Scaling and Luescher Term in a non-Abelian (2+1)d SU$(2)$ Quantum Link Model

Abstract

We investigate a non-Abelian SU$(2)$ quantum link model in 2+1 dimensions on a hexagonal lattice using tensor network methods. We determine the static quark potential for a wide range of bare coupling values and find that the theory is confining. We also probe the existence of a Luescher term and find a clear signal, however, the value of the dimensionless constant $γ$ strongly deviates from the expected universal value $-π/24$ for almost all values of the coupling $g^2$ we investigated. The width of the strings scales logarithmically with the string length again for all $g^2$-values, providing evidence for a rough string, with no indication for a roughening transition.

Figures (16)

• Figure 1: All eight possible plaquette environments for the $\{5\}$ representation, each with their two supported configurations. The fractions represent the charges external to the plaquette at the six sites.
• Figure 2: Exemplary plot of the simulated geometry for $N_x=5$ and $N_y=8$. The black triangles indicate the even site basis sites and the black arrow signifies the way in which the $1$d MPS is inlaied into the $2$d system.
• Figure 3: Exemplary plot of a flux string ground state at $g^2=1.5$. For each link, the expectation value of $\mathbb{1}-\ket{0}\bra{0}$ is plotted signifiying the total flux carried by the link. Dark blue is $1$, yellow is $0$.
• Figure 4: $\text{Var}(H)/(\Delta m)^2 \approx (1-f)$ for the empty systems (top) and the systems with strings (bottom).
• Figure 5: Total runtime for each coupling both with and without a string in a log-log plot for $N_x=5$.