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Truncated Hilbert space approach to the 2d $φ^{4}$ theory

Z. Bajnok, M. Lajer

TL;DR

The paper tackles the non-integrable two-dimensional $φ$-four theory in the broken phase by applying a massive version of the truncated Hilbert space approach to finite-volume systems. It constructs a renormalized finite-volume Hamiltonian, diagonalizes it in a truncated free-boson basis, and extracts the finite-volume spectrum, which is interpreted through Bethe-Yang analysis to obtain infinite-volume masses and scattering data. The study reveals two neutral excitations (breathers) at moderate couplings, a kink sector whose mass follows from the vacuum energy splitting, and a critical point in the Ising universality class, while highlighting deviations of leading exponential finite-volume corrections from Lüscher’s formula due to inelastic processes. Overall, the work demonstrates the effectiveness of the truncated Hilbert space method for non-integrable QFTs, providing nonperturbative insight into masses, scattering, and critical behavior of the two-dimensional φ^4 theory.

Abstract

We apply the massive analogue of the truncated conformal space approach to study the two dimensional $φ^{4}$ theory in finite volume. We focus on the broken phase and determine the finite size spectrum of the model numerically. We interpret the results in terms of the Bethe-Yang spectrum, from which we extract the infinite volume masses and scattering matrices for various couplings. We compare these results against semiclassical analysis and perturbation theory. We also analyze the critical point of the model and confirm that it is in the Ising universality class.

Truncated Hilbert space approach to the 2d $φ^{4}$ theory

TL;DR

The paper tackles the non-integrable two-dimensional -four theory in the broken phase by applying a massive version of the truncated Hilbert space approach to finite-volume systems. It constructs a renormalized finite-volume Hamiltonian, diagonalizes it in a truncated free-boson basis, and extracts the finite-volume spectrum, which is interpreted through Bethe-Yang analysis to obtain infinite-volume masses and scattering data. The study reveals two neutral excitations (breathers) at moderate couplings, a kink sector whose mass follows from the vacuum energy splitting, and a critical point in the Ising universality class, while highlighting deviations of leading exponential finite-volume corrections from Lüscher’s formula due to inelastic processes. Overall, the work demonstrates the effectiveness of the truncated Hilbert space method for non-integrable QFTs, providing nonperturbative insight into masses, scattering, and critical behavior of the two-dimensional φ^4 theory.

Abstract

We apply the massive analogue of the truncated conformal space approach to study the two dimensional theory in finite volume. We focus on the broken phase and determine the finite size spectrum of the model numerically. We interpret the results in terms of the Bethe-Yang spectrum, from which we extract the infinite volume masses and scattering matrices for various couplings. We compare these results against semiclassical analysis and perturbation theory. We also analyze the critical point of the model and confirm that it is in the Ising universality class.

Paper Structure

This paper contains 29 sections, 123 equations, 24 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical, semiclassical and perturbative results
  3. Classical analysis
  4. Semiclassical considerations
  5. Related integrable models and perturbative results
  6. Quantum theory
  7. Truncated Hilbert space approach
  8. Mini Hilbert space approach
  9. Technical implementations
  10. Description of the spectrum
  11. Infinite volume data
  12. Polynomial finite size corrections
  13. Leading exponential corrections
  14. Numerical analysis and interpretations
  15. Periodic boundary condition
  16. ...and 14 more sections

Figures (24)

  • Figure 1: The kink--anti-kink $B_{1}$ fusion process, together with the corresponding pole in the kink $B_{1}$ scattering process.
  • Figure 2: The $B_{1}-B_{1}$ fusion process, together with the $B_{1}$-kink fusing to excited kink process.
  • Figure 3: The raw energy spectrum of the model as the function of the volume
  • Figure 4: The ground-state energy density $e_{0}(l)/l$ as the function of the volume for various truncated energy levels together with the extrapolated results. The energy density obtained at $l=7$ is indicated with a dashed line.
  • Figure 5: Data points as a function of the truncation energy together with the fitted function for the ground-state energy density at $g=0.06$ and volume $l=10$.
  • ...and 19 more figures