anomalous)quantum)cri/cality)linking)an/ferromagne/sm)and ...nqs2011/archive/presenfiles/...1....

Anomalous quantum cri/cality linking an/ferromagne/sm and superconduc/vity in organic metals

C. Bourbonnais NQS2011, Kyoto, November 2011

A. Sedeki N. Doiron-‐LeyraudS. René De Cotret L. Taillefer

Coll.

P. Auban-‐SenzierD. Jerome

Outline

1. Organic conductors: what are they ?

2. (TMTSF)2PF6 : paradigm for AF -‐ SC proximity

3. Normal state : anomalous quantum cri/cality

4. RG to Quasi-‐1D electron gas :

AF-‐SC interference and ext. quantum cri/cality

5. Conclusion

PeryleneDMTCNQ

Donors Acceptorse- e-

BEDT-TTF

1. Introducing the physics of quasi-one-dimensional organic conductors 9

FIGURE 7. Variation of the Peierls critical temperature as a function of pressure in TTF-TCNQ. After refs. [9, 4]

FIGURE 8. TMTSF donor molecule with the profile of atomic orbitals that enter in the HOMO. The cousin sulfur basedmolecule TMTTF has a similar form.

TMTSF or TMTTF

Etc.

1. Introducing the physics of quasi-one-dimensional organic conductors 11

0.05 0.10 0.15 0.20 0.25 0.30

0.2

0.4

0.6

0.8

1.0

1.2

1.4TPTP

0

! TP0t⊥2

CDW

Metal

FIGURE 10. Variation of the normalized Peierls mean-field critical temperature as the nesting deviations parameter t⊥2.A similar variation can be found in the RPA treatment of the Metal-SDW transition for a wrapped Fermi surface withnesting deviations (see text).

FIGURE 11. Side view of the crystal structure of the Bechgaard (and Fabre) salts (TMTSF)2X [(TMTTF)2X] series.

The solution leads to the Peierls temperature as a function of nesting deviations is shown in Fig. 10. Comparing

with Fig.9-b, nesting frustration can mimic the pressure dependence and can then provide a reasonable mechanism

for the suppression of the Peierls instability.

4 The Bechgaard and Fabre salts series

4.1 The Bechgaard salts

While TMTSF-DMTCNQ did not show superconductivity up to 13 kbar, its stable metallic state gave the

necessary impetus for chemistry to further explore the synthesis of materials based on the promising TMTSF

molecule. So at nearly the end of 1979, Bechgaard et al.[15], introduced a new series of one-chain cation radialsalts, the (TMTSF)2X, where the choice of the small inorganic (radical) ion X= PF

−6 , AsF

−6 , NO

−3 , . . ., leads to

a series of isostructural compounds, soon christened as the Bechgaard salts series (Fig. 11). The zig-zag stackingof the TMTSF molecules creates cavities for the anions which together with the triclinic structure favors a slight

dimerization of the organic stacks. At ambient pressure the metallic character is well marked at room temperature

and for compounds like X= PF−6 and AsF

−6 with centro-symmetrical anions, it carries on for temperature as low

as 12K or so where a sharp metal-insulator transition occurs – the case of (TMTSF)2PF6 is shown in Fig. 12.

Initially believed as a Peierls phase transition, X-ray experiments fail to detect any lattice superstructure [13].

The insulating state was quickly found to be the result of a magnetic superstructure, a spin-density-wave (SDW)

state. The SDW state will be discussed in more details below. In parallel pressure studies were undertaken on the

(TMTSF)2PF6 compound by Jerome et al., at Orsay [17]; the insulating state was found to be rapidly suppressedand ultimately giving rise to superconductivity at Tc = 0.9K under 12 kbar of pressure (Fig. 13). The zerofield phase diagram of the first organic superconductor (TMTSF)2PF6 is shown in Fig. 14. A comparable phase

diagram was found for (TMTSF)2AsF6 with a similar centro-symmetrical anion [18]. Substituting PF6 with the

non centrosymmetrical anion ClO4 yielded first ambient pressure organic superconductor (TMTSF)2ClO4 below

(TMTSF)2X

Bechgaard salts Series: one-‐chain ca/on-‐radical salts

X = PF6 , AsF6 , ClO4 , ..... (radicals, δ=0.5e)

Quasi-‐1D Fermi surface

Γ

1 hole/2 mol + weak dimeriza/on ~ ‘1/2-‐filled’

band

~ 12 KK. Bechgaard et al., Solid State Comm. 33, 1119 (1980)

Resis/vity vs T

P= 1bar

P=12 kbar

D. Jerome et al., J. Phys. Lee. 41, L95 (1980).

AF -‐ SC proximity

D. Jerome, H. J. Schulz, Adv. Phys. 31, (1982)

N. Doiron-‐Leyraud, P. Auban-‐Senzier et al., Phys. Rev. B 80, 214531 (2009)

Normal phase: NMR T−11

S. Brown et al., 2008

F. Creuzet et al., J. Phys. Lee. 45, (1984);Also,

Synthe/c Metals 19, 277 (1987);

Wu et al., PRL 94, 97004 (2005)

T−11 ∼T

T + Θ

Curie-‐Weiss enhancement

: probing spin fluctuations

T−11 ∝ T (FL)

(T1T )−1 ∼ ξ2 (d = 2) ξ ∼ (T + Θ)−1/2

Anomalous T−11 enhancement

No Fermi liquid recovery T−11 ∝ T (Korringa law)

Y. Kimura et al., PRB 84,045123 (2011).

(TMTSF)2PF613C NMRT−11 CW enhancement vs Tc

(T1T )−1 = C0 +C

T + Θ

SDW fluct. correlated to pairing (Tc )

C vs Tc :

N. Doiron-‐Leyraud, P. Auban-‐Senzier et al., Phys. Rev. B 80, 214531 (2009); EPJB, 2010 78, 23 (2010); P.Auban-‐Senzier et al., J. Phys: Cond. Mat. 23,

345702 (2011).

Anomalous scaeering: linear resis/vity

ʹ′

AT + BT2 ~T 2

ρ(T ) ≈ AT + BT 2

ʹ′ Fermi Liquid T2 at Pc ʹ′

Anomalous scaeering: linear resis/vity

-‐ Link between scaeering and pairing

ρ(T ) = AT + BT 2

N. Doiron-‐Leyraud, P. Auban-‐Senzier et al., Phys. Rev. B 80, 214531 (2009)

FLSDW

PPc

Quantum Cri/cality and the phase diagram

FLSDW

Pc PSCd

Quantum order parameter (SDW) fluct.

NFL → FL at P > Pc at T ≠0

P’c

NFL

NFL

NFL → FL at P > P’cRole of SC ? Excita/ons ?

Shio in the QCP paradigm

(e-‐h + e-‐e)

T T

Landau-‐Ginzburg-‐Wilson paradigm

T−11 , ρ

T−11 ∼ T ρ ∼ T 2

Hertz, Millis (1976,1993)Moriya, Ueda (1990,2002), Abanov et al., (2003)

ρ ∼ aT + bT 2

(T1T )−1 ∼ (T + Θ)−1

(e-‐h)

ρ ∼ T 2

SDW modula/on wave vector q0 (NMR) : the best calculated nes/ng vector

L. Ducasse et al., J. Phys. C. Solid State Phys. 18, L947 (1985).

Mechanism for SDW order: Nes/ng of the Fermi surface + repulsive interac/ons

Mechanism of the transi/ons -‐ SDW state

T. Takahashi et al., J.Phys. Soc. Jpn 55, 1364 (1986). J. M. Delrieu et al., J. Physique 47, 839 (1986).

13C NMR : M. Misawa et al., (2010)

q0 ≈ (0.5 a*, 0.24 b*)

From 1H NMR lineshape

Quasi-‐1D electron gas model : SDW -‐ SC phases

quasi-‐1D Fermi surface

V.J. Emery et al., Phys. Rev. Lee. (1982))

g1 ↔ χσ

g3 ↔ ∆D

g2 ↔ TSDW

(≈ 0.2.....0.4)

(≈ 0.02...0.04)

(≈ 0.5....0.7)

g1, g3

g2

q0

⊥

q0

e-h pairing q0

⊥

e-e pairing

⊥

χ0P (0, T ) ∼ lnEF

max{T, 4t�⊥}

Many-‐body physics of the q-‐1D electron gas (interference of pairings)

χ0C(0, T ) ∼ lnEFT

DW-‐Cooper pairing mixing present at every order

etc.P C P

e-h ↔ e-e

P C

5

∂! Σ+ = + + · · ·

∂! = +

+ + · · ·

∂! = + · · ·

FIG. 3: Flow equations for the one-particle Matsubara self-energy, Σ+, the g1,2 (open square) and Umklapp g3 (fullsquare) scattering amplitudes for right (continuous line) andleft (dashed line) moving electrons. The crossed and slashedlines refer to the high energy interval and outer energy shellrespectively (permutations between crossed and slashed lineare not shown).

∂!z⊥(k̄⊥) = z(k̄⊥)−1 1

2

∫∫d k′⊥2π

d q⊥2π

×{(

g2(k̃⊥1)g1k̃⊥1) − g22(k̃⊥1) − g

21(k̃⊥1)

)I ′1(k̃⊥1, iων)

−(g22(k̃⊥2) + g

21(k̃⊥2) − g2(k̃⊥2)g1(k̃⊥2)

)I ′2(k̃⊥2, iων)

−(g23(k̃⊥3) + g

23(k̃⊥4) − g3(k̃⊥3)g3(k̃⊥4)

)I ′3(k̃⊥3, iων)

}

(11)

where ∂! ≡ ∂/∂$, and k̃⊥i are defined in (A4).The temperature dependent coefficients Ii and I ′i are

given in Appendix A. The integration of (10-11) is carriedout up to $ → ∞, which leads to the z and z⊥ factors attemperature T . It worth stressing that both two-loop andone-loop diagrams are calculated using free propagators(see Appendix A).

IV. RESULTS

A. Quasi-particle weight

An important quantity entering in the description ofquasi-particles and that can be extracted from the one-particle self-energy is the ‘angle resolved’ quasi-particleweight, z(k⊥) ≡ z

(kF (k⊥), iων=0

), defined on the Fermi

surface. It is obtained from the solution of Eq. 10 atων=0 = πT . This variation of z(k⊥) on the FS as afunction of k⊥ and T in the SDW and SC parts of thephase diagram (Fig. 2) is shown in Fig. 4.

0

0.2

0.4

0.6

−π −π/2 0 π/2 π

k⊥

0.8

t′⊥ = 26.8K

10

5

2

0.8

0.9 T [K] = 300

100

(b)

0

0.2

0.4

0.6

z(k ⊥

)

−π −π/2 0 π/2 π

k⊥

12

20

16

14

0.8

0.9

z(k ⊥

)

t′⊥ = 25.0K

T [K] = 300

(a)

100

FIG. 4: Variation of the quasi-particle weight on the Fermisurface as a function of k⊥ at different temperatures. (a):SDW (t′⊥ = 25 K < t

′∗

⊥, TSDW ! 12 K); (b): SC (t′

⊥ =26.8 K > t′∗⊥, Tc ! 0.8 K).

In the SDW region at high temperature, namely wellabove the scale of one-particle transverse coherence,TX ∼ t⊥, where the system is essentially 1D, z(k⊥) dis-plays little minima at k⊥ = ±π/2. According to (4), thet⊥ part of the spectrum vanishes at those points, whichfrom a perturbation viewpoint of t⊥ implies that 1D ef-fects are the strongest there. Such a high temperaturemodulation, thought small, agrees with earlier investiga-tions based on perturbative and mean field treatments oft⊥18–24, which found the same location for the the spec-tral weight minima on the FS. Above TX , Fig. 5 showsthat z(k⊥) ∼ T α decays as a non universal power law intemperature

(α ∼ O(g2)

), in accordance with the sum-

mation of next-to-leading (two-loop) logarithmic singularself-energy diagrams of Fig. 3 in the limit of 1D electrongas model.30,38,39(dashed line of Fig. 5).

As T goes below TX , the influence of t⊥ becomesclearly non perturbative. The temperature decay of z,thought still present, becomes less rapid than the powerlaw above TX ; an indication of weakening of the two-loopsingularity and modified Cooper and Peierls channels in-terference by the coherent warping of the FS. connectionwith marginal Fermi liquid.

Moreover, the position of minima in z(k⊥) graduallyshifts to k⊥ = ±π/4 and k⊥ = ±3π/4 as the details ofthe Fermi surface becomes coherent. This shift resultsfrom the nesting condition of the whole spectrum on theFS,

E+(kF + q0) = −E−(kF ) + δ(k⊥), (12)

where δ(k⊥) = 4t′⊥ cos 2k⊥. The minima coincide withthe loci, k⊥ = ±π/4, and ±3π/4 on the FS whereδ(±π/4) and δ(±3π/4) vanish and perfect nesting con-ditions prevail. Conversely, at k⊥ = ±π/2, 0, and ±π,δ(k⊥) = ±4t′⊥, and deviations reach their maximum. Asthe temperature comes close to TSDW, z(k⊥) enters in thecritical SDW regime and falls off toward zero due to sin-

C P

P

P

g1,2

g3

k-‐resolved flow of scaeering amplitudes

Instability lines vs nes/ng altera/ons (‘pressure’)

15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

SDW fluct. induce d-‐wave pairing (Interference)

Singulari/es↔ T scales for instabili/es via

Scaling theory (RG) of both pairing channels

χSDW(T ; g2, g3)χSCd(T ; g1, g2){

R.Duprat & C.B., Eur. Phys. J. B, 21, 219(2001)C.Nickel et al., Phys. Rev. Lee. 95, 247001 (2005) C.B. & A. Sedeki, PRB 80, 085105 (2009)

Normal phase: extended Curie Weiss regime15 20 25 30

⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

C. B. and A. Sedeki, PRB 80, 085105 (2009)

-‐ d-‐Cooper pairing boosts (interferes construc/vely with) SDW

-‐ Enhancement Fits a Curie-‐Weiss law

15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

χSDW(q0) ∼ (T + Θ)−1

Impact on the normal phase : NMR

C. B. and A. Sedeki, PRB 80, 085105 (2009)

15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

S. Brown et al., (2005-‐2008)

N. Doiron-‐Leyraud et al., Phys. Rev. B 80, 214531 (2009); EPJB (2010)

RG

T−11 ≈ C0T + CT

T + ΘP > Pc

5

∂! Σ+ = + + · · ·

∂! = +

+ + · · ·

∂! = + · · ·


∂!z⊥(k̄⊥) = z(k̄⊥)−1 1

2

∫∫d k′⊥2π

d q⊥2π

×{(

g2(k̃⊥1)g1k̃⊥1) − g22(k̃⊥1) − g

21(k̃⊥1)

)I ′1(k̃⊥, iων)

−(g22(k̃⊥2) + g

21(k̃⊥2) − g2(k̃⊥)g1(k̃⊥2)

)I ′2(k̃⊥, iων)

−(g23(k̃⊥3) + g

23(k̃⊥4) − g3(k̃⊥3)g3(k̃⊥4)

)I ′3(k̃⊥, iων)

}

(11)

where ∂! ≡ ∂/∂$, k̃⊥i and k̃⊥ are defined in (A4) and(A7), respectively.

The temperature dependent coefficients Ii and I ′i aregiven in Appendix A. The integration of (10-11) is carriedout up to $ → ∞, which leads to the z and z⊥ factors attemperature T . It worth stressing that both two-loop andone-loop diagrams are calculated using free propagators(see Appendix A).

IV. RESULTS



(kF (k⊥), iων=0


surface. It is obtained from the solution of Eq. 10 atων=0 = πT . This variation of z(k⊥) on the FS as a

0

0.2

0.4

0.6

−π −π/2 0 π/2 π

k⊥

0.8

t′⊥ = 26.8K

10

5

2

0.8

0.9 T [K] = 300

100

(b)

0

0.2

0.4

0.6

z(k ⊥

)

−π −π/2 0 π/2 π

k⊥

12

20

16

14

0.8

0.9

z(k ⊥

)

t′⊥ = 25.0K

T [K] = 300

(a)

100


′∗

⊥, TSDW ! 12 K); (b): SC (t′

⊥ =26.8 K > t′∗⊥, Tc ! 0.8 K).

function of k⊥ and T in the SDW and SC parts of thephase diagram (Fig. 2) is shown in Fig. 4.


(α ∼ O(g2)





E+(kF + q0) = −E−(kF ) + δ(k⊥), (12)

where δ(k⊥) = 4t′⊥ cos 2k⊥. The minima coincide withthe loci, k⊥ = ±π/4, and ±3π/4 on the FS whereδ(±π/4) and δ(±3π/4) vanish and perfect nesting con-ditions prevail. Conversely, at k⊥ = ±π/2, 0, and ±π,δ(k⊥) = ±4t′⊥, and deviations reach their maximum. As

Two-‐loop RG : one-‐parCcle self-‐energy

One-particle spectral quantities : z(k⊥), A(kF (k⊥), ω), m∗, τ−1k⊥ , ...

Quasi-‐par/cle weight vs T and ‘pressure’15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

0

0.2

0.4

0.6

0.8

1

z(k

⊥=0)

0 2 4 6 8 10

T [K]

0

0.2

0.4

0.6

0.8

1

z(k

⊥=π/2)

0 2 4 6 8 10

T [K]

8

0

0.2

0.4

0.6

0.8

1

z(k

⊥)

0 2 4 6 8 10

T [K]

t′⊥[K] T c[K] g(0)35.0 0.12 0.0232.0 0.23 0.0329.0 0.45 0.08

27.7 0.62 0.1726.8 0.80 0.4026.0 1.07 1.33

k⊥ = 0 (•), π/2 (◦)

t′⊥[K] T c[K] g(0)

FIG. 6: Low temperature dependece quasi-particle weightz(k⊥) for different antinesting parameters t

′⊥ for k⊥ = 0 (full

circles), and π/2 (open circles). The continuous lines corre-spond to a fit to Eq. (18) of the marginal liquid theory atk⊥ = 0.

phase digram. We see from Fig. 5 that not too far fromt′∗⊥ at the intermediate t

′⊥ = 26.8K, z(k⊥) exhibits a sig-

nificative decay in the metallic state extending far abovethe critical domain, comprising the entire Curie-Weissregime of spin correlations. The amplitude of the z de-cay correlates with the one of SDW fluctuations in thenormal phase as t′⊥ is tuned away from t

′∗⊥. When t

′⊥ is far

upward, at 35 K, for instance, one has Tc = 0.12K ! T ∗c ,and a very weak temperature dependence is found for zabove Tc, which is very close to that of a Fermi liquid(Fig. 5).

If we now concentrate on the Curie-Weiss temperatureinterval, Tc

Normal phase quasi-‐par/cle scaeering rate 5

∂! Σ+ = + + · · ·

∂! = +

+ + · · ·

∂! = + · · ·


∂!z⊥(k̄⊥) = z(k̄⊥)−1 1

2

∫∫d k′⊥2π

d q⊥2π

×{(

g2(k̃⊥1)g1k̃⊥1) − g22(k̃⊥1) − g

21(k̃⊥1)

)I ′1(k̃⊥, iων)

−(g22(k̃⊥2) + g

21(k̃⊥2) − g2(k̃⊥)g1(k̃⊥2)

)I ′2(k̃⊥, iων)

−(g23(k̃⊥3) + g

23(k̃⊥4) − g3(k̃⊥3)g3(k̃⊥4)

)I ′3(k̃⊥, iων)

}

(11)

where ∂! ≡ ∂/∂$, k̃⊥i and k̃⊥ are defined in (A4) and(A7), respectively.

The temperature dependent coefficients Ii and I ′i aregiven in Appendix A. The integration of (10-11) is carriedout up to $ → ∞, which leads to the z and z⊥ factors attemperature T . It worth stressing that both two-loop andone-loop diagrams are calculated using free propagators(see Appendix A).

IV. RESULTS



(kF (k⊥), iων=0


surface. It is obtained from the solution of Eq. 10 atων=0 = πT . This variation of z(k⊥) on the FS as a

0

0.2

0.4

0.6

−π −π/2 0 π/2 π

k⊥

0.8

t′⊥ = 26.8K

10

5

2

0.8

0.9 T [K] = 300

100

(b)

0

0.2

0.4

0.6

z(k ⊥

)

−π −π/2 0 π/2 π

k⊥

12

20

16

14

0.8

0.9

z(k ⊥

)

t′⊥ = 25.0K

T [K] = 300

(a)

100


′∗

⊥, TSDW ! 12 K); (b): SC (t′

⊥ =26.8 K > t′∗⊥, Tc ! 0.8 K).

function of k⊥ and T in the SDW and SC parts of thephase diagram (Fig. 2) is shown in Fig. 4.


(α ∼ O(g2)





E+(kF + q0) = −E−(kF ) + δ(k⊥), (12)

where δ(k⊥) = 4t′⊥ cos 2k⊥. The minima coincide withthe loci, k⊥ = ±π/4, and ±3π/4 on the FS whereδ(±π/4) and δ(±3π/4) vanish and perfect nesting con-ditions prevail. Conversely, at k⊥ = ±π/2, 0, and ±π,δ(k⊥) = ±4t′⊥, and deviations reach their maximum. As

One-‐par/cle self-‐energy : τ−1(∝ ρ), z, ....

Within the Curie-‐Weiss domain:

C. B. and A. Sedeki, C. R. Physique 12, 532 (2011), A. Sedeki and C.B. (2011)

7

0.001

0.01

0.1

1

AA

−30 −10 0 10 30ωω

20100300

t�⊥ = 25 K

−30 −10 0 10 30ωω

20100300

t�⊥ = 25 K

−30 −10 0 10 30ωω

20100300

t�⊥ = 25 K

0.001

0.01

0.1

1

AA

−30 −10 0 10 30ωω

3100300

t�⊥ = 26.8 K

−30 −10 0 10 30ωω

3100300

t�⊥ = 26.8 K

−30 −10 0 10 30ωω

3100300

t�⊥ = 26.8 K

0.001

0.01

0.1

1

10

100

AA

−30 −10 0 10 30ωω

2100300

t�⊥ = 35 K

k⊥ = 0

−30 −10 0 10 30ωω

2100300

t�⊥ = 35 K

k⊥ = π/2

−30 −10 0 10 30ωω

2100300

t�⊥ = 35 K

k⊥ = π/4

FIG. 9: Low frequency dependence of the spectral weightof the one-particle self-energy at t�⊥ = 25, 26.8 and 35 K ondifferent regions of the Fermi surface

Fig. 9 for different t�⊥ and k⊥. In general, the size ofthe peak at the Fermi level grows as T decreases, whileits width in frequency is reduced; note that at the peak,

A(k⊥, 0) coincides with −2Σ��(k⊥, 0), which defines thescattering rate that will be discussed separately below

(Sec. IV C 2).

As the temperature decreases, its rounding effect onthe peak of the spectral weight is reduced and A(k⊥, ω)develops a cusp like singularity at small ω, as T is loweredwell below TX . The spectral weight follows the power lawform A(k⊥, ω) ∼ |ω|η−1, with the exponent η = insteadof η =? for a Fermi liquid. This ....

However, at the quantitative level, this occurs nonuni-

formly along the Fermi surface, which differs from whatis expected. At k⊥ = ±π/4 (or ±3π/4), for example, thepeak is slightly more pronounced and narrower despite

a steeper slope in Σ� at the origin (Fig. 8), and thus asmaller value of the quasi-particle weight (Fig. 5). This

behavior at zero frequency is with the one of the Imagi-

nary part ω = 0, as discussed earlier.

2 Effective mass and electron-electron scattering rate

In the superconducting sector t�⊥ > t�∗⊥, the k⊥− re-

solved scattering rate shows strong anisotropy on the

Fermi surface. The absolute maximum is found in the

longitudinal direction k⊥ = 0, with secondary maximataking place at k⊥ = ±π, namely where the edges of theopen Fermi surface cross the Brillouin zone in the per-

pendicular direction. These points markedly differ fromthe expected ‘hot’ spots at k⊥ = ±π/4 and ±3π/4, asdeduced from E(k) at the best nesting conditions forthe antiferromagnetic wave vector q0 = (2kF , π). Theanisotropy that is actually found results from the inter-

ference of electron-hole with electron-electron scattering,

FIG. 10: Temperature dependence of the singular part of theelectron-electron scattering rate at different t�⊥

0.1

0.2

0.3

0.4

0.5

0.6

aa

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

k⊥k⊥

26.827.227.7

3032

0

0.005

0.01

0.015

aa

3540

50

FIG. 11: Anisotropy of the linear coefficient of the electron-electron scattering rate at different t�⊥

which moves the maxima in the regions where the super-

conducting SCd order parameter or the gap is expected

to take its largest values below Tc.For all k⊥, however, τ

−1k⊥

shows the same anomalous

temperature dependence, which fits pretty well the non

Fermi liquid polynomial form

τ−1k⊥ ≈ a(k⊥)T + b(k⊥)T2, (15)

in the temperature interval Tc t�∗⊥, the k⊥− re-

solved scattering rate shows strong anisotropy on the

Fermi surface. The absolute maximum is found in the

longitudinal direction k⊥ = 0, with secondary maximataking place at k⊥ = ±π, namely where the edges of theopen Fermi surface cross the Brillouin zone in the per-

pendicular direction. These points markedly differ fromthe expected ‘hot’ spots at k⊥ = ±π/4 and ±3π/4, asdeduced from E(k) at the best nesting conditions forthe antiferromagnetic wave vector q0 = (2kF , π). Theanisotropy that is actually found results from the inter-

ference of electron-hole with electron-electron scattering,

FIG. 10: Temperature dependence of the singular part of theelectron-electron scattering rate at different t�⊥

0.1

0.2

0.3

0.4

0.5

0.6

aa

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

k⊥k⊥

26.827.227.7

3032

0

0.005

0.01

0.015

aa

3540

50

FIG. 11: Anisotropy of the linear coefficient of the electron-electron scattering rate at different t�⊥

which moves the maxima in the regions where the super-

conducting SCd order parameter or the gap is expected

to take its largest values below Tc.For all k⊥, however, τ

−1k⊥

shows the same anomalous

temperature dependence, which fits pretty well the non

Fermi liquid polynomial form

τ−1k⊥ ≈ a(k⊥)T + b(k⊥)T2, (15)

in the temperature interval Tc

On the nature of the normal state above Tc

15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

|t⊥ EF

| TTc| 1D:LL

∼∼ 1 K ∼ 100 K

. . .

∼ 3000 K

SDW

SCd

NFL

Interfering e-‐e & e-‐h pairings :

DisCnct electron liquid (NFL)

Extended QC region

Summary

RG of Q-‐1D electron gas at

T � t⊥, t�⊥Interfering Cooper pairing with SDW:

-‐ TSDW → Tct�⊥

-‐ Extended Curie-‐Weiss domain

-‐ Not a Fermi liquid at low T : T−11 , τ−1, z, ....

-‐ Extended quantum cri/cality

Linear resis/vity correlated with Tc

C-‐W SDW fluctua/ons ( ) vs Tc

SDW + SC dome under pressure

T−11

15 20 25 30⊥b(K)

100

102T(K)

SDW

SCd

t'

15 20 25 30⊥b(K)

100

102T(K)

Θ

SDW

SCd

t'

C-W

F.L. Ning et al., Phys. Rev. Lee. 104, 37001 (2010)

Ba(Fe1-‐x Cox )2As2

Y. Nakai et al., Phys. Rev. Lee. 105, 107003 (2010)

BaFe2(As1-‐x Px )2

75As

anomalous)quantum)cri/cality)linking)an/ferromagne/sm)and ...nqs2011/archive/presenfiles/...1....

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