Hard Core Exclusion Models on Lattices: Rods, Rectangles and Discs - PowerPoint PPT Presentation

Hard Core Exclusion Models on Lattices: Rods, Rectangles and Discs Joyjit Kundu ( Institute of Mathematical Sciences, Chennai ) ! Deepak Dhar ( Tata Institute of Fundamental Research, Mumbai ) ! Jürgen Stilck ( Universidade Federal Fluminense, Niterói, Brazil ) ! R. Rajesh ( Institute of Mathematical Sciences, Chennai )

Hard Core Systems: Spheres

Hard Core Systems: Long Rods Isotropic phase • Long rods in three dimensions interacting through excluded volume interaction ! ★ Onsager, Flory, Zwanzig ! • Virial expansion for free energy ! • Exact for infinite aspect ratio ! • Liquid crystals Nematic phase

Two dimensions • Mermin Wagner theorem ! • Phases with quasi long range order ! • Two step freezing of hard discs ! liquid-hexatic transition ! • hexatic-solid transition ! • • Hard rods: long range correlations

Gas of squares (example) Zhao et. al., PNAS, 2011

Hard Core Lattice Gas Models

Hard Core Lattice Gas Models 1-NN

Hard Core Lattice Gas Models 2-NN

Hard Core Lattice Gas Models 3-NN

Hard rods on a lattice k-mers X-mers Y-mers Hard core exclusion As ρ is increased from 0 to 1 , what are the different phases possible? What is the nature of the phase transitions?

ρ → 0 Rods are far from each other ! randomly oriented ! ! Isotropic phase: h | ρ x � ρ y | i = 0

ρ = 1 (fully packed) Disordered

ρ = 1 (fully packed) Disordered

ρ = 1 (fully packed) Disordered

ρ = 1 (fully packed) Disordered

ρ = 1 (fully packed) Disordered

ρ = 1 (fully packed) Disordered Ω ≥ 2 ( L/k ) 2 L 2 ≥ ln(2) S > 0 k 2

ρ = 1 (fully packed) Disordered Nematic Ω ≥ 2 ( L/k ) 2 L 2 ≥ ln(2) S > 0 k 2

ρ = 1 (fully packed) Disordered Nematic Ω ≥ 2 ( L/k ) 2 L 2 ≥ ln(2) S > 0 k 2

ρ = 1 (fully packed) Disordered Nematic Ω ≥ 2 ( L/k ) 2 L 2 ≥ ln(2) S > 0 k 2

ρ = 1 (fully packed) Disordered Nematic Ω ≥ 2 ( L/k ) 2 Ω = 2 k L L 2 ≥ ln(2) S L 2 = ln( k ) S > 0 → 0 k 2 L Disordered phase: h | ρ x � ρ y | i = 0

Low and high densities ρ = 1 ρ =0 Disordered What happens at intermediate densities?

Dimers (k=2) • Fully packed ! Kastelyn, 1961 Heilmann, Lieb, 1970 ! • Isotropic at all densities ! Kunz, 1970 • Power law correlations when fully packed ! • What about k>2?

Monte Carlo simulation h | n v � n h | i ρ Ghosh, Dhar, EPL, 2007 Nematic phase exists for k ≥ 7 Nematic phase exists for k ⪼ 1 Disertori, Giuliani, Commun. Math. Phys. 2013

ρ = 1 - ε Entropy for nematic phase Each row has L ✏ holes and L (1 − ✏ ) rods k A simple combinatorial problem S nem = − ✏ ln( k ✏ ) + ✏ + . . . L 2

ρ = 1 - ε Entropy for disordered phase Number of holes = L 2 ✏ Number of rods to be removed = L 2 ✏ k Remove randomly L 2 = ln( k ) + 1 S dis k [ − ✏ ln( ✏ ) − (1 − ✏ ) ln(1 − ✏ )] k 2

ρ = 1 - ε � c ≈ a k 2

First transition ⇒ second transition h | n v � n h | i ρ Nematic phase exists for k ≥ 7

First transition ⇒ second transition h | n v � n h | i ρ Nematic phase exists for k ≥ 7

Questions • What is the nature of the first transition? ! • Does the second transition exist? ! • If it exists, what is the nature of the second transition, high density phase? ! • Is it possible to find an exact solution to the problem? ! • What is the phase diagram for rectangles?

Nature of first transition • Low density: isotropic ! • Intermediate density: nematic phase ! ★ vertical ! ★ horizontal ! • Universality class?

Critical Phenomena • Diverging correlation length ξ ! • Order parameter m ! • Characterised by critical exponents ⇠ ∼ ✏ − ν m ∼ ✏ β � ∼ ✏ − γ

Isotropic-Nematic transition Ising 3 state Potts D M.-Fernandez et.al., EPL, 2008

Second transition? • Occurs at high densities ( ≈ 0.92 for k=7) ! • Evaporation, deposition Monte Carlo gets jammed ! • Is there an efficient algorithm? ρ =0.86

An efficient algorithm

An efficient algorithm

An efficient algorithm A 1 -d problem

An efficient algorithm 1 -d problem L − 1 L − k = + Z ( L ) = zZ ( L − k ) + Z ( L − 1) Prob = zZ ( L − k ) Z ( L ) Equilibrates Efficient Parallelizable

Equilibration ρ≈ 0.96

Equilibration ρ≈ 0.96

Existence of high density disordered phase 1 µ =3.89, L=252 1 µ =7.60 0.8 0.8 µ =6.91 µ =6.57 0.6 Q 0.4 0.6 0.2 Q 0 0.4 L=126 1 × 10 6 2 × 10 6 3 × 10 6 0 L=154 t L=210 0.2 L=336 µ =7.60 L=448 L=952 0 fit to Eq.(1) 1 × 10 6 2 × 10 6 3 × 10 6 0 t h 3 ✏ v 2 t 3 i − ⇡ Q ( t ) = exp

Continuous transition? Q = | n v � n h | h n v + n h i 1 L=98 0.9 L=126 L=154 0.8 0.7 0.6 0.5 Q 0.4 0.3 0.2 0.1 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 �

Binder Cumulant U h Q 4 i U = 1 � 3 h Q 2 i 2 . 0.7 L=154 L=210 0.6 L=336 L=448 0.5 L=952 0.52 0.61 0.4 U 0.3 0.2 0.1 0 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 µ

Order parameter Q ' L − β / ν f q ( ✏ L 1 / ν ) � =0.90; � / � =0.22 2.5 L=154 L=210 L=336 2 L=448 L=952 1.5 Q L � / � 1 0.5 0 -20 -15 -10 -5 0 5 10 15 20 � L 1/ �

Susceptibility � ' L γ / ν f χ ( ✏ L 1 / ν ) � =0.90; � / � =1.56 6 L=154 L=210 5 L=336 L=448 4 L=952 � L - � / � 3 2 1 0 -20 -10 0 10 20 � L 1/ �

Compressibility  ' L α / ν f κ ( ✏ L 1 / ν ) � =0.90; � / � =0.22 0.1 L=154 L=210 L=336 0.09 L=448 L=952 0.08 � L - � / � 0.07 0.06 0.05 -20 -15 -10 -5 0 5 10 15 20 � L 1/ �

Ising? • Transition appears not to be in Ising universality ! • But two symmetric ordered states ! • High density disordered phase different from low density isotropic phase? An order parameter?

High density phase What it is not

Correlations 10 0 µ =7.60 µ =6.91 10 -1 µ =6.50 10 -2 10 0 C QQ (r) 10 -3 10 -2 C QQ (r) 10 -4 10 -4 L=154 L=252 L=490 10 -5 L=980 10 -6 10 0 10 1 10 2 r 10 -6 10 0 10 1 10 2 r A power law?

Susceptibility 400 0.018 µ =7.60 L=154 µ =6.91 L=182 350 µ =6.50 L=210 0.012 P(Q) L -1 L=448 L=952 300 0.006 250 0 � -100 0 100 200 Q L 150 100 50 200 300 400 L No divergence with L. ! If power law, then exponent > 2

Stacks

Stack distribution µ =7.600 µ =6.910 10 -4 µ =6.570 µ =5.585 µ =3.476 10 -6 µ =1.386 µ =0.200 D(s) 10 -8 10 -10 10 -12 0 50 100 150 200 250 s Exponential at all chemical potentials

Binding-unbinding transition?

Binding-unbinding transition?

Binding-unbinding transition? 1 2 3

Binding-unbinding transition? 7 i=j i � j 6 5 d ij 4 3 2 1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 � No evidence for bound state

Geometric Clusters Replace x-mers by 1 ! Rest by 0

Geometric Clusters Replace x-mers by 1 ! Rest by 0

Geometric Clusters 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Replace x-mers by 1 ! Rest by 0

Geometric Clusters 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Replace x-mers by 1 ! Rest by 0

Cluster size distribution 10 0 L= 448 10 -1 L= 560 F cum (s) L= 896 L=1568 L=2016 L=2576 A s 1- � 10 -2 10 -3 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 s Cutoff ∼ 10 6 A crossover length scale ξ≈ 1500

Nature of high density phase • Circumstantial evidence for long range correlations ! • A large crossover length scale ! • What happens at larger length scales?

Bethe Approximation • Beyond numerics ! • Onsager solution exact for ∞ aspect ratio ! • Bethe approximation treats nearest neighbour interactions exactly ! • What is the Bethe approximation for finite length rods? ! • Is there a second transition?

Bethe Lattice Each site connected to q nbrs No loops Perimeter → constant Volume Cayley tree: dominated by perimeter Bethe lattice: Core of the Cayley tree

Some issues with Bethe lattice Consider coordination number 6 Suppose ρ red > ρ green = ρ blue

Some issues with Bethe lattice Consider coordination number 6 Interchange red and green Suppose ρ red > ρ green = ρ blue

Some issues with Bethe lattice Consider coordination number 6 Interchange red and green Suppose ρ red > ρ green = ρ blue Then, ρ red = ρ green Contradiction ⇒ no nematic order possible

Random Locally Tree-like Lattice (RLTL) 1 m − 1 m M Each site connected to two ! sites in next layer by a x- and y- bond

Solution ✓ ◆ ✓ ◆ 1 − k − 1 1 − k − 1 s ( ρ x , ρ y ) = ln ρ x ρ x k k ✓ ◆ ✓ ◆ 1 − k − 1 1 − k − 1 + ln ρ y ρ y k k (1 − ρ ) ln(1 − ρ ) − ρ x k ln ρ x k − ρ y k ln ρ y − k Keep ρ fixed and maximize entropy � < � c 0.36 � = � c 0.34 0.32 s( � ) ψ = ρ x − ρ y 0.3 0.28 � > � c 0.26 0.24 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 �