Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective Hamiltonians and Wilson--Polchinski renormalisation

Ricky Li, Benoit Vicedo

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.

Effective Hamiltonians and Wilson--Polchinski 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 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.
Paper Structure (75 sections, 330 equations, 1 figure)

This paper contains 75 sections, 330 equations, 1 figure.

Table of Contents

  1. Introduction and overview
  2. Approaches to renormalisation
  3. Rigorous perturbative renormalisation
  4. Non-perturbative renormalisation: path-integrals
  5. Non-perturbative renormalisation: Hamiltonians
  6. The Hamiltonian Polchinski equation
  7. Free field realisations and smooth regularisations
  8. Effective Hamiltonian and Polchinski's equation
  9. Main example and future directions
  10. Background and motivation
  11. Current algebras
  12. Chiral and anti-chiral currents
  13. Fourier mode decompositions
  14. Currents on the cylinder.
  15. Local operators.
  16. ...and 60 more sections

Figures (1)

  • Figure 1: A plot of the $2$-loop renormalisation group flow of the anisotropic deformation of the $\mathfrak{su}_2$ WZW model at level $1$, in the marginal $(g_2, g_1)$-plane near the Kosterlitz–Thouless point $(g_2, g_1) = (0,0)$. The gray arrows indicate the direction of the flow. The blue/orange curves correspond to different negative/positive values of the $2$-loop renormalisation group invariant $C = g_1^2 - g_2^2$. The green lines are the separatrices $g_2 = - |g_1|$ corresponding to the Berezinskii--Kosterlitz--Thouless transition which separates the massless phase, where the theory flows to a massless free boson in the IR, from the massive phase where the mass grows in the IR. The shaded region where $C < 0$ and $g_2 > 0$ contains the renormalised trajectories emanating from the one parameter family of conformal field theories given by the free compactified boson with radius $R=2/\beta$, depicted by the red line along the positive real axis $g_1 = 0$ and $g_2 \geq 0$.

Theorems & Definitions (2)

  • Example 3.1
  • Example 3.2