, x) is induced by the pres-
ence of the interface.
The
p-wave
component
of
the
pair
amplitude
Fp(iωn, x) is defined by [87, 88]
Fp(iωn, x) =
�
ˆGp(iωn, x)
�
12.
(31) | [
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] |
of
the
pair
amplitude
Fp(iωn, x) is defined by [87, 88]
Fp(iωn, x) =
�
ˆGp(iωn, x)
�
12.
(31)
We note that, from the Fermi-Dirac statistics, Fs(iωn, x)
is an odd (even) function of ωn for bogolons (s-wave SC),
and Fp(iωn, x) is an even (odd) function of ωn for bo-
golons (s-wave SC).
It is also worthwhile to explore the diagonal com-
ponents in Nambu space of ˆGp(iωn, x).
The trace of
ˆGp(iωn, x) can be regarded as quasiparticle contribution
of current [98], which is given by
J(x) = i
�
α†(x) ∂
∂xα(x) − | [
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] |
of current [98], which is given by
J(x) = i
�
α†(x) ∂
∂xα(x) −
� ∂
∂xα†(x)
�
α(x)
�
= 1
β
�
n
j(iωn, x)
(32)
7
with
j(iωn, x) =
i
nd
Tr ˆGp(iωn, x),
(33)
and nd is a factor correcting double counting: nd = 2 for
bogolon and nd = 1 for s-wave SC.
III.
SUMMARY OF S-WAVE
SUPERCONDUCTOR JUNCTION
Before showing the results for the bogolon junction,
here we summarize the result of the s-wave SC junc-
tion for reference.
The detailed setup and expressions
are listed in Appendix C. In the following of this sec- | [
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] |
The detailed setup and expressions
are listed in Appendix C. In the following of this sec-
tion, we assume |∆L| = |∆R| corresponding to the dis-
cussion in Sec. II E. For all figures in this section, we
choose |∆L|/µ = 0.01 with µ = ℏ2k2
F/2m and set µ = 1
and kF = 1. The specific form of the Green’s function
is shown in Eqs. (C3)–(C5). We note that the results in
this section are not completely original.
A.
Semi-infinite superconductor
First, we discuss the result for the semi-infinite SC | [
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] |
A.
Semi-infinite superconductor
First, we discuss the result for the semi-infinite SC
(x < 0), where the edge is located at x = 0. We take
the limit Z → ∞ in Eqs. (C3)–(C5). The pair amplitude
[ ˆG(iωn, x, x′)]12 is given by
[ ˆG(iωn, x, x′)]12 =
m∆L
2ikFℏ2�
(iωn)2 − |∆L|2
×
�
eik+
L |x−x′| − e−ik+
L (x+x′) + e−ik−
L |x−x′| − eik−
L (x+x′)�
.
(34)
[ ˆG(iωn, x, x′)]12 is the even function of frequency, which
is the same frequency dependence as the bulk.
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] |
is the same frequency dependence as the bulk.
This
frequency dependence implies the absence of Andreev
reflection in Eq. (34).
In this way, s-wave pair po-
tential, which does not have spatial dependence, does
not induce the p-wave component at the edge [106].
Although odd-frequency pairs are not induced by the
constant pair potential, spatially varying pair potential,
∆(x) ̸= Const., induces the odd-frequency p-wave pair at
the surface [87, 88].
B.
Superconductor junction without barrier
potential | [
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0.9094603061676025,
0.23007214069366455,
-0.18385040760040283,
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0.13774193823337555,
0.5271782875061035,
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0.5373722910881042,
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0.32496437430381775,
0.15250037610530853,
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0.27568507194519043,
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0.5646992325782776,
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0.13283264636993408,
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0.28075361251831055,
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0.3289434015750885,
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0.2956008315086365,
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0.5871798396110535,
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0.3353216350078583,
0.5539067983627319,
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0.31984230875968933,
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0.023234473541378975,
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0.7792451977729797,
0.5716584324836731,
0.03766390681266785,
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0.3195960223674774,
1.2207046747207642,
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0.5313622355461121,
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0.5453619360923767,
0.12801194190979004,
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-0.6226314902305603,
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0.4815133213996887,
0.4625204801559448,
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-0.04096032306551933,
0.11957752704620361,
-0.45982351899147034,
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-0.08495113998651505,
0.0265695471316576,
0.028737762942910194,
0.18944251537322998
] |
the surface [87, 88].
B.
Superconductor junction without barrier
potential
Next, we show the result of the SC junction without the
barrier potential, i.e., Z = 0. We assume x, x′ < 0 in the
following of this section. The expressions of the physical
quantities are listed below.
The s-wave component of
FIG. 3.
Physical quantities at x = 0 for s-wave SC junc-
tion without barrier potential. (a) The s-wave component of
the pair amplitude, (b) the p-wave component of the pair | [
0.0028602643869817257,
-0.3926820755004883,
0.7837051153182983,
-0.13397027552127838,
-0.3451155722141266,
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0.04849182069301605,
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0.573274552822113,
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0.8066231608390808,
0.0491144172847271,
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0.3524716794490814,
0.6912891268730164,
0.2145814299583435,
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0.11337263137102127,
0.43654170632362366,
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1.09158456325531,
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0.10073824971914291,
-0.028086284175515175,
-0.18084080517292023,
-0.5211480855941772,
0.08408722281455994,
-0.0026608158368617296,
0.3997044265270233,
-0.5486352443695068,
0.1482241451740265,
0.02089025266468525,
0.29833513498306274,
-0.2960173785686493,
-0.35888150334358215,
0.41774284839630127,
-0.2899214029312134,
-0.3155932128429413,
0.3805943429470062,
0.4417024552822113,
-0.38021594285964966,
0.6164857745170593,
0.22454190254211426,
-0.27838003635406494,
-0.27762356400489807,
-0.12807758152484894,
0.14623486995697021,
-0.3439958393573761,
0.13617047667503357,
-0.07585924863815308,
0.0804712325334549,
0.15980587899684906,
-0.6045713424682617,
0.18359719216823578,
-0.34027808904647827,
-0.27279478311538696,
0.002210108330473304,
0.049466826021671295,
0.014065262861549854,
0.177387997508049,
-0.08184833824634552,
0.006128513719886541,
0.5970933437347412,
-1.0727676153182983,
0.6359199285507202,
0.07098997384309769,
-0.0013170137535780668,
-0.6603776216506958,
0.185890331864357,
-0.2738417983055115,
0.1780831515789032,
-0.3043712377548218,
-0.40695351362228394,
0.19867275655269623,
-0.18265390396118164,
0.01849161833524704,
0.05330665409564972,
0.2741905748844147,
-0.2769622802734375,
0.6894333362579346,
-0.1882305145263672,
0.07426444441080093,
-0.18500545620918274,
0.0925271287560463,
1.165938138961792,
0.6090743541717529,
-0.12135213613510132,
-0.6267529129981995,
0.04860975220799446,
0.017449883744120598,
0.03254902362823486,
0.9960905909538269,
-0.12086213380098343,
0.325859010219574,
0.00015421614807564765,
0.18619787693023682,
-0.15687471628189087,
0.28087249398231506,
0.6939945816993713,
-0.22521406412124634,
-0.5369682908058167,
0.08835960924625397,
0.5153563022613525,
-0.18148237466812134,
0.4362184703350067,
0.09652318060398102,
-0.06508374214172363,
-0.06545152515172958,
0.18918751180171967,
-0.45587170124053955,
-0.6309643387794495,
-0.43867436051368713,
-0.3354777693748474,
0.04203961789608002,
0.008152059279382229,
-0.3642800450325012,
0.16722571849822998,
0.5729690790176392,
-0.29910629987716675,
-0.30047285556793213,
-0.08992444723844528,
0.17352032661437988,
-0.3703116178512573,
0.18597598373889923,
-0.23620717227458954,
0.10146841406822205,
0.4320807158946991,
0.2042769342660904,
-0.31917741894721985,
0.258165568113327,
0.10365559905767441,
-0.3801959156990051,
0.06766866147518158,
0.06107179820537567,
-0.2818232774734497,
-0.08296399563550949,
-0.14337870478630066,
-0.5991530418395996,
-0.13900505006313324,
0.062030352652072906,
-0.006084710359573364,
-0.1632758527994156,
0.3219170570373535,
0.02686554566025734,
-0.5428943037986755,
-0.31102174520492554,
0.1368565410375595,
0.9682644605636597,
-0.5783070921897888,
-0.412549763917923,
0.0029405870009213686,
0.4522843360900879,
-0.3261980414390564,
-0.5999558568000793,
-0.536274790763855,
0.9136747717857361,
0.42526617646217346,
-0.2771207392215729,
-0.4292967915534973,
0.04223176836967468,
0.4555705785751343,
-0.5132543444633484,
-0.022674571722745895,
-0.5133587121963501,
-0.29250892996788025,
0.11036163568496704,
0.1401386559009552,
-0.21693505346775055,
-0.8512603640556335,
0.62386554479599,
-0.14871351420879364,
0.1378614902496338,
-0.010524818673729897,
0.4966452717781067,
-0.09422209858894348,
-0.1093258336186409,
0.25125598907470703,
-0.6991128921508789,
0.2524913251399994,
0.0952128991484642,
-0.3129708766937256,
-0.538903534412384,
-0.2781306803226471,
0.23951083421707153,
0.22633543610572815,
0.6881679892539978,
-0.023082852363586426,
-0.03678273409605026,
0.2243812084197998,
-0.411331444978714,
0.26344048976898193,
-0.11927716434001923,
-0.32438766956329346,
-0.9111965894699097,
-0.7052267789840698,
0.14998000860214233,
0.33712244033813477,
-0.022963859140872955,
0.08388406038284302,
-0.43456020951271057,
-0.06063566356897354,
-0.2773137390613556,
0.2162046879529953,
0.2737366557121277,
-0.6948242783546448,
0.014386310242116451,
-0.03606576845049858,
0.43372491002082825,
0.059890516102313995,
-0.15827344357967377,
-0.2931922674179077,
0.5804625153541565,
0.16598182916641235,
-0.5229155421257019,
0.31023913621902466,
-0.5824079513549805,
-0.22452619671821594,
0.5751980543136597,
-0.21623757481575012,
0.5061048269271851,
0.48192358016967773,
0.09686985611915588,
0.49109041690826416,
0.5603522062301636,
-0.4463269114494324,
-0.724743127822876,
0.7964890599250793,
-0.21696089208126068,
-0.4107188880443573,
-0.3929823040962219,
0.3666095435619354,
0.7961618900299072,
0.07700802385807037,
-0.4805544316768646,
-0.017135892063379288,
0.22858625650405884,
-0.07957968860864639,
0.10239211469888687,
0.23221421241760254,
-0.29087039828300476,
0.5122880339622498,
0.14875368773937225,
1.0811426639556885,
-0.5241024494171143,
0.1153823658823967,
-0.1003003865480423,
0.6089062094688416,
0.20863014459609985,
-0.0059965830296278,
-0.19723181426525116,
0.18530049920082092,
0.5300800800323486,
-0.10292773693799973,
-0.23152807354927063,
-0.04715055227279663,
-0.8950607776641846,
0.35473328828811646,
0.34883347153663635,
-0.13643354177474976,
0.018790777772665024,
0.6005333065986633,
0.22943854331970215,
1.1517484188079834,
-0.1288827657699585,
0.11133682727813721,
-0.38956978917121887,
0.006624692585319281,
0.09102462232112885,
0.17469434440135956,
-0.10116581618785858,
-0.33336201310157776,
0.007423983421176672,
0.34149423241615295,
-0.3839128017425537,
0.5116578340530396,
-0.6765802502632141,
-0.6623426079750061,
0.3896166980266571,
0.4845138192176819,
-0.29362815618515015,
-0.002369989175349474,
0.029350947588682175,
0.7738617062568665,
-0.48617038130760193,
-0.6439390778541565,
0.07088913768529892,
-0.24180181324481964,
0.11031679809093475,
0.26247623562812805,
0.005627133883535862,
0.1797255575656891,
0.5613723993301392,
-0.4503895342350006,
-0.4210372269153595,
0.025387167930603027,
-0.020070623606443405,
-0.05044674873352051,
-0.2545638680458069,
0.8007506728172302,
0.5860878825187683,
-0.13898740708827972,
0.3101171851158142,
-0.09000638127326965,
0.18714311718940735,
0.9464688897132874,
-0.8175172805786133,
0.6877529621124268,
-0.24080108106136322,
0.47272855043411255,
-0.5947452783584595,
-0.0894438773393631,
0.4995175898075104,
0.4391377568244934,
-0.056624602526426315,
0.2211935967206955,
-0.28306815028190613,
0.14693373441696167,
0.01867309957742691,
-0.34132838249206543,
-0.11257277429103851,
-0.46720001101493835,
-0.5907356142997742,
-0.6572962999343872,
0.32998213171958923,
0.5418263077735901,
0.04853016510605812,
-0.19732822477817535,
0.355455219745636,
0.46682268381118774,
-0.09404314309358597,
0.15126663446426392,
0.2400389015674591,
-0.2817515730857849,
0.15013058483600616,
0.7796357274055481,
-0.37625929713249207,
0.57859206199646,
-0.49860793352127075,
0.28533366322517395,
-0.17491400241851807,
-0.33842068910598755,
-0.009234542027115822,
-0.17336569726467133,
-0.6273378133773804,
0.48810824751853943,
0.010170120745897293,
-0.14835737645626068,
-0.5567438006401062,
-0.5160799026489258,
-0.5733317732810974,
0.445370078086853,
0.2898971438407898,
0.023822983726859093,
-0.048395752906799316,
-0.11845579743385315,
-0.22305811941623688,
-0.10994735360145569,
-0.040628641843795776,
-0.015305735170841217,
-0.05488881096243858,
0.10878343880176544
] |
the pair amplitude, (b) the p-wave component of the pair
amplitude, (c) the LDOS normalized by its value in nor-
mal state (∆L = 0), (d) the LDOS in the complex plane
for θ = 2π/3, which is defined by the extension of ω to the
complex energy plane z, i.e., D(ω, x = 0) → D(z, x = 0),
(e) j(iωn) = j(iωn, x = 0), and (f) the Josephson current
J(x = 0). Figure legend of (c) is the same as that of (a), and
the legend of (e) is the same as that of (b).
pair amplitude defined by Eq. (28) is given by
Fs(iωn, x) = | [
-0.3010362684726715,
-0.4910213053226471,
0.7898716330528259,
-0.4589335322380066,
-0.451712965965271,
0.3198092579841614,
0.2557014226913452,
-0.9823470115661621,
0.21481066942214966,
-0.5524342060089111,
-0.22304590046405792,
0.8037827610969543,
-0.4848995506763458,
0.14929670095443726,
0.1079820916056633,
0.8797426223754883,
0.7114466428756714,
0.1507074236869812,
-1.1090987920761108,
-0.7438502907752991,
0.5781484842300415,
0.03737446665763855,
0.47649887204170227,
0.1344906985759735,
0.0488717295229435,
0.0691576898097992,
0.6077859401702881,
-0.09602367132902145,
0.6806219816207886,
0.3873266279697418,
-0.12384594976902008,
0.5535449385643005,
-0.14234672486782074,
-0.02481801249086857,
0.15217341482639313,
0.376803457736969,
0.39364802837371826,
-0.06632847338914871,
0.4157070815563202,
-0.1507638692855835,
0.43550604581832886,
-0.5334384441375732,
0.1131594181060791,
0.10062741488218307,
-0.4832514524459839,
-0.41906237602233887,
-1.204745888710022,
-0.05864705890417099,
-0.4040359556674957,
-0.3559292256832123,
0.17000342905521393,
0.49506890773773193,
-0.006453973241150379,
0.678141176700592,
-0.24677029252052307,
-0.11735745519399643,
-0.5047142505645752,
-0.3186327815055847,
0.7796531319618225,
-0.3943988084793091,
0.49112728238105774,
0.040113989263772964,
0.21823693811893463,
0.5664957165718079,
-0.2110021561384201,
0.8861758708953857,
-0.13133732974529266,
0.1564255654811859,
-0.4875585436820984,
0.0523114949464798,
0.3263465464115143,
0.21810472011566162,
-0.4564533531665802,
-0.3037717044353485,
-0.048639118671417236,
0.5527846813201904,
-0.40263620018959045,
-0.04202987626194954,
0.22110065817832947,
-0.09139768779277802,
-0.1251886636018753,
0.15917836129665375,
0.2652183771133423,
-0.4200498163700104,
0.05812649801373482,
-0.18640032410621643,
0.04013703390955925,
0.8410089612007141,
-0.5627732276916504,
0.8196976780891418,
0.22766807675361633,
-0.22436368465423584,
-0.05369643121957779,
0.2382211834192276,
0.3097800016403198,
0.8708455562591553,
0.08942125737667084,
-0.41632184386253357,
0.6385853290557861,
0.10135317593812943,
0.1317148208618164,
-0.01905846782028675,
-0.2594018876552582,
-0.5497891306877136,
-0.005115407984703779,
-0.7593782544136047,
-0.6688141822814941,
-1.1549431085586548,
-0.4291401207447052,
-0.03989661484956741,
-1.0823296308517456,
0.31183603405952454,
-0.03750880807638168,
-0.24992290139198303,
-0.1476588398218155,
-0.24770969152450562,
-0.16264352202415466,
-0.06513527035713196,
0.3087168335914612,
0.5260718464851379,
0.000671625544782728,
0.6093016266822815,
-0.08135165274143219,
0.6036815643310547,
0.5059481263160706,
-0.29811573028564453,
0.12109589576721191,
0.5580320954322815,
-1.016189455986023,
-0.11317145824432373,
0.3378375768661499,
0.45650550723075867,
0.4316284954547882,
0.5885598063468933,
-0.4114640951156616,
-0.6770069003105164,
-0.318019837141037,
-0.416745126247406,
-0.6997195482254028,
0.059732869267463684,
-0.4866848587989807,
-0.28833892941474915,
-0.6700490117073059,
-0.26086753606796265,
0.2613426148891449,
-0.20894336700439453,
-0.3778133988380432,
0.12519629299640656,
0.15216241776943207,
0.2517991364002228,
-0.4919314682483673,
-0.01680224947631359,
-0.3399699926376343,
0.08851998299360275,
-0.5355545282363892,
-0.5230619311332703,
0.5830084681510925,
0.13415615260601044,
-0.10387139767408371,
0.08728906512260437,
0.2532142400741577,
-0.1401105374097824,
0.12538395822048187,
-0.3427965044975281,
0.557714581489563,
-0.6138351559638977,
0.07795822620391846,
-0.06924962252378464,
-0.1150776669383049,
-0.04096194729208946,
0.07531812787055969,
-0.390689492225647,
-0.08053407818078995,
0.1904411017894745,
-0.3295460641384125,
-0.6240590810775757,
-0.025006385520100594,
0.19279763102531433,
0.25681817531585693,
-0.01880926452577114,
-0.6561583876609802,
-0.7434176802635193,
-0.009198919869959354,
-0.06058241054415703,
0.3271203339099884,
0.2828902304172516,
0.0026326372753828764,
0.16357986629009247,
0.33194640278816223,
0.4456638693809509,
-0.17007535696029663,
0.17730313539505005,
0.23163653910160065,
-0.7554672956466675,
1.0229969024658203,
0.024433642625808716,
-0.6730733513832092,
0.14402778446674347,
0.30382055044174194,
-0.09144715219736099,
-0.07945451140403748,
-0.1086796447634697,
0.4195556342601776,
0.20369821786880493,
-0.3781783878803253,
0.5655854940414429,
0.13749514520168304,
0.2523023784160614,
-0.5868646502494812,
0.252224326133728,
0.1501580774784088,
-0.6490141153335571,
0.03802015632390976,
0.13487082719802856,
0.24553680419921875,
-0.18416869640350342,
-0.21355901658535004,
-0.34933626651763916,
0.4673704206943512,
0.39098164439201355,
0.5127638578414917,
-0.16163738071918488,
0.14401891827583313,
0.4786534905433655,
-0.04993348568677902,
0.7777858376502991,
0.7549082636833191,
0.5022083520889282,
0.5545809268951416,
-0.4395391345024109,
-0.07431939244270325,
-0.18844377994537354,
0.28433069586753845,
0.5799110531806946,
0.3927164375782013,
0.42406609654426575,
0.011770794168114662,
0.15455099940299988,
0.4409337043762207,
-0.5076125264167786,
0.16054661571979523,
0.6311478614807129,
-0.36454203724861145,
0.16326674818992615,
-0.07605082541704178,
-0.1107984110713005,
-0.06236318498849869,
-0.14955361187458038,
-0.0355004221200943,
-0.24357183277606964,
0.3220007121562958,
0.026014583185315132,
-0.1180175393819809,
-0.6944928169250488,
0.0884048193693161,
-0.11717148125171661,
0.06564164161682129,
-0.07532130926847458,
0.09381505846977234,
0.16895486414432526,
-0.48809853196144104,
-0.6229988932609558,
0.1568441092967987,
-0.6817804574966431,
-0.05714202672243118,
-0.35214635729789734,
-0.9986971616744995,
-0.03407401964068413,
0.5893200039863586,
0.08778636902570724,
0.24897827208042145,
0.0834900438785553,
-0.47862666845321655,
-0.08931034803390503,
-0.8661666512489319,
-0.5001598000526428,
-0.16181083023548126,
-0.30118662118911743,
-0.8619112372398376,
-0.39860057830810547,
0.2879295349121094,
0.48121488094329834,
0.699387788772583,
0.035350605845451355,
-0.24025556445121765,
-0.6267696022987366,
0.18156716227531433,
-0.452151894569397,
-0.5112013816833496,
-0.5239472389221191,
-0.17525343596935272,
0.3739185631275177,
0.3178988993167877,
-0.5804989337921143,
-0.0032472223974764347,
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] |
pair amplitude defined by Eq. (28) is given by
Fs(iωn, x) =
∆Lm
ikFℏ2ΩL(iωn)
�
1 − e−i(k+
L −k−
L )x
× (iωn)2 sin2(θ/2) + iΩL(iωn)2 sin(θ/2) cos(θ/2)
(iωn)2 − |∆L|2 cos2(θ/2)
�
.
(35)
The numerical result at x = 0 is shown in Fig. 3 (a).
We note that the phase of Fs(iωn, x = 0) is indepen-
dent of ωn. For the plot, we choose the phase such that
Fs(iωn, x = 0) becomes real [this choice also applies to
Figs. 3 (b), 4 (a), and 4 (b)].
The functional form is | [
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] |
Figs. 3 (b), 4 (a), and 4 (b)].
The functional form is
substantially modified from bulk as the relative phase
θ = θR − θL is increased. The peak of the pair ampli-
tude becomes sharper with increasing θ and shows delta-
function-like behavior at θ = π − δ with δ ≪ 1. The
8
specific form of the pair amplitude at x = 0 is given by
Fs(iωn, x = 0) = i∆Lm
�
ω2n + |∆L|2
kF|∆L|ℏ2
δ|∆L|/2
ω2n + δ2|∆L|2/4
(36)
for θ = π − δ with δ ≪ 1. The right-hand side shows the
presence of the Lorentzian with the width δ|∆L|/2. | [
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0.3771449327468872,
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-0.31079521775245667,
-0.0014509970787912607,
0.1113877221941948,
0.051377926021814346,
0.6222189664840698,
0.17751477658748627,
-0.2726038694381714,
0.22143694758415222,
0.23320449888706207,
-0.440768301486969,
0.17408163845539093,
-0.016458110883831978,
0.5699447989463806,
-0.39456185698509216,
-0.6248942613601685,
-0.29817304015159607,
-0.2612554728984833,
0.37631070613861084,
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0.012770594097673893,
0.45064815878868103,
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-0.10930594801902771,
0.23207303881645203,
0.22068779170513153,
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0.8928903341293335,
0.5918363928794861,
0.19254153966903687,
0.1831466555595398,
0.01869744434952736,
0.3040068447589874,
0.5792468786239624,
-0.17617252469062805,
0.10351996123790741,
-0.26774996519088745,
0.5323796272277832,
-0.46396327018737793,
-0.09442925453186035,
0.8808118104934692,
0.18329092860221863,
0.10262523591518402,
0.33532819151878357,
0.0739729031920433,
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0.23598749935626984,
-0.18519313633441925,
-0.2621913254261017,
-0.3914514482021332,
-0.4094628095626831,
-0.6261454224586487,
0.46791982650756836,
0.20941826701164246,
0.1706511527299881,
-0.07700598239898682,
0.25276660919189453,
0.40668633580207825,
0.21633833646774292,
0.4823981523513794,
0.00656727934256196,
-0.19523648917675018,
0.2547791302204132,
0.5114169716835022,
-0.9253844618797302,
-0.11872637271881104,
0.018413344398140907,
0.5239356756210327,
0.12525634467601776,
-0.19741125404834747,
-0.1726391315460205,
-0.35231757164001465,
-0.5437019467353821,
0.47269514203071594,
0.16246812045574188,
-0.19384761154651642,
-0.2761980891227722,
-0.5154516696929932,
-0.054145194590091705,
0.26416251063346863,
0.36741331219673157,
0.004269643221050501,
-0.48497095704078674,
-0.09018845111131668,
-0.18629100918769836,
0.04276905953884125,
0.06280594319105148,
-0.06495411694049835,
-0.14405407011508942,
-0.04104771092534065
] |
presence of the Lorentzian with the width δ|∆L|/2.
The p-wave component of pair amplitude defined by
Eq. (31) becomes finite due to translational symmetry
breaking [87, 88]:
Fp(iωn, x) = 2i∆Lm
ℏ2
iωn sin(θ/2)e−iθ/2
(iωn)2 − |∆L|2 cos2(θ/2)e−i(k+
L −k−
L )x.
(37)
Fp(iωn, x = 0) is shown in Fig. 3 (b). For θ = π, the pair
amplitude diverges at ωn → 0, because the denominator
in Eq. (37) becomes zero at θ → π, ωn → 0.
Let us also discuss diagonal components of the Green’s
function. The LDOS is given by | [
-0.3316064178943634,
-0.41646286845207214,
0.8122774958610535,
-0.5259851813316345,
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0.013740811496973038,
0.012447219341993332,
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0.06818096339702606,
0.6663683652877808,
1.0168533325195312,
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0.409606397151947,
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0.5462842583656311,
0.3084639310836792,
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0.39579713344573975,
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0.7249003052711487,
0.40463510155677795,
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0.7089049220085144,
0.15364930033683777,
0.06128883734345436,
0.4661323130130768,
0.08911504596471786,
0.5492247343063354,
0.1737610250711441,
0.4020689129829407,
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0.1132623478770256,
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0.5226582288742065,
0.3825218081474304,
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0.38132503628730774,
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0.45287448167800903,
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0.04787280783057213,
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0.327115535736084,
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1.1778061389923096,
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0.12214440107345581,
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0.380929559469223,
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1.0663326978683472,
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] |
Let us also discuss diagonal components of the Green’s
function. The LDOS is given by
D(ω, x) = Dbulk(ω)
−
2m
kFℏ2π Re
�ω + i0+
Ωret(ω)
|∆L|2 sin2(θ/2)e−i(k+
L −k−
L )x
(ω + i0+)2 − |∆L|2 cos2(θ/2)
�
(38)
with Ωret(ω) = ΩL(ω + i0+)sgn ω. We choose the branch
cut of square root function in Ω along the negative side
of real axis. We have introduced the bulk DOS by
Dbulk(ω) =
2m
kFℏ2π Re ω + i0+
Ωret(ω) .
(39)
The LDOS of the s-wave SC shown in Fig. 3 (c) has di- | [
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] |
Dbulk(ω) =
2m
kFℏ2π Re ω + i0+
Ωret(ω) .
(39)
The LDOS of the s-wave SC shown in Fig. 3 (c) has di-
vergence peaks, which correspond to the Andreev bound
states [106, 107].
In Sec. IV B, we will discuss a generalized LDOS by
changing ω ∈ R → z ∈ C: D(ω, x) → D(z, x), which
will allow us to understand the relation between LDOSs
of s-wave SC and bogolon junctions. Figure 3 (d) shows
the LDOS in the complex plane. The positions of the
peaks originated from Andreev bound states are given
by ωABS(θ) = ±|∆L|
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] |
peaks originated from Andreev bound states are given
by ωABS(θ) = ±|∆L|
�
1 − sin2(θ/2), which appear on
the real axis. The comparison with the bogolon junction
will be discussed in Sec. IV B.
We turn to the discussion of J(x) defined by Eq. (32).
We start the discussion with j(iωn, x), which shows a
contribution at the frequency ωn [see Eq. (33)]:
j(iωn, x) = −4m|∆L|2
ℏ2
sin(θ/2) cos(θ/2)e−i(k+
L −k−
L )x
(iωn)2 − |∆L|2 cos2(θ/2)
.
(40)
The numerical result is shown in Fig. 3 (e) for the x = 0 | [
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] |
L )x
(iωn)2 − |∆L|2 cos2(θ/2)
.
(40)
The numerical result is shown in Fig. 3 (e) for the x = 0
case. The peak of j(iωn, x = 0) becomes sharper as the
relative phase θ increases and delta-function-like behav-
ior at θ = π−δ with δ ≪ 1, which is similar to the s-wave
FIG. 4.
Frequency-dependence of physical quantities of s-
wave SC at x = 0 for several values of Z. (a) The s-wave
component of the pair amplitude (even-frequency), (b) the
p-wave component of the pair amplitude (odd-frequency), (c) | [
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] |
p-wave component of the pair amplitude (odd-frequency), (c)
the LDOS normalized by its value in normal state (∆L = 0),
and (d) j(iωn) = j(iωn, x = 0). The relative phase is chosen
as θ = 2π/3.
component of pair amplitude. Especially at x = 0, we
can perform the summation of ωn in Eq. (32) and obtain
[108]
J(x = 0) = 2m|∆L|
ℏ2
sin θ
2 tanh
�β|∆L|
2
cos θ
2
�
.
(41)
For zero-temperature limit β → ∞ with θ ̸= π, we obtain
the simpler form:
J(x = 0) = 2m|∆L|
ℏ2
sin θ
2.
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] |
the simpler form:
J(x = 0) = 2m|∆L|
ℏ2
sin θ
2.
(42)
The current-phase relation is plotted in Fig. 3 (f) for
−π < θ < π.
Actually, Eq. (41) is related to the Josephson (conserv-
ing) current [108]. According to Ref. [98], the conserving
current I is evaluated as I = J(x = 0) for s-wave SC
junction case, which is derived from Heisenberg equation.
C.
Effect of barrier potential
We consider the effect of the barrier potential con-
trolled by the parameter Z defined in Eq. (27). Figures 4 | [
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] |
trolled by the parameter Z defined in Eq. (27). Figures 4
(a) and (b) show the Z dependence of the s-wave and
p-wave components of pair amplitudes at x = 0, respec-
tively. The relative phase is chosen as θ = 2π/3. The
delta-function-like behavior in Fs(iωn, x = 0) near θ = π
can be seen only at Z = 0 as shown in the inset of (a).
Below, we discuss the characteristic features of the
LDOS and J(x) at x = 0 in detail. The LDOS at x = 0
9
is expressed as follows:
D(ω, x = 0)
=
2m
πkFℏ2 Re
(ω + i0+)Ωret(ω) | [
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] |
9
is expressed as follows:
D(ω, x = 0)
=
2m
πkFℏ2 Re
(ω + i0+)Ωret(ω)
Z2Ωret(ω)2 + (ω + i0+)2 − |∆|2 cos2(θ/2).
(43)
Figure 4 (c) shows the LDOS for θ = 2π/3. The position
of the peak for ω > 0 in the LDOS moves to the higher
energy as Z increases. The positions of the peaks are
given by ωAB(θ) = ±|∆L|
�
1 − sin2(θ/2)/(Z2 + 1) [98,
109]. Taking the limit Z → ∞, we obtain
D(ω, x = 0) =
2m
πZ2kFℏ2 Re ω + i0+
Ωret(ω)
= 1
Z2 Dbulk(ω),
(44)
where Dbulk(ω) is the bulk DOS defined in Eq. (39). It | [
-0.2783460021018982,
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0.6317707300186157,
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-0.5208533406257629,
-0.44257092475891113,
0.6223006844520569,
0.38651445508003235,
0.03133310377597809,
-0.2984031140804291,
-0.08035118877887726,
-0.08384103327989578,
0.041328851133584976,
0.48020246624946594,
-0.42192789912223816,
0.18582506477832794,
0.3181079626083374
] |
Ωret(ω)
= 1
Z2 Dbulk(ω),
(44)
where Dbulk(ω) is the bulk DOS defined in Eq. (39). It
is notable that Eq. (44) is independent of θ and propor-
tional to the bulk DOS in this limit. The magnitude of
the LDOS becomes smaller by the factor 1/Z2.
Next, we discuss the result of J(x = 0). Before taking
the summation of ωn in Eq. (32), j(iωn) is suppressed as
Z increases, which is shown in Fig. 4 (d). We perform the
summation of ωn and the expression at zero temperature
limit β → ∞ is obtained as [106]: | [
-0.03995243459939957,
-0.1424911618232727,
0.1990782469511032,
-0.6306748986244202,
-0.27775654196739197,
0.07666338235139847,
0.578053891658783,
-0.9867124557495117,
0.19337902963161469,
-0.28812330961227417,
-0.34022393822669983,
0.637213408946991,
-0.33144837617874146,
0.2925044000148773,
-0.15534049272537231,
0.9365605115890503,
1.0318124294281006,
0.1956741362810135,
-0.5940951108932495,
-0.7106563448905945,
0.6507599353790283,
-0.41488608717918396,
0.3702954053878784,
0.4426499903202057,
0.062469031661748886,
-0.0629211962223053,
0.53327476978302,
-0.14466027915477753,
0.117768794298172,
0.5998220443725586,
-0.45028382539749146,
0.6799416542053223,
-0.1413291096687317,
-0.22597883641719818,
0.2697085738182068,
0.61588054895401,
0.17189247906208038,
0.2503235340118408,
0.0922597199678421,
0.019102638587355614,
0.13744719326496124,
-0.32845446467399597,
0.3241109848022461,
0.37048929929733276,
-0.29363733530044556,
-0.2793135941028595,
-1.3336937427520752,
-0.21424385905265808,
-0.23610147833824158,
-0.5274596214294434,
-0.020209789276123047,
0.11718643456697464,
0.006998404394835234,
0.19998006522655487,
-0.16860508918762207,
-0.11948451399803162,
-1.1460152864456177,
0.03594233840703964,
0.41732296347618103,
-0.2008274644613266,
0.3234589993953705,
-0.035695161670446396,
-0.10987545549869537,
0.40427860617637634,
-0.3673709034919739,
1.050458550453186,
-0.12668822705745697,
0.13729813694953918,
-0.6171610355377197,
0.4512023329734802,
0.28629443049430847,
0.15165813267230988,
-0.45955386757850647,
-0.1045924499630928,
-0.21278005838394165,
0.28029119968414307,
-0.6838073134422302,
-0.2391814887523651,
0.00301527907140553,
0.551696240901947,
-0.01326205674558878,
-0.16453485190868378,
0.5182274580001831,
-0.8135230541229248,
0.3074915111064911,
-0.14833901822566986,
-0.07203298807144165,
0.997242271900177,
-0.42778363823890686,
0.3831687867641449,
0.3812149167060852,
-0.004947904031723738,
0.2867472767829895,
0.27627822756767273,
-0.09674384444952011,
0.6330757141113281,
-0.06859508156776428,
0.18812553584575653,
0.23923242092132568,
0.15554174780845642,
0.13785111904144287,
-0.0052268011495471,
-0.05999283492565155,
-0.38500919938087463,
-0.3342035710811615,
-0.8397524356842041,
-0.47827041149139404,
-0.5898397564888,
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0.3015473186969757,
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0.6163597106933594,
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0.9463501572608948,
0.1609053909778595,
1.1548411846160889,
0.32449308037757874,
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0.16216504573822021,
0.5282673239707947,
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] |
summation of ωn and the expression at zero temperature
limit β → ∞ is obtained as [106]:
J(x = 0) = m|∆|
ℏ2
1
Z2 + 1
�
Z2 + 1
Z2 + cos2(θ/2) sin θ.
(45)
We can confirm that the above expression reduces to
Eq. (42) in Z → 0 limit. On the other hand, for Z → ∞
limit, we obtain
J(x = 0) = m|∆|
ℏ2Z2 sin θ,
(46)
whose θ-dependence is determined by the factor sin θ
[110].
We note that J(x = 0) is accompanied by the
factor 1/Z2, which is the same feature as that of LDOS
in the strong barrier limit.
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] |
factor 1/Z2, which is the same feature as that of LDOS
in the strong barrier limit.
IV.
RESULT FOR BOGOLON JUNCTION
With the knowledge of the conventional SC junction
explained in Sec. III, we are now ready to discuss the
bogolon junction. In the following subsections, we ap-
ply the Green’s function in Eq. (24) to the specific cases.
Firstly, in the next subsection (Sec. IV A) we will provide
the result of the semi-infinite system as the simplest case.
This case corresponds to the Z → ∞ limit of Eq. (24). | [
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] |
This case corresponds to the Z → ∞ limit of Eq. (24).
Secondly, in Sec. IV B, we will discuss the result of the bo-
golon junction without the barrier potential, i.e. Z = 0,
to explore the physics of Andreev reflection of bogolon.
Finally in Sec. IV C, we will consider the bogolon junc-
tion for Z ̸= 0 and clarify the effect of the barrier po-
tential. The comparison of bogolon junction with s-wave
SC junction is summarized in Table I. For all figures in
this section, we choose Γ1L/µ = 0.01 with µ = ℏ2k2 | [
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] |
this section, we choose Γ1L/µ = 0.01 with µ = ℏ2k2
F/2m
and set µ = 1 and kF = 1.
We mainly discuss the physical quantities at the inter-
face (x = 0) in this section. The x-dependence can be
seen more clearly using the quasiclassical Green’s func-
tion, which will be discussed in the next section (Sec. V).
A.
Semi-infinite superconductor with Bogoliubov
Fermi surface
In this subsection, we consider the semi-infinite SC
(x < 0) as the simplest non-uniform system, where the | [
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0.6219495534896851,
0.39551714062690735,
0.06309196352958679,
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0.051655299961566925,
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0.1177479699254036,
0.2929154336452484
] |
(x < 0) as the simplest non-uniform system, where the
edge is located at x = 0. We take the limit Z → ∞ in
Eqs. (24)–(26) [111]. The pair amplitude [ ˆG(iωn, x, x′)]12
is given in the following form:
[ ˆG(iωn, x, x′)]12 =
−iΓ2Lsgn ωnm
2ikFℏ2�
(iωn + iΓ1Lsgn ωn)2 + |Γ2L|2
×
�
eik+
L |x−x′| − e−ik+
L (x+x′) + e−ik−
L |x−x′| − eik−
L (x+x′)�
.
(47)
Since there are no Andreev reflections, the pair ampli-
tude, i.e. [ ˆG(iωn, x, x′)]12, exhibits a pure odd-frequency | [
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] |
tude, i.e. [ ˆG(iωn, x, x′)]12, exhibits a pure odd-frequency
pairing in Eq. (47), which is the same frequency depen-
dence as the bulk. Namely, the p-wave component is not
induced at the edge in the case of the bogolon junction
without spatial dependence in the self-energy.
In order to highlight distinctive properties arising from
the translational symmetry breaking at interfaces, the
following subsections focus on the SC junctions and stud-
ies the physics of Andreev reflection of bogolon.
B. | [
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] |
ies the physics of Andreev reflection of bogolon.
B.
Bogolon junction without barrier potential
In this subsection, we consider the bogolon junction
with Z = 0. From Eq. (26), we have the relation b(±)
out = 0,
which implies that there are no normal reflections and
transmissions. Then, we can focus on the contribution of
the Andereev reflection to the physical properties of the
bogolon junction.
1.
Off-diagonal quantities
In contrast to the semi-infinite case discussed in | [
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] |
bogolon junction.
1.
Off-diagonal quantities
In contrast to the semi-infinite case discussed in
Sec. IV A, the pair amplitude is the mixed function of
even- and odd-frequencies. The s-wave component of the
pair amplitude defined by Eq. (28) is given by
Fs(iωn, x) = −iΓ2Lsgn ωnm
ikFℏ2ΩL(iωn)
�
1 − e−i(k+
L −k−
L )x
× (iωn + iΓ1Lsgn ωn)2 sin2(θ/2) + iΩL(iωn)2 sin(θ/2) cos(θ/2)
(iωn + iΓ1L)2 + |Γ2L|2 cos2(θ/2)
�
.
(48)
10
FIG. 5.
Physical quantities at x = 0 for bogolon junction | [
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] |
�
.
(48)
10
FIG. 5.
Physical quantities at x = 0 for bogolon junction
without barrier potential. (a) The s-wave component of the
pair amplitude, (b) the p-wave component of the pair am-
plitude, (c) the LDOS of bogolons normalized by its value
of clean limit (Γ1L = Γ2L = 0), (d) the LDOS of bogolons
in the complex energy plane for θ = 2π/3, which is de-
fined by the extension of ω to the complex plane z, i.e.,
D(ω, x = 0) → D(z, x = 0), (e) j(iωn) = j(iωn, x = 0), and
(f) J(x = 0) = (1/β) � | [
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] |
D(ω, x = 0) → D(z, x = 0), (e) j(iωn) = j(iωn, x = 0), and
(f) J(x = 0) = (1/β) �
n j(iωn). Figure legends of (b)-(d) are
the same as that of (a). We set |Γ2L|/Γ1L = 0.9 in (a)–(e).
The first term is the contribution from bulk and the sec-
ond one is from the Andreev reflection.
The numeri-
cal result of Fs(iωn, x = 0) is shown in Fig. 5 (a). The
phase of Fs(iωn, x = 0) is independent of ωn, which is
the same feature as the s-wave SC junction. We choose | [
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the same feature as the s-wave SC junction. We choose
the phase such that Fs(iωn, x = 0) becomes real for the
plot [this choice also applies to Figs. 5 (b), 6 (a), and 6
(b)]. The odd-frequency dependence of Fs(iωn, x) can be
directly verified from Eq. (48) by noting that ΩL(iωn) is
an even function of ωn. The height of the pair amplitude
gradually decreases as the relative phase θ increases and
reaches zero at θ = π.
In the present bogolon junction, the even-frequency | [
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] |
reaches zero at θ = π.
In the present bogolon junction, the even-frequency
pairing is induced near the interface due to the trans-
lational symmetry breaking.
To see this, we evaluate
the p-wave component of the pair amplitude defined by
Eq. (31) as
Fp(iωn, x) = 2Γ2Lsgn ωnm
ℏ2
×
(iωn + iΓ1Lsgn ωn) sin(θ/2)e−iθ/2
(iωn + iΓ1Lsgn ωn)2 + |Γ2L|2 cos2(θ/2)e−i(k+
L −k−
L )x.
(49)
In Eq. (49), the bulk terms vanish, and then only the
term from Andreev reflection contributes. The results at | [
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] |
term from Andreev reflection contributes. The results at
x = 0 is shown in Fig. 5 (b). In contrast to the s-wave
SC, the pair amplitude of bogolons does not diverge near
θ = π due to the presence of finite Γ1L in the denominator
in the second line of Eq. (49).
2.
Diagonal quantities
Now we turn to the diagonal quantities such as the
LDOS and J(x). The specific form of the LDOS of bo-
golons is given by
D(ω, x) = Dbulk(ω)
+
2m
kFℏ2π Re
�ω + iΓ1L
Ωret(ω)
|Γ2L|2 sin2(θ/2)e−i(k+
L −k−
L )x | [
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0.3136403262615204,
0.45565876364707947,
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] |
D(ω, x) = Dbulk(ω)
+
2m
kFℏ2π Re
�ω + iΓ1L
Ωret(ω)
|Γ2L|2 sin2(θ/2)e−i(k+
L −k−
L )x
(ω + iΓ1L)2 + |Γ2L|2 cos2(θ/2)
�
,
(50)
where Dbulk(ω) is a bulk DOS of bogolons defined by
Dbulk(ω) =
2m
kFℏ2π Re ω + iΓ1L
Ωret(ω)
(51)
with Ωret(ω) = Ω(ω + i0+)sgn ω.
The second line of
Eq. (50) is the Andreev reflection part, which takes a
finite value in the non-uniform case (θ ̸= 0). We list in
Table I the expressions of the LDOS of bogolons in the
low-ω limit for θ = 0 [row (c)] and θ = π [row (d)]. | [
-0.24022282660007477,
-0.10226660221815109,
0.2628806531429291,
-0.3381575047969818,
-0.25123336911201477,
0.4644971787929535,
0.042886655777692795,
-0.6883330941200256,
0.2056911736726761,
-0.46975177526474,
-0.3246206045150757,
0.4329087436199188,
-0.4703627824783325,
0.2850320339202881,
0.14584502577781677,
1.142546534538269,
0.672694206237793,
0.14772233366966248,
-0.7481896877288818,
-1.0933696031570435,
0.4286767244338989,
-0.5235785841941833,
0.3066162168979645,
0.3043428659439087,
-0.32555484771728516,
0.2495744228363037,
0.6190395951271057,
0.3894830644130707,
0.21282267570495605,
0.5834652781486511,
-0.21310372650623322,
0.32732051610946655,
0.3569440245628357,
0.15708951652050018,
0.2208660989999771,
0.2615123391151428,
0.3150770366191864,
0.1544840931892395,
0.07159795612096786,
-0.04731031507253647,
0.10144784301519394,
-0.17197571694850922,
0.073125459253788,
0.6104050874710083,
-0.19142065942287445,
-0.26261353492736816,
-1.1094589233398438,
0.17268918454647064,
-0.016733281314373016,
-0.6825904250144958,
-0.3184935450553894,
-0.12550804018974304,
0.1345233917236328,
0.2908618450164795,
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] |
low-ω limit for θ = 0 [row (c)] and θ = π [row (d)].
Figure 5 (c) shows the LDOS of bogolons at x = 0. In
the uniform case (θ = 0), the LDOS has the zero-energy
peak, which is consistent with the previous calculations
in bulk [21, 37]. As θ increases, the pseudo-gap appears
at zero energy. The gap formation in the LDOS is in-
complete for any θ, i.e., the value of the LDOS at ω = 0
is always finite.
As discussed above, the LDOS behaviors of s-wave SC
and bogolon junctions are quite different. Nevertheless, | [
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] |
and bogolon junctions are quite different. Nevertheless,
the correspondence between the two junctions can be vi-
sualized by considering the LDOS in a complex frequency
space. Let us extend ω in the LDOS of bogolons onto the
complex plane as D(ω, x) → D(z, x) with z ∈ C. Fig-
ure 5 (d) shows the LDOS in the complex plane at x = 0
for the bogolon junction. The positions of the peaks are
given by ωABS(θ) = −iΓ1L±i|Γ2L|
�
1 − sin2(θ/2) located
on the imaginary axis. On the other hand, for the s-wave | [
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] |
�
1 − sin2(θ/2) located
on the imaginary axis. On the other hand, for the s-wave
SC case, these peak positions appear on the real axis as
discussed in Sec. III. The LDOS of bogolons [Fig. 5 (c)]
corresponds to the s-wave SC junction with 90◦-rotation
in the complex plane [Fig. 3 (d)].
11
Next, we show the results for quasiparticle current
J(x). We first consider the contribution at ωn defined
by Eq. (33):
j(iωn, x) = 2m|Γ2L|2
ℏ2
×
sin(θ/2) cos(θ/2)e−i(k+
L −k−
L )x
(iωn + iΓ1Lsgn ωn)2 + |Γ2L|2 cos2(θ/2). | [
-0.2971433401107788,
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] |
ℏ2
×
sin(θ/2) cos(θ/2)e−i(k+
L −k−
L )x
(iωn + iΓ1Lsgn ωn)2 + |Γ2L|2 cos2(θ/2).
(52)
Here, Andreev reflection only contributes to j(iωn, x)
similar to Eq. (49). The frequency dependence of Eq. (52)
at x = 0 is shown in Fig. 5 (e). j(iωn, x = 0) becomes
zero at θ = 0 or π. The value at ω = 0 for θ = 2π/3 case
is larger than the value at ω = 0 for θ = π/3 case, which
is in contrast to the s-wave SC junction. The absolute
value of j(iωn, x) for ωn → 0 takes the maximum value
when θ = cos−1(Γ2
1L/(2Γ2 | [
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] |
value of j(iωn, x) for ωn → 0 takes the maximum value
when θ = cos−1(Γ2
1L/(2Γ2
1L − |Γ2L|2)).
For zero-temperature limit β → ∞, we can take the
summation of ωn:
J(x = 0) = −m|Γ2L|
πℏ2
sin θ
2 ln
�Γ1L + |Γ2L| cos(θ/2)
Γ1L − |Γ2L| cos(θ/2)
�
.
(53)
Compared with the s-wave SC junction case,
the
θ-dependence of Eq. (53) has the additional factor
ln
�Γ1L + |Γ2L| cos(θ/2)
Γ1L − |Γ2L| cos(θ/2)
�
(see row (e) in Table I). Due to
this logarithmic factor, J(x = 0) is a continuous function | [
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] |
�
(see row (e) in Table I). Due to
this logarithmic factor, J(x = 0) is a continuous function
even at θ = π, which is a clear difference from the s-wave
SC junction. Additionally, in comparison to the s-wave
SC junction case in Eq. (42), the different sign for the
bogolon junction is a consequence of odd-frequency pair
potential, which is referred to as the π-junction. Namely,
the relation [ˆΣ(−iωn, x)]12 = −[ˆΣ(iωn, x)]12, is the origin
of the minus sign. Figure 5 (f) shows J(x = 0) for several | [
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] |
of the minus sign. Figure 5 (f) shows J(x = 0) for several
values of |Γ2L|/Γ1L. The maximum value of |J(x = 0)|
shifts towards θ = 0 as |Γ2L|/Γ1L increases because the
contribution of the factor cos(θ/2) in Eq. (53) becomes
larger.
3.
Correlation between LDOS and pair amplitudes
Let us discuss the correlation between the LDOS of bo-
golon and the pair amplitudes Fs(iωn, x) and Fp(iωn, x).
We here focus on the |Γ2L|/Γ1L-dependence for the uni-
form case (θ = 0) and the non-uniform case (θ = π). | [
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] |
form case (θ = 0) and the non-uniform case (θ = π).
Since the relation Γ1L > |Γ2L| must be satisfied to
guarantee the positive DOS, we consider the case for
0 < |Γ2L|/Γ1L < 1.
First, in the uniform case (θ = 0), the LDOS and s-
wave pair amplitude are shown in Fig. 6 (a) and (b),
respectively.
Note that the p-wave component of pair
amplitude is zero in this uniform case.
The LDOS at
ω = 0 in (a) increases as |Fs(iωn → 0, x = 0)| in (b)
increases. Especially in the limit of |Γ2L|/Γ1L → 1, both | [
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] |
increases. Especially in the limit of |Γ2L|/Γ1L → 1, both
LDOS at ω = 0 and Fs(iωn → 0, x = 0) diverge. We will
revisit the relation between the LDOS and Fs in Sec. V.
FIG. 6.
Frequency-dependence of LDOS of bogolons nor-
malized by its value of clean limit (Γ1L = Γ2L = 0) and pair
amplitudes at Z = 0. (a) LDOS at θ = 0 and (b) s-wave
component of pair amplitude at θ = 0. (c) LDOS at θ = π
and (d) p-wave component of pair amplitude at θ = π. The
lines in (b)-(d) are shared with those in (a). | [
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] |
lines in (b)-(d) are shared with those in (a).
Second, in the θ = π case, the LDOS and p-wave pair
amplitude are shown in Fig. 6 (c) and (d), respectively.
The s-wave component of pair amplitude is zero in this
case. The depth of the pseudogap of the LDOS in (c)
becomes larger as Fp(iωn, x = 0) in (d) increases. Hence,
the depth of pseudogap in the LDOS is correlated with
the magnitude of the p-wave pair amplitude.
Therefore, the zero-energy LDOS is correlated with
pair amplitudes.
C. | [
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0.020605774596333504
] |
Therefore, the zero-energy LDOS is correlated with
pair amplitudes.
C.
Effect of barrier potential
In this subsection, we study the effect of the bar-
rier potential. The barrier-potential dependence of the
Green’s function, as expressed in Eqs. (24)–(26) [see also
Eqs. (C3)–(C5)], is controlled by the parameter Z defined
in Eq. (27). In the limit of Z → ∞, Z-dependence of ˜a(±)
out
given by Eq. (25) is roughly expressed as ˜a(±)
out ∼ 1/Z2,
while ˜b(±)
out is expressed as ˜b(±)
out ∼ 1. Thus, the contri- | [
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] |
out ∼ 1/Z2,
while ˜b(±)
out is expressed as ˜b(±)
out ∼ 1. Thus, the contri-
bution of Andreev reflection becomes smaller than the
normal reflection.
Figure 7 shows the physical quantities of bogolons at
x = 0. The s-wave pair amplitude, p-wave pair ampli-
tude, the LDOS of bogolons, and j(iωn) are shown in
(a), (b), (c), and (d), respectively. The relative phase
is chosen as θ = 2π/3. The absolute values of the pair
amplitudes at x = 0 in (a) and (b) become smaller as Z | [
-0.07304845005273819,
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0.6591033339500427,
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-0.9029265642166138,
0.28642725944519043,
0.12663067877292633,
-0.4183915853500366,
0.4810461103916168,
0.7719258666038513,
-0.046789392828941345,
-0.03469422832131386,
-0.13458071649074554,
-0.23057420551776886,
-0.06965089589357376,
0.8235238790512085,
-0.16764403879642487,
-0.3493046164512634,
0.18092983961105347,
0.2311811000108719,
-0.297049343585968,
-0.17893698811531067,
0.1679190993309021,
-0.4796888530254364,
-0.3938613831996918,
-0.8536615967750549,
0.0005898477975279093,
-0.22203867137432098,
0.5502923727035522,
-0.14878635108470917,
0.37904584407806396,
0.17731113731861115,
0.23926764726638794,
-0.2164750099182129,
-0.49054718017578125,
0.16693982481956482,
-0.46484747529029846,
-0.025054344907402992,
-0.058720529079437256,
-0.12740331888198853,
-0.5605013370513916,
0.6812939047813416,
0.12618201971054077,
0.14981192350387573,
-0.2963509261608124,
-0.17914052307605743,
0.17807123064994812,
0.19236263632774353,
0.011106996797025204,
-0.3584584593772888,
0.31504783034324646,
0.4808675944805145,
-0.35492631793022156,
0.1591484099626541,
-0.25146982073783875,
0.06731172651052475,
0.34987643361091614,
0.329279363155365,
-0.20478154718875885,
0.38762539625167847,
-0.47896692156791687,
0.38915592432022095,
0.44524696469306946,
-0.8057050704956055,
-0.22300118207931519,
0.16562968492507935,
-0.5121971368789673,
-0.4268384277820587,
0.07407880574464798,
0.3446674644947052,
0.21161189675331116,
-0.013095464557409286,
-0.4484364092350006,
0.08787500113248825,
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0.06796643137931824,
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-0.4539344310760498,
0.09298999607563019,
-0.1295495629310608,
0.4092309772968292,
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-0.05618973821401596,
1.0099138021469116,
0.736990213394165,
0.21966524422168732,
-0.5258207321166992,
-0.0941476821899414,
-0.41173651814460754,
-0.3590923845767975,
0.4740518033504486,
-0.03654975816607475,
0.18291759490966797,
-0.14092537760734558,
0.2100486010313034,
-0.17934781312942505,
0.3729512393474579,
0.9073227643966675,
-0.4854583442211151,
-0.08855976164340973,
0.09316705912351608,
0.2758930027484894,
-0.5110546946525574,
0.22998358309268951,
0.04126203432679176,
-0.16277752816677094,
0.1500350683927536,
-0.006777280475944281,
-0.1512092500925064,
-0.5084645748138428,
0.09484821557998657,
-0.48507317900657654,
-0.6039665341377258,
0.18472595512866974,
0.22072537243366241,
0.36079004406929016,
0.2570941150188446,
-0.7011563777923584,
-0.2821762263774872,
-0.24997177720069885,
0.219486802816391,
-0.358311265707016,
0.2320704460144043,
0.10558291524648666,
-0.19252634048461914,
0.20682525634765625,
-0.14030757546424866,
-0.42239800095558167,
0.1461254507303238,
0.12270911782979965,
0.14397890865802765,
0.13633567094802856,
-0.2164430171251297,
0.24432271718978882,
-0.5098036527633667,
-0.3209058344364166,
-0.6767316460609436,
-0.06863144040107727,
-0.260996013879776,
0.18803057074546814,
0.223703533411026,
-0.02015296369791031,
-0.001992523903027177,
-0.34934863448143005,
-0.10067158192396164,
0.2577730119228363,
-0.049294620752334595,
-0.020614102482795715,
-0.1522444635629654,
0.38977953791618347,
0.4296661615371704,
-0.029908524826169014,
-0.10328718274831772,
-0.18722409009933472,
0.5305785536766052,
-0.1490160971879959,
0.35344183444976807,
-0.4164424538612366,
0.09913242608308792,
0.8403609395027161,
-0.20280268788337708,
0.32427000999450684,
0.013774563558399677,
-0.2633017301559448,
-0.2882458567619324,
0.2674003839492798,
-0.6166142225265503,
-0.479715496301651,
0.4332008957862854,
0.08449093252420425,
-0.27210983633995056,
0.07287365943193436,
0.4238850176334381,
0.18881437182426453,
-0.25167316198349,
0.435823529958725,
-0.38626962900161743,
0.17498579621315002,
-0.5067381262779236,
0.49839022755622864,
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0.00672802422195673,
0.022545702755451202,
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0.1713842749595642,
0.3766438066959381,
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0.26346534490585327,
0.11883936077356339,
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0.12505704164505005,
0.414518803358078,
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0.184200257062912,
0.09905298054218292,
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0.015716714784502983,
0.05393175780773163,
0.613509476184845,
-0.14059506356716156,
-0.5976383090019226,
-0.14979201555252075,
0.48009610176086426,
0.18379050493240356,
-0.27879613637924194,
0.4998319745063782,
-0.29113122820854187,
-0.18175192177295685,
0.44657838344573975,
-0.11810304969549179,
0.10773571580648422,
0.3748570382595062,
0.015752144157886505,
0.17846353352069855,
0.23647360503673553,
-0.057970356196165085,
-0.6046258211135864,
0.04406960681080818,
0.08831167221069336,
-0.09089326858520508,
-0.270207017660141,
0.33845019340515137,
0.906566858291626,
-0.03245072811841965,
-0.3704710006713867,
0.33707600831985474,
0.03204190358519554,
0.10126962512731552,
0.14003755152225494,
-0.17020471394062042,
-0.012765157967805862,
0.34212416410446167,
0.5667337775230408,
1.077893853187561,
-0.29602572321891785,
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0.2285306453704834,
0.13951429724693298,
0.14762058854103088,
-0.02002784051001072,
0.02810594253242016,
0.04152597486972809,
0.0408630408346653,
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0.47720780968666077,
0.26920637488365173,
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0.13422812521457672,
0.26846131682395935,
0.746339738368988,
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0.20372362434864044,
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0.0999084934592247,
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0.2794594466686249,
0.0271509550511837,
0.29322147369384766,
-0.15762943029403687,
-0.10210496932268143,
0.47656646370887756,
0.20000356435775757,
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0.059926506131887436,
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0.6811778545379639,
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0.45199087262153625,
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0.15943752229213715,
0.22541649639606476,
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0.42820215225219727,
-0.5652686357498169,
-0.4974210262298584,
0.6738646626472473,
0.7426904439926147,
0.2880754768848419,
0.34342727065086365,
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-0.016062894836068153,
0.5280853509902954,
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0.360493540763855,
-0.3359896242618561,
0.4024249017238617,
-0.3822793662548065,
0.08339253067970276,
1.272773027420044,
0.07959312945604324,
0.043962325900793076,
0.1010051965713501,
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0.5181927680969238,
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-0.5402238368988037,
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0.1944369226694107,
0.4671461880207062,
0.08815484493970871,
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0.3066707253456116,
0.02630700170993805,
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0.22558750212192535,
0.2738319933414459,
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-0.16864646971225739,
0.6109476089477539,
-0.6623812913894653,
0.38418787717819214,
0.08056625723838806,
0.4565292000770569,
0.08406560122966766,
-0.2763528823852539,
0.006506161764264107,
-0.15254263579845428,
-0.38517308235168457,
0.6375731229782104,
0.14671015739440918,
0.2630273103713989,
-0.11900489032268524,
-0.17260579764842987,
-0.2282681167125702,
0.40798699855804443,
-0.10391600430011749,
-0.39976927638053894,
-0.21543623507022858,
0.17393819987773895,
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0.012143341824412346,
-0.20682816207408905,
-0.13193362951278687,
0.05500035732984543
] |
amplitudes at x = 0 in (a) and (b) become smaller as Z
increases due to high barrier potential.
12
Bogolon junction
s-wave SC junction
(a)
Bulk pair (s-wave)
Odd-freq. pair
Even-freq. pair
(b)
Induced pair at the interface (p-wave)
Even-freq. pair
Odd-freq. pair
(c) Zero-energy LDOS
at (θ, Z) = (0, 0) (bulk) [Dbulk(ω = 0)]
2m
kFℏ2π
1
�
1 − (|Γ2L|/Γ1L)2
0 (gapped)
(d)
Zero-energy LDOS
at (θ, Z) = (π, 0) [D(ω = 0, x = 0)]
2m
kFℏ2π
�
1 − (|Γ2L|/Γ1L)2
2m|∆L|
kFℏ2π δ(ω)
(e) Current phase relation for −π < θ < π | [
-0.4127625823020935,
-0.42691484093666077,
0.6504743695259094,
-0.17137357592582703,
-0.5677758455276489,
0.17427407205104828,
0.23585589230060577,
-0.8144683241844177,
-0.1906469613313675,
-0.3790319859981537,
0.12282212823629379,
0.6156772971153259,
-0.4044261872768402,
0.08283886313438416,
0.42918479442596436,
0.9672682285308838,
0.7841519713401794,
0.05694499611854553,
-0.9793105721473694,
-0.7918568253517151,
0.5147593021392822,
-0.26871493458747864,
1.0535155534744263,
0.18370191752910614,
0.017220498993992805,
0.008125930093228817,
0.41128334403038025,
-0.052757006138563156,
0.5551454424858093,
0.410879909992218,
-0.17190879583358765,
1.1627593040466309,
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0.09230374544858932,
0.24981285631656647,
0.39447304606437683,
0.22572016716003418,
0.11742957681417465,
0.12396117299795151,
-0.44266921281814575,
0.34457674622535706,
-0.5019177794456482,
-0.18614763021469116,
0.27826374769210815,
-0.2510395050048828,
-0.6419219970703125,
-1.073113203048706,
0.015912462025880814,
-0.48659682273864746,
-0.44224363565444946,
0.14534106850624084,
0.5150829553604126,
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0.35053932666778564,
-0.11374669522047043,
-0.1795063465833664,
-0.5579683780670166,
-0.10878148674964905,
0.6094737648963928,
-0.2482631951570511,
0.38668718934059143,
-0.0008352365694008768,
-0.19586344063282013,
0.4257201552391052,
-0.17540761828422546,
1.1004993915557861,
0.060914382338523865,
0.01075584627687931,
-0.12167973816394806,
-0.045534852892160416,
0.22229516506195068,
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0.22431467473506927,
0.3248826861381531,
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0.27704939246177673,
0.30794429779052734,
0.038343098014593124,
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0.680370569229126,
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0.2006780356168747,
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0.7847117185592651,
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0.6591114401817322,
0.04390683025121689,
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0.4219517409801483,
0.712468147277832,
0.43594786524772644,
0.5713852643966675,
-0.16956265270709991,
0.0036056481767445803,
0.49966514110565186,
0.10705792903900146,
0.29139700531959534,
0.34271785616874695,
-0.3591916561126709,
-0.18783527612686157,
-0.1668175905942917,
-0.5973947644233704,
-0.5622960329055786,
-0.5666120648384094,
-0.4777884781360626,
-0.1304452121257782,
-1.4330964088439941,
0.13771095871925354,
0.14300796389579773,
-0.45696133375167847,
-0.2407337874174118,
-0.24396617710590363,
0.09775137156248093,
-0.1080196276307106,
0.5835590362548828,
0.5471826195716858,
0.1445881426334381,
0.8241119980812073,
-0.24909855425357819,
0.5390931963920593,
0.3873949944972992,
0.04824415594339371,
0.14226952195167542,
0.459013432264328,
-0.5806775093078613,
0.4645831882953644,
-0.41798892617225647,
0.755375325679779,
0.25273042917251587,
0.23094436526298523,
0.0791763961315155,
-0.6552103161811829,
-0.19405728578567505,
-0.08524597436189651,
-0.012704145163297653,
-0.06760723143815994,
-0.27625736594200134,
0.04620898514986038,
-0.5376554727554321,
-0.1682945191860199,
0.16827712953090668,
-0.5019996166229248,
-0.32355421781539917,
-0.19150766730308533,
0.040115389972925186,
-0.1955740749835968,
-0.46409347653388977,
-0.10340026766061783,
-0.591105043888092,
-0.014321886003017426,
-0.338194340467453,
-0.5172826647758484,
0.36846619844436646,
-0.22057010233402252,
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0.09414417296648026,
0.0704856812953949,
0.055452894419431686,
0.15314452350139618,
-0.07903621345758438,
0.5051848888397217,
-0.3492957651615143,
-0.024587174877524376,
-0.26077398657798767,
0.06426107883453369,
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-0.34250685572624207,
0.12230656296014786,
0.09723598510026932,
0.3078400790691376,
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0.42237773537635803,
0.5554363131523132,
0.2903289496898651,
0.2820204794406891,
0.502458930015564,
0.40557870268821716,
0.24273480474948883,
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0.246305450797081,
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1.1320562362670898,
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0.3458317220211029,
0.4608725607395172,
0.02035292237997055,
0.5036779642105103,
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0.26907655596733093,
0.27460941672325134,
-0.5307300686836243,
0.34675133228302,
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0.17073163390159607,
0.33300527930259705,
-0.7141838669776917,
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0.03911370784044266,
-0.4318491518497467,
0.07454457879066467,
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0.2617909014225006,
0.6089147329330444,
-0.2634698152542114,
0.5441462397575378,
0.5581440329551697,
0.26154762506484985,
0.5861261487007141,
-0.7220259308815002,
-0.3148142099380493,
-0.28644824028015137,
0.17189060151576996,
0.15534792840480804,
0.2490113377571106,
0.4911453127861023,
0.09616569429636002,
-0.04686274752020836,
0.6356286406517029,
-0.19940686225891113,
-0.15002362430095673,
0.6430100798606873,
-0.3106534779071808,
0.25231948494911194,
0.006403063423931599,
-0.1688610166311264,
0.011049207299947739,
0.10677433013916016,
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0.20956528186798096,
0.46494317054748535,
0.13578471541404724,
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0.10652615875005722,
0.18116770684719086,
0.19612783193588257,
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-0.04412828013300896,
0.520620584487915,
-0.04811634123325348,
0.48350945115089417,
0.2633882462978363,
-0.45460647344589233,
-0.1003892719745636,
-0.5931658744812012,
-0.33414745330810547,
0.09037676453590393,
0.23303261399269104,
-0.6261862516403198,
-0.40437060594558716,
0.5114826560020447,
0.8639264106750488,
1.062522530555725,
-0.14795808494091034,
-0.5123004913330078,
-0.40561386942863464,
0.17500753700733185,
-0.10762069374322891,
-0.4497484862804413,
-0.6072239279747009,
-0.24262477457523346,
0.0822397843003273,
0.16258469223976135,
0.03420735150575638,
-0.052338261157274246,
0.7021737098693848,
0.07101226598024368,
0.5471433401107788,
0.645541787147522,
-0.1641915738582611,
-0.44444936513900757,
-0.1807224452495575,
-0.7449697256088257,
0.2906084358692169,
-0.2510438561439514,
-0.5777184367179871,
-0.6497300863265991,
0.01266509760171175,
-5.82570743560791,
0.2756805121898651,
-0.5633283853530884,
0.4797194004058838,
-0.37347686290740967,
0.2673289477825165,
0.9310601949691772,
-0.16494198143482208,
-0.025644119828939438,
0.060160376131534576,
-0.41573184728622437,
0.16273000836372375,
-0.16904312372207642,
-0.16349250078201294,
-0.077694371342659,
0.22303716838359833,
0.30597981810569763,
0.3444861173629761,
0.2919270098209381,
-0.13542994856834412,
0.4521481990814209,
0.14779210090637207,
-0.08851267397403717,
0.02363644354045391,
0.5299720764160156,
0.002151625230908394,
-0.16325560212135315,
0.17811867594718933,
-0.27118295431137085,
-0.6309493184089661,
-0.439598947763443,
-0.6797099709510803,
-0.2180715948343277,
-0.1722143590450287,
0.48087236285209656,
0.16882584989070892,
0.62008136510849,
0.049661509692668915,
0.20136433839797974,
-0.5332701802253723,
0.44913145899772644,
0.5372021794319153,
-0.011531984433531761,
-0.12926419079303741,
-0.01620233803987503,
-0.22216834127902985,
0.05845937132835388,
0.3223980963230133,
0.10586602240800858,
0.6267500519752502,
-0.0736798569560051,
-0.2349381148815155,
0.020221112295985222,
0.31881850957870483,
0.20377273857593536,
0.02255167067050934,
-0.34609994292259216,
0.118728406727314,
-0.329695463180542,
0.3788636326789856,
-0.12067747861146927,
0.08200662583112717,
-0.39584678411483765,
-0.3728656768798828,
0.20040050148963928,
0.4415489137172699,
-0.6573969721794128,
-0.2168421596288681,
0.8117901086807251,
-0.10864302515983582,
-0.41571468114852905,
-0.009298708289861679,
0.26634231209754944,
-0.46997585892677307,
0.661514937877655,
-0.41001096367836,
0.4946823716163635,
0.05187961831688881,
-0.4121973514556885,
0.10836860537528992,
-0.6558279991149902,
0.0267934650182724,
0.46358516812324524,
0.19068998098373413,
0.2833106517791748,
0.4597703814506531,
-0.014521860517561436,
0.08416587114334106,
-0.0896572470664978,
0.20892232656478882,
0.4449484944343567,
-0.04362912103533745,
0.2168135792016983,
0.07257998734712601,
0.46038904786109924,
-0.10270316153764725,
0.025619439780712128,
0.021939294412732124,
0.4524322748184204,
0.03812012821435928,
0.6358857750892639,
-0.15846388041973114,
-0.06675811856985092,
-0.8531298041343689,
-0.3474922776222229,
0.48797711730003357,
-0.37453004717826843,
-0.07484479248523712,
0.7357950210571289,
-0.5491206645965576,
0.2463705688714981,
-0.35709142684936523,
-0.6366625428199768,
0.06694095581769943,
0.7139893174171448,
-0.045107271522283554,
-0.7147401571273804,
0.31878742575645447,
-0.15194496512413025,
-0.21474461257457733,
0.09877204149961472,
0.08892887085676193,
-0.29998791217803955,
-0.008328326046466827,
-0.2064201384782791,
0.17010262608528137,
-0.1954193115234375,
0.5779935717582703,
-0.3753671646118164,
0.10907356441020966,
0.3051612079143524,
0.06529035419225693,
-0.36579641699790955,
-0.04163900017738342,
0.35439929366111755,
-0.6411136984825134,
-0.2933778166770935,
0.1371401697397232,
0.6897138357162476,
-0.030817441642284393,
0.54099440574646,
0.1877005398273468,
0.06839344650506973,
-0.3408302962779999,
0.23065747320652008,
0.3030276596546173,
-0.13957928121089935,
-0.4781099855899811,
-0.48528918623924255,
-0.016265133395791054,
0.21653705835342407,
-0.4338114559650421,
0.2175176441669464,
-0.3602042496204376,
-0.20294933021068573,
0.5553057789802551,
0.34214216470718384,
-0.08796153962612152,
0.23865237832069397,
-0.38761863112449646,
0.16805565357208252,
0.618525505065918,
-0.8401845693588257,
-0.034218680113554,
0.008874491788446903,
0.04197682440280914,
-0.6261570453643799,
-0.21274542808532715,
0.1261482983827591,
0.15337416529655457,
-0.29504838585853577,
-0.4137877821922302,
-0.07633404433727264,
-0.28597843647003174,
0.10182419419288635,
0.1871127337217331,
-0.29814308881759644,
-0.34137532114982605,
0.43052154779434204,
-0.3133896589279175,
0.1553412675857544,
0.10691525042057037,
-0.3603167235851288,
1.2488211393356323,
0.648048996925354,
0.011074083857238293,
-0.388408899307251,
0.35803940892219543,
-0.342660516500473,
-0.19101281464099884,
0.49284201860427856,
-0.07747051119804382,
0.41193708777427673,
-0.21482160687446594,
0.6243578195571899,
-0.3164532482624054,
0.5472753643989563,
0.6351547241210938,
-0.5456092357635498,
-0.2293229103088379,
0.3030908703804016,
0.5068410038948059,
-0.1743623912334442,
0.6176154017448425,
-0.17574140429496765,
0.08949082344770432,
-0.21095341444015503,
-0.08808804303407669,
-0.19610996544361115,
-0.2354653775691986,
0.05976736918091774,
-0.6402463912963867,
-0.48705384135246277,
0.5454548001289368,
0.376819372177124,
0.16911733150482178,
0.17915047705173492,
-0.49471399188041687,
-0.46616923809051514,
-0.09837309271097183,
0.2180209904909134,
-0.6319120526313782,
0.15975452959537506,
0.21892403066158295,
-0.7059451937675476,
0.13618572056293488,
0.019961193203926086,
-0.6051135063171387,
0.00002179145667469129,
0.18901613354682922,
-0.12149921804666519,
0.1843084841966629,
-0.10840803384780884,
0.31813713908195496,
-0.4464305341243744,
-0.15024609863758087,
-0.943598747253418,
-0.4786088764667511,
-0.4022655189037323,
0.16331230103969574,
-0.23520764708518982,
0.36004436016082764,
-0.040393926203250885,
-0.6439135074615479,
-0.25844818353652954,
0.00034829878131859004,
0.8711900115013123,
-0.4692377746105194,
-0.48973140120506287,
0.32605093717575073,
0.43273499608039856,
-0.23996669054031372,
-0.3768279552459717,
-0.28538957238197327,
0.9709708094596863,
-0.4612370431423187,
-0.29340407252311707,
-0.7800451517105103,
0.11268217116594315,
0.8839461803436279,
-0.5047663450241089,
-0.08782701939344406,
-0.6369061470031738,
-0.48342421650886536,
-0.2113507091999054,
0.18806132674217224,
-0.16310390830039978,
-0.6749209761619568,
0.34828680753707886,
-0.11055673658847809,
-0.18172751367092133,
-0.3559130132198334,
0.6916711926460266,
0.05462505295872688,
-0.5151809453964233,
0.32128962874412537,
-0.4664490222930908,
0.15638160705566406,
0.06267649680376053,
0.04410213977098465,
-0.26727861166000366,
-0.5211907625198364,
-0.23626548051834106,
-0.12813159823417664,
0.28191787004470825,
-0.10546079277992249,
-0.2561253011226654,
-0.11708816140890121,
-0.5289697647094727,
0.04779195785522461,
-0.06551779061555862,
-0.002820778638124466,
-0.8536521196365356,
-0.3348334729671478,
0.051455140113830566,
0.399850070476532,
0.23524869978427887,
-0.042563166469335556,
-0.2857075333595276,
-0.2233933061361313,
-0.6295130252838135,
0.4405807554721832,
-0.21206343173980713,
-0.5412998795509338,
-0.0640350729227066,
-0.18053887784481049,
0.4174291789531708,
-0.3119179606437683,
-0.4527090787887573,
-0.5413684248924255,
0.4514801800251007,
0.112608902156353,
-0.051462698727846146,
0.5566221475601196,
-0.16676470637321472,
-0.15359658002853394,
0.510562539100647,
-0.06743840128183365,
0.1230488270521164,
0.3042973279953003,
0.08996123820543289,
0.8623831868171692,
0.666658341884613,
-0.09928601235151291,
-0.6760536432266235,
0.283069372177124,
-0.10481707006692886,
-0.10407550632953644,
-0.2065245658159256,
0.5631035566329956,
0.5835458636283875,
0.2986653745174408,
-0.3846147358417511,
0.10986711829900742,
0.32725951075553894,
-0.09503190964460373,
0.021383674815297127,
0.24471023678779602,
-0.41620591282844543,
0.28002357482910156,
0.5894878506660461,
1.1681758165359497,
-0.25588876008987427,
0.3902689516544342,
0.13174298405647278,
-0.012228360399603844,
0.4411603510379791,
0.084796242415905,
0.041288428008556366,
0.056883204728364944,
0.10245770961046219,
-0.22046972811222076,
-0.08970820158720016,
-0.7517493963241577,
-0.615688145160675,
0.2849426567554474,
0.3242621123790741,
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0.07405059039592743,
0.12601009011268616,
0.11801787465810776,
0.5619121193885803,
-0.16662482917308807,
-0.19566582143306732,
-0.7577988505363464,
-0.37105312943458557,
0.13239602744579315,
0.04575308412313461,
-0.3545585870742798,
-0.6596493721008301,
0.06797482818365097,
0.0427878312766552,
0.010661839507520199,
0.6579805612564087,
-0.027018874883651733,
-0.6706174612045288,
0.19299399852752686,
0.14238116145133972,
-0.3622021973133087,
0.6743651032447815,
0.13358443975448608,
0.6264069080352783,
-0.1908646821975708,
-0.6900457143783569,
0.25215643644332886,
-0.4774109423160553,
0.49711868166923523,
-0.01628914289176464,
0.1495763510465622,
-0.18435361981391907,
0.3991033732891083,
-0.43716874718666077,
-0.48135706782341003,
0.21390998363494873,
0.2111620157957077,
-0.1872214376926422,
-0.43663153052330017,
0.549329400062561,
0.38503214716911316,
0.2941851317882538,
0.3190414309501648,
0.07095767557621002,
0.26428577303886414,
0.633480429649353,
-0.5128211379051208,
0.300991028547287,
-0.4293184280395508,
0.07966820895671844,
-0.7013161182403564,
-0.06703264266252518,
1.1422449350357056,
0.47787079215049744,
-0.015705419704318047,
0.2961942255496979,
0.06362005323171616,
-0.3974391222000122,
0.4746621251106262,
-0.4462985098361969,
-0.4518667757511139,
-0.7540876269340515,
-0.4843335747718811,
-1.0065240859985352,
0.4678909480571747,
0.1918269395828247,
0.11408673971891403,
-0.06796940416097641,
0.4037666618824005,
0.01641472429037094,
0.17308460175991058,
0.10283446311950684,
-0.2527427673339844,
-0.342244416475296,
-0.10700032860040665,
0.6206819415092468,
-0.8063130378723145,
0.3537493050098419,
-0.16158168017864227,
0.2855466604232788,
0.4065975844860077,
-0.1294478476047516,
0.3918302059173584,
-0.4751881957054138,
-0.28329119086265564,
0.21133394539356232,
0.12400713562965393,
0.11724703758955002,
-0.4338846504688263,
-0.24668057262897491,
-0.15557821094989777,
0.5053638815879822,
0.3444439768791199,
-0.4774153530597687,
-0.1996300369501114,
-0.07318779826164246,
-0.1649436056613922,
-0.07008294761180878,
-0.3979831635951996,
-0.16130168735980988,
-0.19634714722633362,
-0.1534193605184555
] |
2m
kFℏ2π
�
1 − (|Γ2L|/Γ1L)2
2m|∆L|
kFℏ2π δ(ω)
(e) Current phase relation for −π < θ < π
at T = 0 (Z = 0) [J(x = 0)]
−m|Γ2L|
πℏ2
sin θ
2 ln
�Γ1L + |Γ2L| cos(θ/2)
Γ1L − |Γ2L| cos(θ/2)
�
2m|∆L|
ℏ2
sin θ
2
(f) Current phase relation for −π < θ < π
at T = 0 (Z → ∞) [J(x = 0)]
− m|Γ2L|
2πℏ2Z2 ln
�Γ1L + |Γ2L|
Γ1L − |Γ2L|
�
sin θ
m|∆L|
ℏ2Z2 sin θ
TABLE I. Comparison of functional forms between bogolon junction (the third column) and s-wave SC junction (the fourth | [
-0.5648281574249268,
-0.40709546208381653,
0.7082269787788391,
-0.29410138726234436,
-0.1993841528892517,
-0.019493628293275833,
0.22424732148647308,
-0.6412931084632874,
-0.08083071559667587,
-0.4253253638744354,
0.031679172068834305,
0.412861168384552,
-0.5049824118614197,
0.3975524306297302,
0.3922194242477417,
0.8668554425239563,
0.766097903251648,
-0.00038413985748775303,
-0.38192877173423767,
-0.7669047117233276,
0.27959510684013367,
0.00932665728032589,
0.7892904877662659,
0.07690667361021042,
-0.16733236610889435,
0.10651473701000214,
0.4579826891422272,
-0.025992998853325844,
0.46039026975631714,
0.37260255217552185,
-0.26759693026542664,
0.10828640311956406,
0.10726572573184967,
0.03660308197140694,
0.26104211807250977,
0.24608585238456726,
0.1306810975074768,
0.3004756569862366,
0.1404847353696823,
-0.14580173790454865,
-0.1154530867934227,
0.0065836780704557896,
0.3108888566493988,
0.5438588857650757,
0.11104618012905121,
-0.33428817987442017,
-1.3914755582809448,
0.1339496374130249,
-0.11833973228931427,
-0.17346101999282837,
-0.11066783964633942,
0.2079702764749527,
0.3980286121368408,
0.12651856243610382,
-0.2315194308757782,
0.4453337788581848,
-0.23325558006763458,
-0.051283106207847595,
0.45841968059539795,
-0.36723363399505615,
0.2718692421913147,
0.29301366209983826,
-0.05322030186653137,
0.5295846462249756,
-0.32718753814697266,
0.9925351142883301,
-0.23296287655830383,
0.25577637553215027,
-0.2821623980998993,
-0.10183122009038925,
0.8127226829528809,
0.40079283714294434,
-0.028259089216589928,
0.13342289626598358,
0.3105560541152954,
0.08954419940710068,
-0.4497074484825134,
-0.26798582077026367,
0.2758125364780426,
-0.002064470201730728,
-0.13876894116401672,
-0.29510995745658875,
0.12121743708848953,
-0.1845635622739792,
0.22311395406723022,
0.18112921714782715,
0.07691148668527603,
0.8671524524688721,
-0.5776181817054749,
0.5399518013000488,
-0.30385836958885193,
-0.3938903510570526,
0.010896808467805386,
0.3839820921421051,
0.6526175141334534,
0.3872354328632355,
-0.26868492364883423,
-0.19464603066444397,
0.5874219536781311,
0.25150197744369507,
0.3039151132106781,
-0.24693597853183746,
-0.6269786357879639,
-0.15230655670166016,
-0.4333924949169159,
-1.1799492835998535,
-0.8536351919174194,
-0.730916440486908,
-0.5983344316482544,
-0.058690574020147324,
-1.165917158126831,
0.0727219358086586,
-0.23409786820411682,
-0.8785717487335205,
-0.1591612696647644,
0.1383787989616394,
-0.2666354179382324,
-0.5735594630241394,
0.07662323862314224,
0.7131320834159851,
-0.04449291527271271,
0.5463544130325317,
-0.2946312427520752,
0.3677457571029663,
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] |
column). We show the frequency dependence of pair amplitudes in (a) bulk and (b) interface. The zero-energy LDOS is shown
for (c) θ = 0 and (d) θ = π at x = 0, and the quasiparticle current J(x = 0) for (e) Z = 0 and (f) Z → ∞ at zero temperature.
FIG. 7.
Frequency-dependence of physical quantities of
bogolon junction at x = 0 for several values of Z with
|Γ2L|/Γ1L = 0.9. (a) The s-wave component pair amplitude,
(b) the p-wave component pair amplitude, (c) the LDOS of | [
0.050918061286211014,
-0.29808980226516724,
0.8996404409408569,
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0.3328930735588074,
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0.4924383759498596,
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0.6154007315635681,
0.43034839630126953,
1.0214694738388062,
0.4535527527332306,
0.23956750333309174,
-0.46213844418525696,
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0.4135630428791046,
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0.8293513655662537,
0.09216286987066269,
0.2643664479255676,
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0.4524150788784027,
0.04663722589612007,
0.32706624269485474,
0.6341243386268616,
-0.2566719949245453,
0.6613316535949707,
-0.11423338204622269,
0.13812796771526337,
0.21934257447719574,
0.12581278383731842,
0.12876293063163757,
0.10745516419410706,
0.11970224976539612,
-0.310977965593338,
0.06327047199010849,
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0.057443831115961075,
0.38555943965911865,
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0.3959212303161621,
0.3528591990470886,
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0.27421775460243225,
0.5722068548202515,
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0.3714320957660675,
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0.30963170528411865,
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0.5745853185653687,
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0.27739331126213074,
0.4371033310890198,
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0.41140544414520264,
0.6009435057640076,
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0.3059713840484619,
0.0829874649643898,
0.6914837956428528,
0.44469594955444336,
0.1282501220703125,
-0.3220708668231964,
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0.06641007214784622,
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0.08898934721946716,
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0.22179126739501953,
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0.12447801977396011,
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0.24648305773735046,
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0.32983094453811646,
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0.3574611246585846,
0.40420404076576233,
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0.3105052709579468,
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0.38075682520866394,
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0.14802218973636627,
0.0495525598526001,
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0.039568569511175156,
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0.12767814099788666,
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0.39510786533355713,
0.3557703495025635,
0.11336659640073776,
0.23467642068862915,
0.23978634178638458,
0.14919398725032806,
0.3549739718437195,
0.14671261608600616,
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1.4893043041229248,
0.3903469443321228,
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0.4712865948677063,
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0.3252674639225006,
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0.19913993775844574,
0.38409045338630676,
0.12482555210590363,
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0.10520283877849579,
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0.43762442469596863,
0.4894569516181946,
0.8613898754119873,
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0.14592523872852325,
0.5520731210708618,
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0.5028762817382812,
0.787900447845459,
0.42887184023857117,
0.3096269369125366,
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0.6213156580924988,
0.23111122846603394,
0.02022220939397812,
0.21865984797477722,
0.09094073623418808,
0.1956436038017273,
0.39999905228614807,
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0.13723506033420563,
0.6597627997398376,
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0.49245357513427734,
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0.1618342101573944,
0.039714887738227844,
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0.22350488603115082,
0.14840364456176758,
0.12721344828605652,
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0.013753754086792469,
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0.272356778383255,
0.7668256759643555,
1.0931787490844727,
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0.34686121344566345,
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0.025891408324241638,
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0.4562135934829712,
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0.9014352560043335,
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0.1342843770980835,
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0.11276553571224213,
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0.22309136390686035,
0.33641862869262695,
0.4249275028705597,
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0.2824830412864685,
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0.10557017475366592,
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0.7271983623504639,
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0.32843542098999023,
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1.0424708127975464,
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0.02165086567401886,
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] |
(b) the p-wave component pair amplitude, (c) the LDOS of
bogolons normalized by its value of clean limit (Γ1L = Γ2L =
0), and (d) j(iωn). The lines in (b)-(d) are shared with those
in (a). The relative phase is chosen as θ = 2π/3.
Let us take a closer look at the LDOS of bogolons,
which is given by
D(ω, x = 0)
=
2m
πkFℏ2 Re
(ω + iΓ1L)Ωret(ω)
Z2Ωret(ω)2 + (ω + iΓ1L)2 + |Γ2L|2 cos2(θ/2).
(54)
Figure 7 (c) shows the LDOS at x = 0. The pseudo-gap
for a small value of Z [Z = 0 and 1 in Fig. 7 (c)] changes | [
-0.06985165923833847,
-0.45903903245925903,
0.6174457669258118,
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0.5880187153816223,
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0.2725662887096405,
0.24008136987686157,
1.0918418169021606,
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0.47496992349624634,
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0.26503559947013855,
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] |
for a small value of Z [Z = 0 and 1 in Fig. 7 (c)] changes
into the zero energy peak for a larger value of Z [Z = 3
in Fig. 7 (c)], which is proportional to the bulk value as
shown in Eq. (55). Indeed, in the limit of Z → ∞, we
obtain
D(ω, x = 0) =
2m
πZ2kFℏ2 Re ω + iΓ1L
Ωret(ω)
= 1
Z2 Dbulk(ω, x = 0),
(55)
where Dbulk(ω) is the bulk DOS defined in Eq. (51). In
this limit, Eq. (55) is independent of θ and proportional
to the bulk DOS and decays with the factor 1/Z2. The | [
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] |
to the bulk DOS and decays with the factor 1/Z2. The
appearance of the bulk DOS in Z → ∞ limit is a common
feature with the s-wave SC junction [see Eq. (44)].
Let us proceed to consider J(x = 0), which is given
by the frequency summation of Fig. 7 (d). To see the Z-
dependence, we evaluate J(x = 0) for zero-temperature
limit in the analytical expression:
J(x = 0) = −m|Γ2L|
2πℏ2
1
Z2 + 1
�
Z2 + 1
Z2 + cos2(θ/2)
× sin θ ln
�
Γ1L
√
Z2 + 1 + |Γ2L|
�
Z2 + cos2(θ/2)
Γ1L
√
Z2 + 1 − |Γ2L|
�
Z2 + cos2(θ/2)
�
.
(56) | [
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] |
× sin θ ln
�
Γ1L
√
Z2 + 1 + |Γ2L|
�
Z2 + cos2(θ/2)
Γ1L
√
Z2 + 1 − |Γ2L|
�
Z2 + cos2(θ/2)
�
.
(56)
Specifically in the limit of Z → ∞, we obtain
J(x = 0) = − m|Γ2L|
2πℏ2Z2 sin θ ln
�Γ1L + |Γ2L|
Γ1L − |Γ2L|
�
.
(57)
The θ-dependence is determined by the factor sin θ, and
J(x = 0) is proportional to 1/Z2 in the limit of β → ∞.
D.
Comparison between bogolon junction and
s-wave SC junction
We have studied the bogolon junction with the bulk
odd-frequency pair and made a contrast against the con- | [
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] |
odd-frequency pair and made a contrast against the con-
ventional s-wave SC with bulk even-frequency pair. We
summarize the main results in Table I. The third column
13
shows the properties of the bogolon junction, and the
fourth column shows those of the s-wave SC junction.
At the interface, the even-frequency p-wave pair is in-
duced by the translational symmetry breaking for the
bogolon junction, and the odd-frequency p-wave pair is
induced for the s-wave SC junction [rows (a) and (b)]. | [
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] |
induced for the s-wave SC junction [rows (a) and (b)].
The bogolon junction has a zero-energy peak in the
LDOS in bulk (θ = 0), whose height is given by ∼
[1 − (|Γ2L|/Γ1L)2]−1/2, while the s-wave SC has a su-
perconducting gap [row (c)]. For θ = π, the LDOS of
the bogolon junction at the interface has a pseudo-gap
with the depth of ∼ [1 − (|Γ2L|/Γ1L)2]1/2, which is the
inverse of the bulk DOS, while the s-wave SC has a delta-
function behavior due to the Andreev bound state [row
(d)]. | [
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] |
function behavior due to the Andreev bound state [row
(d)].
The rows (e) and (f) show the current phase relations
of the two cases, which have opposite signs due to the
different frequency dependence of the pair potential. For
Z = 0 shown in row (e), the bogolon junction includes an
additional logarithmic factor compared to the s-wave SC
junction. The θ-dependence becomes the same for both
the junctions in Z → ∞ limit as shown in row (f), except
for the presence of a minus sign in the bogolon junction. | [
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] |
for the presence of a minus sign in the bogolon junction.
In this way, the bogolon junction has the characteris-
tic features compared with the conventional s-wave SC
junction.
V.
QUASICLASSICAL GREEN’S FUNCTION
In the above, we have mainly focused on the Green’s
function at x = 0.
On the other hand, since we con-
sider the non-uniform system, the unique properties of
bogolons can also be captured in the x dependence of
the Green’s function. While the McMillan Green’s func- | [
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] |
the Green’s function. While the McMillan Green’s func-
tion includes the rapidly oscillating components with a
characteristic scale of k−1
F , we focus on the quasiclassical
Green’s function with slowly-varying spatial components.
We construct the quasiclassical Green’s function by
combining the method in Refs. [112, 113] and McMillan
Green’s function derived in Sec. II. We start by decom-
posing the Green’s function as
ˆG(iωn, x, x′) =
�
αα′=±
ˆGαα′(iωn, x, x′)eikF(αx−α′x′). (58) | [
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] |
posing the Green’s function as
ˆG(iωn, x, x′) =
�
αα′=±
ˆGαα′(iωn, x, x′)eikF(αx−α′x′). (58)
The x dependence of each component ˆGαα′(iωn, x, x′)
does not include rapidly oscillating contribution with a
characteristic scale of k−1
F , which is consistent with the
picture of the quasiclassical Green’s function. However,
ˆGαα′(iωn, x, x′) is discontinuous at x = x′. Therefore,
to satisfy the Eilenberger equation [114], we define the
continuous semiclassical Green’s function as | [
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continuous semiclassical Green’s function as
ˆg±±(iωn, x) = ±iˆ1 − 2vFˆτ z ˆG±±(iωn, x − 0+, x),
(59)
where vF = ℏkF/m is a Fermi velocity [112, 113].
Since the overall phase does not change the physical
properties, we fix the phase on the left side as θL = 0.
Then, from Eq (59), we obtain the quasiclassical Green’s
function for x < 0 as
ˆg±±(iωn, x) =
i
ΩL(iωn)
�
Γ2Lsgn ωnˆτ y + (iωn + iΓ1Lsgn ωn)ˆτ z
+ ˜a(∓)
oute−ikF(ΩL(iωn)/µ)x�
± ΩL(iωn)ˆτ x − i(iωn + iΓ1Lsgn ωn)ˆτ y + i|Γ2L|sgn ωnˆτ z��
.
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] |
+ ˜a(∓)
oute−ikF(ΩL(iωn)/µ)x�
± ΩL(iωn)ˆτ x − i(iωn + iΓ1Lsgn ωn)ˆτ y + i|Γ2L|sgn ωnˆτ z��
.
(60)
The first line is a bulk part, and the second line, which
is proportional to ˜a(∓)
out, describes the contributions from
the Andreev reflection. Using Eq. (60), we can evaluate
the physical quantities defined in Sec. II F in the quasi-
classical representations, whose specific forms are listed
in Appendix D. One can confirm that Eq. (60) satisfies
the following Eilenberger equation [106, 112, 113]:
−ivF
∂ | [
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] |
the following Eilenberger equation [106, 112, 113]:
−ivF
∂
∂x ˆg±±(iωn, x) = ±(iωnˆ1 − ˆΣ)ˆτ zˆg±±(iωn, x)
∓ ˆg±±(iωn, x)(iωnˆ1 − ˆΣ)ˆτ z.
(61)
We can also check that ˆg±±(iωn, x) satisfies the normal-
ization condition
ˆg±±(iωn, x)2 = −ˆ1.
(62)
One can extract the factor of spatial dependence from
Eq. (60) as
e−ikF(ΩL(iωn)/µ)x
= exp(2
�
(ωn + Γ1Lsgn ωn)2 − |Γ2L|2x/ℏvF).
(63)
Since
�
(ωn + Γ1Lsgn ωn)2 − |Γ2L|2 is real, Eq. (63) does
not exhibit an oscillating part and is suppressed with | [
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] |
not exhibit an oscillating part and is suppressed with
increasing distance from the interface (The s-wave SC
junction has similar feature because
�
ω2n + |∆L|2 in
Eqs. (C7) is also real.). Then, the Green’s function con-
verges to the bulk value for the limit of x → −∞.
On the other hand, the spatial dependence of the re-
tarded Green’s function is extracted as
exp(−2i
�
(ω + iΓ1L)2 + |Γ2L|2 sgn ω x/ℏvF).
(64)
14
In contrast to Matsubara form, Eq. (63), the above quan- | [
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] |
(ω + iΓ1L)2 + |Γ2L|2 sgn ω x/ℏvF).
(64)
14
In contrast to Matsubara form, Eq. (63), the above quan-
tity includes both damping and oscillating parts, be-
cause
�
(ω + iΓ1L)2 + |Γ2L|2sgn ω with Γ1L > 0 exhibits
both real and imaginary part. Especially for small en-
ergy [ω ≪ (Γ2
1L − |Γ2L|2)/Γ1L], the oscillating part of
Eq. (64) reduces to exp(−2iΓ1L|ω|x/ℏvF
�
Γ2
1L − |Γ2L|2).
Namely, the quasiclassical Green’s function has oscil-
lating components with the frequency-dependent length
scale k−1
F
�
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] |
lating components with the frequency-dependent length
scale k−1
F
�
Γ2
1 − |Γ2L|2µ/Γ1L|ω|, even though we have ex-
cluded rapidly oscillating components with a period of
k−1
F .
In the s-wave SC case [Eq. (C8)],
�
ω2 − |∆L|2sgn ω is
real for |ω| > |∆L| and purely imaginary for |ω| < |∆L|.
This indicates that the LDOS has oscillation without
dumping in |ω| > |∆L| while has dumping without os-
cillation in |ω| < |∆L|.
We define the characteristic length by taking the limit | [
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] |
cillation in |ω| < |∆L|.
We define the characteristic length by taking the limit
of ω → 0 in Eq. (64), which is expressed as ξ =
ℏvF/
�
Γ2
1L − |Γ2L|2. We have evaluated a similar quan-
tity for the bulk in Ref. [21], which is defined by the
relative position of two bogolons and hence corresponds
to the pair radius. In contrast, the length ξ in this pa-
per is defined by the center of mass coordinate of Green’s
function, which corresponds to the coherent length. Nev- | [
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] |
function, which corresponds to the coherent length. Nev-
ertheless, we obtain the same form for the two length
scales.
Finally, we comment on the correlation between the
LDOS and the pair amplitude of bogolon in bulk limit.
The second line of Eq. (60) for θ = 0 is zero.
Then,
Eq. (60) reduces to
ˆg±±(iωn, x) = f(iωn, x)ˆτ y + g(iωn, x)ˆτ z
(65)
with
f(iωn, x) =
Γ2Lsgn ωn
�
(ωn + Γ1L)2 − |Γ2L|2 ,
(66)
g(iωn, x) =
iωn + iΓ1Lsgn ωn
�
(ωn + Γ1L)2 − |Γ2L|2 .
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] |
Γ2Lsgn ωn
�
(ωn + Γ1L)2 − |Γ2L|2 ,
(66)
g(iωn, x) =
iωn + iΓ1Lsgn ωn
�
(ωn + Γ1L)2 − |Γ2L|2 .
(67)
Note that g(iωn, x) is purely imaginary, and f(iωn, x) is
real, which is a consequence of odd-frequency pairing. In
this case, the normalization condition is given by
f(iωn, x)2 + g(iωn, x)2 = −1.
(68)
Since f(iωn, x) is real and g(iωn, x) is purely imaginary, if
|f(iωn, x)| increases, |g(iωn, x)| must also increase to sat-
isfy Eq. (68). Such correlation is consistent with the be- | [
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] |
isfy Eq. (68). Such correlation is consistent with the be-
havior of LDOS in Fig. 6. On the other hand, in the case
of s-wave SC with even-frequency pairing, both g(iωn, x)
and f(iωn, x) are purely imaginary. Hence, if |f(iωn, x)|
increases, |g(iωn, x)| decreases. This tradeoff relation cor-
responds to the gap in the LDOS in the presence of pair
amplitude. Although the correlation between the LDOS
and pair amplitude is demonstrated for bulk, we expect | [
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] |
and pair amplitude is demonstrated for bulk, we expect
that it holds also for non-uniform cases as supported by
the numerical results.
VI.
SUMMARY AND DISCUSSION
In this paper, we have studied superconductor Joseph-
son junctions with the Bogoliubov Fermi surface, uti-
lizing the low-energy effective model of bogolon. Since
the bogolon Cooper pair arises from the self-energy ef-
fect, it is necessary to determine the Green’s func-
tion satisfying the Gor’kov equation.
For the evalua- | [
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] |
tion satisfying the Gor’kov equation.
For the evalua-
tion of the Green’s function, we applied the McMillan’s
method, with which the non-Hermitian effective Hamil-
tonian ˆH0(x) + ˆΣ(iωn, x) needs to be analyzed. We also
calculated the quasiclassical Green’s function by elimi-
nating rapidly oscillating components with a scale k−1
F
and revealed spatial dependencies.
We compared the results with those of a conventional
spin-singlet s-wave superconductor and examined the | [
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