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

• 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

Isotropic phase 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

As ρ is increased from 0 to 1, what are the different phases possible? What is the nature

• f the phase transitions?

Hard core exclusion

k-mers Y-mers X-mers

ρ → 0

Rods are far from each other! randomly oriented!

!

Isotropic phase: h|ρx ρy|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 S L2 ≥ ln(2) k2 > 0

ρ = 1 (fully packed)

Disordered Nematic

Ω ≥ 2(L/k)2 S L2 ≥ ln(2) k2 > 0

ρ = 1 (fully packed)

Disordered Nematic

Ω ≥ 2(L/k)2 S L2 ≥ ln(2) k2 > 0

ρ = 1 (fully packed)

Disordered Nematic

Ω ≥ 2(L/k)2 S L2 ≥ ln(2) k2 > 0

ρ = 1 (fully packed)

Disordered Nematic

S L2 = ln(k) L → 0 Ω = 2kL Ω ≥ 2(L/k)2 S L2 ≥ ln(2) k2 > 0

Disordered phase:h|ρx ρy|i = 0

Low and high densities

ρ=0 ρ=1

Disordered What happens at intermediate densities?

Dimers (k=2)

• Fully packed!
• Isotropic at all densities !
• Power law correlations

when fully packed!

• What about k>2?

Heilmann, Lieb, 1970! Kunz, 1970 Kastelyn, 1961

Monte Carlo simulation

Ghosh, Dhar, EPL, 2007

Nematic phase exists for k≥7

h|nv nh|i ρ

Disertori, Giuliani, Commun. Math. Phys. 2013

Nematic phase exists for k⪼1

ρ = 1-ε

Entropy for nematic phase

Snem L2 = −✏ ln(k✏) + ✏ + . . .

Each row has L✏ holes and L(1−✏)

k

rods

A simple combinatorial problem

ρ = 1-ε

Entropy for disordered phase

Sdis L2 = ln(k) k2 + 1 k [−✏ ln(✏) − (1 − ✏) ln(1 − ✏)] Number of holes = L2✏ Number of rods to be removed = L2✏

k

Remove randomly

ρ = 1-ε

c ≈ a k2

Nematic phase exists for k≥7

h|nv nh|i ρ

First transition ⇒ second transition

Nematic phase exists for k≥7

h|nv nh|i ρ

First transition ⇒ second transition

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

m ∼ ✏β ⇠ ∼ ✏−ν ∼ ✏−γ

Isotropic-Nematic transition

D M.-Fernandez et.al., EPL, 2008

Ising 3 state Potts

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

Z(L) = zZ(L − k) + Z(L − 1) Prob = zZ(L − k) Z(L)

Equilibrates Efficient Parallelizable = +

L − k L − 1

Equilibration

ρ≈0.96

Equilibration

ρ≈0.96

Existence of high density disordered phase

0.2 0.4 0.6 0.8 1 1×106 2×106 3×106 Q t µ=7.60 µ=3.89, L=252 L=126 L=154 L=210 L=336 L=448 L=952 fit to Eq.(1)

0.2 0.4 0.6 0.8 1 1×106 2×106 3×106 Q t µ=7.60 µ=6.91 µ=6.57

Q(t) = exp h −⇡ 3 ✏v2t3i

Continuous transition?

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Q

• L=98

L=126 L=154

Q = |nv nh| hnv + nhi

Binder Cumulant U

U = 1 hQ4i 3hQ2i2 .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 U µ L=154 L=210 L=336 L=448 L=952 0.52 0.61

Order parameter

Q ' L−β/νfq(✏L1/ν)

0.5 1 1.5 2 2.5

• 20
• 15
• 10
• 5

5 10 15 20 Q L/ L1/ =0.90; /=0.22 L=154 L=210 L=336 L=448 L=952

Susceptibility

' Lγ/νfχ(✏L1/ν)

1 2 3 4 5 6

• 20
• 10

10 20 L-/ L1/ =0.90; /=1.56 L=154 L=210 L=336 L=448 L=952

Compressibility

0.05 0.06 0.07 0.08 0.09 0.1

• 20
• 15
• 10
• 5

5 10 15 20 L-/ L1/ =0.90; /=0.22 L=154 L=210 L=336 L=448 L=952

 ' Lα/νfκ(✏L1/ν)

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

A power law?

10-6 10-5 10-4 10-3 10-2 10-1 100 100 101 102 CQQ(r) r µ=7.60 µ=6.91 µ=6.50

10-6 10-4 10-2 100 100 101 102 CQQ(r) r L=154 L=252 L=490 L=980

Susceptibility

No divergence with L.! If power law, then exponent > 2

50 100 150 200 250 300 350 400 200 300 400

• L

µ=7.60 µ=6.91 µ=6.50

0.006 0.012 0.018

• 100

100 P(Q) L-1 Q L L=154 L=182 L=210 L=448 L=952

Stacks

Stack distribution

Exponential at all chemical potentials

10-12 10-10 10-8 10-6 10-4 50 100 150 200 250 D(s) s µ=7.600 µ=6.910 µ=6.570 µ=5.585 µ=3.476 µ=1.386 µ=0.200

Binding-unbinding transition?

Binding-unbinding transition?

Binding-unbinding transition?

1 2 3

Binding-unbinding transition?

1 2 3 4 5 6 7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 dij

• i=j

ij

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

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

Geometric Clusters

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

Cluster size distribution

Cutoff ∼ 106

10-3 10-2 10-1 100 100 101 102 103 104 105 106 107 Fcum(s) s L= 448 L= 560 L= 896 L=1568 L=2016 L=2576 A s1-

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

Perimeter Volume → constant

No loops Cayley tree: dominated by perimeter Bethe lattice: Core of the Cayley tree

Some issues with Bethe lattice

Consider coordination number 6 Suppose ρred > ρgreen = ρblue

Some issues with Bethe lattice

Consider coordination number 6

Interchange red and green

Suppose ρred > ρgreen = ρblue

Some issues with Bethe lattice

Consider coordination number 6

Interchange red and green

Contradiction ⇒ no nematic order possible

Suppose ρred > ρgreen = ρblue Then, ρred = ρgreen

Random Locally Tree-like Lattice (RLTL)

Each site connected to two! sites in next layer by a x- and y- bond

m m−1 M 1

Solution

s(ρx, ρy) = ✓ 1 − k − 1 k ρx ◆ ln ✓ 1 − k − 1 k ρx ◆ + ✓ 1 − k − 1 k ρy ◆ ln ✓ 1 − k − 1 k ρy ◆ − (1−ρ) ln(1−ρ) − ρx k ln ρx k − ρy k ln ρy k

Keep ρ fixed and maximize entropy

0.24 0.26 0.28 0.3 0.32 0.34 0.36

• 1
• 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1 s()

• < c

= c > c

ψ = ρx − ρy

Results:order parameter

No second transition

q=4

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 | |

• k=6

k=5 k=4

Order parameter q>4

0.1808 0.1812 0.1816 0.2 0.4 0.6 0.8 1 s()

• < c

= c > c 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• q=6; k=4

q=6; k=5 q=6; k=6

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

• k=5; q=10

k=5; q=8 k=5; q=6

ψ = ρ|| − ρ⊥

Results: kmin

k q

4 6 5 18 6 44 7 110 8 266 9 654 10 1612 11 3994 12 9968 13 25028

100 101 102 103 104 105 4 6 8 10 12 14 qmax k

qmax ∼ exp(k) kmin ∼ ln(q)

Is there a second transition?

No

Interacting rods

A site allowed to have two monomers,! but only perpendicular intersection allowed

Weight = u

u = 0: hard rods u 6= 0: promotes disorder

Results: order parameter

q=4; k=4

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 ||

• u=0.10

u=0.15 u=0.20

Results: phase diagram

q=4

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 u

• k=4

k=5 k=6 =0 =0

Results: phase diagram

q=4

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 u

• k=4

k=5 k=6 =0 =0

LDD=HDD

Hard Rectangles

m × mk k → aspect ratio

Rectangles 2x14

Nematic

Rectangles 2x14

Columnar

Phase Diagram for Rectangles

1 2 3 4 5 6 7 8 0.7 0.75 0.8 0.85 0.9 0.95 1

k

• HD

I C N

1 2 3 4 5 6 7 0.75 0.8 0.85 0.9 0.95 1

k

• C

N I D H

2 × 2k 3 × 3k

Phase Diagram for Rectangles

1 2 3 4 5 6 7 8 0.7 0.75 0.8 0.85 0.9 0.95 1

k

• HD

I C N

1 2 3 4 5 6 7 0.75 0.8 0.85 0.9 0.95 1

k

• C

N I D H

2 × 2k 3 × 3k

1 − a mk2 a k + b(m) k3 c(m) + d(m) k

Summary

• Introduced a Monte Carlo algorithm!

★ Overcame jamming! ★ Efficient and easily parallelised! ★ Works for all shapes!

• A continuous second transition!
• Square:α∕ν≈0.22; β∕ν≈0.22; γ∕ν≈1.55; ν≈0.90!
• Triangle: indistinguishable from q=3 Potts model!
• High density phase indistinguishable from low

density phase

Summary

• Bethe lattice not useful for studying systems

with orientational order!

• Introduced a new lattice: RLTL!

★ existence of a nematic phase! ★ kmin is a function of coordination number!

• Introduction of finite repulsive interaction

results in two transitions; HDD=LDD

Outlook

7 6

I N D k ρ

Outlook

7 6

I N D k ρ

Outlook

7 6

I N D k ρ

Outlook

7 6

I N D k ρ

Outlook

7 6

I N D ? k ρ

Outlook

7 6

I N D ? k ρ Continuum model but with oriented rectangles Zeroes of partition function

Outlook

• Rectangles with non-integer aspect ratio!
• Fully packed problem!
• Three dimensions!
• Poly-dispersed systems

ρ ρ ρ D D I N ρ D I N