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The renormalized Hamiltonian truncation method in the large $E_T$ expansion

J. Elias-Miro, M. Montull, M. Riembau

TL;DR

This work advances the renormalized Hamiltonian truncation approach by deriving the exact $\Delta H_n$ corrections to arbitrary order in the large-$E_T$ expansion and implementing them in 2D scalar theories. By separating high-energy contributions and expressing local-renormalization coefficients in terms of phase-space functions $\Phi_m(E)$, the authors construct a practical framework with explicit diagrammatic rules and fast local approximations. They test the method on the solvable $\phi^2$ perturbation and the nontrivial $\phi^4$ theory, showing improved convergence and closer agreement with perturbative results, including at strong coupling. The results open avenues for RG-inspired resummations and extensions to higher dimensions, offering a scalable route to accurate nonperturbative spectra in strongly coupled QFTs.

Abstract

Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact $g^2$ and some of the $g^3$ contributions for the $φ^4$ theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and implement them to compute the lowest energy levels at strong coupling. A simple diagrammatic representation of the corrections and various tests are also introduced.

The renormalized Hamiltonian truncation method in the large $E_T$ expansion

TL;DR

This work advances the renormalized Hamiltonian truncation approach by deriving the exact corrections to arbitrary order in the large- expansion and implementing them in 2D scalar theories. By separating high-energy contributions and expressing local-renormalization coefficients in terms of phase-space functions , the authors construct a practical framework with explicit diagrammatic rules and fast local approximations. They test the method on the solvable perturbation and the nontrivial theory, showing improved convergence and closer agreement with perturbative results, including at strong coupling. The results open avenues for RG-inspired resummations and extensions to higher dimensions, offering a scalable route to accurate nonperturbative spectra in strongly coupled QFTs.

Abstract

Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact and some of the contributions for the theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and implement them to compute the lowest energy levels at strong coupling. A simple diagrammatic representation of the corrections and various tests are also introduced.

Paper Structure

This paper contains 21 sections, 107 equations, 7 figures.

Table of Contents

  1. Introduction and review
  2. Calculation of $\Delta H$ at any order
  3. Scalar theories
  4. Case study $\hbox{\boldmath{$\phi$}}^{2}$ perturbation
  5. Two-point correction
  6. Three-point correction
  7. A numerical test
  8. Spectrum and convergence
  9. The $\hbox{\boldmath{$\phi$}}^4$ theory
  10. Two-point correction
  11. Local expansion and the phase-space functions
  12. Spectrum and convergence
  13. Three point correction and further comments
  14. Summary of the method and comparison with Ref. Rychkov:2014eea
  15. Conclusion and outlook
  16. ...and 6 more sections

Figures (7)

  • Figure 1: Comparison of both sides of Eqs. (\ref{['ch1']}) and (\ref{['ch22']}).
  • Figure 2: Left: comparison of the exact vacuum energy with the numerical result as a function of the coupling constant $g_2$ (for $V=g_2\int dx \phi^2$). Right: left plot with the $y$-axis zoomed in a factor $\times 20$.
  • Figure 3: Left: comparison of the exact vacuum energy with the numerical result as a function of the truncation energy $E_T$. Right: left plot zoomed in.
  • Figure 4: Left: comparison of the exact energy difference ${\cal E}_1-{\cal E}_0$ with respect the numerical result as a function of the truncation energy $E_T$. Right: left plot zoomed in. On both plots we have taken the absolute value of the curve corresponding to the $VV$ corrections, in blue.
  • Figure 5: Left: The vacuum energy ${\cal E}_0^i$ as a function of the truncation energy $E_T$ for a coupling of $g=0.1$. Right: Energy difference between the first $\mathds{Z}_2$-odd excited state and the vacuum energy ${\cal E}_0^i$ as a function of the truncation energy for $g=0.1$. In both plots, the dotted curves are computed with the truncated Hamiltonian while the solid and dashed curves are computed with the renormalized hamiltonian at order $VV$. Dashed and dotted lines correspond to the cutoffs $E_W=E_T/2$ and $E_W=E_T/5$. We have overlaid two dashed black lines corresponding to the calculation in perturbation theory, see. Eqs. (\ref{['spert1']}) and (\ref{['spert2']}).
  • ...and 2 more figures