Superfluid to normal phase transition in strongly correlated bosons in two and three dimensions

Using quantum Monte Carlo simulations, we investigate the finite-temperature phase diagram of hard-core bosons ($XY$ model) in two- and three-dimensional lattices. To determine the phase boundaries, we perform a finite-size-scaling analysis of the condensate fraction and/or the superfluid stiffness. We then discuss how these phase diagrams can be measured in experiments with trapped ultracold gases, where the systems are inhomogeneous. For that, we introduce a method based on the measurement of the zero-momentum occupation, which is adequate for experiments dealing with both homogeneous and trapped systems, and compare it with previously proposed approaches.

Figure 1

Finite-temperature phase diagram in two and three dimensions, and the mean-field (MF) prediction. In all dimensions, the phase diagram contains a superfluid (SF) lobe surrounded by the normal fluid (NF) phase.

Figure 2

(a) Superfluid stiffness in 2D for $\mu =0$ and several values of $L$. The error bars (not shown) are smaller than the point size used in the plot. (b) Data collapse according to the relation in Eq. (4). The inset in (b) shows the rescaled superfluid stiffness vs $T$.

Figure 3

(a) Derivative of the zero-momentum occupation $n_0$ with respect to the temperature for different values of $L$. The inset shows $n_0$ vs $T$. (b) Finite-size scaling of the height of the negative peak in $d{n}_0/dT$. The continuous line is a fit to the function $g\left(L\right)=a_0+a_1{L}^{7/4}{\ln }^3\left(a_{2}L\right)$. The inset shows the finite-size scaling of $T^{*}\left(L\right)$.

Figure 4

(a) Superfluid stiffness of the 3D system for $\mu =0$ and several values of $L$. (b) Data collapse according to the relation in Eq. (12). The inset shows the rescaled superfluid stiffness as a function of $T$.

Figure 5

(a) Condensate fraction in 3D for $\mu =0$ and several values of $L$. (b) Data collapse according to the relation in Eq. (14). The inset shows the rescaled condensate fraction as a function of $T$.

Figure 6

(a) $d{n}_0/dT$ for different values of $L$. The inset shows $n_0$ vs $T$. (b) Finite-size scaling of the height of the negative peak in $d{n}_0/dT$. The continuous line is a fit to the function $g\left(\ln L\right)=a_0+a_1\ln L$. The inset shows the finite-size scaling of $T^{*}\left(L\right)$.

Figure 7

(a) Two-dimensional density at $T/t=0.2012$ and a trapping potential $V_0/t=0.0003$, for $\mu =0$ in the center of the trap. (b) The corresponding density profile $\rho \left(r\right)$, as well as the local compressibility ${\kappa }_{\mathrm{diff}}\left(r\right)$, as a function of the distance from the center of the trap $r$. All distances $x$, $y$, and $r$ are measured in units of $a$, while the local compressibility is measured in units of $1/t$.

Figure 8

(a) $n_0$ as a function of $T$ and $\beta$ in a trapped 2D system with $V_0/t=0.00125$, $\mu =0$ in the center of the trap, and $L=128$. (b) Derivatives of $n_0$ with respect to $\beta$ and with respect to $T$.

Figure 9

Estimate of the critical points based on the local compressibility (black dots; based on a system with $L=256$) as well as the derivative of the zero-momentum occupation with respect to $\beta$ (red diamonds; based on a system with $L=128$). At the tip of the superfluid lobe, we include further results for different system sizes and trap strengths (yellow pentagon: $L=32$; blue triangle: $L=64$; violet empty circle: $L=256$). The phase diagram of the homogeneous system is also shown.

Figure 10

(a) The quantity $Q\left(T\right)$ as a function of temperature extracted from the momentum distributions shown in the inset. The exponent in Eq. (35) has been set to $s=3$. (b) Derivatives of $n_0$ with respect to $\beta$ and with respect to $T$. The three-dimensional system is prepared with $V_0/t=0.04$, $\mu =0$, and $L=32$.

Figure 11

Estimates of the critical points based on the local compressibility (black dots; based on a system with $L=64$) as well as the derivatives of the zero-momentum occupation with respect to $\beta$ (red diamonds; based on a system with $L=32$). At the tip of the superfluid lobe, we include further results for different system sizes and trap strengths (yellow pentagon: $L=16$; blue triangle: $L=20$). The estimates based on Eq. (35) are shown by purple empty triangles. The phase diagram of the homogeneous system is also drawn with green empty squares.