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Worldvolume Hybrid Monte Carlo algorithm for group manifolds

Masafumi Fukuma

Abstract

The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is a reliable and versatile algorithm for addressing the numerical sign problem. It resolves the ergodicity issues commonly encountered in Lefschetz thimble-based approaches while maintaining low computational costs. In this paper, as a general framework for applying WV-HMC to lattice gauge theories, we extend the algorithm to systems defined on compact group manifolds. The key is to introduce a symplectic structure on the tangent bundle of the worldvolume and formulate molecular dynamics upon it. The validity of the proposed algorithm is demonstrated using the one-site model with a purely imaginary coupling constant.

Worldvolume Hybrid Monte Carlo algorithm for group manifolds

Abstract

The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is a reliable and versatile algorithm for addressing the numerical sign problem. It resolves the ergodicity issues commonly encountered in Lefschetz thimble-based approaches while maintaining low computational costs. In this paper, as a general framework for applying WV-HMC to lattice gauge theories, we extend the algorithm to systems defined on compact group manifolds. The key is to introduce a symplectic structure on the tangent bundle of the worldvolume and formulate molecular dynamics upon it. The validity of the proposed algorithm is demonstrated using the one-site model with a purely imaginary coupling constant.

Paper Structure

This paper contains 50 sections, 2 theorems, 341 equations, 12 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Complex analysis on complexified groups
  3. Complexification of a compact group
  4. Derivatives of functions on $G^\mathbb{C}$
  5. Cauchy's theorem for group manifolds
  6. Outline of GT/WV-HMC for group manifolds
  7. Anti-holomorphic gradient flows in $G^\mathbb{C}$
  8. Outline of GT-HMC for group manifolds
  9. Outline of WV-HMC for group manifolds
  10. Unconstrained molecular dynamics on $G^\mathbb{C}$
  11. Continuum version (Hamiltonian dynamics)
  12. Discrete version (molecular dynamics)
  13. Constrained molecular dynamics on $G^\mathbb{C}$
  14. Continuum version (Hamiltonian dynamics)
  15. Discrete version (molecular dynamics)
  16. ...and 35 more sections

Key Result

Theorem 1

Let $\mathcal{D}$ be a domain in $\mathbb{C}^N$ (viewed as $\mathbb{R}^{2N}$) and $f(z)$ be a holomorphic function on $\mathcal{D}$. Then, the integral $I_\Sigma$ of $f(z)$ over a real $N$-dimensional oriented submanifold $\Sigma\subset\mathcal{D}$, depends only on the boundary of $\Sigma$.

Figures (12)

  • Figure 1: Cauchy's theorem for $\mathbb{C}^N$.
  • Figure 2: Cauchy's theorem for $G^\mathbb{C}$.
  • Figure 3: Deformation of $\Sigma_0 = G$ into a submanifold $\Sigma$ of $G^\mathbb{C}$. The deformed surface $\Sigma$ approaches a Lefschetz thimble $\mathcal{J}$, which consists of points flowing out from a critical point $U_\ast$. $v \in T_U \Sigma$ ($n \in N_U \Sigma$) is the tangent (normal) vector at $U\in\Sigma$ lifted from a tangent (normal) vector $v_0 \in T_{U_0} \Sigma_0$ ($n_0 \in N_{U_0} \Sigma_0$) at $U_0\in\Sigma_0$.
  • Figure 4: $\mathbb{R}$-linear map $A:\,w_0=v_0+n_0 \mapsto w = E v_0 + F n_0$.
  • Figure 5: WV-HMC over the worldvolume $\mathcal{R}$. Measurements are performed for configurations in a subregion $\tilde{\mathcal{R}}$ consisting of configurations with $t \in [\tilde{T}_0, \tilde{T}_1]$. We set $T_0 = 0$ in the figure.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof