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The k-folded sine-Gordon model in finite volume

Z. Bajnok, L. Palla, G. Takacs, F. Wagner

TL;DR

This work studies the $k$-folded sine-Gordon model $ ext{SG}(eta,k)$ in finite volume, focusing on vacuum splitting, the low-lying multi-particle spectrum, and vacuum expectation values of exponential fields.The authors combine the generalized twisted NLIE, the Truncated Conformal Space Approach (TCSA), and semiclassical instanton calculations to cross-check and validate the finite-volume predictions, including a test of the Lukyanov–Zamolodchikov vacuum expectation value formula in the folded theory.They establish that the leading finite-size vacuum splitting matches between NLIE and instanton analyses, confirm the $k$-dependent degeneracies and volume dependence of multi-particle levels via the KM3 $S$-matrices and Bethe–Yang equations, and provide strong numerical support for the LZ VEVs in finite volume.Overall, the $k$-folded model serves as a precise laboratory for instanton effects beyond the dilute gas approximation and for testing exact integrable structures in finite volume, with implications for understanding twisted NLIE, kink/breather spectra, and exact VEV formulas.The results underscore the robustness of integrability under $k$-fold folding and reinforce the role of finite-volume techniques in validating nonperturbative field-theory predictions.

Abstract

We consider the k-folded sine-Gordon model, obtained from the usual version by identifying the scalar field after k periods of the cosine potential. We examine (1) the ground state energy split, (2) the lowest lying multi-particle state spectrum and (3) vacuum expectation values of local fields in finite spatial volume, combining the Truncated Conformal Space Approach, the method of the Destri-de Vega nonlinear integral equation (NLIE) and semiclassical instanton calculations. We show that the predictions of all these different methods are consistent with each other and in particular provide further support for the NLIE method in the presence of a twist parameter. It turns out that the model provides an optimal laboratory for examining instanton contributions beyond the dilute instanton gas approximation. We also provide evidence for the exact formula for the vacuum expectation values conjectured by Lukyanov and Zamolodchikov.

The k-folded sine-Gordon model in finite volume

TL;DR

This work studies the $k$-folded sine-Gordon model $ ext{SG}(eta,k)$ in finite volume, focusing on vacuum splitting, the low-lying multi-particle spectrum, and vacuum expectation values of exponential fields.The authors combine the generalized twisted NLIE, the Truncated Conformal Space Approach (TCSA), and semiclassical instanton calculations to cross-check and validate the finite-volume predictions, including a test of the Lukyanov–Zamolodchikov vacuum expectation value formula in the folded theory.They establish that the leading finite-size vacuum splitting matches between NLIE and instanton analyses, confirm the $k$-dependent degeneracies and volume dependence of multi-particle levels via the KM3 $S$-matrices and Bethe–Yang equations, and provide strong numerical support for the LZ VEVs in finite volume.Overall, the $k$-folded model serves as a precise laboratory for instanton effects beyond the dilute gas approximation and for testing exact integrable structures in finite volume, with implications for understanding twisted NLIE, kink/breather spectra, and exact VEV formulas.The results underscore the robustness of integrability under $k$-fold folding and reinforce the role of finite-volume techniques in validating nonperturbative field-theory predictions.

Abstract

We consider the k-folded sine-Gordon model, obtained from the usual version by identifying the scalar field after k periods of the cosine potential. We examine (1) the ground state energy split, (2) the lowest lying multi-particle state spectrum and (3) vacuum expectation values of local fields in finite spatial volume, combining the Truncated Conformal Space Approach, the method of the Destri-de Vega nonlinear integral equation (NLIE) and semiclassical instanton calculations. We show that the predictions of all these different methods are consistent with each other and in particular provide further support for the NLIE method in the presence of a twist parameter. It turns out that the model provides an optimal laboratory for examining instanton contributions beyond the dilute instanton gas approximation. We also provide evidence for the exact formula for the vacuum expectation values conjectured by Lukyanov and Zamolodchikov.

Paper Structure

This paper contains 27 sections, 134 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The $k$-folded sine--Gordon model
  3. $\mathrm{SG}(\beta ,k)$: Lagrangian and symmetries
  4. The spectrum of local operators
  5. Truncated Conformal Space for $\mathrm{SG}(\beta ,k)$
  6. $\mathrm{SG}(\beta ,k)$ in the NLIE framework
  7. Vacuum structure and instantons in finite volume
  8. Symmetry considerations
  9. Leading finite size corrections to the vacuum energy
  10. Instantons in finite volume
  11. Multi-particle energy levels in finite volume
  12. Particle spectrum in the classical $\mathrm{SG}(\beta ,k)$ in infinite volume
  13. The particle spectrum of the quantum $\mathrm{SG}(\beta ,k)$
  14. The Bethe--Yang equations
  15. Comparison with the TCSA data
  16. ...and 12 more sections

Figures (11)

  • Figure 1: The potential with the identification (\ref{['period']}) for $k=4$
  • Figure 2: The one-instanton contribution to the vacuum energy
  • Figure 3: TCSA data versus predictions for $E_1$, $E_2$ and $E_3$ at $p=2/7$
  • Figure 4: The $p$ (in)dependence of $E_1$ and $E_2$ in various 4-folded models
  • Figure 5: Breather masses in the 2-folded model with $p=2/7$
  • ...and 6 more figures