Hierarchical Lattices: from Potts Model to

Directed Polymers

in memory of Bambi the scientist and educator

Tongji University, June 4th, 2016

Opening of Physics Academic Week

Department Christmas Party, 1998

胡先生关于相变与非线性动力学方面的研究始于1977年，那时他刚结束法国的博士后阶段并来到布朗大学作为Leo Kadanoff的博士后，当时正值混沌动力学研究蓬勃发展的阶段，受Kadanoff的鼓励，胡斑比转向关于相变与非线性动力学方面的研究。

The renormalization-group theory of phase transitions and critical

phenomena represented one of the greatest triumphs in

theoretical physics in the last decade.

The method of the renormalization group has since been applied

to a wide variety of physical problems. Recently there is an

intense interest in the study of nonlinear dynamics and

transitions to chaos. Here again the renormalization group has

proved a very useful tool.

When Wilson first developed the renormalization-group theory of critical

phenomena, extensive use of techniques borrowed from quantum field

theory was made.

This approach was later streamlined by Brézin et al., who employed the

full apparatus of field theory and the Callan—Symanzik equation.

In the mean time the basic ideas of the renormalization group became

quite clear, and attempts were made to implement the ideas directly

without recourse to field theory. Such efforts resulted in the so-called

“real-space renormalization group”.

In this approach, calculations are performed directly in position space, in

contradistinction to the field-theoretic ε-expansion in momentum space.

The advent of the real-space renormalization group not only rendered

possible a transparent and elegant implementation of the basic ideas of

the renormalization group but also introduced new and effective

methods of calculations.

To underline the physical content of the renormalization group and to

facilitate its applicability to problems for which no field-theoretic

transcription is readily available, I will confine my discussion to the real-

space approach.

The similarity between critical and chaotic phenomena is a theme

threading the entire article. The purpose of this work is primarily

pedagogical, and the hope is to initiate the neophyte to the rudiments so

that he will be adequately equipped to pursue a more in-depth study.

The Ising model:

H J

T i j

H

T i

i

ij

Partition function:

Z H exp H i

Tr exp H

Renormalization group transformation:

exp H ' Tr exp T , H

Transformation on finite-dimensional

lattices is carried out using various

approximate schemes

B. Hu, Problem of universality in phase transitions on hierarchical

lattices, Phys. Rev. Lett. 55, 2316 (1985).

B. Hu, Extended universality and the question of spin on hierarchical

lattices, Phys. Rev. B 33, 6503 (1986).

Y. K. Wu and B. Hu, Phase transitions on complex Sierpinski carpets,

Phys. Rev. A 35, 1404 (1987).

To gain a better understanding of universality, Bambi then turned his attention to models defined on hierarchical lattices…

Potts model on the Berker lattice

1

2

1

2

3 4

K

K

tracing out 3 and 4

3 4 : w1 e2K q 1 2

x2 q 1 2

3 4 : w0 2eK q 2 2

2x q 2 2

Hence,

%x e%K w1

w0

x2 q 1

2x q 2

2

RG transformation

Lk 2k

Nk 4k Lk2

A real piece of art!

Potts model on laced Sierpinski gasket

Liang Tian et al., EPJB 86: 197 (2013)

Peculiar transition between two disordered phases Pathology of the lattice where some sites have diverging number of neighbors

Is the real-space RG doomed?

Localization transition of directed polymer in a disordered medium

ASEP (Asymmetric Simple Exclusion Process)

“Ising model” of nonequilibrium 1D transport

J p q 1

Particles on a ring with a weak bond (constriction) Wolf and Tang, PRL 65, 1591 (1990) Lebowitz and Janowsky, PRA 45, 618 (1992)

Q: will there be a reduction of current as soon as r < 1?

A: yes (MF) “no” (simulation)

Mappings

ASEP, Burger’s Eq.

Dx2

1

2x

2 random current

Surface growth, KPZ Eq.

h x,t D2h

2h

2 x,t

t

x

h T lnZ

Directed polymer in a disordered medium with end fixed at (x, t)

H x0 , x dt

2

dx

dt

2

V x,t

S(t )

slow bond = localizing potential

Scaling argument

ASEP, Burger’s Eq.

Dx2

1

2x

2 random current

t

x

h T lnZ

Directed polymer in a disordered medium with end fixed at (x, t)

H x0 , x dt

2

dx

dt

2

V x,t

S(t )

slow bond = localizing potential

Will the directed polymer localize under an arbitrarily weak attractive potential?

Energy gain from visiting the attractive potential Energy loss from not visiting favorable sites at a distance

Marginal dimension: d 1

: 0t td

E : t

Directed polymer localization on the hierarchical lattice

Consider directed paths from A to B Path energy = energy

assigned to bonds

Energy on dashed bonds have a different distribution

Iterative relation for the ground state energy distribution

1) each branch: Ek1

b Ek(1) Ek

(2)

2) optimization: Ek1 minb Ek1

b

Given realization:

Normal branch:

Defect branch:

Weak attraction

Pk (x) ; Pk (x k )

Define: uk k

k, k1 2 k

Then

In marginal dimension, correlation length diverges exponentially under a weak attraction

Particles on a ring with a weak bond (constriction) Wolf and Tang, PRL 65, 1591 (1990) Lebowitz and Janowsky, PRA 45, 618 (1992)

Q: will there be a reduction of current as soon as r < 1?

A: yes (MF) “no” (simulation)

ASEP in 1D with a weak bond

DNA Melting

L.-H. Tang and H. Chaté, “Rare-event induced binding transition of heteropolymers,”

Phys. Rev. Lett. 86, 830 (2001).

T. Hwa, E. Marinari, K. Sneppen, and L.-H. Tang, “Localization of denaturation

bubbles in random DNA sequences,” Proc. National Academy of Sciences, USA. 100,

4411-4416 (2003).

Short sequences differential

Long Chains

Resurgence of KPZ KITP Program: New approaches to nonequilibrium and random systems, Jan-Mar 2016

Stretched exponential tail

Homework assignment

n

b Take the limit n,b1, keeping

d 1b 1

n 1 constant

dP(x,l)

dl L1 P̂(s,l)ln P̂ s,l

(x) d 1 P x,l 1 ln dyP y,l

x

0

L1 P̂(s,l)ln P̂ s,l

(x)

1

2idsesxP̂(s,l)ln P̂ s,l

i

i

Special cases: 1) d 1: P(x,t) t 1/2x ct

t1/2

, gaussian

2) d : P(x,l) (x cl), (x) 1

cex/cee

x/c

, Gumbel

Conjecture: Pk k1d

E k k

, d u : exp A u

, 1 1

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