[Paper Review] Extracting Physics from Topologically Frozen Markov Chains

Urs Gerber, Irais Bautista+3 more|arXiv (Cornell University)|Oct 2, 2014

Topological and Geometric Data Analysis3 citations

TL;DR

This paper proposes methods to extract physical observables and topological susceptibility from Monte Carlo simulations trapped in a single topological sector due to long auto-correlation times. Using the Brower-Chandrasekharan-Negele-Wiese (BCNW) formula and topological charge density correlations, it demonstrates accurate estimation of observables like action density and magnetic susceptibility, with high precision for $\chi_{\rm t}$ when $\langle Q^2 \rangle \gtrsim 1.5$.

ABSTRACT

In Monte Carlo simulations with a local update algorithm, the auto-correlation with respect to the topological charge tends to become very long. In the extreme case one can only perform reliable measurements within fixed sectors. We investigate approaches to extract physical information from such topologically frozen simulations. Recent results in a set of sigma-models and gauge theories are encouraging. In a suitable regime, the correct value of some observable can be evaluated to a good accuracy. In addition there are ways to estimate the value of the topological susceptibility.

Motivation & Objective

• Address the challenge of long auto-correlation times in local update algorithms that trap simulations in fixed topological sectors.
• Develop reliable methods to extract physical expectation values $\langle \Omega \rangle$ and topological susceptibility $\chi_{\rm t}$ from simulations restricted to fixed $Q$.
• Assess the viability of the BCNW formula and topological charge density correlation method in non-trivial models like the 2d $O(3)$ model and 1d $O(2)$ model.
• Validate the accuracy of these methods in regimes where $\langle Q^2 \rangle \gtrsim 1.5$, ensuring convergence to physical results.
• Explore applicability to 4d gauge theories and QCD, where topological freezing is a growing concern with finer lattices.

Proposed method

• Apply the Brower-Chandrasekharan-Negele-Wiese (BCNW) formula to reconstruct $\langle \Omega \rangle$ from topologically restricted measurements $\langle \Omega \rangle_{|Q|}$, assuming $|Q| \leq 2$.
• Use the Aoki-Fukaya-Hashimoto-Onogi formula for the correlation of topological charge density: $\langle q_0 q_x \rangle_{|Q|, |x| \gg 1} \approx -\frac{\chi_{\rm t}}{V} + \frac{Q^2}{V^2}$, to estimate $\chi_{\rm t}$.
• Perform simulations using the Wolff cluster algorithm to enable efficient sampling and direct measurement of $\chi_{\rm t}$ for validation.
• Test the methods on the 1d $O(2)$ model (quantum rotor) and 2d $O(3)$ model with standard, Manton, and constraint actions.
• Fit observables like action density $s = \langle S \rangle / V$ and magnetic susceptibility $\chi_{\rm m}$ across different $L$-ranges to assess convergence and accuracy.
• Compare fitting results from $|Q| \leq 2$ sectors with directly measured values in full $Q$-ensembles to validate the method.

Experimental results

Research questions

• RQ1 Can physical expectation values $\langle \Omega \rangle$ be accurately reconstructed from measurements restricted to fixed topological charge $Q$?
• RQ2 To what extent does the BCNW formula provide reliable estimates of $\langle \Omega \rangle$ and $\chi_{\rm t}$ in finite volumes with $\langle Q^2 \rangle \lesssim 10$?
• RQ3 How effective is the topological charge density correlation method for estimating $\chi_{\rm t}$ in practical simulations with moderate $V$?
• RQ4 What is the required minimum $\langle Q^2 \rangle$ for reliable results using the BCNW formula or charge density correlation?
• RQ5 Can these methods be extended to 4d gauge theories and QCD simulations with fine lattices where topological freezing is severe?

Key findings

• The BCNW formula enables accurate reconstruction of $\langle \Omega \rangle$ for observables like action density and magnetic susceptibility, with results converging to the full-ensemble value when $\langle Q^2 \rangle \gtrsim 1.5$.
• For the 2d $O(3)$ model with $\beta = 1$, the fitted action density $s$ from $|Q| \leq 2$ sectors at $L=32$ matches the directly measured value $1.24008(5)$ within $1\sigma$.
• The magnetic susceptibility $\chi_{\rm m}$ is extracted with exceptional precision: fits yield $36.57(2)$ at $L=128$, matching the directly measured value $36.57(2)$.
• Topological susceptibility $\chi_{\rm t}$ estimated via the BCNW formula agrees with direct measurements within less than $2\sigma$, e.g., $0.00259(14)$ vs. $0.002790(5)$ at $L=128$.
• The charge density correlation method remains viable for $\langle Q^2 \rangle \gtrsim 2/3$, though signal-to-noise degrades rapidly in larger volumes, limiting its use in 4d models.
• The methods are promising for QCD simulations, where typical $\langle Q^2 \rangle \sim 10$, and have been validated in related models including the Schwinger model and 4d $SU(2)$ gauge theory.