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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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
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
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
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 !!