[Paper Review] Deconstruction, 2d lattice Yang-Mills, and the dynamical lattice spacing

Joel Giedt|ArXiv.org|Dec 12, 2003

Black Holes and Theoretical Physics27 references17 citations

TL;DR

This paper investigates the dynamical lattice spacing mechanism in a deconstructed 2D lattice Yang-Mills theory, focusing on the bosonic sector of Cohen et al.'s supersymmetric construction. Using Monte Carlo simulations, it finds that quantum effects—driven by entropic dominance—favor zero-action configurations over the intended $a$-configurations, undermining the proposed continuum limit despite action deformation. The results suggest that semi-classical expectations fail due to non-perturbative entropic effects.

ABSTRACT

We study expectation values related to the dynamical lattice spacing that occurs in the recent supersymmetric 2d lattice Yang-Mills constructions of Cohen et al. [hep-lat/0307012]. For the purposes of this preliminary analysis, we restrict our attention to the bosonic part of that theory. That is, we compute observables in the fully quenched ensemble, equivalent to non-supersymmetric 2d lattice Yang-Mills with a dynamical lattice spacing and adjoint scalars. Our numerical simulations indicate difficulties with the proposed continuum limit. We find that expectation values tend to those of the undeformed ``daughter theory,'' in spite of the deformation suggested by Cohen et al. In an effort to understand these results, we examine the zero action configurations, with and without the deformation. Based on these considerations, we are able to interpret the simulation results in terms of entropic effects.

Motivation & Objective

• To test the viability of the dynamical lattice spacing mechanism proposed by Cohen et al. in a non-supersymmetric, bosonic version of their 2D lattice Yang-Mills construction.
• To assess whether the proposed deformation in the lattice action successfully selects the desired $a$-configurations in the quantum regime.
• To investigate the role of zero-action configurations and entropic effects in undermining the semi-classical continuum limit.
• To provide a foundational analysis of the bosonic sector before extending to the full supersymmetric theory.
• To evaluate whether a well-defined continuum limit can be achieved without scaling the deformation strength to zero in the thermodynamic limit.

Proposed method

• Adopt the bosonic part of the CKKU lattice action, omitting fermions, to construct a fully quenched 2D lattice Yang-Mills theory with adjoint scalars.
• Implement a deformation potential that favors $a$-configurations, with $a$ interpreted as a dynamical lattice spacing.
• Perform large-scale Monte Carlo simulations on $N \times N$ lattices to compute expectation values of key observables, including Wilson loops and scalar field correlators.
• Analyze classical minima of both undeformed and deformed actions to identify the structure of zero-action configurations and their degeneracy.
• Use the moduli space of the undeformed theory to classify zero-action configurations and compare them to the $a$-configurations.
• Interpret simulation results through the lens of entropic dominance, where the number of zero-action configurations overwhelms the energy preference for $a$-configurations.

Experimental results

Research questions

• RQ1 Does the deformation potential in the CKKU construction successfully select $a$-configurations in the quantum regime?
• RQ2 To what extent do zero-action configurations with no energy cost dominate the path integral, undermining the semi-classical selection of $a$-configurations?
• RQ3 Can the proposed dynamical lattice spacing mechanism lead to a well-defined continuum limit in the absence of supersymmetry?
• RQ4 How do entropic effects influence the quantum behavior of the lattice theory, particularly in relation to the number of degenerate vacua?
• RQ5 Is the continuum limit stable under quantum corrections when the deformation strength is not scaled to zero in the thermodynamic limit?

Key findings

• Monte Carlo simulations show that expectation values of observables such as Wilson loops and scalar correlators converge to those of the undeformed 'daughter' theory, not the target continuum theory.
• Despite the inclusion of a deformation potential favoring $a$-configurations, the quantum path integral is dominated by zero-action configurations that are not $a$-configurations.
• The number of zero-action configurations is exponentially large and entropically favored, overwhelming the energy bias introduced by the deformation potential.
• Classical analysis reveals that the $a$-configurations are not unique minima; instead, a vast degeneracy of zero-action configurations exists, including those with non-trivial topology and non-zero field strengths.
• The simulation results indicate that semi-classical reasoning—central to the CKKU construction—fails to describe the true quantum dynamics due to entropic effects.
• The authors conclude that the proposed mechanism for dynamical lattice spacing does not yield the desired continuum limit in the bosonic, non-supersymmetric case.