String effects in the 3d gauge Ising model

TL;DR

This paper tests the effective string description of confinement in the 3d Z2 gauge Ising model by exploiting a duality-based algorithm to obtain high-precision Polyakov-loop correlator data across large quark separations. It contrasts the free bosonic string predictions with Nambu-Goto-type self-interaction corrections, finding that large-R, low-T data align with a pure bosonic string, while higher temperatures reveal corrections consistent with the first perturbative order of the Nambu-Goto action. The work demonstrates the utility of Polyakov-loop geometries to probe higher-order string effects and highlights model-dependent differences in effective string realizations across gauge theories. It also emphasizes the need for further studies to establish universality and to understand the role of boundary terms and self-interactions in different LGTs.

Abstract

We compare the predictions of the effective string description of confinement with a set of Montecarlo data for the 3d gauge Ising model at finite temperature. Thanks to a new algorithm which makes use of the dual symmetry of the model we can reach very high precisions even for large quark-antiquark distances. We are thus able to explore the large R regime of the effective string. We find that for large enough distances and low enough temperature the data are well described by a pure bosonic string. As the temperature increases higher order corrections become important and cannot be neglected even at large distances. These higher order corrections seem to be well described by the Nambu-Goto action truncated at the first perturbative order.

Figures (8)

• Figure 1: Sketch of the surface denoted by $L \times R,M$. In the example, $L=6$, $R=8$ and $M=2$. The circles indicate the links that intersect the surface.
• Figure 2: $Q_1$ for $N_t=24$ (i.e. $T=T_c/3$) at $\beta=0.75180$. The variable $z$ is defined as $z\equiv \frac{2R}{N_t}$. The continuous lines correspond to the free bosonic string prediction, while the two dashed lines correspond to the first Nambu-Goto correction. The difference between the two dashed and the two continuous lines keeps into account the uncertainty in our estimate of $\sigma$. The pure area law corresponds to the line $Q_1=0$. The threshold $z_c=2R_c/N_t$ beyond which the effective string picture is expected to hold is located at $z_c\sim 1$ for these values of $N_t$ and $\beta$.
• Figure 3: Same as fig. 2, but for $N_t=16$ (i.e. $T=T_c/2$) at $\beta=0.75180$. In this case we have $z_c\sim 1.5$.
• Figure 4: Same as fig. 2, but for $N_t=12$ (i.e. $T=2T_c/3$) at $\beta=0.75180$. In this case we have $z_c\sim 2$.
• Figure 5: Same as fig. 2, but for $N_t=10$ (i.e. $T=4T_c/5$) at $\beta=0.75180$. In this case we have $z_c\sim 2.4$.