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

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions March 27, 2009

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Outline Modulated patterns in physics 1 Phenomenological approach 2 Large- n approach 3 The selection rule 4 Some remarks 5

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Modulated patterns in physics Modulated patterns in physics superconductors ( ≈ 10 µ m ) phospholipides ( ≈ 25 nm ) chemical reactions ( ≈ . 3 mm ) convection ( ≈ 1 cm )

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

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach Phenomenological approach Ginzburg-Landau functional � + b d 2 r |∇ φ ( r ) | 2 F φ = F 0 [ φ ] 2 � �� � � �� � “mexican hat” surface tension

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach Phenomenological approach Ginzburg-Landau functional � + b d 2 r |∇ φ ( r ) | 2 F φ = F 0 [ φ ] 2 � �� � � �� � “mexican hat” surface tension �� Q d 2 r d 2 r ′ φ ( r ) g ( r − r ′ ) φ ( r ′ ) − 2 � �� � long-range int.

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach Phenomenological approach Ginzburg-Landau functional � + b d 2 r |∇ φ ( r ) | 2 F φ = F 0 [ φ ] 2 � �� � � �� � “mexican hat” surface tension �� Q d 2 r d 2 r ′ φ ( r ) g ( r − r ′ ) φ ( r ′ ) − 2 � �� � long-range int. � κ d 2 r ( ∇ 2 φ ( r )) 2 + 2 � �� � curvature effects

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach Phenomenological approach Ginzburg-Landau functional � + b d 2 r |∇ φ ( r ) | 2 F φ = F 0 [ φ ] 2 � �� � � �� � “mexican hat” surface tension �� Q d 2 r d 2 r ′ φ ( r ) g ( r − r ′ ) φ ( r ′ ) − 2 � �� � long-range int. � κ d 2 r ( ∇ 2 φ ( r )) 2 + + . . . 2 � �� � curvature effects

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Phenomenological approach One more picture Langmuir Magnetic film of garnet film DMPA and as T ր cholesterol period as T ր increases period decreases

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large- n approach Large- n model H = 1 � S ( � x ) V ( � x ,� y ) S ( � y ) 2 � x ,� y Spins satisfy the mean spherical constraint � x ) 2 � = N x � S ( � �

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large- n approach Large- n model H = 1 � S ( � x ) V ( � x ,� y ) S ( � y ) 2 � x ,� y Spins satisfy the mean spherical constraint � x ) 2 � = N x � S ( � � Inroduce a Lagrange multiplier µ to enforce the constraint Fourier transform of the interaction kernel v ( � k ) � d d k 1 1 = k B T (2 π ) d µ + v ( � k ) Critical temperature T c (if any!) � d d k 1 ( k B T c ) − 1 = µ c + v ( � (2 π ) d k ) where µ c = − min q ∈ BZ v ( q ).

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large- n approach Correlation functions e i � � d d k k · � x G ( � x ) ≡ � S (0) S ( � x ) � = k B T v ( � (2 π ) d k ) + µ Normalization of G ( � x = 0) = 1 fixes µ . Rotationally invariant system: v ( � k ) is a function of k 2

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large- n approach Correlation functions e i � � d d k k · � x G ( � x ) ≡ � S (0) S ( � x ) � = k B T v ( � (2 π ) d k ) + µ Normalization of G ( � x = 0) = 1 fixes µ . Rotationally invariant system: v ( � k ) is a function of k 2 Correlation functions calculation ⇔ finding poles of the integrand!

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions Large- n approach Correlation functions e i � � d d k k · � x G ( � x ) ≡ � S (0) S ( � x ) � = k B T v ( � (2 π ) d k ) + µ Normalization of G ( � x = 0) = 1 fixes µ . Rotationally invariant system: v ( � k ) is a function of k 2 Correlation functions calculation ⇔ finding poles of the integrand! Assume k s [ v ( k ) + µ ] is a polynomial M � a m z m P ( z ) = m =0 in z = k 2 = ⇒ correlator displays a net of M correlation and modulation lengths. Different length scales arise for any M ≥ 2

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 ✝ ✆

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 [ k 2 + λ 2 ] − p . E.g. Coulomb screened � 1 1 y | e − λ | � x − � y | d = 3 , V = k ) = [ k 2 + λ 2 ] − 1 ⇒ v ( � 8 π | � x − � = 1 y | e − λ | � x − � y | d = 2 , V = 4 π ln | � x − �

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 [ k 2 + λ 2 ] − p . E.g. Coulomb screened � 1 1 y | e − λ | � x − � y | d = 3 , V = k ) = [ k 2 + λ 2 ] − 1 ⇒ v ( � 8 π | � x − � = 1 y | e − λ | � x − � y | d = 2 , V = 4 π ln | � x − � ✞ ☎ Short range ✝ ✆

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 [ k 2 + λ 2 ] − p . E.g. Coulomb screened � 1 1 y | e − λ | � x − � y | d = 3 , V = k ) = [ k 2 + λ 2 ] − 1 ⇒ v ( � 8 π | � x − � = 1 y | e − λ | � x − � y | d = 2 , V = 4 π ln | � x − � ✞ ☎ Short range ✝ ✆ Lattice Laplacian: ∆( � k ) = 2 � d l =1 (1 − cos k l ) In real space: � 2 d for � x = � y � � x | ∆ | � y � = − 1 for | � x − � y | = 1 In the continuum limit ∆ → k 2

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 Q v ( k ) = k 2 + k 2 + λ 2 (screened model of frustrated phase separation in cuprates). Pole dynamics controls evolution of correlation lengths.

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 ∗ ) = λ 2 + 2 √ Q ) 1 x ) = k B T β 2 − α 2 [( λ 2 − α 2 ) e − α | � x | − ( λ 2 − β 2 ) e − β | � x | ] G ( � 4 π | � x | For T < T ∗ k B T x | e − α 1 | � x | G ( � x ) = 8 α 1 α 2 π | � [( λ 2 − α 2 1 + α 2 2 ) sin α 2 | � x | + 2 α 1 α 2 cos α 2 | � x | ] where α 2 , β 2 = λ 2 + µ ∓ ( µ − λ 2 ) 2 − 4 Q p 2 and α = α 1 + i α 2 = β ∗ ✞ ☎ Poles located at k = i α, i β Pole dynamics ✝ ✆

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 ) = a 4 k 4 − a 2 k 2 Teubner-Stray correlator k ) = a 2 k 4 − a 1 k 2 + µ G − 1 ( � x ) ≈ sin κ | � x | Pole dynamics x | e −| � x | /ξ G ( � κ | � where �� κ = µ/ 4 a 2 + a 1 / 4 a 2 �� ξ = µ/ 4 a 2 − a 1 / 4 a 2

Thermally Increasing Correlation/Modulation Lengths and Other Selection Rules in Systems with Long Range Interactions The selection rule . . . summarizing LR-SR x d − 2 ( A 1 e − x /ξ 1 + A 2 e − x /ξ 2 + . . . ) 1 High- T limit G ( x ) = where at least one ξ i diverges x d − 2 e − x /ξ + . . . i.e. At low temperatures G ( x ) ≈ cos( κ x ) correlations turn into modulation lengths modulation length increase as T is raised