Half-vortex unbinding and Ising transition in constrained superfluids

We analyze the thermodynamics of the atomic and (nematic) pair superfluids appearing in the attractive two-dimensional Bose-Hubbard model with a three-body hard-core constraint that has been derived as an effective model for cold atoms subject to strong three-body losses in optical lattices. We show that the thermal disintegration of the pair superfluidity is governed by the proliferation of fractional half vortices leading to a Berezinskii-Kosterlitz-Thouless transition with an unusual jump in the helicity modulus. In addition to the (conventional) Berezinskii-Kosterlitz-Thouless transition out of the atomic superfluid, we furthermore identify a direct thermal phase transition separating the pair and the atomic superfluid phases and show that this transition is continuous with critical scaling exponents consistent with those of the two-dimensional Ising universality class. We exhibit a direct connection between the partial loss of quasi-long-range order at the Ising transition between the two superfluids and the parity selection in the atomic winding number fluctuations that distinguish the atomic from the pair superfluid.

Figure 1

Vortex-antivortex pair (left) and half-vortex–antivortex pair (right) in a classical $XY$ model, where the phase is represented by the direction of the shown arrows. Whereas the arrows along the line separating the vortex-antivortex cores (denoted in the figure by $+$ and $-$) are aligned, they form a domain wall (string) of antiparallel spins in the case of half vortices.

Figure 2

Finite temperature phase diagram of constrained bosons on the square lattice for $t/\left|U\right|=0.08$ and $\mu /\left|U\right|=-0.5$. The ASF-DSF transition belongs to the two-dimensional Ising universality class (dashed line). The blue and red lines indicate the BKT transitions of the ASF and DSF, respectively, and are distinguished by the unbinding of vortices (${\nu }_V=1$) and half vortices (${\nu }_V=1/2$). Insets: different possible scenarios for the region where the two BKT lines and the Ising transition line meet.

Figure 3

Atomic ($G_1$) and dimer ($G_2$) equal-time Green's function at the largest distance $L/2$ for different values of the dimer hopping strength $t^{\prime }$, at fixed $T/t=0.625$.

Figure 4

Energy $E$, filling $n$, and helicity $Y$ as functions of the temperature $T$ for $t^{\prime }/t=0.6875$, providing no indication for first-order behavior at the ASF-DSF transition at ${(T/t)}_c\approx 0.73$, indicated by the dashed line.

Figure 5

Data collapse for the odd helicity $Y_{\mathrm{odd}}$ for $t^{\prime }/t=0.6875$ using $\kappa =0$ and $\nu =1$. Inset: $Y_{\mathrm{odd}}$ for different system sizes across the critical point at ${(T/t)}_c\approx 0.73$.

Figure 6

Data collapse for the condensate density $C_1$ for $t^{\prime }/t=0.6875$ using standard Ising critical exponents ($\beta =1/8$ and $\nu =1$). Inset: rescaled condensate density $C_1$ in the vicinity of the critical point exhibiting its algebraic decay $C_1\propto L^{-1/4}$.

Figure 7

Helicity $Y$ as a function of temperature $T/t$ for different system sizes for $t^{\prime }=0$ (left) and $t^{\prime }/t=0.875$ (right) at $t/\left|U\right|=0.08$ and $\mu /\left|U\right|=-0.5$. The solid (dashed) line relates to the usual (unusual) universal helicity jump $2T/\left(\pi \right|U\left|\right)$, $8T/\left(\pi \right|U\left|\right)$.

Figure 8

$A$ vs $T/t$ for $t^{\prime }=0$, $t/\left|U\right|=0.08$, and $\mu /\left|U\right|=-0.5$. The inset shows ${\chi }^2/\mathrm{DOF}$ vs $A$ for the fit of the finite-size data using system sizes $L=10,20,30,...,200$ to Eq. (7). The goodness of the fit shows a clear minimum at $A=1$, indicating the unbinding of integer vortices.

Figure 9

Goodness of the fit (${\chi }^2/\mathrm{DOF}$) vs $A$ using different fitting ranges for $t^{\prime }/t=0.375$ (left) and $t^{\prime }/t=0.75$ (right) at $t/\left|U\right|=0.08$ and $\mu /\left|U\right|=-0.5$. The minimum of the goodness, ${\chi }^2/\mathrm{DOF}$, is shifted toward $A=1$ or 4 as smaller system sizes are omitted from the fitting procedure.