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<p>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.</p><p>As a primary example of our general method, we apply it to the marginal anisotropic deformation of the <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="fraktur">su</mi> <mn>2</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathfrak{su}}_2 \)</EquationSource> </InlineEquation> 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.</p>

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Effective Hamiltonians and Wilson-Polchinski renormalisation

  • 期刊论文

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  • 发表日期：

    2026年7月2日

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  • Ricky Li,
  • Benoît Vicedo

摘要

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 su 2 \( {\mathfrak{su}}_2 \) 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.