The SU(3)/Z_3 QCD(adj) deconfinement transition via the gauge theory/"affine" XY-model duality

TL;DR

This work studies the deconfinement transition of SU(3)/${\mathbb Z}_3$ QCD(adj) by exploiting a duality to an affine two-component XY-spin model on a 2d lattice. Using Monte Carlo simulations, the authors measure two Z3 order parameters, their susceptibilities, vortex density, energy, and specific heat across varying volumes, temperatures, and bion fugacities, finding a first-order deconfinement transition for moderate fugacities. Finite-size scaling and the energy distribution near the critical temperature exhibit clear signs of phase coexistence (double-peak histograms), supporting a discontinuous transition and yielding an estimate of latent heat per spin. The results illuminate the interplay between electric and magnetic degrees of freedom in the finite-temperature gauge theory, validate the spin-model dual description, and help chart the beta-L phase diagram for different flavor numbers, with implications for the connection between small-$L$ semiclassical dynamics and large-$L$ confinement. Overall, the study provides nonperturbative confirmation of a first-order deconfinement transition in the dual framework and advances understanding of electric-magnetic Coulomb-gas structure in QCD(adj).

Abstract

Earlier, two of us and M. Unsal [arXiv:1112.6389] showed that some 4d gauge theories, compactified on a small spatial circle of size L and considered at temperatures 1/beta near deconfinement, are dual to 2d "affine" XY-spin models. We use the duality to study deconfinement in SU(3)/Z_3 theories with n_f>1 massless adjoint Weyl fermions, QCD(adj) on R^2 x S^1_beta x S^1_L. The"affine" XY-model describes two "spins" - compact scalars taking values in the SU(3) root lattice, with nearest-neighbor interactions and subject to an "external field" preserving the topological Z_3^t and a discrete Z_3^chi subgroup of the chiral symmetry of the 4d gauge theory. The equivalent Coulomb gas representation of the theory exhibits electric-magnetic duality, which is also a high-/low-temperature duality. A renormalization group analysis suggests - but is not convincing, due to the onset of strong coupling - that the self-dual point is a fixed point, implying a continuous deconfinement transition. Here, we study the nature of the transition via Monte Carlo simulations. The Z_3^t x Z_3^chi order parameter, its susceptibility, the vortex density, the energy per spin, and the specific heat are measured over a range of volumes, temperatures, and "external field" strengths (in the gauge theory, these correspond to magnetic bion fugacities). The finite-size scaling of the susceptibility and specific heat we find is characteristic of a first-order transition. Furthermore, for sufficiently large but still smaller than unity bion fugacity (as can be achieved upon an increase of the S^1_L size), at the critical temperature we find two distinct peaks of the energy probability distribution, indicative of a first-order transition, as has been seen in earlier simulations of the full 4d QCD(adj) theory. We end with discussions of the global phase diagram in the beta-L plane for different numbers of flavors.

Figures (12)

• Figure 1: The two simplest possible phase diagrams connecting the small-$L$ and large-$L$ behavior for theories which are confining in the infinite-$L$ zero-$T$ (infinite-$\beta$ limit). The thick red lines in the lower left hand corner denote the location of the discontinuous phase transition studied here. In both diagrams, here and in Figure 2, "C" and "D" refer to confined and deconfined phases, while "$\chi$" and "$d\chi$" denote the realization of the $SU(n_f)$ continuous and $Z_{2 N_c n_f}$ discrete chiral symmetries in the various regions. $\beta_D$ and $\beta_\chi$ are the deconfinement and $SU(n_f)$-restoration temperatures, respectively, in the infinite-$L$ theory. $L_\chi$ denotes a critical radius $L$ beyond which $SU(n_f)$ is broken (at zero temperature).
• Figure 2: The $n_f=5$ theory is (most likely) conformal with a, presumably, sufficiently weak-coupling Banks-Zaks like fixed point $\alpha_*$. If so, upon increasing $L$, the semiclassical analysis is always applicable. In particular, it shows that the conformal zero-temperature theory confines at any finite $L$ with an exponentially small mass gap Poppitz:2009uq. There is a thermal deconfinement and $d\chi$-breaking transition with critical temperature given, for $SU(2)$, by $T_c(L) ={ \alpha(L) \over 2 L}$Anber:2011gn. The transition is second order for $SU(2)$. The line drawn on the figure corresponds to $T_c(L)$ drawn with the two-loop $SU(2)$ coupling. The transition for $SU(3)$ is---probably, see Section 2.3---discontinuous and occurs at temperatures $T' \sim 1$, parametrically giving $T_c(L) \sim {g_4^2(L)\over L}$, similar to the $SU(2)$ case. The semiclassical theory, now expected to be valid at all $L$, predicts that the discrete chiral symmetry, but not $SU(n_f)$, also breaks at the deconfinement transition.
• Figure 3: Order parameter vs. temperature $T'$ of the ${\mathbb Z}_3 \times {\mathbb Z}_3$ model with symmetry-breaking fields of strength (a): $h=0.1$, and (b): $h=1.0$ in lattices of width $N=8$ (green crosses), $N=16$ (blue stars), and $N=32$ (red circles).
• Figure 4: Energy per spin vs. temperature $T'$ of the ${\mathbb Z}_3 \times {\mathbb Z}_3$ model with symmetry-breaking fields of strength (a): $h=0.1$, and (b): $h=1.0$ in lattices of width $N=8$ (green crosses), $N=16$ (blue stars), and $N=32$ (red circles).
• Figure 5: Vortex density vs. temperature $T'$ of the ${\mathbb Z}_3 \times {\mathbb Z}_3$ model with symmetry-breaking fields of strength (a): $h=0.1$, and (b): $h=1.0$ in lattices of width $N=8$ (green crosses), $N=16$ (blue stars), and $N=32$ (red circles).