Addressing
Add
i soft-matter
ft
tt
questions using
q
gq
quantum
many-body physics tools
D Zeb Rocklin
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Paul M Goldbart, Shina Tan
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332
Approach
Develop
p analogy
gy between statistical
mechanics of directed polymers in two
dimensions and q
quantum one-dimensional
systems (cf. de Gennes 1968)
y Apply
pp y modern quantum
q
techniques
q
to
elucidate behavior of directed polymer
systems
y
y
Mission
y
Characterize densityy fluctuations,
thermodynamics, and effect of geometry
and topology
system
p gy in directed polymer
p y
y
Outline
Part I: Polymer
y
system
y
and mapping
pp g
y Part II: Interactions and density structure
aand correlations
co e at o s
y Part III: Geometrically and topologically
constrained systems
y
Part I: System
Clean system
y
of ideal polymers
p y
y 2-dimensional: a substrate or a thin sheet
y Directed via tension or directional
potential
y
x
τ
Collective excitations in low
dimensions
Interactingg particles
p
in 1D, or lines in 2D,
yield only collective excitations
yS
Single-polymer
g e po y e dynamics
y a cs suppressed
supp esse
y Emergent polymer fluid has new
properties
y
Quantum particles to classical lines
x
τ
From thermal lines to quantum
particles
y
y
y
y
Partition function for N linelike objects with
some interaction
Imaginary-time matrix element of quantum
particles
Generic quantum system taken to be bosonic:
noncrossing polymers are fermionic
P h integrall over polymer
Path
l
conformations
f
xi(τ)
( )
Parameter relationships
Quantum System
Polymer System
Mass
Tension (units of temperature)
Position
Lateral direction
Time
Longitudinal direction
Inverse temperature
System length
System size
System width
x
τ
Part II:
P
II IInteractions and
d Density
D
Structure and Correlations
de Gennes (1968):
noncrossing polymers
Noncrossingg p
paths = hardcore bosons
y In ID, hardcore bosons = free fermions
y Path integrals for free fermions
y
ME Fisher (1984)
de Gennes (1968):
noncrossing polymers
Ground state dominance far from system
y
boundaries
y Su
Sum ove
over ssingle-particle
g e pa t c e eexcitations
c tat o s
y X-ray form factor displays logarithmic
divergence corresponding to Kohn
form
anomaly of Fermi gas X-ray
factor
y
n(k)
k
-KF
KF
q/2KF
Behavior of noncrossing polymers
Friedel oscillations
near edges
y Density-density
e s ty e s ty
correlations
y Interpolymer width
distribution
x KF
Width Distribution
Probabilitty
y
ρ(x)
x
τ
width
Barelt et al. 1990
Lieb-Liniger
Lieb
Liniger (1963) model
y
y
y
c = ∞ case: hardcore
h d
bosons/
b
/
free fermions/noncrossing
polymers
Bethe Ansatz solution:
Single-particle excitations
extinguished
h d
Results via Lieb
Lieb-Liniger
Liniger
x
τ
Free Energy
System free energy
y Lateral correlations
found from
Lieb-Liniger ground
state
y Friedel
F d l oscillations
ll
y More general
correlations
l i
require
i
Lieb-Liniger excitation
spectrum
y
Strength of Interaction
Lateral Distance
Alternative technique: bosonization
Universal field theory for 1D systems
y Characterized by two T
T-LL parameters u, K
y Conjugate fields represent density, phase
fluctuations
y
Correlation results via bosonization
x
τ
K=∞
K>1
K=1
K<1
Free bosons
Contact
bosons
Hardcore
bosons
Long-range
bosons
Attractive
fermions
Free fermions
Repulsive
fermions
Part II: Geometrical and
Topological Restrictions
Topological impurities
Weak external p
potential V(x,τ)
( ) can be
handled perturbatively
y St
Strong
o g potential
pote t a ca
can restrict
est ct number
u be o
of
polymers NL passing to left, right
y Potential can pull polymers to one side
y
NL
NR
Characterizing the restricted
system
Calculate p
partition function of system
y
with topological restriction
y Determine
ete
e polymer
po y e density
e s ty response
espo se to
constriction
y Connect thermal polymer system to
nonequilibrium quantum system
y
Entropic force
y
Ground state of Fermi system:
y
Polymers experience “level repulsion”
y In
I thermodynamic
h
d
i limit,
li i O(N2)
contribution to free energy is the
maximum-probability
i
b bili configuration
fi
i ρ(x)
( )
y
Free energy minimization
τ
x
Polymer density: gaps and
singularities
Polymer
density
τ
x
a
g
Lateral
position
Polymer density: gaps and
singularities
Polymer
density
a
g
Lateral
position
Why a gap?
Emergent
g
long-range
g
g force
y Contact between two regions of
polymers
po
y e s at later
ate “time”
t e
Gapp size
y
τ
Pin position
x
Force law for topological pin
Force on p
pin ~ N2 T/L
y For NL = 0,
y Small displacement: Hooke
Hooke’ss law
y Tight constriction: logarithmic divergence
y
Conclusions
Interactions strongly modify 2D polymer
behavior
y Topological constraints can generate longrange effects in polymer system and
connect to nonequilibrium quantum
systems
t
y Polymer distributions and correlations
can be described using quantum manymany
body techniques
y
Support from an NDSEG Fellowship and NSF