Boson Pairing and Unusual Criticality in a Generalized $XY$ Model

We discuss the unusual critical behavior of a generalized $XY$ model containing both $2\pi$-periodic and $\pi$-periodic couplings between sites, allowing for ordinary vortices and half-vortices. The phase diagram of this system includes both single-particle condensate and pair-condensate phases. Using a field theoretic formulation and worm algorithm Monte Carlo simulations, we show that in two dimensions it is possible for the system to pass directly from the disordered (high temperature) phase to the single particle (quasi)condensate via an Ising transition, a situation reminiscent of the “deconfined criticality” scenario.

Figure 1

Schematic phase diagram of the model Eq. (2). Solid lines are continuous transitions of the type indicated; the dashed line is first order. The dotted line indicates the region where the nature of the transition is unknown. The renormalization group analysis developed here applies in the vicinity of the solid dot. Inset: an illustration of a half-vortex pair and the associated string.

Figure 2

(Left) RG flow in the $\kappa -z_1$ plane for $J=\frac{31}{4\pi }$. Here the $\mu \cos (2\pi \tilde{h})$ perturbation is irrelevant, even though $⟨\mu ⟩\cos (2\pi \tilde{h})$ is relevant. Note that the Ising fixed point at $\kappa =0$, $z_1=0$ is stable. (Right) RG flow for $J=\frac{29}{4\pi }$. The Ising fixed point is now unstable.

Figure 3

Phase diagram near the meeting of the Ising (◇) and $\frac{1}{2}KT$ ($\times$) transitions determined by numerical simulation. The transitions are determined as described in the text with the Ising line stopping at the last point where we are confident of the nature of the transition. The dotted line is a sketch of the (presumably) first order line connecting to the usual KT transition (see Fig. 1).

Figure 4

The ratio Eq. (8) of partition functions is used to locate the Ising transition from the crossing point for different system sizes. Here ◇, $+$, □, $\times$ symbols are for systems sizes $L=20$, 40, 80, 160, respectively, $J=2.8$, and the (small) error bars are included. The dotted line corresponds to the pure Ising value ${\zeta }_c=0.3729$.

Figure 5

$\zeta$ plotted vs $\kappa L^{1/\nu }$. For the same $L$ as in Fig. 4, the best collapse is achieved for $\nu =1.00\pm 0.02$.