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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