[PPT] - Thermally Increasing Correlation/Modulation Lengths and Other PowerPoint Presentation

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions

March 27, 2009

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Outline

1

Modulated patterns in physics

2

Phenomenological approach

3

Large-n approach

4

The selection rule

5

Some remarks

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Modulated patterns in physics

Modulated patterns in physics

superconductors (≈ 10µm) phospholipides (≈ 25nm) chemical reactions (≈ .3mm) convection (≈ 1cm)

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Modulated patterns in physics

Stripes, fingers, bubbles and the like

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach

Phenomenological approach

Ginzburg-Landau functional Fφ = F0[φ]

“mexican hat”

+ b 2

d2r|∇φ(r)|2
surface tension

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach

Phenomenological approach

Ginzburg-Landau functional Fφ = F0[φ]

“mexican hat”

+ b 2

d2r|∇φ(r)|2
surface tension

− Q 2

d2r d2r′φ(r)g(r − r′)φ(r′)
long-range int.

SLIDE 7

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach

Phenomenological approach

Ginzburg-Landau functional Fφ = F0[φ]

“mexican hat”

+ b 2

d2r|∇φ(r)|2
surface tension

− Q 2

d2r d2r′φ(r)g(r − r′)φ(r′)
long-range int.

+ κ 2

d2r(∇2φ(r))2
curvature effects

SLIDE 8

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach

Phenomenological approach

Ginzburg-Landau functional Fφ = F0[φ]

“mexican hat”

+ b 2

d2r|∇φ(r)|2
surface tension

− Q 2

d2r d2r′φ(r)g(r − r′)φ(r′)
long-range int.

+ κ 2

d2r(∇2φ(r))2
curvature effects

+ . . .

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach

One more picture

Magnetic garnet film as T ր period increases Langmuir film of DMPA and cholesterol as T ր period decreases

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Large-n model

H = 1 2

x,

y

S( x)V ( x, y)S( y) Spins satisfy the mean spherical constraint

xS(

x)2 = N

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Large-n model

H = 1 2

x,

y

S( x)V ( x, y)S( y) Spins satisfy the mean spherical constraint

xS(

x)2 = N Inroduce a Lagrange multiplier µ to enforce the constraint Fourier transform of the interaction kernel v( k) 1 = kBT

ddk

(2π)d 1 µ + v( k) Critical temperature Tc (if any!) (kBTc)−1 =

ddk

(2π)d 1 µc + v( k) where µc = − minq∈BZ v(q).

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Correlation functions

G( x) ≡ S(0)S( x) = kBT

ddk

(2π)d ei

k· x

v( k) + µ Normalization of G( x = 0) = 1 fixes µ. Rotationally invariant system: v( k) is a function of k2

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Correlation functions

G( x) ≡ S(0)S( x) = kBT

ddk

(2π)d ei

k· x

v( k) + µ Normalization of G( x = 0) = 1 fixes µ. Rotationally invariant system: v( k) is a function of k2 Correlation functions calculation ⇔ finding poles of the integrand!

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Correlation functions

G( x) ≡ S(0)S( x) = kBT

ddk

(2π)d ei

k· x

v( k) + µ Normalization of G( x = 0) = 1 fixes µ. Rotationally invariant system: v( k) is a function of k2 Correlation functions calculation ⇔ finding poles of the integrand! Assume ks[v(k) + µ] is a polynomial P(z) =

M

m=0

amzm in z = k2 = ⇒ correlator displays a net of M correlation and modulation lengths. Different length scales arise for any M ≥ 2

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Short and long range interactions

✞ ✝ ☎ ✆

Long range

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Short and long range interactions

✞ ✝ ☎ ✆

Long range screened LR interaction may be emulated by the FT kernel [k2 + λ2]−p. E.g. Coulomb screened d = 3, V =

1 8π 1 | x− y|e−λ| x− y|

d = 2, V =

1 4π ln |

x − y|e−λ|

x− y|

=

⇒ v( k) = [k2 + λ2]−1

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Short and long range interactions

✞ ✝ ☎ ✆

Long range screened LR interaction may be emulated by the FT kernel [k2 + λ2]−p. E.g. Coulomb screened d = 3, V =

1 8π 1 | x− y|e−λ| x− y|

d = 2, V =

1 4π ln |

x − y|e−λ|

x− y|

=

⇒ v( k) = [k2 + λ2]−1

✞ ✝ ☎ ✆

Short range

SLIDE 18

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Short and long range interactions

✞ ✝ ☎ ✆

Long range screened LR interaction may be emulated by the FT kernel [k2 + λ2]−p. E.g. Coulomb screened d = 3, V =

1 8π 1 | x− y|e−λ| x− y|

d = 2, V =

1 4π ln |

x − y|e−λ|

x− y|

=

⇒ v( k) = [k2 + λ2]−1

✞ ✝ ☎ ✆

Short range Lattice Laplacian: ∆( k) = 2 d

l=1(1 − cos kl)

In real space:

x|∆|

y = 2d for x = y −1 for | x − y| = 1 In the continuum limit ∆ → k2

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

LR-SR competition: an example

short-range attractive interaction long-range screened Coulomb interaction Fourier transform of the interaction kernel v(k) = k2 + Q k2 + λ2 (screened model of frustrated phase separation in cuprates). Pole dynamics controls evolution of correlation lengths.

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

Correlation function (d = 3)

At high temperatures (T > T ∗ where µ(T ∗) = λ2 + 2√Q) G( x) = kBT 4π| x| 1 β2 − α2 [(λ2 − α2)e−α|

x| − (λ2 − β2)e−β| x|]

For T < T ∗ G( x) = kBT 8α1α2π| x| e−α1|

x|

[(λ2 − α2

1 + α2 2) sin α2|

x| + 2α1α2 cos α2| x|] where α2, β2 = λ2 + µ ∓ p (µ − λ2)2 − 4Q 2 and α = α1 + iα2 = β∗

✞ ✝ ☎ ✆

Poles located at k = iα, iβ Pole dynamics

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large-n approach

SR-SR competition

FT of the interaction kernel v(k) = a4k4 − a2k2 Teubner-Stray correlator G −1( k) = a2k4 − a1k2 + µ G( x) ≈ sin κ| x| κ| x| e−|

x|/ξ

where κ =

µ/4a2 + a1/4a2

ξ =

µ/4a2 − a1/4a2

Pole dynamics

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions The selection rule

. . . summarizing

LR-SR High-T limit G(x) =

1 xd−2 (A1e−x/ξ1 + A2e−x/ξ2 + . . .)

where at least one ξi diverges At low temperatures G(x) ≈ cos(κx)

xd−2 e−x/ξ + . . . i.e.

correlations turn into modulation lengths modulation length increase as T is raised

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions The selection rule

. . . summarizing

LR-SR High-T limit G(x) =

1 xd−2 (A1e−x/ξ1 + A2e−x/ξ2 + . . .)

where at least one ξi diverges At low temperatures G(x) ≈ cos(κx)

xd−2 e−x/ξ + . . . i.e.

correlations turn into modulation lengths modulation length increase as T is raised SR-SR modulation and correlation mantain their identity as T is varied modulation length decrease as T is raised

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions The selection rule

. . . summarizing

LR-SR High-T limit G(x) =

1 xd−2 (A1e−x/ξ1 + A2e−x/ξ2 + . . .)

where at least one ξi diverges At low temperatures G(x) ≈ cos(κx)

xd−2 e−x/ξ + . . . i.e.

correlations turn into modulation lengths modulation length increase as T is raised SR-SR modulation and correlation mantain their identity as T is varied modulation length decrease as T is raised

✞ ✝ ☎ ✆

SELECTION RULE

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Some remarks

Some remarks

1/n corrections do not alter substantially this picture lattice effects raise Tc from 0 to a finite value

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Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Some remarks

References

Z. Nussinov, ”Thermally Increasing Correlation/Modulation

Lengths and Other Selection Rules in Systems with Long Range Interactions” cond-mat/0506554 (June 22, 2005).

M. Seul and D. Andelman, ”Domain Shapes and Patterns:

The Phenomenology of Modulated Phases” Science 267, no. 5197 (January 27, 1995): 476-483.

Z. Nussinov et al., ”Avoided Critical Behavior in O(n)