Nonequilibrium Kondo problem with spin-dependent chemical potentials: Exact results

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by  Hosho Katsura

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Nonequilibrium Kondo problem with spin-dependent chemical potentials:
Exact results
7
Hosho Katsura ∗
0
0
2 Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

n
(Received February 6, 2008)
a
J

0 We give a brief review of the Kondo effect and exactly solve the nonequilibrium Kondo
1
problematthespecialpointintheparameterspaceofthemodelwithspin-dependentchem-

] ical potentials. Using the obtained solution, we compute several experimentally observable
l
e
- quantities: charge current, spin current and magnetic properties. Applications to the phys-
r
t ically important cases such as the case where the impurity spin is placed in between two
s
t. ferromagnetically polarized leads are discussed.
a
m
KEYWORDS: kondo problem, spin current, nonequilibrium steady state, bosonization
-
d
n
o
c
[ 1. Introduction

1
In most metals, the resistivity monotonically decreases with the decrease of temperature
v
6 because it is dominated by phonon scattering. On the other hand, the metals with magnetic
9
1 impurities such as Fe have the resistance minimum at a certain temperature, known as Kondo
1
0 temperature TK, and under TK the resistivity approaches unitary limit as the temperature
7
approachesabsolutezero.Theexperimentaldiscovery ofthiseffectdatesbacktotheearly30s.
0
/
t Thereasonfortheresistanceminimumwasnotknownatthattime.Inthe60s,however,avery
a
m significant advance of the theory for the effect was developed by Kondo.1 Kondo’s calculation
-
of the resistivity, which was to explain this minimum, was based on the s-d model where the
d
n
local impurity spin S is coupled via an exchange interaction J with the conduction electrons
o
c of the host metal. Using third order perurbation theory in coupling J, he showed that this
:
v
s-d interaction leads to singular scattering of the conduction electrons near the Fermi surface
i
X
and a lnT contribution to the resistivity. The lnT term increases at low temperatures for an
r
a
antiferromagnetic coupling and when this term is included with the phonon contribution to
the resistivity, i.e., T5 term, it is sufficient to explain the observed resistance minimum. Since
his perturbation calculations could not be valid at low temperatures, a more comprehensive
theory was needed to explain the low temperature behavior. This task attracted a lot of theo-
retical interests in the late 60s and early 70s. More extensive perturbative calculations which
sum the most divergent terms were performed by Abrikosov.2 The scaling idea introduced by
Anderson provided a new theoretical framework for the Kondo problem.3 This idea was taken
∗E-mail address: [email protected]
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over by Wilson’s renormalization group method discussed below. Another perturbative ap-
proach was developed inthestudyof theAndersonmodel,whichis moremicroscopic thans-d
model,whereperturbativecalculations forCoulombrepulsionU havenodivergence.4 Nonper-
turbative approaches to the Kondo problem will be overviewed below. From the present point
of view, the Kondo effect is understood as a precursory phenomenon of the formation of the
boundstate between theconduction electrons and the impurity spin,known as Kondo singlet.
The theoretical structure underlying the Kondo problem is similar to the one in Quantum
Chromodynamics (QCD) and the Kondo effect corresponds to the confinement of quarks and
gluons.
Recently, the Kondo effect is revived in the context of mesoscopic physics such as semi-
conductor quantum dot devices where the quantum dot spin and the electrons in electrodes
corresnpondtotheimpurity spinandtheconduction electrons in theoriginal Kondoproblem,
respectively.5 The crucial difference between the usual Kondo effect and that in mesoscopic
systems is the nonequilibrium effect, i.e., the voltage vias between electrodes. It is very diffi-
cult to approach theoretically to the nonequilibrium situation of the Kondo problem. One of
the standard method to treat nonequilibrium problems perturbatively is the Keldysh Green’s
function method (discussed in ref. 6 in connection with the Kondo effect). Nonperturbative
approaches will be discussed in detail below.
BeforediscussingthenonequilibriumKondoproblem,let usbrieflyoverview thenonperturba-
tiveapproaches totheequilibriumKondoproblem(moredetailed information can befoundin
ref.7and8).ThefirstimportantcontributiondevelopedbyWilsoninthe70sistheNumerical
Renormalization Group (NRG) method9 which provides a way to numerically perform scale
transformations proposedby Anderson.By usingNRG, Wilson has obtained definitive results
for the ground state and low temperature behavior for the spin half s-d model. The second
developmentinthelate70sandearly80sisthediscoveryofexactsolutionstotheKondoprob-
lem. By using Bethe Ansatz (BA) method which is a powerful tool to solve one-dimensional
integrable models such as the Heisenberg model and the Hubbard model, Andrei10 and Wieg-
mann11 solved the spin half s-d model independently of each other. The critical idea, already
used in Wilson’s NRG, in BA approach is the mapping to the equivalent one-dimensional
model. After these works, Wiegmann showed that BA approach could beextended to give ex-
act results for the Anderson model.12 The most recent nonperturbative approaches by Affleck
and Ludwig in the 90s are based on Conformal Field Theory (CFT).13 They have formulated
the Kondo problem as a boundary quantum critical phenomenon and have applied CFT to
it. First, they found the nontrivial strong-coupling fixed point where the absorption of the
impurity spin is elegantly described by the deformation of Kac-Moody algebra, infinite di-
mensional Lie algebra, for the spin part. Second, they assumed that the absorption process at
the point away from the fixed point is governed by Kac-Moody fusion rules, the analogue of
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addition of angular momentum in CFT, and extracted the anomalous non-Fermi liquid(NFL)
behavior of the multichannel Kondoproblem introduced by A. Blandin and P. Nozi´eres.After
the works of Affleck and Ludwig, Emery and Kivelson have mapped the two-channel Kondo
problem into the resonant-level model, which is equivalent to noninteracting fermions for a
particular value of the z component of the exchange similar to the Toulouse limit,14 by using
the standard Abelian bosonization technique.15 Since the resultant resonant-level model is a
quadratic form in electrons and impurity spin operators at this special Fermi Liquid (FL)
fixed point, the absorption of the impurity spin is described by the unitary transformation
which diagonalizes the above-mentioned quadratic form.
Weshallnowleave theequilibriumcaseandturntothenonperturbativeapproachestothe
nonequilibriumKondoproblem.Thefirstcontribution is given by Schiller andHershfield.16,17
They combined the Y-operator method18 developed by Hershfield with the Emery-Kivelson
solution discussed above and then obtained the exact solution for the nonequilibrium Kondo
problem at a special point in the parameter space. After this work, Delft et al have de-
veloped the CFT approach combined with Hershfield’s Y-operator formulation.19 Recently,
MehtaandAndreihaveappliedtheNonequilibriumBetheAnsatz(NEBA)tothecomputation
for steady state properties of quantum impurity problems such as interacting resonant level
model(IRLM).20 Here we should note that the comprehensive nonperturbative understanding
of the nonequilibrium Kondo problem is still an open problem and requires more extensive
studies.
In this article, we extend the Schiller-Hershfield solution to the nonequilibrium Kondo
problem to include the case where the chemical potential of each lead depends on the spin
of conduction electrons. Actually, these situations are often discussed in the area such as
spintronics. It is different from Ref.16 and 17 that we employ a description of nonequilibrium
steady states (NESS),21,22 developed in the context of mathematically rigorous theory such
as C∗-algebra, instead of using Hershfield’s Y-operator. The obtained results given below are
generalization of those in ref. 16 and 17 and reproduce their results as limiting cases.
2. Model and mapping
The physical system under consideration is shown schematically in Fig.(1). It consists of
left(L) andright(R) leads of noninteracting spin-1/2electrons, which interact via an exchange
coupling with a spin-1/2 impurity spin placed in between the two leads. Following Ref.,13
the conduction-electron channels that couple to the impurity are reduced to one-dimensional
fields ψ (x), where α = L/R and σ = , are the lead and spin indices respectively. In what
ασ
↑ ↓
follows, we assume that the conduction-electron dispersion around the Fermi level is linear,
i.e., ǫ = ~v k, where ǫ and k are measured relative to the Fermi level and Fermi wave
k F k
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JRR JLL
L R
(Ve(cid:14)Vs)/2 (cid:38)
(cid:87)
(Ve(cid:16)Vs)/2
((cid:16)Ve(cid:14)Vs)/2
((cid:16)Ve(cid:16)Vs)/2
JLR
Fig. 1. Schematicdescriptionofoursystem.Atunneljunctionconsistsoftwoleadsofnoninteracting
spin-1/2electronsandaspin-1/2impurityspinplacedinbetweenthetwoleads.Theenergylevels
denote the spin-dependent chemical potentials of the conduction electrons.
number respectively. The most general form of the s-d Hamiltonian H is given by
∞ ∂
H = i~v ψ† (x) ψ (x)dx
F ασ ∂x ασ
∞
α=L,Rσ=↑,↓Z
X X
+ Jαβsλ τλ µ g Hτz, (1)
λ αβ − B i
α,β=L,Rλ=x,y,z
X X
where ~τ denotes the impurity spin and sλ is defined as
αβ
σλ
sλ ψ† (0) σσ′ψ (0), (2)
αβ ≡ ασ 2 βσ′
σ,σ′
X
where σλ, (λ = x,y,z) are a Pauli matrices. And µ and g are the Bohr magneton and the
B i
impurity Land´e g factor, respectively. In addition to the above Hamiltonian H, we consider
the following nonequilibrium conditions:
Y = Ye+Ys,
0 0 0
eVe ∞
Ye = [ψ† ψ ψ† ψ ]dx,
0 2 Lσ Lσ − Rσ Rσ
σ Z−∞
X
eVs
Ys = [σψ† ψ σψ† ψ ]dx, (3)
0 2 Lσ Lσ − Rσ Rσ
σ
X
where Ve and Vs correspond to a usual electronic voltage and a spin voltage respectively.
In Ref. 16 and 17, Schiller and Hershfield have considered the case where only V exists. On
e
the other hand, there are some situations, such as ferromagnetic leads cases, where the spin
voltage V exists and plays a key role in the spin transport phenomena such as spincurrent.
s
Therefore, the nonequilibrium Kondo problem with the above more general nonequilibrium
conditions needs to be newly formulated.
Weshalldiscusshowtodiagonalize theaboveHamiltonian. First,werestricttheregionofthe
Jαβ parameter space to the case where Jαα = Jαα Jα, JLR JLR JLR, JLR = JRL = 0,
λ x y ≡ ⊥ x y ≡ ⊥ z z
andJLL = JRR J takes aparticularvalue(discussedbelow).WenotethatsettingJLR = 0
z z z z
≡
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produces no significant difference on the low-energy physics since, in the renormalization
group sense, JLR = 0 is generated on scaling from weak coupling.23 Next, we bosonize eq.(1)
z
6
and eq.(3) respectively by following the Emery-Kivelson solution of the two-channel Kondo
problem.15 We introduce four different boson fields to account for four different fermion fields
as
1
ψ (x) = ( 1)NL↓+NR↑+NR↓e−iΦL↑(x), (4)
L↑
√2πa −
1
ψ (x) = ( 1)NR↑+NR↓e−iΦL↓(x), (5)
L↓
√2πa −
1
ψ (x) = ( 1)NR↓e−iΦR↑(x), (6)
R↑
√2πa −
1
ψ (x) = e−iΦR↓(x), (7)
R↓
√2πa
where a is a high-energy cut-off(lattice spacing), and number operators for fermion N (i =
i,σ
L,R,σ = , ) are defined as
↑ ↓
∞ 1 ∞ ∂Φ (x)
† iσ
N = dx :ψ (x)ψ (x) := dx , (8)
iσ iσ iσ 2π ∂x
Z−∞ Z−∞
andsatisfy thecanonical commutation relations [Φ (x),N ]= iδ δ .Thebosonfields Φ
iσ jσ ij σσ′ i,σ
also satisfy standarad commutation relations
[Φ (x),Φ (y)] = iπδ δ sgn(x y), (9)
iσ jσ′ ij σσ′
− −
and hence we can make sure of the following correspondence between the fermion fields and
boson fields:
∂Φ (x)
† iσ
2π : ψ (x)ψ (x) := (10)
iσ iσ ∂x
∞ ∂ v ∞ ∂Φ (x)
iv dx : ψ† (x) ψ (x) := F dx :( iσ )2 : (11)
F iσ ∂x iσ 4π ∂x
Z−∞ Z−∞
where : AB... : denotes the normal ordering of operator products AB.... Furthermore, we
introduce four new boson fields constructed by the above four boson fields in eq.(7) as
1
Φ = (Φ +Φ +Φ +Φ ),
c L↑ L↓ R↑ R↓
2
1
Φ = (Φ Φ +Φ Φ ),
s L↑ L↓ R↑ R↓
2 − −
1
Φ = (Φ +Φ Φ Φ ),
f L↑ L↓ R↑ R↓
2 − −
1
Φ = (Φ Φ Φ +Φ ), (12)
sf L↑ L↓ R↑ R↓
2 − −
where subscripts c,s,f,sf correspond to collective charge, spin, flavor (left minus right), and
spin-flavor modes respectively. Now, the Hamiltonian and nonequilibrium conditions can be
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rewritten in terms of the boson fields in eq.(12) as
~v ∞ ∂Φ 2
F ν
H = dx
4π ∂x
ν=c,s,f,sfZ−∞ (cid:18) (cid:19)
X
J+
+ [ τxsin(χ )+τycos(χ )]cos(χ )
s s sf
πa −
J−
[τxcos(χ )+τysin(χ )]sin(χ )
s s sf
− πa
JLR
⊥ [τxcos(χ )+τysin(χ )]sin(χ )
s s f
− πa
J ∂Φ
+ zτz s µ g Hτz, (13)
x=0 B i
2π ∂x | −
and
eVe ∞ ∂Φ (x)
Ye = dx f ,
2π ∂x
Z−∞
eVs ∞ ∂Φ (x)
Ys = dx sf , (14)
2π ∂x
Z−∞
where J± are the even and odd combinations of JLL and JRR, i.e., J± (JLL+JRR)/2 and
⊥ ⊥ ≡ ⊥ ⊥
χ ,χ and χ are defined as
s f sf
π π
χ Φ (0) (N N )+ ,
s s c s
≡ − 2 − 2
π
χ Φ (0) (2N N N ),
f f c f sf
≡ − 2 − −
π
χ Φ (0) (N N ). (15)
sf sf f sf
≡ − 2 −
The next step is to employ the canonical transformation H′ = UHU† and Y′ = UY U† with
0
U = exp(iχ τz). Since χ commutes with both χ and χ , the transformed Hamiltonian H′
s s f sf
is written as
~v ∞ ∂Φ (x) 2 J+ J−
H′ = F dx ν + τycos(χ ) τxsin(χ )
sf sf
4π ∂x πa − πa
ν=c,s,f,sfZ−∞ (cid:18) (cid:19)
X
JLR Jz ∂Φ (0)
⊥ τxsin(χ )+ ~v s τz µ g Hτz, (16)
f F B i
− πa 2π − ∂x −
(cid:20) (cid:21)
where we have used the commutation relation
∂Φ (x)
eiχsτz, ν = 2πiδ δ(x)eiχsτz, (17)
νs
∂ −
x
(cid:20) (cid:21)
andhaveneglecteddivergentconstant(thevalidityhasbeendiscussedinref.24).Thenonequi-
librium conditions Ye and Ys are unaffected by the above canonical transformation, i.e.,
0 0
Y′ = Y . The last step is refermionization of the above bosonized operators. To this end,
0
we first express the spin ~τ in terms of fermion operator as d = iτ+. The four boson fields
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Φ (ν = c,s,f,sf) are refermionized according to
ν
eiπd†d
ψf(x) = e−i(Φf(x)−π2(2Nc−Nf−Nsf)),
√2πa
eiπd†d
ψsf(x) = e−i(Φsf(x)−π2(Nf−Nsf)), (18)
√2πa
etc. Here the phase iπd†d, similar to the Jordan-Wigner transformation, guarantees the anti-
commutation relations between d and ψ (x).
ν
Once these steps are completed, we can obtain the refermionized forms for H′ and Y′:
∞ ∂ J+
H′ = i~v dxψ†(x) ψ (x)+ [ψ† (0)+ψ (0)](d† d)
F ν ∂x ν 2√2πa sf sf −
c,s,f,sfZ−∞
X
JLR J−
+ ⊥ [ψ†(0) ψ (0)](d† +d)+ [ψ† (0) ψ (0)](d† +d)
2√2πa f − f 2√2πa sf − sf
1
+[µ g H (J 2π~v ): ψ†(0)ψ (0) :] d†d , (19)
B i − z − F s s − 2
(cid:18) (cid:19)
and
∞
(Ye)′ = eVe dxψ†(x)ψ (x),
0 f f
Z−∞
∞
(Ys)′ = eVs dxψ† (x)ψ (x). (20)
0 sf sf
Z−∞
Here we can immediately notice that (19) would be a quadratic form in fermion operators
if we set J = 2π~v . Since the nonequilibrium condition Y′ is originally a quadratic form,
z F
both H′ and Y′ reduce to quadratic form at this special point. Hence the strongly interacting
nonequilibrium problem maps to a noninteracting one. For convenience, we introduce new
Majorana fermions as
d+d† d† d
aˆ = , ˆb = − . (21)
√2 i√2
Here we note that Majorana fermions satify aˆ2 =ˆb2 = 1/2. After Fourier transformation, H′
and Y for J = 2π~v can be rewritten as
z F
J+
H′ = ǫ ψ† ψ iµ g Haˆˆb+i (ψ† +ψ )ˆb
k ν,k ν,k − B i 2√πaL sf,k sf,k
ν=f,sf k k
X X X
JLR J−
+ ⊥ (ψ† ψ )aˆ+ (ψ† ψ )aˆ, (22)
2√πaL f,k − f,k 2√πaL sf,k − sf,k
k k
X X
Y′ = eVe ψ† ψ +eVs ψ† ψ , (23)
k f,k f,k k sf,k sf,k
where L is the size of the system, ǫP= ~v k and we hPave neglected the terms involve the ψ
k F c
and ψ which are decoupled from aˆ and ˆb.
s
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3. Solution and observables
In this section, we shall show how to diagonalize eqs.(22) and (23) simultaneously. After
obtaining the exact solution, we will calculate experimentally observable quantities such as
charge current, spin current and magnetic properties.
Since the precise procedure for diagonalization of H′ is given in ref. 17, we shall only outline
†
the main steps briefly. The first step is to construct scattering states c which diagonalize
ν,k
H′ and are defined by the operator equation
[c† ,H′]= ǫ c† +iη(ψ† c† ) (ν = f,sf), (24)
ν,k − k ν,k ν,k − ν,k
wherethepositiveinfinitesimalη isintroducedtoguaranteeappropriateboundaryconditions.
Detailed derivation of the scattering-states operators are given in the APPENDIX B in ref.
17. Using c , our Hamiltonian H′ can be written in a diagonal form as
ν,k
H′ = ǫ c† c (25)
k ν,k ν,k
ν=f,sf k
X X
Next we need to expand original operators, ψ(†),aˆ and ˆb, in terms of scattering-states opera-
ν,k
(†)
tors c . TABLE II in ref. 17 summarizes the results of expansion. At this point, we employ a
ν,k
description of nonequilibrium steady states (NESS) instead of the Y-operator method devel-
oped by Hershfield. In NESS approach, once we can obtain steady state density matrix ρ ,
+
we impose the following conditions on the two-point function:
†
c c = F (ǫ )δ δ (µ,ν = f,sf), (26)
h µ,k ν,k′i µ k µν kk′
where ... Tr(...ρ ) stands for the NESS average and F (ǫ) are defined by the ordinary
+ µ
h i ≡
Fermi distribution function f(ǫ) as F (ǫ) f(ǫ eVe) and F (ǫ) f(ǫ eVs) respectively.
f sf
≡ − ≡ −
Weshouldnoteherethatthereis asomewhatsubtlepointto useC∗-algebraic approach,since
we involved non-local operators such as N in our bosonization procedure.
i
Now, we are ready to calculate the observable quantities. The physical observables of
interestsuchaschargecurrentI ,spincurrentI andimpuritymagnetizationMz arerewritten
c s
after the canonical transformation U:
eJLR
I′ = i ⊥ (ψ† +ψ )aˆ, (27)
c 2~√πaL f,k f,k
k
X
J− J+
I′ = i (ψ† +ψ )aˆ (ψ† ψ )ˆb, (28)
s 2~√πaL sf,k sf,k − 2~√πaL sf,k − sf,k
k k
X X
Mz = iµ g aˆˆb, (29)
B i
(the derivation can be found in ref. 17). From now on, we focus on the average of transformed
operators, since ... = lim Tr(...e−iHtρ eiHt) = lim Tr(U...U† Ue−iHtU† Uρ U†
t→+∞ 0 t→+∞ 0
h i · · ·
UeiHtU†) = lim Tr(U...U†e−iH′tρ′eiH′t) = U...U† , where ρ is the density matrix for
t→+∞ 0 h i 0
the initial state and we have used the cyclic property of the trace. Using the condition for
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Symbol Definition
Γ [(JLR)2+(J−)2]/4πa~v
a ⊥ F
Γ (J+)2/4πa~v
b F
Γ (JLR)2/4πa~v
1 ⊥ F
Γ (J−)2/4πa~v
2 F
Γ (JLL)2/4πa~v
L ⊥ F
Γ (JRR)2/4πa~v
R ⊥ F
Γ JLRJ−/4πa~v
m ⊥ F
Γ JLRJ+/4πa~v
p ⊥ F
Table I. Symbols and their definitions.
the two-point correlation function (26), we can obtain the average of I′,I′ and Mz due to
c s
the quadratic nature of them. (Generally speaking, we can also compute higher-order terms
by using Wick’s theorem for c ).
ν,k
Charge Current and Spin Current
First, let us discuss the charge current and spin current in our system. Since the charge
current obtained here is exactly equal to the result in,17 we only write down the result:
eΓ ∞
I (V) = I′(V) = 1 dǫA (ǫ)[f(ǫ eVe) f(ǫ+eVe)], (30)
c h c i 2π~ a − −
Z−∞
where
ǫ+iΓ
b
A (ǫ) = Im , (31)
a − (ǫ+iΓ )(ǫ+iΓ ) (µ g H)2
(cid:26) a b − B i (cid:27)
and Γ ,(α = 1,a,b) are tabularized below. Here, we note that I (V) depends only on Ve and
α c
is independent of the spin voltage Vs.
We will now leave the discussion of charge current and turn to that of spin current.
We obtain the following result which includes the one in ref. 17 as a specific case:
I (V)= I′
s s
h i
1 ∞ f(ǫ eVe)+f(ǫ+eVe) f(ǫ eVs)+f(ǫ+eVs)
= dǫ − −
2π~ 2 − 2
Z−∞ (cid:20) (cid:21)
(µ g H)(Γ Γ )Γ ǫ
B i L R 1
−
× (ǫ+iΓ )(ǫ+iΓ ) (µ g H)2 2
a b B i
| − |
1 ∞
+ dǫ[f(ǫ eVs) f(ǫ+eVs)][Γ A (ǫ)+Γ A (ǫ)]
2π~ − − 2 a b b
Z−∞
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1
Gs(Vs,T) kBT/(cid:42)b Gs(Vs,0) (cid:42)a/(cid:42)b
0.8
2e/h 0.8 0 2e/h 0.2
0.5 0.6 1.0
0.6
1.0 5.0
0.4 2.0 0.4
0.2 0.2
-7.5 -5 -2.5 2.5 5 7.5 -7.5 -5 -2.5 2.5 5 7.5
eVs/(cid:42) eVs/(cid:42)
b b
s
Fig. 2. The spin differential conductance Gs(V ,T) as a function of V (left) for zero magnetic field
and different temperatures and (right) at T = 0 and µBgiH = 2Γa, for different ratios of Γa to
Γb.
2
1 ∞ dǫ[f(ǫ eVs) f(ǫ+eVs)]2(ǫ2 +ΓaΓb+(µbgiH)2) ΓL−2ΓR ,
− 2π~ − − (ǫ+iΓ )(ǫ+iΓ ) (µ(cid:16)g H)2 2(cid:17)
Z−∞ | a b − B i |
(32)
where A is defined as
b
ǫ+iΓ
a
A (ǫ) = Im . (33)
b − (ǫ+iΓ )(ǫ+iΓ ) (µ g H)2
a b B i
(cid:26) − (cid:27)
We can immediately notice that eq.(32) reproduces eq.(6.7) in ref. 17 if we set Vs = 0. We
also notice that we have finite spin current I even though the case where JLL = JRR, i.e.,
s ⊥ ⊥
symmetric case due to the existence of Γ .
b
Let us now discuss the spin current for specific cases, (i)symmetric, zero magnetic field, at
finite temperature and (ii)symmetric, finite magnetic field, at T = 0 in full detail. If we set
JLL = JRR, Γ = Γ and Γ = 0, then we can write down I as
⊥ ⊥ L R 2 s
Γ ∞
I (Vs) = b dǫA (ǫ)[f(ǫ eVss) f(ǫ+eVs)]. (34)
s 2π~ b − −
Z−∞
Contrast to the charge current case (30), I involves A , i.e., the spectral function for the
s b
Majorana fermion ˆb . We can obtain a closed form expression for (34) using the digamma
function ψ(z)25 which is defined by the gamma function Γ(z) as
d
ψ(z) = lnΓ(z). (35)
dz
For example, case (i), the spin current is rewritten as
Γ 1 Γ +ieVs
b b
I = Im ψ + , (36)
s π~ 2 2πk T
B
(cid:26) (cid:18) (cid:19)(cid:27)
where k is Boltzmann constant. For finite magnetic field case, we can also explicitly rewrite
B
I using the digamma function, however it takes a more complicated form.
s
Next, we will show the differential conductance for the spin current G (V ,T) dI /dVs as
s s s
≡
10/13