Effective Hamiltonians and Wilson--Polchinski renormalisation

TL;DR

This paper introduces a non-perturbative Hamiltonian approach to Wilsonian renormalisation in 2D quantum field theories, successfully deriving RG flows for models like sine-Gordon using vertex operator algebras.

Contribution

It develops a novel Hamiltonian Wilsonian renormalisation method employing vertex operator algebras, applicable to marginal deformations of conformal field theories.

Findings

Reproduces sine-Gordon RG flow near Kosterlitz--Thouless point

Derives Hamiltonian analogue of Polchinski's equation

Provides a non-perturbative framework for 2D QFT renormalisation

Abstract

We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski's equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the $\mathfrak{su}_2$ Wess--Zumino--Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz--Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.