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Precision estimates of large charge RG exponents $Y_q$ in the 3D XY universality class

Martin Hasenbusch

TL;DR

This work determines the RG exponents $Y_q=3-D_q$ for large charge $q$ at the 3D $O(2)$ fixed point by combining an iterative large-$q$ strategy with a worm algorithm on an improved 3N XY lattice model that suppresses leading corrections to scaling. The key methodological advance is the use of non-trivial weights in the worm update and the computation of ratios $R_q(r)=C_q(r)/C_{q-1}(r)$ to extract $\Delta_q=D_q-D_{q-1}$ with high precision up to $q=64$, achieving close agreement with the large-$q$ EFT down to $q\approx 4$. The analysis yields precise coefficients $c_{3/2}=0.33163(9)$ and $c_{1/2}=0.2765(8)$ in the expansion $D_q=c_{3/2} q^{3/2}+c_{1/2} q^{1/2}-0.0937256+\cdots$, and shows that residual corrections are significantly smaller in the 3N model than in the standard XY model. These results provide robust benchmarks for large-charge EFTs in the 3D XY universality class and clarify systematic error sources in MC studies of $D_q$ at large $q$.

Abstract

We accurately compute the RG exponents $Y_q$ of large $q$ fields at the $O(2)$ invariant fixed point in three dimensions. We build on an iterative approach that has been previously proposed and is implemented by using the worm algorithm. We simulate an improved XY model, that has next-to-next-to-nearest couplings in addition to nearest ones. In the worm update we incorporate weights, which allows us to obtain accurate results up to $q=64$. For example we get $Y_q=1.76370(12)$, $0.89167(23)$, and $-0.11203(34)$ for $q=2$, $3$, and $4$, respectively. The comparison with the large $q$ effective field theory gives an excellent agreement down to $q=4$ and provides accurate estimates of the parameters of the effective field theory.

Precision estimates of large charge RG exponents $Y_q$ in the 3D XY universality class

TL;DR

This work determines the RG exponents for large charge at the 3D fixed point by combining an iterative large- strategy with a worm algorithm on an improved 3N XY lattice model that suppresses leading corrections to scaling. The key methodological advance is the use of non-trivial weights in the worm update and the computation of ratios to extract with high precision up to , achieving close agreement with the large- EFT down to . The analysis yields precise coefficients and in the expansion , and shows that residual corrections are significantly smaller in the 3N model than in the standard XY model. These results provide robust benchmarks for large-charge EFTs in the 3D XY universality class and clarify systematic error sources in MC studies of at large .

Abstract

We accurately compute the RG exponents of large fields at the invariant fixed point in three dimensions. We build on an iterative approach that has been previously proposed and is implemented by using the worm algorithm. We simulate an improved XY model, that has next-to-next-to-nearest couplings in addition to nearest ones. In the worm update we incorporate weights, which allows us to obtain accurate results up to . For example we get , , and for , , and , respectively. The comparison with the large effective field theory gives an excellent agreement down to and provides accurate estimates of the parameters of the effective field theory.

Paper Structure

This paper contains 15 sections, 50 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The model and the two-point function
  3. The worm algorithm for the XY model with 3N couplings
  4. Ratios $R_q(r)=C_{q}(r)/C_{q-1}(r)$
  5. Gain by using non-trivial weights
  6. Numerical results
  7. Derivative with respect to $K_1$
  8. Residual leading corrections to scaling
  9. Fit Ansätze
  10. The standard XY model
  11. 3N XY model
  12. Fitting by the large $q$-expansion
  13. Summary and outlook
  14. Acknowledgement
  15. Ratio of partition functions

Figures (7)

  • Figure 1: Data for $R_4(L)$ for the standard XY model at $K=0.45416475$ are analyzed. Estimates of $\Delta_4$ are plotted versus the minimal linear lattice $L_{min}$ that is taken into account in the fit. The estimates are obtained by using the Ansätze (\ref{['fit0']},\ref{['fit0om']},\ref{['fit2om']}). In the caption we denote the fits by fit 1, fit 2, and fit 3, respectively. Details are discussed in the text.
  • Figure 2: Data for the 3N XY model at $K_1=0.37069947$ and $K_3=0.0415$ are analyzed. Estimates of $\Delta_2$ obtained by fitting $R_2(L)$ using the Ansätze (\ref{['fit0']},\ref{['fit2']},\ref{['fit4']}) are plotted versus the minimal linear lattice size $L_{min}$ that is taken into account. In the caption we denote the fits by fit 1, fit 2, and fit 3, respectively. The solid line indicates our final estimate, while the dashed lines give the error. The values of $L_{min}$ are slightly shifted to avoid overlap of the symbols. Details are discussed in the text.
  • Figure 3: Same as FIG. \ref{['DeltaQ2']} but for $q=4$ instead of $q=2$.
  • Figure 4: Same as FIG. \ref{['DeltaQ2']} but for $q=8$ instead of $q=2$.
  • Figure 5: Same as FIG. \ref{['DeltaQ2']} but for $q=16$ instead of $q=2$.
  • ...and 2 more figures