Universal Properties of Self-Avoiding Walks from Two-Dimensional Field Theory

Abstract

We use the recently conjectured exact $S$-matrix of the massive $\mathrm{O}(n)$ model to derive its form factors and ground state energy. This information is then used in the limit $n\to 0$ to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio $C/D$ which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over $m$-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to $m\le 2$ gives results which are accurate to at least one part in $10^3$. This remarkable fact is traced to a softening of the $m>2$ branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body $S$-matrix, $S(0)=-1$. Since this is a general property of interacting 2d field theories, it suggests that similar approximations may well hold for other models. However, we also study the moments of the area of self-avoiding loops,

48 pages, UCSBTH-93-12, ISAS-93-75

References in corpus (2)

• Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails
• N=2 Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers

Cited by in corpus (61)

• Non-integrable Quantum Field Theories as Perturbations of Certain Integrable Models
• Linear response theory for a pair of coupled one-dimensional condensates of interacting atoms
• The Spin-Spin Correlation Function in the Two-Dimensional Ising Model in a Magnetic Field at $T=T_c$
• Polymer depletion effects near mesoscopic particles
• Infinite-time Average of Local Fields in an Integrable Quantum Field Theory after a Quantum Quench
• Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus
• Scattering Theory and Correlation Functions in Statistical Models with a Line of Defect
• Excitons in one-dimensional Mott insulators
• Universal amplitude ratios in the two-dimensional $q$-state Potts model and percolation from quantum field theory
• Finite temperature correlations in the one-dimensional quantum Ising model
• Transport in Quantum Dots from the Integrability of the Anderson Model
• Haldane Gapped Spin Chains: Exact Low Temperature Expansions of Correlation Functions
• Correction-to-scaling exponents for two-dimensional self-avoiding walks
• Polymers Confined between Two Parallel Plane Walls
• Correlation Functions in the Two-dimensional Ising Model in a Magnetic Field at $T=T_c$
• Finite Temperature Dynamical Correlations in Massive Integrable Quantum Field Theories
• Form factors approach to current correlations in one dimensional systems with impurities
• Optical conductivity of one-dimensional Mott insulators
• Off-Shell Dynamics of the O(3) Nonlinear Sigma-Model -- Beyond Monte-Carlo and Perturbation Theory
• Bosonic-type S-Matrix, Vacuum Instability and CDD Ambiguities
• Self-avoiding walks and polygons on the triangular lattice
• Universal amplitude ratios in the two-dimensional Ising model
• Mass scales and crossover phenomena in the Homogeneous Sine-Gordon Models
• New families of flows between two-dimensional conformal field theories
• Massive and massless phases in self-dual $Z_N$ spin models: some exact results from the thermodynamic Bethe ansatz
• Mapping between the Sinh-Gordon and Ising Models
• Finite-Volume Form Factors in Semiclassical Approximation
• Universal Amplitude Ratios of The Renormalization Group: Two-Dimensional Tricritical Ising Model
• Vacuum Expectation Values from a variational approach
• On the Short Distance Behavior of the Critical Ising Model Perturbed by a Magnetic Field
• Dynamical Spin Response of Doped Two-Leg Hubbard-like Ladders
• Short distance behaviour of correlators in the 2D Ising model in a magnetic field
• Mimicking interacting relativistic theories with stationary pulses of light
• Susceptibility amplitude ratios in the two-dimensional Potts model and percolation
• Correction induced by irrelevant operators in the correlators of the 2d Ising model in a magnetic field
• Local density of states of 1D Mott insulators and CDW states with a boundary
• Dynamical density correlation function of 1D Mott insulators in a magnetic field
• Integrable O(n) model on the honeycomb lattice via reflection matrices : Surface critical behaviour
• Universal Ratios in the 2-D Tricritical Ising Model
• Form Factors of Exponential Operators and Exact Wave Function Renormalization Constant in the Bullough-Dodd Model
• Boundary effects on the local density of states of one-dimensional Mott insulators and charge density wave states
• Dynamical Correlations of the Spin-1/2 Heisenberg XXZ Chain in a Staggered Field
• Boundary form factors of the sinh-Gordon model with Dirichlet boundary conditions at the self-dual point
• Random Bond Ising Model and Massless Phase of the Gross-Neveu Model
• Perturbative versus Non-perturbative QFT -- Lessons from the O(3) NLS Model
• Duality and Form Factors in the Thermally Deformed Two-Dimensional Tricritical Ising Model
• Massless flows between minimal W models
• Dynamical response functions in the quantum Ising chain with a boundary
• Correlators in integrable quantum field theory. The scaling RSOS models
• Exact Matrix Elements in Supersymmetric Theories
• Varying the Unruh Temperature in Integrable Quantum Field Theories
• Spectral properties of one-dimensional spiral spin density wave states
• Integrability, Non-integrability and confinement
• An exact universal amplitude ratio for percolation
• Long Polymers Near Wedges and Cones
• Local density of states of a quarter-filled one-dimensional Mott insulator with a boundary
• Form Factors in Off--Critical Superconformal Models
• A universal amplitude ratio for the q<4 Potts model from a solvable lattice model
• Some approximate analytical methods in the study of the self-avoiding loop model with variable bending rigidity and the critical behaviour of the strong coupling lattice Schwinger model with Wilson fermions
• Form Factors of the Tricritical Three-state Potts Model in its Scaling Limit
• Random Walk with a Boundary Line as a Free Massive Boson with a Defect Line