Finite-temperature phase structures of hard-core bosons in an optical lattice with an effective magnetic field

We study finite-temperature phase structures of hard-core bosons in a two-dimensional optical lattice subject to an effective magnetic field by employing the gauged CP${}^1$ model. Based on the extensive Monte Carlo simulations, we study their phase structures at finite temperatures for several values of the magnetic flux per plaquette of the lattice and mean particle density. Despite the presence of the particle number fluctuation, the thermodynamic properties are qualitatively similar to those of the frustrated $XY$ model, with only the phase as a dynamical variable. This suggests that cold-atom simulators of the frustrated $XY$ model are available irrespective of the particle filling at each site.

Figure 1

The ground-state energy per site of the CP${}^1$ model Eq. (28) as a function of the magnetic flux $f$ for the averaged density (a) $\rho =0.5$ and (b) 0.95, obtained through simulated annealing. The solid curve denotes the energy of the CP${}^1$ model without the chemical-potential term, while the dashed curve denotes that of the XY model with setting $J=t\rho (1-\rho )$ for comparison. The two curves overlap completely in (a); see the text for the reason. (c) and (d) The distribution over the lattice ($L=12$ and 10 for $f=1/2$ and 2/5, respectively) of the mean density $\langle {\rho }_{\overline{i}}\rangle$ (the upper panels) and the vorticity $m_{\overline{i}}$ (the lower panels) at each plaquette $\overline{i}$ (the site of the dual lattice) for $\rho =0.95$; (c) $f=1/2$ and (d) $f=2/5$.

Figure 2

The mean density $\rho$ and the particle number fluctuation ${\Delta }_{\rho }$ as a function of $\beta \mu$ at the high-temperature or zero-hopping limit $\beta t\to 0$; the behavior is thus independent of $f$.

Figure 3

The particle number fluctuation ${\Delta }_{\rho }$ as a function of $\beta t$ for (a) $\mu =0$ and (b) $\mu =0.7$. The solid, dashed, and dotted curves correspond to $f=0$, 1/2, and 2/5, respectively.

Figure 4

Several thermodynamic quantities for $f$=0: (a) the specific heat $C$, (b) the in-plane susceptibility $\chi$, and (c) the helicity modulus ${\rm Y}$ for the CP${}^1$ model with $\mu =0$ and $f=0$ as a function of $\beta t$. The system size for each curve is $L=12$, 24, 36, and 48. An additional curve ${\rm Y}=8/\pi \beta t$ in (c) indicates the universal jump of a BKT transition.

Figure 5

Several thermodynamic quantities for $f$=1/2: (a) the specific heat $C$, (b) the in-plane susceptibility $\chi$, and (c) the helicity modulus ${\rm Y}$ for the CP${}^1$ model with $\mu =0$ and $f=1/2$ as a function of $\beta t$. The system size for each curve is $L=12$, 24, 36, 48, and 60. An additional curve ${\rm Y}=8/\pi \beta t$ in (c) indicates the universal jump of a BKT transition.

Figure 6

The left panels show the power-law scaling collapse of the specific heat data for (a) FFXYM and (c) gauged CP${}^1$ model. The right panels show the power-law and logarithmic fitting of the specific heat peak with respect to sizes $L$ for (b) FFXYM and (d) the gauged CP${}^1$ model. The power-law fitted value is $\alpha /\nu =0.439$ for (a) and $\alpha /\nu =0.397$ for (c).

Figure 7

Data of the BKT transition of the FFXYM [(a) and (c)] and the gauged CP${}^1$ model for $f=1/2$ [(b) and (d)]. In (a) and (b) we show $T_{\mathrm{BKT}}$ obtained by method (i) (see the text) by the horizontal line; (a) $T=0.437J/k_{\mathrm{B}}$ and (b) $T=0.363J/k_{\mathrm{B}}$. In addition, the temperature corresponding to the crossing point in method (ii) is plotted as a function of $\eta$. Figures (c) and (d) show the best fit of the scaling relation Eq. (39) at the corresponding $T_{\mathrm{BKT}}$, which is obtained by finding the minimum of ${\chi }^2$-fit error shown in the inset.

Figure 8

Several thermodynamic quantities for the CP${}^1$ model with $\mu =0$ and $f=2/5$ as a function of $\beta t$. (a) The specific heat $C$, (b) the in-plane susceptibility, and (c) the helicity modulus ${\rm Y}$. The system size for each curve is $L=20$, 30, 40, and 50. (d) Specific heat vs $L^2.$

Figure 9

Several thermodynamic quantities for the CP${}^1$ model with $\mu =0.7$ and $f=1/2$ as a function of $\beta t$: (a) mean particle number $\rho$, (b) the specific heat $C$, (c) the in-plane susceptibility $\chi$, and (d) the helicity modulus ${\rm Y}$. The system size for each curve is $L=12$, 24, 36, and 48.