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Lattice Field Theory Analysis of the Chiral Heisenberg Model

Simon Hands, Johann Ostmeyer

Abstract

Motivated by ongoing interest in the universal behaviour of the Hubbard model of spinning electrons on honeycomb and $π$-flux lattices at the semi-metal -- Mott insulator phase transition, we formulate the \threeD~chiral Heisenberg model, a theory of relativistic fermions in three spacetime dimensions, as a lattice field theory using domain wall fermions. The contact interaction term preserves an SU(2) global symmetry. We perform numerical simulations using the Rational Hybrid Monte Carlo algorithm on system sizes $L^3\times L_s$ with $L\in\{8,\ldots,24\}$ and domain wall separation $L_s\in\{8,16,24\}$. We locate the phase transition corresponding to spontaneous SU(2)$\to$U(1) breaking, yielding critical exponent estimates $ν^{-1}=0.63(3)$, $η_Φ=1.42(8)$. These values are considerably removed from estimates obtained from simulations performed in (2+1)D, ie. with the time and spatial directions treated differently, but align more closely with analytic estimates obtained using 3D covariant field theory. We also present first results for the fermion correlator, ultimately needed for the determination of the exponent $η_Ψ$, highlighting the need to rotate the fermion source to a common reference direction in isospace in order to obtain a signal.

Lattice Field Theory Analysis of the Chiral Heisenberg Model

Abstract

Motivated by ongoing interest in the universal behaviour of the Hubbard model of spinning electrons on honeycomb and -flux lattices at the semi-metal -- Mott insulator phase transition, we formulate the \threeD~chiral Heisenberg model, a theory of relativistic fermions in three spacetime dimensions, as a lattice field theory using domain wall fermions. The contact interaction term preserves an SU(2) global symmetry. We perform numerical simulations using the Rational Hybrid Monte Carlo algorithm on system sizes with and domain wall separation . We locate the phase transition corresponding to spontaneous SU(2)U(1) breaking, yielding critical exponent estimates , . These values are considerably removed from estimates obtained from simulations performed in (2+1)D, ie. with the time and spatial directions treated differently, but align more closely with analytic estimates obtained using 3D covariant field theory. We also present first results for the fermion correlator, ultimately needed for the determination of the exponent , highlighting the need to rotate the fermion source to a common reference direction in isospace in order to obtain a signal.

Paper Structure

This paper contains 11 sections, 35 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Formulation and Symmetry
  3. Numerical Results
  4. Finite Volume Scaling
  5. Comparison with Previous Work
  6. Fermion Correlator
  7. Specification of Observable
  8. THC fit results
  9. Summary
  10. Sign of the Determinant
  11. Gauge fixing (rotations into $z$-basis)

Figures (9)

  • Figure 1: Scalar field histories over 1000 RHMC trajectories on $12^3\times8$. Coloured lines show the component $\phi_{i=1\ldots3}$; the black line is $\vert\Phi\vert$ (see eq. \ref{['eq:pseudo']}).
  • Figure 2: Pseudo-order parameter $\vert\Phi\vert$ (see eq. \ref{['eq:pseudo']}) vs. coupling $\beta$ for 5 spacetime volumes $L^3$ with $L_s=16$. Data from $L_s=8,24$ are virtually indistinguishable.
  • Figure 3: Binder cumulant $B$ as a function of the coupling $\beta$ for $L_s=16$. The plots look very similar for $L_s=8,24$. Error bars for $L\le 12$ are too small to be visible on this scale, results for $L=24$ are very noisy and have been omitted.
  • Figure 4: Data collapse for the FVS \ref{['eq:FVS']} using cubic fits. Results are provided in Tab. \ref{['tab:fsscubic']}. The panels differ in domain wall separation $L_s$.
  • Figure 5: Critical exponents $\nu^{-1}$, $\eta_\Phi$ from Refs. Ladovrechis:2022aofZerf:2017zqiGracey:2018qbaKnorr:2017yzeJanssen:2014geaLang:2025cwlLang:2018cskWang:2026jwfOtsuka:2020lhcOtsuka:2015ibaLiu:2021npkLiu:2018swwParisenToldin:2014nkkXu:2020qbjOstmeyer:2021efsOstmeyer:2020uovBuividovich:2018crqBuividovich:2018yar as listed in Tab. \ref{['tab:compendium']}. Points labelled "3D" (including this work) correspond to covariant methods with fully preserved Lorentz symmetry in 3 dimensions and are listed in the upper part of the table. Points labelled "(2+1)D" correspond to non-covariant methods that distinguish 2 spatial dimensions from one separate temporal dimension and are listed in the lower part of the table. The absence of error bars indicates that no uncertainties were provided in the references, respectively.
  • ...and 4 more figures