[Paper Review] Kosterlitz-Thouless theory and lattice artifacts

János Balog|arXiv (Cornell University)|Nov 23, 2000

Theoretical and Computational Physics1 references9 citations

TL;DR

This paper investigates lattice artifacts in the massive continuum limit of the 1+1 dimensional O(2) nonlinear sigma model (XY model), showing they vanish only as inverse powers of the logarithm of the correlation length, not as standard Symanzik powers of the lattice spacing. Using the exact S-matrix bootstrap for the Sine-Gordon model and renormalization group-improved perturbation theory, the authors derive universal, calculable expressions for leading lattice artifacts in phase shifts and the Noether current correlation function, with a parameter-free two-loop formula provided for the latter.

ABSTRACT

The massive continuum limit of the 1+1 dimensional O(2) nonlinear $σ$-model (XY model) is studied using its equivalence to the Sine-Gordon model at its asymptotically free point. It is shown that leading lattice artifacts are universal but they vanish only as inverse powers of the logarithm of the correlation length. Such leading artifacts are calculated for the case of the scattering phase shifts and the correlation function of the Noether current using the bootstrap S-matrix and perturbation theory respectively.

Motivation & Objective

• To understand the nature of lattice artifacts in the massive continuum limit of the 2D O(2) nonlinear sigma model (XY model).
• To determine whether these artifacts follow the standard Symanzik power-law behavior or a slower, logarithmic convergence.
• To calculate leading lattice artifacts for physical observables such as scattering phase shifts and the Noether current correlation function.
• To establish universality and calculability of these artifacts using exact S-matrix data and renormalization group-improved perturbation theory.
• To provide a parameter-free two-loop formula for lattice artifacts in the Noether current correlation function.

Proposed method

• Mapping the XY model to the Sine-Gordon model via known duality relations, exploiting its exact solvability.
• Using the exact bootstrap S-matrix of the Sine-Gordon model to compute leading lattice artifacts in scattering phase shifts.
• Applying renormalization group-improved perturbation theory to compute the two-point correlation function of the Noether current.
• Introducing an external field coupled to the Noether charge to non-perturbatively determine a universal constant required for the current correlation function.
• Deriving a two-loop formula for lattice artifacts in the current correlation function using the exact value of the universal constant obtained from the free energy in an external field.
• Employing asymptotic expansions in the running coupling and logarithmic scaling to analyze the continuum limit behavior.

Experimental results

Research questions

• RQ1 Do lattice artifacts in the 2D O(2) model vanish as inverse powers of the lattice spacing, as in standard Symanzik theory?
• RQ2 What is the functional form of the leading lattice artifacts in the massive continuum limit of the O(2) model?
• RQ3 Can the leading lattice artifacts be calculated universally and non-perturbatively using the exact S-matrix of the Sine-Gordon model?
• RQ4 How do the lattice artifacts in the Noether current correlation function behave, and can they be expressed in a parameter-free form?
• RQ5 What is the role of the universal constant $ \kappa $, and how is it determined non-perturbatively in this context?

Key findings

• Lattice artifacts in the 2D O(2) model vanish as inverse powers of the logarithm of the correlation length, not as integer powers of the lattice spacing, indicating a much slower approach to the continuum limit than in most lattice field theories.
• The leading lattice artifacts are universal and calculable, with explicit expressions derived for the scattering phase shifts using the exact S-matrix bootstrap method.
• For the Noether current correlation function, a parameter-free two-loop formula for lattice artifacts is derived, with the universal constant $ \kappa $ determined non-perturbatively via the free energy in an external field.
• The value of the universal constant $ \kappa $ is found to be $ \kappa = -\frac{1}{4} - \ln\sqrt{2\pi} $, which ensures consistency with the exact S-matrix result for the free energy in an external field.
• The asymptotic expansion of the current correlation function's leading correction is shown to contain no free parameters, with the leading term expressed as $ \tilde{I}_1(p/M) = \frac{2}{3\pi} \left( \frac{1}{2\lambda} - 2.419 \right) $, where $ \lambda $ is the running coupling.
• The agreement between the perturbative expansion and the exact result for the free energy in an external field confirms the consistency of the method and the correctness of the derived universal constant.