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A study of truncation effects in boundary flows of the Ising model on a strip

Gabor Zsolt Toth

TL;DR

This work investigates whether truncation in the truncated conformal space approach (TCSA) can be understood as renormalizing Hamiltonian coefficients, by studying the critical Ising model on a strip with a boundary magnetic field. It develops both an exact field-theoretic description and two truncation schemes—mode truncation and TCSA—and derives explicit forms for renormalization functions $s_0$ and $s_1$ to relate truncated spectra to the exact ones. Perturbative and numerical results show that the renormalization picture works well at leading orders but exhibits truncation-scheme-dependent behavior, especially at large coupling, with the mode truncation and TCSA producing different large-$h$ dynamics and second flows. The findings have implications for interpreting TCSA data on boundary flows and motivate extensions to other perturbed minimal models and deeper analyses of the boundary field theory structure in finite volume.

Abstract

We investigate the idea that the effect of the truncation applied in the TCSA method on the spectrum coincides with the effect of a suitable changing of the coefficients of the terms in the Hamiltonian operator. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. A detailed quantum field theoretical description of this model is also given, and we propose a description as a perturbation of the infinite coupling limit. The investigation is also carried out for a truncation method which preserves the solvability of the model. The results of perturbative and numerical calculations presented support the above idea and show that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method.

A study of truncation effects in boundary flows of the Ising model on a strip

TL;DR

This work investigates whether truncation in the truncated conformal space approach (TCSA) can be understood as renormalizing Hamiltonian coefficients, by studying the critical Ising model on a strip with a boundary magnetic field. It develops both an exact field-theoretic description and two truncation schemes—mode truncation and TCSA—and derives explicit forms for renormalization functions and to relate truncated spectra to the exact ones. Perturbative and numerical results show that the renormalization picture works well at leading orders but exhibits truncation-scheme-dependent behavior, especially at large coupling, with the mode truncation and TCSA producing different large- dynamics and second flows. The findings have implications for interpreting TCSA data on boundary flows and motivate extensions to other perturbed minimal models and deeper analyses of the boundary field theory structure in finite volume.

Abstract

We investigate the idea that the effect of the truncation applied in the TCSA method on the spectrum coincides with the effect of a suitable changing of the coefficients of the terms in the Hamiltonian operator. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. A detailed quantum field theoretical description of this model is also given, and we propose a description as a perturbation of the infinite coupling limit. The investigation is also carried out for a truncation method which preserves the solvability of the model. The results of perturbative and numerical calculations presented support the above idea and show that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method.

Paper Structure

This paper contains 21 sections, 94 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The exact spectrum
  3. The free model
  4. The perturbed model
  5. Reverse description
  6. The free model
  7. The perturbed model
  8. Exact spectrum in the Mode Truncated version
  9. The free model
  10. The perturbed model
  11. Perturbative results
  12. Mode Truncation scheme
  13. TCS scheme
  14. Numerical results
  15. Mode Truncation Scheme
  16. ...and 6 more sections

Figures (12)

  • Figure 1: Exact (dashed lines) and mode truncated (solid lines) energy gaps ($E_i-E_0$) in the $v$ and $u$ sectors respectively as a function of $\ln(h)$ at truncation level $n_c=9$
  • Figure 2: Exact (dashed lines) and mode truncated (solid lines) normalized spectra in the $v$ and $u$ sectors respectively as a function of $\ln(h)$ at truncation level $n_c=9$
  • Figure 3: The function $s_0(h)$ for the $v$ sector in the ranges $h\in [0,3]$, $h\in [0,20]$, $h\in [0,400]$ and $s_0\in [0.95,1.05]$ at truncation level $n_c=9$
  • Figure 4: The function $s_1(h)$ for the $v$ sector in the ranges $h\in [0,2]$, $h\in [0,20]$, $h\in [0,400]$ and $s_1\in [0,2]$ at truncation level $n_c=9$
  • Figure 5: The function $s_1(h)/s_0(h)$ for the $v$ sector in the ranges $h\in [0,2]$, $h\in [0,20]$, $h\in [0,400]$ and $s_1/s_0\in [0,2]$ at truncation level $n_c=9$
  • ...and 7 more figures