Deconfinement in N=1 super Yang-Mills theory on R^3 x S^1 via dual-Coulomb gas and "affine" XY-model

TL;DR

The paper analyzes the thermal deconfinement transition of N=1 SU(2) super Yang-Mills on R^3 × S^1 with a small spatial circle. It develops two equivalent effective descriptions—the dual-Coulomb gas of electric and magnetic charges (coupled to a light modulus) and an affine XY-model with a symmetry-breaking perturbation—and validates them through Monte Carlo simulations. The results indicate a continuous deconfinement transition, with Z_2^(L) center symmetry remaining unbroken at Tc and the discrete R-symmetry restored, resembling the transition in SU(2) QCD(adj) with continuously varying critical exponents. The work provides a controlled framework for studying confinement and deconfinement in 2D Coulomb-gas-like systems and highlights the role of neutral bions and the modulus in supersymmetric settings. It also opens avenues to connect these gauge-theory dynamics with 2D condensed-matter analogs and lattice studies of related theories.

Abstract

We study finite-temperature N=1 SU(2) super Yang-Mills theory, compactified on a spatial circle of size L with supersymmetric boundary conditions. In the semiclassical small-L regime, a deconfinement transition occurs at T_c <<1/L. The transition is due to a competition between non-perturbative topological "molecules"---magnetic and neutral bion-instantons---and electrically charged W-bosons and superpartners. Compared to deconfinement in non-supersymmetric QCD(adj) arXiv:1112.6389, the novelty is the relevance of the light modulus scalar field. It mediates interactions between neutral bions (and W-bosons), serves as an order parameter for the Z_2^{L} center symmetry associated with the non-thermal circle, and explicitly breaks the electric-magnetic (Kramers-Wannier) duality enjoyed by non-supersymmetric QCD(adj) near T_c. We show that deconfinement can be studied using an effective two-dimensional gas of electric and magnetic charges with (dual) Coulomb and Aharonov-Bohm interactions, or, equivalently, via an XY-spin model with a symmetry-breaking perturbation, where each system couples to the scalar field. To study the realization of the discrete R-symmetry and the Z_2^{beta} thermal and Z_2^{L} non-thermal center symmetries, we perform Monte Carlo simulations of both systems. The dual-Coulomb gas simulations are a novel way to analyze deconfinement and provide a new venue to study the phase structure of a class of two-dimensional condensed matter models that can be mapped into dual-Coulomb gases. Our results indicate a continuous deconfinement transition, with Z_2^{L} remaining unbroken at the transition. Thus, the SYM transition appears similar to the one in SU(2) QCD(adj) arXiv:1112.6389 and is also likely to be characterized by continuously varying critical exponents.

Figures (13)

• Figure 1: The scales in the finite-temperature problem. The bion size is much smaller than the inverse temperature, which, in turn, is much smaller than inter-bion separation, i.e., $r_{*} < \beta \ll {\Delta R}_{\rm bion}$.
• Figure 2: LEFT: Magnetic and electric charge densities as a function of the temperature. RIGHT: Aharonov-Bohm phase contribution to the partition function (see text for definition and interpretation). Two volumes, $N=16,32$ were only considered in the dual-Coulomb gas simulation.
• Figure 3: Dual Coulomb gas scalar field observables, Eq. (\ref{['scalarobservables']}). LEFT: the average of $|\phi|$. RIGHT: susceptibility of $\phi$. We interpret these results as showing that, for $T>2$, the field strongly fluctuates around $\phi=0$; see text and Fig. \ref{['fig:CoulombScalar2']}.
• Figure 4: LEFT to RIGHT panel: Histograms of distributions of values of $\phi$, for $N=16$, for $T=0.4, 2$ and $5$. The histograms for $N=32$ are identical. These confirm our interpretation of Fig. \ref{['fig:CoulombScalar']}. (The histograms are normalized, i.e., the area under each curve equals one. In this and the following histograms, $10000$ Monte Carlo sweeps of the lattice were made at each temperature. Configurations were taken at every sweep, with the first 2000 neglected for equilibration.)
• Figure 5: $W$-boson fugacities $\xi_W(\phi)$ for $\kappa = {g^2 \over 2 \pi} = 4\pi$. LEFT: $T=0.4$, MIDDLE: $T=2$, RIGHT: $T=5$. The one-loop expression of (\ref{['the W fugacity as function of phi']}) is shown by a thick line. The modified fugacity of Eq. (\ref{['positivecore']}), used in the dual-Coulomb gas simulation to keep core energies positive, is shown by a dashed line. At low-$T$, the fugacities vary strongly with $\phi$---and would prefer values of $\phi$ near the edge of the Weyl chamber, where $W$-bosons become massless and our abelian description is not appropriate. However $\phi$ is a dynamical variable, whose value is determined by balancing the $W$-boson and neutral bion ($\cosh 2 \phi$) contribution. The latter is more important at low $T$ and favors $\phi \sim 0$, where the $W$'s are massive. At higher $T$, the fugacities flatten out as functions of $\phi$. See section \ref{['extrapolationsection']} for discussion of the weak coupling behavior of the fugacity.