Flux confinement-deconfinement transition of dimer-loop models on three-dimensional bipartite lattices

Abstract

Motivated by recent work that mapped the low-temperature properties of a class of frustrated spin $S=1$ kagome antiferromagnets with competing exchange and single-ion anisotropies to the fully-packed limit (with each vertex touched by exactly one dimer or nontrivial loop) of a system of dimers and nontrivial (length $s > 2$) loops on the honeycomb lattice, we study this fully-packed dimer-loop model on the three-dimensional bipartite cubic and diamond lattices as a function of $w$, the relative fugacity of dimers. We find that the $w \rightarrow 0$ O($1$) loop-model limit is separated from the $w \rightarrow \infty$ dimer limit by a geometric phase transition at a nonzero finite critical fugacity $w_c$: The $w>w_c$ phase has short loops with an exponentially decaying loop-size distribution, while the $w<w_c$ phase is dominated by large loops whose loop-size distribution is governed by universal properties of the critical O($1$) loop soup. This transition separates two {\em distinct} Coulomb liquid phases of the system: Both phases admit a description in terms of a fluctuating divergence-free polarization field $P_μ(\mathbf{r})$ on links of the lattice and are characterized by dipolar correlations at long distances. The transition at $w_c$ is a flux confinement-deconfinement transition. Equivalently, and independent of boundary conditions, half-integer test charges $q=\pm 1/2$ are confined for $w>w_c$, but become deconfined in the small-$w$ phase. Although both phases are unstable to a nonzero fugacity for the charge $\pm 1/2$ excitations, the destruction of the $w >w_c$ Coulomb liquid is characterized by an interesting slow crossover, since test charges with $q=\pm 1/2$ are confined in this phase.

Figures (23)

• Figure 1: An example of a valid fully-packed configuration of the dimer-loop model on a $4\times3\times3$ cubic lattice with open boundary conditions. Note that the representation of dimers emphasizes their equivalence to trivial length $s=2$ loops that traverse a single link of the lattice in both directions. In such a fully-packed configuration, each site is touched by a single loop, which can be either a dimer (trivial loop) or a nontrivial loop of length $s>2$.
• Figure 2: ($a$) Links connecting a site to its neighbors are represented as ($\vec{r},\mu$) where $\mu$ is the orientations of the link and $\vec{r}$ the coordinate of the site. ($b$) Two examples of the local configuration at a site---a nontrivial loop passing through the site and a trivial loop touching the site---are mapped to the corresponding polarization field $P_{\mu}(\vec{r})$ defined on the lattice links (Eq. \ref{['eq:Pmu']}). In both cases, net divergence of this polarization field at $\vec{r}$ is $0$. ($c$). The three orientations of the principal planes through which we define the flux of the polarization field on the cubic lattice are shown in three colors and labeled by the direction of the normal to these planes. The construction of the polarization field and the definition of the fluxes for the diamond lattice are entirely analogous, except that the normals to the three principal planes are not orthogonal to each other, but are chosen to be along the three principal directions of the diamond lattice, along which periodic boundary conditions are imposed in our computational work. See Sec. \ref{['subsec:Pmu']}, Sec. \ref{['subsec:fluxandloopsize']}, and Sec. \ref{['subsec:fluxdistribution']} for details.
• Figure 3: When the fugacity $f_{1/2}$ for half-charge defects is nonzero, the partition function of the generalized dimer-loop model has contributions from configurations that are not fully packed. An example of such a configuration is shown here for a $4\times3\times3$ cubic lattice with open boundary conditions. Sites colored yellow are touched by a single segment of a nontrivial loop, and thus host half-charge defects, i.e., the corresponding configuration of the polarization field has a nonzero divergence at these vertices, corresponding to a charge $q=\pm 1/2$ located at these vertices. These defects always come in pair which terminates an open string of arbitrary length, including length $s=2$ open strings. There are thus two kinds of objects that occupy a single link of the lattice: trivial loops (dimers) of loop length $s=2$ that traverse a single link in both directions, and open strings of length $s=2$.
• Figure 4: a), b), c) Inequivalent local environments of a square lattice site. In general, a worm starts with a random site (the black circle) which can have one of these local environments. In the first two cases (a,b) the worm tail is fixed at this site, which becomes the entry site $e_0$ for the first pivot encountered in the worm construction. In (a) this first pivot is chosen randomly from the two possibilities displayed, with a probability $1/2$ for each. In (b), the choice of this pivot $\pi_0$ is uniquely determined by the orientation of the dimer, but is used only with a probability $1/2$, while the worm construction is aborted without doing anything with probability $1/2$. In (c), if the randomly placed site is associated with a half-integer charge, the worm update is either aborted with probability $1/2$ without doing anything, or this site is itself chosen as the first pivot with probability $1/2$. In the latter case, the entry to this first pivot is said to be from an "off-lattice" entry point $e_0=0$.
• Figure 5: Given that a pivot was entered from entry $e_k$, the exit $x_k$ via which the pivot is exited by the worm head is chosen from the allowed possibilities using probabilities assigned to each choice. (a) ---(f) This figure provides a pictorial illustration of the local information required to determine the relative weights (Eq. \ref{['eq:weight']}) that appears in detailed balance equation set (Eq. \ref{['eq:detailed_balance']}) that these probabilities must satisfy. Thus, the "off-lattice" entrance/exit choices are depicted with two half-integer charges at the position of the pivot. This is consistent with the fact that a factor of $f_{1/2}^2$ is incorporated in the corresponding relative weights. In case (d) the off-lattice exit is forbidden since that would lead to a unit-vortex of vorticity $\pm 1$ at $\pi_k$, whose fugacity $f_{1}$ has been set to zero in the present study.