questions using quantum many-body physics tools

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1 Addressing soft-matter tt questions using quantum many-body physics tools D Zeb Rocklin Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois Paul M Goldbart, Shina Tan School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332

2 Approach Develop analogy between statistical mechanics of directed polymers in two dimensions and quantum one-dimensional systems (cf. de Gennes 1968) Apply modern quantum techniques to elucidate behavior of directed polymer systems

3 Mission Characterize density fluctuations, thermodynamics, and effect of geometry and topology in directed polymer system

4 Outline Part I: Polymer system and mapping Part II: Interactions and density structure and correlations eato Part III: Geometrically and topologically constrained systems

5 Part I: System Clean system of ideal polymers 2-dimensional: a substrate or a thin sheet Directed via tension or directional potential x τ

6 Collective excitations in low dimensions Interacting gparticles in 1D, or lines in 2D, yield only collective excitations Single-polymer ge e dynamics suppressed Emergent polymer fluid has new properties

7 Quantum particles to classical lines x τ

8 From thermal lines to quantum particles 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 Path integral over polymer conformations x i (τ)

9 Parameter relationships Quantum System Mass Position Time Inverse temperature System size Polymer System Tension (units of temperature) Lateral direction Longitudinal direction System length System width x τ

10 Part II: Interactions and Density Structure and Correlations

11 de Gennes (1968): noncrossing polymers Noncrossing paths = hardcore bosons In ID, hardcore bosons = free fermions Path integrals for free fermions ME Fisher (1984)

12 de Gennes (1968): noncrossing polymers Ground state dominance far from system boundaries Sum over single-particle e excitations X-ray form factor displays logarithmic divergence corresponding to Kohn anomaly of Fermi gas n(k) X-ray form factor k q/2kf -KF KF

13 Behavior of noncrossing polymers Friedel oscillations near edges Density-density sty e sty correlations Interpolymer width distribution ρ(x) Width Distribution x KF x ty Probabilit τ width Barelt et al. 1990

14 Lieb-Liniger Liniger (1963) model c = case: hardcore bosons/ free fermions/noncrossing polymers Bethe Ansatz solution: Single-particle excitations extinguished

15 Results via Lieb-Liniger Liniger System free energy Lateral correlations found from Lieb-Liniger ground state Friedel oscillations More general correlations require Lieb-Liniger excitation spectrum x e Energy Fre τ Strength of Interaction Lateral Distance

16 Alternative technique: bosonization Universal field theory for 1D systems Characterized by two T-L parameters u, K Conjugate fields represent density, phase fluctuations

17 Correlation results via bosonization x τ K= K>1 K=1 K<1 Free bosons Contact bosons Attractive fermions Hardcore bosons Free fermions Long-range bosons Repulsive fermions

18 Part II: Geometrical and Topological Restrictions

19 Topological impurities Weak external potential V(x,τ) can be handled perturbatively Strong potential t can restrict number of polymers N L passing to left, right Potential can pull polymers to one side N L N R

20 Characterizing the restricted system Calculate partition function of system with topological restriction Determine e polymer density response se to constriction Connect thermal polymer system to nonequilibrium quantum system

21 Entropic force Ground state of Fermi system: Polymers experience level repulsion In thermodynamic limit, i O(N 2 ) contribution to free energy is the maximum-probability bbili configuration i ρ(x) ( )

22 Free energy minimization τ x

23 Polymer density: gaps and singularities Polymer density τ x a g Lateral position

24 Polymer density: gaps and singularities Polymer density a g Lateral position

25 Why a gap? Emergent long-range g force Contact between two regions of polymers at later time p size Gap τ x Pin position

26 Force law for topological pin Force on pin ~ N 2 T/L For N L = 0, Small displacement: Hooke s law Tight constriction: logarithmic divergence

27 Conclusions Interactions strongly modify 2D polymer behavior Topological constraints can generate longrange effects in polymer system and connect to nonequilibrium quantum systems Polymer distributions and correlations can be described using quantum manybody techniques Support from an NDSEG Fellowship and NSF