Orbital+spin multimode fluctuations due to vertex corrections

in Fe-pnictides and high-Tc cuprates

－ nematic orbital order and superconductivity －

Hiroshi Kontani (Nagoya Univ.)

in collaboration with Seiichiro Onari (Okayama Univ.)

Youichi Yamakawa (Nagoya Univ.) Masahisa Tsuchiizu (Nagoya Univ.)

Tetsuro Saito (Nagoya Univ.)

Outline

1. Fe-based superconductors ・orbital order and phase diagram ・orbital fluctuations by C66, χRaman

・ SC gap functions in LiFeAs

Fe-based SC: To explain in the normal state phase diagram, mean-field-level approximations are insufficient. →Vertex correction (VC) must be included!

3. Ruthenates (t2g) ・electronic nematic order in Sr3Ru2O7 ・TSC due to orbital fluctuations in Sr2RuO4

2. high-Tc cuprates (dx2-y2, px, py) ・CDW order in the pseudogap region

Onari and HK: PRL (2012), Onari, Yamakawa and HK, PRL (2014)

HK and Yamakawa, PRL (2014)

Saito et al, PRB (2014)

Tsuchiizu et al: PRL (2013)

Tsuchiizu et al: arXiv(2014)

Yamakawa and HK: arXiv(2014)

since 2011~

orbital-order/fluctuations due to vertex correction (VC) .

1. Fe-based superconductors

・orbital order and phase diagram ・orbital fluctuations by C66, χRaman

・ SC gap functions in LiFeAs

Onari and HK: PRL (2010), Onari, Yamakawa and HK, PRL (2014)

HK and Yamakawa, PRL (2014)

Saito et al, PRB (2014)

on the basis of five/ten orbital Hubbard model

Normal State Phase diagram in Fe-based SC

orbital order

TS > TN

SDW transition at TN

SC

(Eyz-Exz)strain=0.3% in LDA is much smaller than 60meV ↓ 　nxz≠nyz orbital order due to U.

TS

TN

LaFeAsO1-xHx

T [K

]

0 0 0.1 x

50

100

150

0.2 0.3

C4→C2

S. Iimura, et al., Nat. Commun. 3, 943 (2012).

　Orbital physics is important.

ARPES

M. Yi et al., PNAS 108, 6878 (2011)

Eyz-Exz~60meV

x~0

(T < TS, detwinned) BaFe2As2

structure transition at TS

SC

F. Kruger et al, PRB 79, 054504 (2009). W. Lv et al, PRB 80, 224506 (2009); C.-C. Lee et al., PRL 103, 267001 (2009)

　Orbital physics is important.

the most important issue

TS > TN

SDW transition at TN

SC

TS

TN

LaFeAsO1-xHx

0 0 0.1 x

50

100

150

0.2 0.3

C4→C2

S. Iimura, et al., Nat. Commun. 3, 943 (2012).

structure transition at TS

SC

→斜方相転移

M. Yi et al., PNAS 108, 6878 (2011)

Eyz-Exz~600K

x~0 BaFe2As2

Strong orbital fluctuations enlarge 1/C66.

H. Kontani and Y. Yamakawa, PRL (2014) Spin nemtic scenario: Fernandes et. al., PRL 105, 157003 (2010)

fitting by orbital fluctuation mechanism

C66

exp

theory

M. Yoshizawa et al., JPSJ (2012) A. Bohmer et al., PRL (2014)

orbital fluctuations

　Orbital physics is important.

Normal State Phase diagram in Fe-based SC the most important issue

1. C66, Raman susceptibility

A. E. Böhmer, et al., PRL 112, 047001 (2014).

C66 Raman

Y. Gallais, et al., PRL 111, 267001 (2013).

Ba(Fe1-xCox)2As2

𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman : direct observation of charge-orbital fluctuations C66 : indirect observation via orbital-lattice coupling

Both 𝐶↓66   and 1/ 𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman  decrease as T→ 𝑇↓𝑆 .

軌道揺らぎ（電気四重極揺らぎ）の観測

1∕𝜒↓𝑥↑2 −𝑦↑2 ↑Ram

an (0) 

Previous theories (spin-nematic, band-Jahn-Teller) cannot explain the Raman measurement.

Observation of strong orbital fluctuations.

𝐶↓66 

cf. spin-nematic scenario

✓χRaman ? ✓(𝐸↓𝑦𝑧 − 𝐸↓𝑥𝑧 )↓𝐋𝐃𝐀,      𝐬𝐭𝐫𝐚𝐢𝐧=𝟎.𝟑% ≈60K≪(𝐸↓𝑦𝑧 − 𝐸↓𝑥𝑧 )↓ARPES ≈600K

✓applicable only when χs(q) is commensurate.

✓No theory based on microscopic Hamiltonian.

Jc/Jab [%]

TS

TN

TS>TN

0 0.1 0.2

≈ BaFe2As2 >1%

TS=TN

C. Fang, et al., PRB 77, 224509 (2008).

L. W. Harriger, et al., PRB 84, 054544 (2011).

0.3

X Γ Y wave vector

0

-0.1

0.1

~60K

LaFeAsO, T<Ts (WIENk2)

E [eV] (m*/m~3)

s𝐩𝐢𝐧  𝐪𝐮𝐚𝐝𝐫𝐮𝐩𝐨𝐥𝐞  𝐨𝐫𝐝𝐞𝐫  ⟨𝑺↓𝒊 ・𝑺↓𝒋 ⟩≠𝟎 R. M. Fernandes, et al., PRL 105, 157003 (2010).

✓geometrical frustration

J1

J2

a

b

J1 ~ 2J2

⟨𝑆↓𝑖 ・𝑆↓𝑗 ⟩≠0

⟨𝑆↓𝑖 ⟩=0

→structure transition a≠b →orbital order

J1

J2

a

b

強いフラストレーション

J1 ~ 2J2

⟨𝑆↓𝑖 ・𝑆↓𝑗 ⟩≠0

⟨𝑆↓𝑖 ⟩=0

0.64

0.60

0.56

0.68

Tem

pera

ture

(non-local)

→a≠b

✓unstable for 𝐽↓𝑐 ∕𝐽↓𝑎𝑏  >0.2%

orbital order scenario orbital order by Coulomb int. Eyz ≠ Exz

→structure transition a≠b

As3-

As3- As3-

As3- nxz>nyz

a

b Fe

𝑂↓𝑥↑2 − 𝑦↑2  

= 軌道秩序 Eyz ≠ Exz

Raman suscep.

Results 　・Orbital order/fluctuations are induced by U. (due to Orbital-spin interference generated by the VC)

　・Both 𝐶↓66  and 𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman   are explained by the orbital fluctuation scenario. 　

F. Krüger, et al., PRB 79, 054504 (2009). W. Lv, J. Wu, and P. Phillips, PRB 80, 224506 (2009) C.-C. Lee, et al., PRL 103, 267001 (2009). S. Onari and H. Kontani, PRL 109, 137001 (2012).

Y. Gallais, et al., PRL 111, 267001 (2013).

charge quadruploe

We analyzed the multiorbital Hubbard model using self-consistent vertex correction (SC-VC) method. Onari-Kontani, PRL (2012)

+ - - +

a

b = strain 0.3%

}',{' JJnnUnnU ++ ∑∑≠

↓↑νµ

νµµ

µµ

Orbital fluctuations due to vertex corrections (VC)

' 2U U J= +J = J’

inter-orbital repulsion orbital order (ex. nxz ≠ nyz) when U’ >U

multi-orbital Coulomb interaction：

unrealistic condition!

1. PRA:! Takimoto (2002)!

2. vertex correction (VC): U'eff is enlarged by the VC at q~0, (π,0) ! Onari-Kontani (2012) near magnetic QCP!

orbital!fluc.

orbital! fluc.

spin fluc.

spin fluc.

AL-VC

' ( / 0.15)U U J U< =

Aslamazov -LarkinVC for charge susceptibility

orbital-spin interference

C66 & 𝜒↓𝑥2−𝑦2↑Raman : linear response theory

全感受率 (U+g) 𝜒↑total (𝒌,𝜔)= 𝜒↑SC−VC (𝒌,𝜔)/1− 𝑔↓ac (𝑘∕𝜔 )𝜒↑SC−VC (𝒌,𝜔)  𝑔↓𝑎𝑐 (𝑘∕𝜔 )= 𝑔↓𝑎𝑐↑0 𝑣↓𝑎𝑐↑2 (𝑘∕𝜔 )↑2 /𝑣↓𝑎𝑐↑2 (𝑘∕𝜔 )↑2 −1 

電子・音響フォノン相互作用

𝜆  = 2𝜋∕𝑘 <𝐿

𝐶↓66↑−1 ∝𝜒↑𝑄 /1− 𝑔↓ac 𝜒↑𝑄  

static lattice deformation

𝑔↓ac : due to e-ph interaction

✓Both   𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman   and 𝐶↓66↑−1   increases  with  𝜒↑𝑄 . ✓𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman < 𝐶↓66↑−1   due to small 𝑔↓𝑎𝑐 .

𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman ∝𝜒↑𝑄 

𝜆  ≫𝐿

Note: Acoustic phonon cannot be excited by photon.

due to Coulomb interaction

ω/k = c≫vac

1.Enhancement of 𝜒↑𝑄   (charge  ^luc.) due to Coulomb interaction (AL-VC).

= contribution by band Jahn-Teller effect

consistent with experiments

2. 𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman : (optical) 3. 𝐶↓66↑ : (ultrasonic)

𝜒↑𝑄 

C66 & 𝜒↓𝑥2−𝑦2↑Raman : fitting of experimental data

Orbital fluctuation scenario (SC-VC) can explain both 𝐶↓66↑   and 𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman . 　　(×spin-nematic scenario)

𝐶↓66↑ 

exp

theory

𝐶↓66↑ (ultrasonic)

theory

𝜒↓𝑥↑2 −𝑦↑2 ↑Ram

an 

𝜒↓𝑥↑2 − 𝑦↑2 ↑Raman (optical)

𝜃↓𝐶↓66  ≈𝜃↓Raman ≈𝜃↓NMR 

=>experimental support for the present study.

Imai group, 2010 obtained by 1/T1T

𝜃↓𝐶↓66  

𝜃↓NMR  𝜃↓Raman 

TS

enhancement of χs(π,0) under TS

✓ 𝐒𝐩𝐢𝐧  𝐟𝐥𝐮𝐜𝐭𝐮𝐚𝐭𝐢𝐨𝐧𝐬  𝐚𝐭  𝒒↓𝑠 =(𝜋,0) are  enlarged  

           under the polarizaiton 𝐸↓𝑥𝑧 − 𝐸↓𝑦𝑧 <0 ⇒ 𝑇↓N > 𝜃↓NMR 

experimental spin structure. 　　　　　　　　　　　　recent neutron experiments.

H. Kontani, T. Saito, and S. Onari, PRB 84, 024528 (2011).

strong orbital-spin correlation

RPA with ΔE = Exz−Eyz Ba(Fe1-xCox)2As2

qx qｙ (0,0)

(π,0)

(π,π)

χs(q) ΔE=−0.04eV

TS TN

θNMR

x

T

consistent with

Strong anisotropy of S under TS with orbital order Theoretical study

S

Sy

Sx [µ

V/K

]

T[K] 500 400 300 200 100 0

TS

TN

Onari et al., unpublished

orbital order Exz<Eyz

S. Jiang et al., PRL 110, 067001 (2013)

１００

BaFe2(As,P)2

Experimental result

Sx

Sy

Large anisotropy Sy≫Sx is well explained by the orbital order scenario!

functional RG(+cRPA): orbital fluctuations due to AL-VC

two-orbital (xz+yz) model

"Importance of the AL-VC" is verified.

M. Tsuchiizu et al., PRL 111, 057003 (2013)

RPA

f-RG

Q

unbiased theoretical method ①n=3.3

②n=2.67 M. Tsuchiizu et al., arXiv (2014)

C2

2. LaFeAsO1-xHx

?

S. Iimura, et al., Nat. Commun. 3, 943 (2012). N. Fujiwara, et al., Phys. Rev. Lett. 111, 097002 (2013).

C'2 C4

C2

Isostructure (C4) transition at x~0.45

0.4>x>0.49: ・C4 symmetry (a=b) under TN→ Spin-lattice coupling is weak. ・upturn of c-axis length: inter-orbital (xz,yz⇔xy) carrier transfer?

(π,0) spin order

|m|~0.8µB

S. Iimura, et al., Nat. Commun. 3, 943 (2012). N. Fujiwara, et al., Phys. Rev. Lett. 111, 097002 (2013).

=b

quadrupole susceptibility in LaFeAsO1-xHx

𝑛↓𝑥𝑧 ≠ 𝑛↓𝑦𝑧  C2 orbital order

C2 C4

𝑛↓𝑥𝑧 = 𝑛↓𝑦𝑧 ≠ 𝑛↓𝑥𝑦  C4 orbital order

M

Γ X

xy

yz

xz

x=0.4

(0,0) (0,0)

(π,0)

(π,π)

(π,0)

(π,π)

X

M

Γ

xy xz

yz

x=0

obtained susceptibilities

Wien2k Fermi surfaces

Both C2, C4 structure transitions are explained.

C2 C4

M

Γ X

xy

yz

xz

x=0.4 X

M

Γ

xy xz

yz

x=0

Wien2k Fermi surfaces

θ

electron-FS

hole-FS (M-point)

hole-FS (Γ point)

nodal S++

θ

electron-FS

hole-FS (M-point)

0

Δ

fullgap S++

S++ state due to ferro- and antoferro-

orbital fluctuations for x~0 and x~0.4

SC gap functions in LaFeAsO1-xHx

3. Gap structure of LiFeAs "fingerprint" of the pairing mechanism

Gap structure is well explained in terms of the orbital-fluctuation-mediated SC

10-orbital model, with SOI 3D structure of the FSs

✓The largest SC gap on the smallest h-FSs

FSs obtained by ARPES (Borisenko et al)

Y. Wang el al., PRB (2013). Yin, Haule, Kotliar, arXiv:1311.1188. Ahn, Eremin et al., PRB (2014) Saito et al, RPB (2014)

kz=π

without  SOI   with  SOI  at  Fe-­‐ion（0.05eV）

xy

yz xz

Fermi  surface  （kz=π）  without  SOI  

xy

yz

xz

change  in  topology  

Effect of Spin-Orbit Interaction (SOI)

kz=π  plane   kz=π  plane  

Fermi  surface  （kz=π）  with  SOI  

two small h-pockets single small h-pockets

splitting

solution of gap equation with SOI

importance of orbital fluctuations in LiFeAs

kz=π plane

Borisenko et al., Symmetry 2012

Δhole1,2

Δhole3 |Δ|

exp.: largest SC gap

Δ on hole-FSs

small h-FS

exp

theory

●spin fluctuation mediated S±

●orbital fluctuation mediated S++

Y. Wang et al., PRB (2013)

2. Sign-reversal between hole-FSs (hole-S+-) Cooperation of orbital-fluctuations (attractive)

and spin-fluctuations (repulsive) ↓

hole-S+- gap state

cf. hole-S+- due to competing repulsions

F. Ahn et al., PRB (2014) Z. P. Yin et al, arXiv:1311.1188.

T. Saito et al, PRB (2014)

attraction +

repulsion

+ +

LiFeAs (kz=π)

Δhole3

Δhole1,2

orbital fluc.

spin

fluc

.

+ -

xy orbital

xz,yz orbitals

→orbital order, enhancement of 1/C66 and χRaman

Orbital+spin mutimode fluctuations due to VC beyond the mean-field-type theories.

Summary (part 1)

→ LaFeAsO1-xHx: C2, C4 structure transitions

S.Onari, Y.Yamakawa and H.Kontani, PRL (2014)

H. Kontani and Y.Yamakawa, PRL (2014)

→ LiFeAs: The largest SC gap on the smallest hole-pockets

T. Saito et al, PRB (2014)

cf. orbital-independent SC gap in BaFe2(As1-xPx)2 Shimojima et a, Science (2010)

xy xz/yz z2

z2

2. Cuprate high-Tc superconductors

・CDW order in the pseudogap region Yamakawa and HK: arXiv(2014)

on the basis of three-orbital d-p Hubbard model

Nearly Fermi liquid picture for cuprates. Te

mpe

ratu

re

hole concentration p 0 ~1/8 0.04

SDW SC

for hole-doped compounds ・𝑝≤0.04 : AFM order at 𝑸𝑆≈(𝜋,𝜋)

・𝑝=0.05~0.2 : d(x2-y2)-wave SC atTc~100K

nearly AFM Fermi Liquid

non-FL behaviors due to spin fluctuations. ・1/𝑇1𝑇 ∝∑𝑞↑▒Im𝜒(𝑞,𝜔)/𝜔 ∝𝜉↑2 ∝1/𝑇−𝜃  ・𝜌(𝑇)∝ImΣ( 𝑞↓cold ,0)∝𝑇↑2 𝜉↑2 ~𝑇

T. Moriya and K. Ueda, Adv. Phys. 49, 555 (2000). K. Yamada: "Electron Correlation in Metals" (Cambridge Univ. Press 2004). D.J. Scalapino, Phys. Rep. 250, 329 (1995). P. Monthoux and D. Pines, PRB 47, 6069 (1993). J. Takeda, T. Nishikawa, and M. Sato, Physica C 231, 293 (1994).

Enhancement of RH(T) due to current-VC

・1/𝑇1𝑇∝1/𝑇 ・𝜌(𝑇)∝𝑇 ・𝑅H(𝑇)∝1/𝑇

RH: breakdown of FL ?

J. Takeda, T. Nishikawa, and M. Sato, Physica C 231, 293 (1994). H. Kontani, K. Kanki, and K. Ueda, PRB 59, 14723 (1999).

= nearly AF Fermi liquid ＋vertex corrections

total current Jk ≠ velocity vk due to current VC RH∝T-1 due to current VC (1999)

Tem

pera

ture

Hg1201 YBCO

CDW(=orbital order) at Q=(δc,0)

Tc Tc

✓nematic CDW in many cuprates Y系 G. Ghiringhelli, et al., Science 337, 821 (2012). Bi系 R. Comin, et al., Science 343, 390 (2014). Hg系 W. Tabis, et al., arXiv:1404.7658. La系 M. Hücker, et al., PRB 83, 104506 (2011).

G. Ghiringhelli, et al., Science 337, 821 (2012).

q// (r.l.u) π/2

○resonant X-ray ○Phase diagram with CDW

𝛿𝑐~𝜋/2

CDW CDW

W. Tabis, et al., arXiv:1404.7658.

R H(T

) [m

m3 /

C]

YBa2Cu3O7-δ

Suppression of RH and S at TCDW Te

mpe

ratu

re

Hg1201 YBCO

N. Doiron-Leyraud, et al., PRX 3, 021019 (2013).

R H(T

)/R H

(200

K)

J. R. Cooper, et al., PRB 44, 12086 (1991).

✓sign-reversals of S, RH at T≪TCDW

S and RH show maxima at T~TCDW

K. Segawa and Y. Ando, PRB 69, 104521 (2004). D. LeBoeuf, et al., PRB 83, 054506 (2011).

RH<0

熱起

電力

S[µ

V/K]

hole-dope

高温の振る舞いから外れるあたり 変曲点？：要確認

Below TCDW: 　① suppression of spin fluctuations 　② Fermi surface reconstruction small e-pocket!

when γspin-fluc ~γimpurity

CDW charge pattern at Q=(δc,0)

px px px px px

py

py

py

K. Fujita, et al., PNAS 111, E3026 (2014).

○STM ○resonant X-ray

R. Comin, et al., arXiv:1402.5415.

p-orbital CDW (period 4a): d-p model has to be studied.

CDW

𝛿𝐶  ~  Δ𝐹𝑆  ~  𝜋/2

nesting between hot-spots?

by studying the VC in term of nearly AF Fermi liquid?

Can we derive the CDW

origin of CDW？

px

py

px px px px py

Theoretical study of d-p hubbard model

・main nesting:Qs~(π,π)

・miner nesting:Qc~(ΔFS,0) χs(q) at Qs~(π,π)

In the RPA, CDW at Qc cannot be reproduced. [CDW at q=0 is obtained only for V>3eV.]

Y. Yamakawa and H. Kontani, arXiv:1406.7520. (d-px-py)-orbital model

Fermi surface

spin susceptibility Ud = 4.1eV

(0,0) (π,0)

(π,π)

δs qx

qy

𝜒↓RPA↑𝑠 (𝒒)

V ~1eV by first principle study

At Fermi level, [p-DOS]:[d-DOS] ~1:2

−4

−2

0

2

Γ X M ΓE

[eV

]

band dispersion

px 軌

道

py軌道 P. Hansmann, et al., New J. Phys. 16, 033009 (2014).

✓intertwining of charge and spin order parameters Davis and Lee, PNAS (2013). Wang and Chubukov, arXiv(2014). Sachdev and Placa, PRL(2013). ・t-J-type model, 　・single-orbital model,

RPA: CDW cannot be obtained

𝜒↓RPA↑𝑐 (𝒒)

(0,0)

(π,0)

(π,π)

qx qy

✓𝜒↑c (𝑞)  given by RPA (V =2.8eV≫first principle value ~1eV)

In the RPA、CDW at Qc~(ΔFS,0) is not obtained！

=>necessary for the VC beyond the RPA!

Coulomb interaction

irreducible suscep.

𝜒↑s(c) (𝑞)= Φ↑𝑠(c) (𝑞)/1− Γ↑s(c) Φ↑𝑠(c) (𝑞)  ✓Spin (Charge) susceptibility

Φ(𝒒)= 𝜒↑0 (𝒒)+𝑋(𝒒) : RPA VC

d-p model (U+V)

self-consistent VC method

✓Aslamazov-Larkin (AL) term for py-orbital

S. Onari and H. Kontani, PRL 109, 137001 (2012). S. Onari, Y. Yamakawa, and H. Kontani, PRL 112, 187001 (2014).

Qc=(ΔFS,0)

d d

d d

d

d

s

s py

py py

py Φ↓𝑦↑𝑐 (𝑸↓𝑐 )= 𝜒↑0 (𝑸↓𝑐 )+

𝜒↓𝑦↑𝑐 (𝑸↓𝑐 )∝1/1−16 𝑉↑2 Φ↓𝑑↑𝑐 (𝑸↓𝑐 )Φ↓𝑦↑𝑐 (𝑸↓𝑐 )  ✓charge susceptibility

Since  Φ↓𝑦↑𝑐 ~Σ{𝜒↓𝑑↑𝑠 }2   ,  Φ↓𝑦↑𝑐 ≫𝑁𝑝𝑥(0)  is  realzed.  

𝑉↑eff ~𝑉√Φ↓𝑦 ∕𝜒↓𝑦↑0   ≫𝑉

V(q) is enhanced by AL-VC at Qc~(ΔFS,0) near magnetic QCP.

⇒CDW at Qc occurs even for V<1eV.

V eff(QC) is enlarged by AL-VC -> nematic CDW even for V<1eV!

py軌道

effective d-p Coulomb int.

k+Qc/2 k'+Qc/2

k'−Qc/2

Qs+Qc/2

Qs−Qc/2 k−Qc/2

k−Qs k−Qs

CeFeAsO1-xFx

three-point vertex is large at Qc=(ΔFS, 0). ⇒  Φ↓𝑦↑𝑐 (𝑞) is large at Qc

spin・charge (orbital) interference!

=1 : CDW

result 1: nematic CDW, Q=(δc,0)

d s

s py py

d

(0,0) (π,0)

(π,π)

δs qx

qy

𝜒↓RPA↑𝑠 (𝒒)

(0,0)

(π,0)

(π,π)

δc qx

qy

𝜒↓𝑑↑𝐶 (𝒒)

𝜒↓𝑦↑𝐶 (𝒒)

𝜒↓𝑑;𝑦↑𝐶 (𝒒)<0 (d, py)-antiphase

1. spin suscep.

2. AL-VC for χc(q)

3. charge susceptibility

Ud= 4.1eV V = 0.65eV T = 0.05eV

CDW wavelength at δc ∼ ΔFS ∼ π/2

V eff(QC) due to the AL-VC -> nematic CDW even for V<1eV!

result 2: nematic CDW, Q=(δc,0)

(0,0) (π,0)

(π,π)

δc qx

qy

𝜒↓𝑑↑𝐶 (𝒒) 𝜒↓𝑦↑𝐶 (𝒒)

𝜒↓𝑑;𝑦↑𝐶 (𝒒)<0

3. charge susceptibility

V = 0.65 eV

py

py

py

px px px px px px Cu Cu Cu Cu Cu Cu

V

4. nematic CDW = (d-py)-antiphase

Orbital order at QC~(π/2,0) is induced by spin-fluctuations-driven VC

significant orbital physics similar to Fe-based SC!

form factor (δnd : δnx : δny) = (-0.56 : 0.21 : 0.80)

−0.4 −0.2 0 0.2 0.40.1

0.2

0.3

0.4

total

Loca

l Den

tisy

of S

tatu

s [eV−1

]

d

_d

Energy [eV]

result 3: STM

d

d d d d d d px px px px px

py

d

py

d

py

d

py

d

py

d

py

d

py

d

py

d

py

d

py

✓𝑅(𝒓,𝐸)= 𝐼(𝒓,+𝐸)/𝐼(𝒓,−𝐸) ∝∫0↑𝐸▒𝑁(𝒓, 𝐸↑′ )𝑑𝐸↑′  /∫−𝐸↑0▒𝑁(𝒓, 𝐸↑′ ) 𝑑𝐸↑′    ⇒

K. Fujita, et al., PNAS 111

, E3026 (2014).

CDW Bi2212

𝑅(𝒓,𝐸) observd by STM is well reproduced.

CDW

R(r,E)>1 : bright

✓local DOS 𝑁(𝒓,𝐸)

with form factor: Δ𝐸𝑙cos(𝜋𝑥∕2 ) 

p = 0

result 4: pseudogap T. Yoshida, et al., JPSJ 81, 011006 (2011).

ΔFS~0.2 5x5-order

Fermi arc structure at T*>Tc observed by ARPES can be explained by CDW at Qc=(δc,0).

①averaging of two-domains Qc=(δc, 0), (0, δc)

double-Q (5x5 order)

Theoretical results

Comin et al, Science 343, 390 (2014). kx

ky

ky

0 π

π

ΔFS=0.51π

hot spot

hot spot

CDW driven Fermi arc

in Bi-based SC

-π

π

kx

0

π 0

0

CDWによるギャップ

−ImGR(k)/π kx

ky

0 π

π

0 −ImGR(k)/π

single-Q (δc,0), (0,δc) averaging

②double-Q CDW

Tem

pera

ture

hole concentration p

TCDW

0 ~1/8 0.04

SDW

CDW

SC

ΔFS

δS

wave vector [r.l.u]

δC 1/4

result 5: doping dependence

δc~ΔFS increases as x→0. consistent with experiments in YBCO, Bi2212, Hg1201

W. Tabis, et al., arXiv:1404.7658.

CDW: Q=(δc,0)

result 6: functional RG(+cRPA) nematic CDW Qc≈(ΔFS, 0) is obtained by functional-RG method

p =0.1 T =0.02eV U =4.4eV V =1.1eV Λ0=0.5eV 64patch

The obtained CDW form factor is different from that given by the SC-VC.

Importance of AL-VC is confirmed by fRG method.

𝜒↓𝑑↑𝐶 (𝒒)

𝜒↓𝑦↑𝐶 (𝒒)

qx

qx

qy

qy

Summary The VC gives orbital-spin "multimode fluctuations"

in Fe-pnictides and high-Tc cuprates:

⇒phase diagram with orbital orders and superconductivity

→orbital order, enhancement of 1/C66 and χRaman

→ LaFeAsO1-xHx: C2, C4 structure transitions

S.Onari, Y.Yamakawa and HK, PRL (2014)

HK and Y.Yamakawa, PRL (2014) in press

→ LiFeAs: The largest SC gap on the smallest hole-pockets T. Saito et al, PRB (2014) in press

1. Fe-pnictides

3. ruthenates →Sr3Ru2O7: nematic orbital order

→Sr2RuO4: orbital+spin mediated TCS

Tsuchiizu et al, PRL (2013)

Tsuchiizu et al, arXiv(2014)

2. cuprates →intra-unit-cell (orbital-antiphase) CDW

Yamakawa and HK, arXiv(2014)

Thank you!