空間反転対称性の破れた超伝導の新奇な物性cond.scphys.kyoto-u.ac.jp/~fuji/noinvsuper2006open.pdf空間反転対称性の破れた超伝導の新奇な物性...

空間反転対称性の破れた超伝導の新奇な物性

Outline1. Introduction

-Fundamental Properties of SC without inversion symmetry

2. Possible pairing state realized in CePt3Si

3. Unique electromagnetic properties and transport phenomena

4. Summary

ThankstoK.Yamada,Y.Onuki,M.Sigrist,D.Agterberg,

S.K.Yip,Y.Matsuda,T.Shibauchi,N.Kimura,T.Takeuchi,R.Settai,H.Ikeda,H.Mukuda,M.YogiandY.Yanasefor

discussions

京大理　藤本聡

Non-centrosymmetric Heavy Fermion Superconductors

Spin-orbit interaction (Rashba interaction)

Asymmetric potential gradient

(from Bauer et al.PRL92,0207003)

Broken inversion sym.

Broken Spin inversion sym.

CePt3Si, UIr, CeRhSi3 , CeIrSi3 (c.f. non HF: Cd2Re2O7 , Li2Pt3B, Li2Pd3B)

(Bauer et al.) (Kobayashi et al.)

(Kimura et al.) (Hiroi et al.) (Takeya et al.)(Sugitani et al.)

!!p!,!"!

!p!,"!

=1

2(| !"| #" $ | #"| !")

+1

2(| !"| #" + | #"| !")

singlet

triplet with

€

Sin -plane = 0

singlet

triplet

Non-centrosymmetric Superconductors (contd.)

Edelstein,JETP68,1244(1989) ; Gor’kov-Rashba(2001); Yip(2002),Frigeri et al.(2004)

Parity non-conserved Mixture of spin singlet and triplet states

Fermi Surface

py px

2!|p|!!p,!"

!p,"

| !"| #"

| !"| #"

“Zeeman energy”depending on !p

!d(k) != !t(k) !L(k)!d(k) = !t(k) !L(k)

( For Rashba int., )

Pairing state d-vector of the spin triplet component

Superconducting gap

HSO = !"L(k) · "#c†kckSO interaction

!!p!,!"!

!p!,"!

!!p,!"

!p,"

For For

Highest Tc , when the attractive int. in this channel is strongest

!!p!,!"!

!p!,"!

!!p,!"

!p,"

Pairing between different Fermi surfaces. Depairing effect !!

NB. The actual pairing state is determined by the interplay between k-dependence of the pairing interaction and the SO interaction.

!L(k) = (sin ky,! sin kx, 0)

!(k) = !s(k)i!y + "d(k) · "!i!y

!d(k) != !t(k) !L(k)!d(k) = !t(k) !L(k)

( For Rashba int., )

Pairing state d-vector of the spin triplet component

Superconducting gap

HSO = !"L(k) · "#c†kckSO interaction

!!p!,!"!

!p!,"!

!!p,!"

!p,"

For For

Highest Tc , when the attractive int. in this channel is strongest

!!p!,!"!

!p!,"!

!!p,!"

!p,"

Pairing between different Fermi surfaces. Depairing effect !!

NB. The actual pairing state is determined by the interplay between k-dependence of the pairing interaction and the SO interaction.

!L(k) = (sin ky,! sin kx, 0)

!(k) = !s(k)i!y + "d(k) · "!i!y

e.g. s+p state

d+f state

g+h state etc.......

(No d+p, g+f, g+p etc)

!s(k) !t(k)In this case, and should be the same irreducible representation of the point group of the system.

!0.1 0 0.1 0.2 0.3

0.0012

0.0014

0.0016

0.0018

0.002

Tc

us

ESO

E_

F=0.1

ESO

E_

F=0.05

(a)

(b)

!0.15!0.1!0.05 0 0.05 0.1 0.15 0.2

!0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

! t! s

us

ESO

EF= 0.1

BCS weak-coupling calculations for , and gap !Tc

2D model with Rashba SO int. and the pairing int.,

Attractive int. in p-wave channel up = !0.15

!0.1 0 0.1 0.2 0.3

0.0012

0.0014

0.0016

0.0018

0.002

Tc

us

ESO

E_

F=0.1

ESO

E_

F=0.05

(a)

(b)

!0.15!0.1!0.05 0 0.05 0.1 0.15 0.2

!0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

! t! s

us

up

V (k, k!) = usi(!y)!"i(!y)#$ + up(k ! n) · (!i!)!"(k! ! n) · (!i!)#$

substantial mixing ofsinglet and triplet states

p-waves-wave

Admixture of spin singlet and triplet states A simple example

Pairing state of the SC state in CePt3Si

Evidence of Line nodes

(Bonalde et al. (2006))

Penetration depth

(Izawa et al. (2005))

Thermalconductivity

NMR 1/T1T (Yogi et al.(2004)) Coherence peak full-gap ? line node at low-T

s+p state ?

Possibility of unusual coherence effect for the p wave state

usual p wave SC

coherence factor of 1/T1T vanishes

(S. Fujimoto, (2005); c.f. Hayashi et al. 2005) 1/T1T (one-particle approximation)

Na(!) =!

k

ImF (k, !)

ImF (k, !) ! !kcoherence factor

! ++

!

!

k

ImF (k, !) != 0

Non-zero forp wave with no inv. !

noncentrosymmetric p wave state

+

!

(BW or chiral p wave)

pypx

Fermi surface

SC gap

Fermi surfacesplitted

+

!

coherence peak of 1/T1T is enhanced !

Na =

Na =!

k

ImF (k, !) = 0

　 Line nodes with small coherence peak of 1/T1T

s+p state ? (Hayashi et al.(2005))

accidental nodes due to the coupling with AF order ?

(Fujimoto (2006))

Is s-wave pairing not suppressed in the highly correlated heavy fermion state of ? CePt3Si

! = !s ±!p sin !

Pairing state realized in CePt3Si

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

m=3.2

m=2.7

m=2.2

m=1.7

m=0.0

Density of States

Case of p-wave pairing dominant

D(!)

!

D(!)

!

0.1

1

0.001 0.01

m=3.2

m=2.7

m=2.2

m=1.7

32*x

Existence of small peak around ! ! ! Small coherence peak of 1/T1T

Line node structure due to the coupling with AF order

!

!!

0

!

2

!!

2kx

ky

kz

AF ordercoexits

!Q = (0, 0,")

( Metoki et al. (2004))

!mQ

!mQ = (mQ, 0, 0)

! = 0.01

! = 0.07

(EF = 1)

!L(!k) = ! !L(k)

Constraint condition from symmetry

• Odd parity

• Point group symmetry

• is a pseudovector!L(k)

Possible gap function for s+p state realized in CePt3SiParity-breaking SO int. HSO = !"L(k) · "#c†kck

C4v symmetry !L(k) = (sin ky,! sin kx,Lz)

Lz = c0 sin kx sin ky sin kz(cos kx ! cos ky) generally, c0 != 0(Samokhin(2005))

(and then,

!d != !t(k) !L(k)deparing effect exists )

!L(!k) = ! !L(k)

Constraint condition from symmetry

• Odd parity

• Point group symmetry

• is a pseudovector!L(k)

Possible gap function for s+p state realized in CePt3SiParity-breaking SO int. HSO = !"L(k) · "#c†kck

C4v symmetry !L(k) = (sin ky,! sin kx,Lz)

Lz = c0 sin kx sin ky sin kz(cos kx ! cos ky) generally, c0 != 0(Samokhin(2005))

(and then,

!d != !t(k) !L(k)deparing effect exists ) SC gap for

Strong repulsion in s-wave channel is harmful for s+p state !!

When c0 = 0

!(k) = !s(k)i!y + !t(k) "L(k) · "!i!y

Lz = 0( )

an extended s-wave channel should be attractive !!!s(k) = a0 + a1(cos kx + cos ky) + a2 cos kz

A1 representation of

!t(k) = b0 + b1(cos kx + cos ky) + b2 cos kz

microscopic calculations Future issue

C4v

s+p state

Unique electromagnetism and tranport phenomenadue to the parity-breaking SO interaction

Magnetoelectric effects in the normal state

Anomalous Hall effect

Charge current induced by Zeeman fieldMagnetization induced by current flows

Magnetoelectric effects in the SC state

Thermal anomalous Hall effect in the SC state

Enhanced byelectron correlation

Magnetization induced by supercurrentParamagnetic supercurrent

Importantin heavy fermion systems !!

Paramagnetism Large Pauli limitting fieldvan-Vleck-like susceptibility

Helical vortex phase

Large Pauli limitting field

H0 = !p ! µ + "(#p " #n) · #$ ! µB#$ · #h

!p± = !p ! µ ±

!

"2[(px +µB

"hy)2 + (py !

µB

"hx)2] + µ2

Bh2

z

perpendicular to the planePauli depairing suppressed

!h

xy

c.f. CePt3Si Hc2 ! 5 T

HPauli ! !/!

2µB " 1 Tweak coupling BCS

HPauli < Hc2 ? exceeding Pauli limit ?

Fermi surface in-plane field

!h

!!p,!"

!p,"!p!,"!

!!p!,!"!

depairing effectPauli limit exists

(poly,Bauer et al.)

!p + !q/2

!!p + !q/2

Rashba model

H!z

c2! 3.2T H

!z

c2 ! 2.7T(single,Yasuda et al.)

not affected by SC gap, if

Fermi Surface

pypx

2!|p|

For CePt3Si ! ! 1K !|p| ! 100K

!h

!! =

xy-plane

!! =!Pauli

2+

!V V

2

xy-plane!h !!h !

!h

For For spherical Fermi surface

!V V

The “van Vleck”-like contribution exists in addition to the usual van Vleck term which stems from the orbital degrees of freedom.

Frigeri et al. (2004), cf. Gorkov-Rashba(2001)

!V V = !

!

p

f("p+) ! f("p!)

"p+ ! "p!

!!/!N

!!/!N

Unusual Paramagnetism

Rashba case (tetragonal)

Knight shift measurement for CePt3Si Yogi et al. (2004,2006)

Both and do not

change below

K!K!

Tc

K!

K!

Theoretical result forspherical Fermi surfaceFrom Frigeri et al.(2004)

How toreconcile

!!/!N

!!/!N

How toreconcile

U/W

!!!!"#$%!!!!!""!!$

xx

xx

c

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5

!SC! (0)

!!(Tc)

Calculating the selfenergy perturbatively up to 2nd order in for 2D Hubbard-like model

U

EF

!

!k-

!k+

DOS

Fujimoto,25pXB-1

Magnetoelectric effects in the normal and SC states

Rashba case (tetragonal)

Dresselhaus case (cubic) L0(k) · !SO interactionL0(k) = (kx(k2

y ! k2

z), ky(k2

z ! k2

x), kz(k2

x ! k2

y))

!h !M

!J

!h !M

!J

SO interaction (k ! n) · ! n = (001)

kz = 0

Magnetoelectric effect in the Normal state

Charge current induced by AC Zeeman field

Magnetization induced by electric field

Mx = !xyEy

( Levitov et al.(1985), for weakly correlated metals )

Jy = !2!xy(dBx/dt)

Electron correlation effects (S.F., cond-mat/0605290)

Rashba case

!xy !

c0!

"0 + AT 2

! Specific heat coefficient

!0 + AT 2 Resistivity

Enhanced by the factor !

Bx,Mx

Dissipative effect !

Ey

A

Magnetoelectric effects in the normal state (contd.)

!kF

EF

! 0.1

B = B0 cos(!t) B0 ! 100 Gauss

Imax ! 1 mA measurable !

!

µB

e

1

v!F

!kF

EF

1

"

dB

dtL2

! ! 10 µ! · cm! ! 100 kHz

Magnetization induced by electric field

Charge current induced by AC Zeeman field

I = JL2

= !2!xy

dB

dtL

2

!n

lead alignedin x-y plane

!I

!B

M = !xyEy ! µB!kF

EFn

n ! 1022 cm!3

measurable !M ! 9 Gauss

v!

F ! 105(cm/s) m!/m0 ! 100( )

Magnetoelectric effect in the SC state

(Edelstein(1989,1995); Yip(2002), one-body approximation)

Supercurrent induced by Zeeman field

Magnetization induced by supercurrent

In the normal state ( T >Tc ), Static magnetic field can not induce dissipative current flows.

( paramagnetic supercurrent )

Rashba case

Bx,Mx

Jy = KyxBx

Mx = !Kyx

m

nx

Jy

Dissipationless effect

Electron correlation effects (S.F., PRB(2005); unpublished)

enhanced by electron correlation effect by the factor (if FM fluctuation is absent)

PM current much more enhanced than diamagnetic supercurrent

!Js = !nse

2

m!c!A + Kyx(!n " !Bx)Total supercurrent

Kyx !eµB

z

!

EF

n0

1/z

The magnetoelectric effects have not yet been detected experimentally !How to detect the Zeeman-field induced (paramagnetic) supercurrent avoiding dissipation ?

However, It is important to discriminate between the Meissner diamagnetic supercurrent and the paramagnetic supercurrent

(i) (ii)

!n!H

!J

lead madeof SC materials

～～～～

Strong electron correlation plays an important role !

Magnetoelectric effect in the SC state

(Edelstein(1989,1995); Yip(2002), one-body approximation)

Supercurrent induced by Zeeman field

Magnetization induced by supercurrent

In the normal state ( T >Tc ), Static magnetic field can not induce dissipative current flows.

( paramagnetic supercurrent )

Rashba case

Bx,Mx

Jy = KyxBx

Mx = !Kyx

m

nx

Jy

Dissipationless effect

Electron correlation effects (S.F., PRB(2005); unpublished)

enhanced by electron correlation effect by the factor (if FM fluctuation is absent)

PM current much more enhanced than diamagnetic supercurrent

!Js = !nse

2

m!c!A + Kyx(!n " !Bx)Total supercurrent

Kyx !eµB

z

!

EF

n0

1/zKyx !

eµB

z

!

EF

n0

Estimation of ME effect

!kF /EF ! 0.1 n ! 1022 cm!3 1/z ! 100vs/v!F ! !/EF ! 0.01

K ! eµB

8!3

"

EF

1z

M ! 0.1 Gauss

Cancellation of paramagentic supercurrent (Yip(2005) )

Js =1

2e!(h!! "

2e

cA) + K(n # B)

M =K

2e[n ! (h"! #

2e

cA)] + MZee

J tot = Js + JM

= Jdia + c!" MZee + cK!(#!xJdia

z , !yJdiaz , !xJdia

x + !yJdiay )

JM = c!" M

F =!

dr[!a|!|2 + b

4|!|4 + 1

2m|D!|2 + K

2en · B " (!(D!)! + !!D!)

+B2

8!! M · B] D = !ih"!

2e

cA

Free energy for Rashba case (Edelstein, Samokhin, Kaur et al.)

due to Parity-breakingSO interaction

In addition, magnetization current exists !Total current

ME effect

paramagnetic supercurrentIn Meissner state, in thermodynamic limit, PM supercurrent vanishes (Yip)

In finite systems, or in mixed state, PM supercurrent is still nonzero !!

c.f. For Cubic system(Dresselhaus case), PM supercurrent always cancels !

Helical vortex statehelical vortex state (Kaur, Agterberg, Sigrist, 2005; Samokhin(2004))

!q = !0eiq·r A kind of Fulde-Ferrel state

Fermi surfacein in-plane field

q = !2m!"n " "h

( inv. sym.-breaking term of free energy)

Same origin as paramagnetic supercurrent

In isolated systems, no bulk current flows, the helical vortex state occurs

The Helical vortex state is more feasible in heavy fermion SC !!

!!

fIB = !"n · "h ! [!(D!)! + !!(D!)]!h€

k + q / 2

€

−k + q / 2

!q

( Kaur et al.(2005), Samokhin(2004))

enhanced

! =Kyx

2ens

enhanced !due to FF state

m!

Hc2 !

!0

2!"2+

!

!0

2!"2

"22!0m

!2#2

!h2

(m!)6

! ! m!( )

Helical vortex statehelical vortex state (Kaur, Agterberg, Sigrist, 2005; Samokhin(2004))

!q = !0eiq·r A kind of Fulde-Ferrel state

Fermi surfacein in-plane field

q = !2m!"n " "h

( inv. sym.-breaking term of free energy)

Same origin as paramagnetic supercurrent

In isolated systems, no bulk current flows, the helical vortex state occurs

The Helical vortex state is more feasible in heavy fermion SC !!

!!

fIB = !"n · "h ! [!(D!)! + !!(D!)]!h€

k + q / 2

€

−k + q / 2

!q

( Kaur et al.(2005), Samokhin(2004))

enhanced

! =Kyx

2ens

enhanced !due to FF state

m!

Hc2 !

!0

2!"2+

!

!0

2!"2

"22!0m

!2#2

!h2

(m!)6

! ! m!( )

! ! µB

16"3

#

EF

1z

!kF /EF ! 0.1 n ! 1022 cm!3

1/z ! 100

v!

F ! 105(cm/s)H(0)c2 =

!0

2!"2! 4 T

HHV =!

!0

2!"2

"2 2!0m!2#2

!!2! 0.4 T

!HV ! 1/q ! 10!6 cm! Inter-vortex

space

q ! µB

16!3

"pF

EF

B

v!F z!h

vortex

!HV

Velocity !v = !"k + #(!n" !$)Rashba SO interaction :

(!!"y,!"x, 0)anomalous velocity causes AHE

Anomalous Hall effect exists only for !H ! c

Strongly enhanced by electron correlation due to the factors,

!AHExy ! C"zzH

!zz !zzspin susceptibility for !H ! c

Anomalous Hall effect (Karplus-Luttinger type)

!Ey

!Jy

!Jx!Ex

!Ey

!Jy

DissipationlessHall current appears

Hz != 0

ky kx

Thermal anomalous Hall effect : Hall effect for heat current

In the SC state, not measurable

Instead, however, we can consider

!xy

!zz = !VVzz given by Van Vleck term only

Van Vleck contribution, not affected by SC transition for !!

SO splittingFermi surface

!! !|p|

Large thermal AHE may exists even in the SC state !!and behaves like in the normal state !!

Precisely,...

!sz(!) = 1! 1µB

""(p)"Hz

2!|k|!AHE

xy ! C!"zzHc

Thermal AHE : Interesting implication for superconducting state

contribute to Re !xy. In the limit of " ! 0,

d

dHz(#2

+(p)#2+(p + q) " #2

!(p)#2!(p + q))|Hz"0

= "1

$|t(p)|(µB "

%!(p)

%Hz). (64)

Using (62), (63), and (64), we end up with,

Re !AHExy

Hz= e2µB

!

!=±

!

k

& tanh('#k!

2T)

"sz('#k! , k)

4$|t('#k! , k)|3

#(%kxtx!%ky ty! " %kx ty!%ky tx! ). (65)

Here %kµt"! $ %kµt"(', k)|#=#!

k!; i.e. %kµ does not operate

on '#k! in the argument of t"(p). Since we have postulatedthat the spin-orbit splitting is much larger than the quasi-particle damping, the anomalous Hall conductivity !AHE

xyis not involved with any relaxation time, and thus deter-mined only by dissipationless processes. It is noted thatthe anomalous Hall conductivity is enhanced by the fac-tor "sz which is equivalent to the enhancement factor of(zz, Eq.(46). In the heavy fermion system CePt3Si, thisenhancement factor is of order % 60, and the detection ofthe anomalous Hall e#ect is feasible in such strongly cor-related electron systems. We would like to stress that inthe expression of the anomalous Hall conductivity (65),not only electrons in the vicinity of the Fermi surface butalso all electrons in the region of the Brillouin zone sand-wiched between the spin-orbit-splitted two Fermi surfacescontribute, in accordance with the fact that (zz is dom-inated by the van-Vleck-like susceptibility.

2. Thermal anomalous Hall e!ect

We now consider the thermal anomalous Hall e#ect,which is the anomalous Hall e#ect for the heat current.To simplify the following analysis, we assume that theenergy current due to the interaction between quasipar-ticles is negligible, and thus the heat current is mainlycarried by nearly independent quasiparticles. We wouldlike to discuss the validity of this assumption in the end ofthis section. Then, we can obtain the thermal anomalousHall conductivity by using a method similar to that usedin the previous section. Using the heat current relatedto the single-particle energy,

JQµ =!

k

c†k1

2[vkµH(p) + ˆH(p)vkµ]ck, (66)

we define the Hall conductivity for the heat current as,

)xy =1

T(L(2)

xy "(L(1)

xy )2

L(0)xy

), (67)

where L(0)xy is equal to the Hall conductivity !xy, and,

L(1)xy = lim

$"0

1

i"K(1)

xy (i"n)|i$"$+i0, (68)

L(2)xy = lim

$"0

1

i"K(2)

xy (i"n)|i$"$+i0, (69)

K(1)xy (i"n) =

" 1/T

0d&&T!{JQx(&)Jy(0)}'ei$n! , (70)

K(2)xy (i"n) =

" 1/T

0d&&T!{JQx(&)JQy(0)}'ei$n! . (71)

Extending the argument in the previous subsectionstraightforwardly to the present case, we have the contri-

butions from the anomalous Hall e#ect to L(1)xy and L(2)

xy ,neglecting the normal Hall e#ect,

L(m)AHExy

Hz= e2!mµB

!

!=±

!

k

&('#k! )m tanh('#k!

2T)

#"sz('#k! , k)

4$|t('#k! , k)|3(%kx tx!%ky ty! " %kxty!%ky tx! ),(72)

with m = 1, 2. Then, the expression for the thermalanomalous Hall conductivity )AHE

xy is given by Eqs.(67)and (72). As in the case of the anomalous Hall e#ectfor the charge current, the thermal anomalous Hall con-ductivity is also dominated by the contributions fromelectrons occupying the momentum space sandwiched be-tween the spin-orbit-splitted Fermi surfaces. This prop-erty brings about a remarkable e#ect in superconductingstates. In the superconducting state, when vortices arepinned in the mixed state, the Hall e#ect for the chargecurrent does not exist. Instead, the thermal Hall e#ect forthe heat current carried by the Bogoliubov quasiparticlesis possible. Since we consider the magnetic field perpen-dicular to the xy-plane, electrons in the normal core donot contribute to the thermal transport in the directionparallel to the plane. Below the superconducting transi-tion temperature !xy is infinite while L(1) is finite. Thusthe thermal Hall e#ect is governed by the first term of the

right-hand side of (67), i.e. the coe$cient L(2)xy . In con-

trast to the normal Hall e#ect for the heat current whichdecreases rapidly in the superconducting state, the co-

e$cient L(2)AHExy is not a#ected by the superconducting

transition when the magnitude of the spin-orbit-splittingis much larger than the superconducting gap as in thecase of CePt3Si and CeRhSi3. Thus, even in the limit ofT ! 0, )AHE

xy /(HzT ) takes a finite value. Moreover inheavy fermion systems, the magnitude of )AHE

xy /(HzT )in the limit of T ! 0 is expected to be much enhancedby the factor "sz.

Finally, we discuss the validity of the disregard for theheat current carried by the interaction between quasi-particles. In the vicinity of the Fermi surface, the quasi-particle approximation is applicable, and the interactionbetween quasiparticles is much reduced by the wave func-tion renormalization factor z2

k! , and may be negligible forheavy fermion systems. However, as seen from Eq.(72),the thermal anomalous Hall conductivity is dominated


d

dHz(#2

+(p)#2+(p + q) " #2

!(p)#2!(p + q))|Hz"0

= "1

$|t(p)|(µB "

%!(p)

%Hz). (64)


Re !AHExy

Hz= e2µB

!

!=±

!

k

& tanh('#k!

2T)

"sz('#k! , k)

4$|t('#k! , k)|3


Here %kµt"! $ %kµt"(', k)|#=#!






JQµ =!

k

c†k1



)xy =1

T(L(2)

xy "(L(1)

xy )2

L(0)xy

), (67)


L(1)xy = lim

$"0

1

i"K(1)

xy (i"n)|i$"$+i0, (68)

L(2)xy = lim

$"0

1

i"K(2)

xy (i"n)|i$"$+i0, (69)

K(1)xy (i"n) =

" 1/T

0d&&T!{JQx(&)Jy(0)}'ei$n! , (70)

K(2)xy (i"n) =

" 1/T

0d&&T!{JQx(&)JQy(0)}'ei$n! . (71)




L(m)AHExy

Hz= e2!mµB

!

!=±

!

k

&('#k! )m tanh('#k!

2T)

#"sz('#k! , k)













d

dHz(#2

+(p)#2+(p + q) " #2

!(p)#2!(p + q))|Hz"0

= "1

$|t(p)|(µB "

%!(p)

%Hz). (64)


Re !AHExy

Hz= e2µB

!

!=±

!

k

& tanh('#k!

2T)

"sz('#k! , k)

4$|t('#k! , k)|3


Here %kµt"! $ %kµt"(', k)|#=#!






JQµ =!

k

c†k1



)xy =1

T(L(2)

xy "(L(1)

xy )2

L(0)xy

), (67)


L(1)xy = lim

$"0

1

i"K(1)

xy (i"n)|i$"$+i0, (68)

L(2)xy = lim

$"0

1

i"K(2)

xy (i"n)|i$"$+i0, (69)

K(1)xy (i"n) =

" 1/T

0d&&T!{JQx(&)Jy(0)}'ei$n! , (70)

K(2)xy (i"n) =

" 1/T

0d&&T!{JQx(&)JQy(0)}'ei$n! . (71)




L(m)AHExy

Hz= e2!mµB

!

!=±

!

k

&('#k! )m tanh('#k!

2T)

#"sz('#k! , k)












Summary

Pairing state of CePt3Si

Accidental line node due to coupling with AF order ?

Unique electromagnetism and transport phenomena

Much enhanced by strong electron correlation !!

Possible pairing state dominant p-wave state with minor fraction of (extended) s-wave state

or due to s+p state ?

Magnetoelectric effects in the SC state and in the normal state

Anomalous Hall effect

Helical vortex phase

Line nodes with small coherence peak of1/T1

Relevant to Heavy Fermion noncentrosymmetric SC !!

空間反転対称性の破れた超伝導の新奇な物性cond.scphys.kyoto-u.ac.jp/~fuji/noinvsuper2006open.pdf空間反転対称性の破れた超伝導の新奇な物性...

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