Natural Optical activity of metals

Natural Optical activity of metals

V.P.Mineev

Commissariat a l’Energie Atomique Grenoble, France

Collaboration: Yu Yoshioka, Graduate School of Osaka University

Contains

• What is the natural optical activity

• Natural optical activity of metals

• Metals without inversion symmetry

• Some properties originating from the band splitting

• Current

• Gyrotropy current

• Gyrotropy conductivity and the Kerr angle

• Conclusion

What is the natural optical activity ?

Natural optical activity of metals

Kerr effect

Metals without inversion symmetry

k _y k _x

k _z

k _y k _x

k _z De Haas - van Alphen

Two band superconductivity

Large residual spin susceptibility at T=0

Current

Gyrotropy current

Gyrotropy conductivity and the Kerr angle

Conclusion

• There was found the current response to the electromagnetic field with finite frequency and wave vector in

noncentrosymmetric metal in normal and in superconducting states.

• The conductivity tensor contains a gyrotropic part responsible for the natural optical activity.

• As an example the Kerr rotation for the polarized light reflected from the surface of noncentrosymmetric metal with cubic

symmetry is calculated. The found value of the Kerr angle is

expressed through the fine structure constant and the ratio of

the light frequency to the spin-orbit band splitting. The result can

be used for the direct experimental determination of the band

splitting.

Hard and Interacting Sphere Suspensions.

Theory versus Experiment: Do we need a theory?

E.I. Kats²

Laue-Langevin Institute, Grenoble, France

L.D.Landau Institute for Theoretical Physics RAS, Moscow, Russia

18th June 2009

1To be published, J.Chem.Phys., (2009)

2Collaboration with A.Muratov, T.Narayanan, A.Moussaid, and acknowledgments to V.Lebedev

I Low limit size: when detailed knowledge of the internal degrees of freedom is not needed: 1 nm ;

I Upper limit size: particles should behave due to Brownian motion (Brownian displacementhhiin a gravity field g is∝ a particle diameterσ) :

hhi ' T

mg,K_B ≡1; πgρσ⁴

6 =T

It givesσ'1µm (for a brick, m=1 kg,hhi '10⁻²⁰cm).

I In time domain, time to move∝particle size τ ' ησ³

T

yields for a 1µm colloid in waterτ '1 s (for a brick it is about 10⁷ years!).

Various colloidal suspensions:

Figure:colloidal suspensions

For deionized suspensions of charged particles, the long-range order appears in extremely dilute dispersions,φ'0.005, whereas for hard spheres it occurs atφ'0.5.

Figure:Temperature control of particle volume fraction.

PNIPAM has a volume transition in which the network in the shell expelling water. Thus the effective volumeφ_eff can be adjusted through the temperature.

Phase Diagram of Hard Spheres

Figure:Phase Diagram of Hard Spheres.

I Other thanφcontrol parameters: aspect ratio, polydispersity but not the size of the sphere.

I φ_HCP ≡π/√ 18.

I Delay prior to nucleation∝to polydispersity.

Experimentally and numericallyφ_RCP is in a window from 0.60 to 0.68.

I Close packed implies that the spheres are in contact with one another with the highest possible coordination number on average. But increasing the degree of coordination, and thus, the bulk system density, comes at the expense of disorder. Thus, ”random” and ”close packed” are at odds with one another.

I Better defined notion is maximally random (or less ordered) jammed (MRJ), where a particle is jammed if it cannot be translated while fixing the positions of all of the other particles in the system (this eliminates ”rattlers”, particles without any contacts!).

I MRJ state to be the one that minimizes order parameter (e.g., scalar crystalline and bond orientational) among all jammed structures.

φ

and average kissing number Z versus aspect ratio

Clays are important examples of non-spherical particles.

Figure:prolate (circles), oblate (squares), biaxial (diamonds) ellipsoids

Crystal close packing shows no such singular behavior and almost independent of aspect ratio for small deviations from 1.

Isostatic system: Z =2f (frictionless particles) and Z =f +1 (strong friction). Number degrees of freedom f =3 for spheres, f =5 for uniaxial ellipsoids and f =6 for biaxial ellipsoids.

Suspensions are scaled-up version of atomic systems:

I A few names: Van der Waals, Einstein, Onsager, Debye....

I Colloidal dispersion with a particle concentration of about 10¹³cm⁻³has elastic constants of the order of about 10 dyn/cm², whereas in atomic solids with atomic density around 10²²cm⁻³, the elastic constants have a value around 10¹²dyn/cm².

I Atomic or molecular systems require several Kilobars pressure for observing structural phase transitions, whereas for colloids it is about 10⁻⁵bar .

I

Hard sphere colloids are genuine soft matter systems!

I

The next step beyond the ideal gas is the hard sphere suspension with entropy driven phase transition, like in Bose gas.

I

Liquid and crystal coexisting for volume

fractions between freezing φ

= 0.49 and

melting φ

= 0.545, and dynamic glass

transition at φ

' 0.56 − 0.58.

I Deborah number De (Old Testament: ”The mountains flowed before the Lord”):

De= τ_in τ_ex

I Practical criteria: viscosity larger than 10¹³Poise, or observation time larger than 100 s. There is very little we can do with to move up or down these criteria! Even then ergodic statistical mechanics can be applied within each individual component ifτcomponent t_obs τsystem I In scattering experiments

τex ∝ 1

D(k)k²; τin ∝ σ² D

and in the range kσ 1, De1. However, nothing is wrong and in this range one measures the short time limit of the self-diffusion coefficient D_s.

Typical time scales

I Solvent relaxation time (solvent degrees of freedom relax to an equilibrium distribution, constrained by a

non-equilibrium configuration of the much slower particles):

10⁻¹³s − 10⁻¹²s

I Longitudinal (sound waves) hydrodynamic time τs 'σ/vs ∝10⁻¹²s − 10⁻¹¹s.

I Shear viscous relaxation time τ_η = ρsσ²

η ∝10⁻¹⁰s − 10⁻⁸s

I Brownian relaxation time

τB ' 2ρp

9ρs

τ_η

I For t τBthe velocities have relaxed and only the particle positions remain as degrees of freedom relaxing by

diffusion

τI = σ²

D₀ ∝10⁻³s

{i} line box of total length L.

I Rescaling of all lengths by a factor 1+: l_i →(1+)l_i, and L→(1+)L. Because excluded volume is irrelevant in this transformation, only entropy of mixing, the reversible work δWtot associated with this processδWtot =−ρTδL, where ρ=N/L, andδL=L.

The same expansion can be done in two steps:

I (a) rod sizes are sequentially rescaled, one at a time l₁→(1+)l₁, then l₂→(1+)l₂, and so on;

I (b) the box size is expanded L→(1+)L.

I In a step (a) any rod that is expanded behaves as a confined wall (”piston”), therefore the work needed to rescale particle i is Pδli =Pli. Consequently the work needed to perform step (a) isδWa=PN

i=1Pδl_i ≡PδLp, whereδL_p ≡δLφ, and p stands for particles, and φ≡P

il_i/L is particle volume fraction. Furthermore, hli=φ/ρ.

I For a step (b) the workδWb =−PδL, and the total work δWtot =δWa+δW_b=PφδL−PδL. From the other hand by its definitionδW_tot =−ρTδL, therefore the equation of state

P = ρT 1−φ

For d=3 (equal sizeσ hard spheres) one still has δW_tot =δW_a+δW_b, withδW_tot =−ρTδV and δWb =−PδV .

I Each particle is surrounded by an excluded volume of diameter 2σ where no other particle’s center of mass may be found. Thus the work needed to expandσintoσ+δσ

δW_σ→σ+δσ =ρTg(σ)δVsweep

whereρTg(σ)is an entropic (kinetic) force per unit area felt by a given particle (g(r)gives probability to find a particle at the distance r ) andδVsweep is the volume change of the excluded volume sphere

δV_sweep =2^d−1S_d(σ)δσ 2

I Summing over all particles δWa=X

i

δWσ →σ+δσ=ρTg(σ)2^d−1δVp

whereδV_p =φδV .

I Combining everything together the equation of state P

ρT =1+2^d−1φg(σ)

I For 1d

g(l) = 1 1−φ

I All virial coefficients for 1d Tonks and Jepsen liquid are unity.

I For 3d the equation of state coincides with that derived by Leibowitz (and he has calculated 5 virial coefficients).

Qualitative speculations

I Spherically symmetric pairwise potential V(r) =Vsr(r) +V_lr(r), where

V_sr(r) =∞, r ≤1, V_sr(r) =0, r >1. (σ≡1).

I The long range part V_lr is a generalized Kac potential (M.Kac studied a model with pure attractive long range potential)

V_lr =−aγ_a³exp(−γar) +rγ_r³exp(−γrr) Both the attractions and repulsions are long ranged, γ_a⁻¹, γ_r⁻¹1, but the repulsions are longerγr ≤γa1, and the both are weak,_rγ_r³, _aγ_a³T .

Figure:Potential V(r)(supplemented by short range attraction) versus interparticle separation.

I Short range part produces rather deep minimum, while long range tails due to frustrating competition between attractions and repulsions might yield to a minimum about 100 times smaller.

I Potential determines the phase behavior through its integral over space, it is multiplied by a factor r²which gives for r '10 the same factor 100. The fact is that there are many more particles of separations between say 11 and 12 particle diameters than between 1 and 2 ( Similar to weak crystallization spirit).

B₂=−1

2 d³r[exp(−V(r)/T)−1]

and for our potential

B₂= 2π

3 −4πa

T −r

T

under asymptotically (atγ_r, γ_a→ 0) exact approximations

I aγ_a³, rγ_r³T ;

I region of integration of V_lr is extended down to r =0, whereas the hard core actually cuts it off below r =1.

I Introducing reduced energy≡a/T and the ratio α=_r/_a

B₂= 2π

3 −4π(1−α)

I When no repulsion, i.e,α=0 in the limitγ→0 the free energy is calculated exactly (Penrose, Leibowitz, Van Kampen)

F

NT = F_HS

NT −4πρ

whereρ=N/V , and F_HS(hard sphere part) is a function only of density and it is convex up to fluid - solid transition.

I Forα6=0 similarly (but even in the limitγa→0 approximately !)

F

NT = F_HS

NT −4π(1−α)ρ and corresponding pressure

P

T = P_HS

T −4π(1−α)ρ²

increasing function of density.

I Ifa> r, thenα <1 and long range tail contribution to P is negative. When T decreases this term becomes more and more negative (via∝1/T ) until the fluid phase is

unstable separating into dilute and dense fluids.

I However ifr > a, the pressure is convex at all

temperatures. Only one fluid phase! Another 1-st order phase transition (liquid - solid) can occur.

Structure factor S(q) =1+ρh(q), where indirect correlation function h(r) =g(r)−1.

I Approximations are easier for direct correlation function c(q)related to h(q)by Ornshtein - Zernike equation

h(q) = c(q) 1−ρc(q) or

S(q) = 1 1−ρc(q)

I S(q)must be positive and finite for all q. It diverges if 1−ρc(q) =0.

I For our potential (a steep short range repulsion plus a weak tail) c(r) =c_sr(r) +c_lr(r), for r <1, c(r)is dominated by hard sphere interaction, whereas for r 1 it is close to

−V_lr/T .

PY

c_lr(q) = 8π

[1+ (q/γa)²]² − 8πα [1+ (q/γr)²]² and

S(q) = 1

1−ρ(cPY(q) +c_lr(q))

Bounded and positive definite interactions

I Potentials do not diverge at the origin and no attractions at all: V(r)≡f(r/σ), where V varies fromat r =0 to zero at r → ∞. f(x)does not have to be analytic!

I Dimensionless temperature t ≡T/, and density φ=πρσ³/6=πρ/6.¯

I The key idea is that at high densityρσ³1 the average interparticle distance a'ρ^−1/3becomes vanishingly small (σ), i.e., the potential is extremely long range.

I In this limit and without short range excluded volume interaction c(r) =−V(r)/T (since c(r)∝δ²F/δρ²).

I Thus

S(q) = 1

1+ ¯ρt⁻¹f(q)

I It has a monotonic decay from the value f(q=0)to zero at q → ∞(M -potentials);

I It has oscillating behavior at q→ ∞, attaining necessary negative values for certain range of q (O potentials).

I Let q^∗ is the value of q at which f(q)attains its minimum negative value. It implies a maximum of S(q)at q^∗and this maximum becomes a singularity at the spinodal line

¯

ρt⁻¹|f(q^∗)|=1

System must reach a crystalline state. If the Fourier transform f(x)has negative component, then an increase in temperature can be compensated by an increase in density. Thus S(q^∗)will have a divergence at all T .

Systems with O potentials freeze at all

temperatures!

Systems with M potentials:

I S(q)is a monotonic function of q at high densities, and one can always find a temperature high enough, so that mean field assumptions are granted and freezing is impossible at such temperatures.

I That does not imply that such systems do not freeze at all.

One merely has to go to low enough temperature and density, where interaction is much larger than T . Then the system will display a hard sphere type of freezing.

I An upper freezing temperature tu must exist for M

potentials, implying that system must remelt at t <t_u upon increasing of the density.

In real space the OZ equation h(x) =c(x) +

Z

d³yh(y)c(x −y)

The direct correlation unction tends to zero with increasing x much more rapidly than the indirect correlation function.

I The PY closure equation

c(x) = (1−exp(V(x)/T))g(x)

For the PY closure c(x) vanishes exactly outside the range of V(x).

I Qualitative basis is in representation of multi-particle distribution function nsas one-particle density n(y)under the imposition of suitable external potential

n_s(y,x₁, ....x_s−1)

n_s−1(x₁, ...x_s−1) =n(y|U) where U(x) =V(x,x₁) +....+V(x₁,x_s−1)

PY - continuation:

I To derive it we consider n(y|U)exp(U(y)/T)as a functional of n(y|U)as U is changed from 0 to its final value. Performing Taylor expansion, taking U(y) =V(y,x), and truncating the expansion at first order, after some algebra, we end up with the PY closure relation:

I

c(x) = (1−exp(V(x)/T))g(x)

I In words: spatial correlations inρin two volume elements have a direct and an indirect component. In the direct component onlyρfluctuations in the two volume elements are considered, whereas the fluctuations in all the

surrounding volume elements are kept fixed. The indirect contribution is added when the fixation is released:

g_total =c+g_indirect

τ(r) =1+n

r<σ

τ(r⁰)d³r⁰−n

r⁰<σ;|r−r⁰|>σ

τ(r⁰)τ(r −r⁰)d³r⁰ Wertheim, Baxter foundτ(r)at r < σ(where it coincides with c(r)), and then g(r)at r > σ, where it coincides withτ(r).

I Wertheim, Baxter solution:

−c(x) =α+βx+γx²+δx³

I

α= (1+2φ)²

(1−φ)⁴ ; β=−6φ(1+0.5φ²) (1−φ)⁴ γ =0; δ = φ(1+2φ)²

2(1−φ)⁴

Similar strategy for interacting hard spheres:

V(r) =

∞,r < σ

V_a(r) +V_r(r), σ <r <b 0,r >b

I To findτ(r)at r < σ, i.e., to solve Ornshtein-Zernike equation with the PY closure relation:

τ(x) =1+n Z

d³y(e^−βV(y)−1)τ(y)(e^{−βV(x−y)}τ(x−y)−1),

I To find g(r)at r >b;

I To interpolateτ(r)fromσto b.

I Depletion: V_a(r)≡0, for r >b_aand for r ≤b_a: V_a(r) =V_a⁰(b_a−r)²(2b_a+r)

(ba−σ)²(2ba+σ)

I Yukawa (attraction or repulsion):

V_r =±V_r⁰expκ(σ−r) r where

V_r⁰= Q (1+κσ)²

Nature is on our side:

The functionτ(r)is a smooth function for r <b, and approaches to 0 at large r

0.5 1 1.5 2 r

5 10 15 20

Τ@rD

0.2 0.4 0.6 0.8 1 1.2 r

-15 -12.5 -10 -7.5 -5 -2.5 2.5

cHrL

In Fourier space a jump at x ≡(r/σ) =1 corresponds to 1/q behavior at q → ∞:

c(q) = 4π q

Z

dxx sin(qx)c(x)

To get better convergence for b<x τ(x)−1= 4π

8π³x Z

dq q sin(qx)h(q) = 1

2π²x Z

dq q sin(qx) c(q) 1−n c(q) = 1

2π²x Z

dq q sin(qx)

c(q) + n c²(q) 1−n c(q)

= c(x) + 1

2π²x Z

dq q sin(qx) n c²(q) 1−n c(q) = n

2π²x Z

dq q sin(qx) c²(q) 1−n c(q) .

τ(x) = ρ 2π²x

Z

dqq sin(qx) c²(q) 1−ρc(q)

I Pressure P = 4πρT

6 σ³τ(σ)− Z b

σ

dr(exp(−V(r)/T)−1)∂r(r³τ(r))

!

I Coordination number N =4π

Z b

σ

drr²g(r)

Note of caution: Two ”routes” connecting P and g (r )

I The virial route

P =ρT − ρ² 6

Z ∞ 0

rdV

dr g(r)4πr²dr

I The compressibility route Tdρ

dP =1+ρ Z ∞

0

h(r)4πr²dr

I It can be verified that B₂and B₃in the PY theory do not depend on the route.

I VdW pressure holds at low densities

P= T

(1/ρ)−σ

I For a weak potential (V <T ),τ(r)has almost the same polynomial form as for HS. For a relatively short ranged (but not small!) V >T , the modification of the HS ansatz is essential only for relatively small r 'b−σ.

I This suggests the ansatz

τ(r) =c₀+c₁r+c₃r³+τs(r) withτs =c₄/(x+c₅)²growing at x →0.

I Then the PY closure allows to calculate c(r), then g(r)for r >b, and eventually theτ(r)is well defined for all r .

Illustrations how it works:

0.5 1 2 5 10 20

qR 0.0001

0.001 0.01 0.1 1 10

IHqRL

Figure:HS with depletive attraction

5 10 15 20 qR 10

100 1000 10000

IHqRL

HS with depletive attraction

0.5 1 2 5 10 20

1 qR 10 100 1000 10000

IHqRL

Sticky HS

1 1.5 2 3 5 7 10 15 qR 10

100 1000 10000

IHqRL

Sticky HS

1 1.5 2 3 5 7 10 15

qR 10

100 1000 10000

IHqRL

1 1.5 2 3 5 7 10 15 qR 10

100 1000 10000

IHqRL

Not everything is so unclouded!

I Increasing polymer concentration (e.g., up to 46 mg/ml) we increase depletion potential up to V_a0 '6T ;

I However, already for V_a0 '3.7T , S(q→0)becomes very large, g(σ)'32 and N '3.55;

I This means that there are strong fluctuations as a

precursor of the two phase state, consisting of liquids with different density;

I OZ equation with PY closure cannot describe this situation;

I It can be done by adding Debye-Bueche function I(q)→F(q)

S(q) + Dl³ (1+ (ql)²)²

where l is a characteristic cluster size, D describes the normalized contribution from the clusters to the scattering intensity, and volume fractionφin the aggregates is larger than in the bulk liquid.

0.5 1 2 5 10 20 qR 0.0001

0.001 0.01 0.1 1 10

IHqRL

Figure:Without Debye correction.

The same but with Debye corrections included

0.5 1 2 5 10 20

qR 0.0001

0.001 0.01 0.1 1 IHqRL

0.5 1 2 5 10 20 qR 10

100 1000 10000 100000.

1.´10⁶ IHqRL

The same but with Debye corrections included

0.5 1 2 5 10 20

qR 10

100 1000 10000 100000.

1.´10⁶ IHqRL

0.1 0.2 0.3 0.4 Φ 3.5

4 4.5 5 -U0k_BT

Figure:Line of limiting stability of liquid phase: blue - b=1.13, red - b=1.25

Optional slide: Where to go further on:

I Close to the freezing transition liquids are more complex than suggested by the pair correlation function!

I Structural shape factorζ=C²/4pS, relied on a Voronoi analysis, C is the circumference, and S is the surface area of the Voronoi cell of each particle (Voronoi cell of a particle a consists of all points closer to a than to any other sites).

I ζ reflects the structural changes close to freezing in more detail than the changes in g(r).

CONDUCTORS TO CONDUCTING POLYMERS

S. Brazovski & N. Kirova CNRS - Orsay, France

Conducting polymers 1978-2008.

electrical conduction and optical activity.

Modern requests for ferroelectric applications and materials.

Existing structural ferroelectricity in a saturated polymer.

Ferroelectric Mott-Hubbard phase and charge disproportionation in quasi 1d organic conductors.

Expectations of the electronic ferroelectricity in

conjugated modified polyenes.

2

Ferroelectricity is a rising demand in

fundamental and applied solid state physics.

Active gate materials and electric RAM in microelectronics, Capacitors in portable WiFi communicators,

Electro-Optical-Acoustic modulators, Electro-Mechanical actuators

Transducers and Sensors in medical imaging.

Request for plasticity – polymer-ceramic composites

but weakening responses – effective ε ~10.

Plastic ferrroelectrics are necessary in medical imaging – low weight :

compatibility of acoustic impedances with biological tissues.

One ferroelectric saturated polymer does exist - Poly(vinylidene flouride) PVDF :

• ferroelectric and pyroelectric,

• efficient piezoelectric if poled – quenched under a high voltage.

Light, flexible, non-toxic, cheap to produce Helps in very costly applications:

• ultrasonic transducers

• hydrophone probes, sonar equipment

• unique as long stretching actuator.

Can we have organic only, particularly polymer only ferroelectric ?

PVDF

substitutes polyethylene – saturated polymer

but: ε ε ε ε~10 – modest efficiency (compare to ε ε ε~500 – 15000 for inorganic FE) ε

Can we go wider, diversely, and may be better with conjugated polymers?

Can we mobilize their fast pi-electrons to make a better job than common ions?

« In the beginning was the Word, …

and without him was not anything made that was made » But was “organic supercondictivity” the only promised land?

Not quite : some of the profet’s visions actually imply a spontaneous electric polarization, hence they are FERROELECTRIC.

Drawing from the PRB 1964 It is a pyroelectric if N ≠ H

Later popular drawing (Sci. Am.) It must be a ferroelectric if R ≠ H also an illustration of

conjugated polymer

R=

phenyl

5

LED display and microelectronic chip made by Phillips Research Lab

Tsukuba, LED TV.

Ferroelectricity in conjugated polymers?

Where does the confidence come?

What may be a scale of effects ?

Proved by success in organic conducting crystals.

conterion

= dopant X

Molecule TMTTF or TMTSF

Arrows show displacements of ions X.

They follow and stabilize the electronic charge disproportionation.

Collinear arrows – ferroelectricity.

Alternating arrows – anti-ferroelectricity.

A single stack is polarized in any case.

Major polarization comes from redistribution of electronic density, hence amplification of polarizability ε by a factor of ( ω

p / ∆ ) ² ~10 ² giving even a background ε ~10 ³

Built-in dimerization of bonds - counterions against each second pair of molecules )

Spontaneous symmetry breaking – displacements of counterions,

nonequivqlence of sites

Spin degrees of freedom are split-off and gapless.

Charge degrees of freedom can be gapful:

chiral phase ϕ = ϕ (x,t) for fermions near +/-K _F :

Gap rigin: Umklapp scattering (Luther and Emery,Dzyaloshinskii & Larkin).

- +

- +

Uexp[i 2 ϕ ] : amplitude of the Umklapp scattering of electrons (-K _F, - K _F ) (+K _F ,+K _F ) is allowed here.

Momentum deficit 4K _F is just compensated by the

reciprocal lattice period. Contineous chiral symmetry lifting: arbitrary translations are forbidden on the lattice.

Amplitude U may have a phase α !

H~ ( h /4 πγ ) [v _ρ ( ∂ _x ϕ ) ² + ( ∂ _t ϕ ) ² /v _ρ ] - Ucos (2 ϕ -2 α )

• Hamiltonian degeneracy ϕ ϕ + π originates current carriers:

±π ^solitons with charges ± e , energy ∆

(= holon = 4K _F CDW discommensuration = Wigner crystal vacancy )

] 2 / exp[

~ ± ⁱ ϕ

Ψ _±

COMBINED MOTT - HUBBARD STATE

2 types of dimerization ⇒

2 interfering sources for two-fold commensurability

⇒ 2 contributions to the Umklapp interaction:

Site dimerization : H _U ^s =-U _s cos 2 ϕ (spontaneous) Bond dimerization : H _U ^b =-U _b sin 2 ϕ (build-in)

At presence of both site and bond types H _U = -U _s cos 2 ϕ -U _b sin 2 ϕ = -Ucos (2 ϕ -2 α )

U _s ≠ 0 α ≠ 0 ^phase ϕ = “mean displacement of all electrons”

shifts from ϕ =0 to ϕ = α , hence the gigantic FE polarization.

From a single stack to a crystal: Macroscopic FerroElectric ground state if the same α is chosen for all stacks,

Anti-FE state if the sign of α alternates - both cases are observed

Then electronic system must also adjust its ground state from α to - α .

Hence the domain boundary U _s ↔ -U _s requires for the phase soliton of the increment δ =-2 α

which will concentrate the non integer charge q=-2 α / π per chain.

φ =- αααα

φ φ φ φ = αααα

alpha- solitons are walls between

domains of opposite FE polarizations

They are on-chain conducting particles only above T _FE . Below T _FE they aggregate into macroscopic walls.

They do not conduct any more,

but determine the FE depolarization dynamics.

10 ReO

SbF

AsF

PF

Real part of dielectric constant of (TMTTF)

X salts

a second order phase transition described by the Curie law

A εεεε

= ---

|||| T- T

||||

P.Monceau F. N and S.B. Phys. Rev. Lett. 86 (2001) 4081

11

Frequency dependence of imaginary part of ε

Comparison of the ε′′ (f) curves at two temperatures near T

:

above - 105K and below - 97K.

T- dependence of relaxation time for the main peak:

Critical slowing down near T

,

Activation law at low T – friction of FE domain walls by charge carriers

Low frequency shoulder - only at T<Tc : pinning of FE domain walls ?

Landau- Khalatnikov

critical relaxation

12 Second harmonic generation λ ( ω )=1400nm

K. Yamamoto et al. JPSJ, 77 (2008) 074709

Problem of identification of the frozen polarization:

through anomalous optical activity - lack of inversion summetry

3 2 2

8 E E

W χ

π

ε ₊

= E

may exist only in case of inversion symmetry breaking

13

Realization: conjugated polymers of the (AB) _x type:

modified polyacetylene (CRCR’) _x

Lift the inversion symmetry, remove the mirror symmetry, do not leave a glide plane.

Keep the double degeneracy to get a ferroelectric.

SOILITONS WITH NONINTEGER VARIABLE CHARGES:

Orthogonal mixing of static and dynamic mass generations.

Realisation: modified polyacetylene (CRCR ′′′′ ) _x

Theories for solitons with variable charges: S.B. & N.K. 1981, M.Rice

2 2

in

ex + ∆

∆

=

∆

Joint effect of extrinsic ∆

ex and intrinsic ∆

in contributions to dimerization gap ∆ _.

∆ ex comes from the build-in site dimerization – inequivalence of sites A and B.

∆ in - from spontaneous dimerization of bonds ∆

in = ∆

b - generic Peierls effect.

2 1 2

0 2

1

2 2 2

1

2 1

, ∆ = ± ∆ − ∆

=

∆

∆

∂ +

∆

−

∆

∆ +

∆

∂

−

cnst

i K i

i Tr i

x x

θ Re∆ s (x) Im∆_s(x)

E ₀

θ θ

Nontrivial chiral angle 0<2 θ < π of the soliton trajectory corresponds to the

noninteger electric charge q= e θ / π

15

(AB)x polymer

0

2 2

i

e

+ ∆

∆

2 2

i

e

+ ∆

∆

−

- ∆

Solitonic intra-gap states

S=0

S=1/2

0

2 2

i

e

+ ∆

∆

2 2

i

e

+ ∆

∆

−

∆

Q e

∆

= 2 tan

∆ π

∆

= e

∆

Q 2 tan

π Special experimental advantage:

ac electric field alternates polarization

by commuting the bond ordering patterns, i.e. moving charged solitons.

Through solitons’ spectral features it opens a special tool of

electro-optical interference.

∆

WILL NOT be spontaneously generated – it is a threshold effect - if ∆

already exceeds the wanted optimal Peierls gap.

Chemistry precaution: make a small

difference of ligands R and R’

16

The necessary polymer does exist:

since 1999 from Kyoto-Osaka-Utah team.

By today – complete optical characterization,

indirect proof for spontaneous bonds dimerization via spectral signatures of solitons.

“Accidental” origin of the success

to get the Peierls effect of bonds dimerization:

weak difference or radicals – only by a distant side group.

Small site dimerisation gap provoke to add the bond dimerisation gap.

Still a missing link : no idea was to check for the Ferroelectricity:

To be tried ? and discovered !

17

Optical results by Z.V. Vardeny group:

Soliton feature, Absorption, Luminescence, Dynamics

Not a polaron, but spin

soliton ?

18

LESSONS and PERSPECTIVES

ππππ -conjugated systems can support the electronic ferroelectricity.

Effect is registered and interpreted in two families of organic crystalline conductors (quasi 1D and quasi 2D).

Mechanism is well understood as combined collective effects of Mott (S.B. 2001) or Peierls (N.K.&S.B. 1981) types.

An example of a must_be_ferroelectric polyene has been already studied (Vardeny et al).

The design is symmetrically defined and can be previewed.

Cases of low temperature phases should not be overlooked.

Solitons will serve duties of re-polarization walls .

19

1. Ferroelectric transition in organic conductors was weakly observed, but missed to be identified, for 15 years before its clarification.

2. Success was due to a synthesis of methods coming from

a. experimental techniques for sliding Charge Density Waves, b. materials from organic metals,

c. ideas from theory of conjugated polymers.

3. Theory guides only towards a single chain polarization.

The bulk arrangement may be also anti-ferroelectric –

still interesting while less spectacular. Empirical reason for optimism:

majority of (TMTTF) ₂ X cases are ferroelectrics.

4. …….

………..

13. High-Tc superconductivity was discovered leading by a “false idea”

of looking for a vicinity of ferroelectric oxide conductors.

Landau Days 2009

Asymptotic freedom in inflationary cosmology with a non-minimally coupled

Higgs field

A.Yu. Kamenshchik

University of Bologna and INFN, Bologna L.D. Landau Institute for Theoretical Physics, Moscow

June 22, 2009

A.O. Barvinsky, A.Yu. Kamenshchik and A.A. Starobinsky, Inflation scenario via the Standard Model Higgs boson and LHC,

Journal of Cosmology and Astroparticle Physics 0911 (2008) 021

A.O. Barvinsky, A.Yu. Kamenshchik, C. Kiefer, A.A. Starobinsky and C. Steinwachs,

Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field,

arXiv: 0904.1698 [hep-ph]

Content

1. Introduction

2. One-loop approximation 3. Renormalization group

4. Inflationary stage versus post-inflationary running

5. Numerical analysis

6. Conclusions and discussion

The task - the construction of a fundamental particle physics model accounting for an inflationary scenario in cosmology.

I A scalar field is very convenient for providing an inflationary stage of the cosmic expansion

I A self-interaction of a scalar field creates problems for inflation

I The inclusion of the non-minimal couplingξRφ² supplies us with an effective potential providing a slow-roll regime for the universe

I Due to quantum effects the early evolution of the universe depends not only on the inflaton-graviton sector, but is strongly effected by the particle content of the theory

I Main quantum effects are encoded in a special

combination of coupling constantsA -anomalous scaling

I The nature of an inflaton scalar field - could it be the Higgs boson ?

I Quantum effects and the renormalization group running

I Theasymptotical freedom effect for the anomalous scaling

I The cosmological model of inflation based on the non-minimally coupled Higgs boson looks as compatible with both : cosmological observations and particle physics bounds, but some details are not yet clear

2 2 4

|Φ|² =Φ^†Φ.

L_int =−X

χ

1

2λ_χχ²ϕ²−X

A

1

2g_A²A²_µϕ² −X

ψ

y_ψϕψψ.¯ Quantum one-loop correction to the potential is

X

particles

(±1)m⁴(ϕ)

64π² lnm²(ϕ)

µ² = λA

128π² ϕ⁴lnϕ² µ² +... .

A= 2 λ

ÃX

χ

λ²_χ+ 3X

A

g_A⁴−4X

ψ

y_ψ⁴

! .

In the context of quantum cosmology the positivity of the coefficient A makes one-loop wave functions of the universe (both no-boundary and tunneling) normalizable.

The anomalous scaling in the case of ξ À 1

determines the quantum rolling force in the effective

equation of the inflationary dynamics and yields the

parameters of the CMB generated during inflation.

A= 3 8λ

³

2g⁴+¡

g²+g⁰²¢₂

−16y_t⁴

´ .

In the conventional range of the Higgs mass 115 GeV≤M_H ≤180 GeV

this quantity at the electroweak scale belongs to the range

−48<A<−20

which strongly contradicts the CMB data which require

−12<A<14.

Taking into account the renormalization group running

A(t ) = 3 8λ(t)

³

2g

(t) + ¡

g

(t) + g

(t) ¢

− 16y

(t)

´ , t = ln(ϕ/µ)

we see that the value of the A on the inflationary

scale is compatible with the CMB data.

F.L. Bezrukov, A. Magnin and M. Shaposhnikov, Standard Model Higgs boson mass from inflation, arXiv:0812.4950 [hep-ph].

F. Bezrukov and M. Shaposhnikov,

Standard Model Higgs boson mass from inflation: two loop analysis,

arXiv:0904.1537 [hep-ph].

A. De Simone, M. P. Hertzberg and F. Wilczek, Running Inflation in the Standard Model, arXiv:0812.4946 [hep-ph].

One-loop approximation

S[g_µν, ϕ] = Z

d⁴x g^1/2 µ

−V(ϕ) +U(ϕ)R(g_µν)− 1

2G(ϕ) (∇ϕ)²

¶

V(ϕ) = λ

4(ϕ²−ν²)²+ λϕ⁴

128π²Alnϕ² µ², U(ϕ) = 1

2(M_P² +ξϕ²) + ϕ² 384π²

µ

Clnϕ² µ² +D

¶ , G(ϕ) = 1 + 1

192π² µ

Flnϕ² µ² +E

¶ .

ˆ

g_µν = 2U(ϕ) M_P² g_µν,

µdϕˆ dϕ

¶₂

= M_P² 2

GU+ 3U⁰² U² . Uˆ =M_P²/2, Gˆ = 1,

Vˆ( ˆϕ) = µM_P²

2

¶₂ V(ϕ) U²(ϕ)

¯¯

¯¯

¯ϕ=ϕ( ˆϕ)

. At the inflation scale with ϕ >MP/√

ξÀv and for large non-minimal couplingξ À1

Vˆ = λM_P⁴ 4ξ²

µ

1 + A 16π² lnϕ

µ

¶ .

Inflationary slow-roll parameters:

ˆ ε≡ M_P²

2 Ã1

Vˆ dVˆ dϕˆ

!₂

= 4 3

µM_P²

ξ ϕ² + A 64π²

¶₂ , ˆ

η≡ M_P² Vˆ

d²Vˆ

dϕˆ² =−4M_P² 3ξϕ² .

Their smallness determines the range of the inflationary stage ϕ > ϕ_end, terminating at the value of ε, which we chose to beˆ ˆ

ε_end= 3/4. Then the inflaton value at the exit from inflation equals

ϕ_end '2M_P/√ 3ξ.

factor e-folding numberN: ϕ²

ϕ²_I =e^x −1 +O µlnN

N

¶

, ϕ²_I = 64π²M_P² ξA , x ≡ NA

48π².

The CMB spectral index n_s, the tensor to scalar ratio r and the spectral index running α:

n_s = 1− 2 N

x e^x−1 , r = 12

N²

µ xe^x e^x −1

¶₂ , α =− 2

N²

x²e^x (e^x −1)².

Renormalization Group improvement

V(ϕ) = λ(t)

4 Z⁴(t)ϕ⁴, U(ϕ) = 1

2

³

M_P² +ξ(t)Z²(t)ϕ²

´ , G(ϕ) = Z²(t).

dλ

dt = β_λ 1−γ, dξ

dt = β_ξ 1−γ, dZ

dt = γ 1−γ. These β-functions depend on running couplings λ and ξ as well as on the rest of the coupling constants in Standard Model.

dg

dt = β_g

1−γ, dg⁰

dt = β_g⁰

1−γ, dg_s

dt = β_gs

1−γ, dy_t

dt = β_yt 1−γ.

In view of the uncertainties of the early Universe model, the inclusion of the two-loop part in the β functions seems to exceed the available precision of the theory. However, the lower Higgs mass bound is very sensitive to the initial conditions for the RG equations at the electroweak scale and to the magnitude of two-loop contributions which might result in 10 ÷ 20% variations of running couplings.

Therefore, where necessary, we will use two-loop

results for β functions.