Phase diagram and thermodynamics of the three-dimensional Bose-Hubbard model

We report results of quantum Monte Carlo simulations of the Bose-Hubbard model in three dimensions. Critical parameters for the superfluid-to-Mott-insulator transition are determined with significantly higher accuracy than has been done in the past. In particular, the position of the critical point at filling factor $n=1$ is found to be at ${(U∕t)}_{\mathrm{c}}=29.34\left(2\right)$, and the insulating gap $\Delta$ is measured with accuracy of a few percent of the hopping amplitude $t$. We obtain the effective mass of particle and hole excitations in the insulating state—with explicit demonstration of the emerging particle-hole symmetry and relativistic dispersion law at the transition tip—along with the sound velocity in the strongly correlated superfluid phase. These parameters are the necessary ingredients to perform analytic estimates of the low temperature $(T⪡\Delta )$ thermodynamics in macroscopic samples. We present accurate thermodynamic curves, including these for specific heat and entropy, for typical insulating $(U∕t=40)$ and superfluid $(t∕U=0.0385)$ phases. Our data can serve as a basis for accurate experimental thermometry, and a guide for appropriate initial conditions if one attempts to use interacting bosons in quantum information processing.

Figure 1

Finite-temperature phase diagram at filling factor $n=1$. Solid circles are simulation results (the line is a guidance for the eye), error bars are plotted. $T=5.591t$ is the critical temperature of the ideal Bose gas with the tight binding dispersion relation. At finite, but low enough temperature, the MI domain is loosely defined as the part of the phase diagram to the right of the gray line. The rest of the nonsuperfluid domain is referred to as normal liquid (NL).

Figure 2

(Color online) Zero-momentum Green function in the Mott phase with the chemical potential $\mu ∕U=0.809$, slightly below the upper phase boundary. Here we show data for the system with $N={10}^3$ bosons at $U∕t=70$ and $T∕t=0.025$. In the inset we plot the energy gap ▵ for linear system sizes $L=10$ and 20. Finite-size errors are within the statistical error bars.

Figure 3

(Color online) Phase diagram of the first MI-SF lobe. Numerical data are shown by open circles. The error bars are shown but are barely visible even in the inset. The dashed lines and the square represent results of Ref. 15 for the strong coupling expansion and the extrapolated position of the diagram tip, respectively.

Figure 4

(Color online) Finite size scaling of the energy gap at the tip of the lobe. $\Delta L∕t$ vs $t∕U$ for system size $L=5$ (solid squares), $L=10$ (open circles), $L=15$ (solid circles), $L=20$ (open squares). Lines represent linear fits used to extract the critical point.

Figure 5

Extrapolation to the thermodynamic limit. We show the intersections (triangles) of the curves ($L=5$, $L=10$), ($L=10$, $L=15$), ($L=15$, $L=20$), vs $L_{\max }^{-2}$. The fit (solid line) yields ${(t∕U)}_{\mathrm{c}}=0.03408\left(2\right)$.

Figure 6

(Color online) Effective mass for hole (solid circles) and particle (open circles) excitations as a function of $t∕U$. The exact results at $t∕U=0$ are $m_{+}=0.25∕t$ and $m_{-}=0.5∕t$. By dashed lines we show the lowest order in $t∕U$ correction to the effective masses. Close to the critical point the two curves overlap, directly demonstrating the emergence of the particle-hole symmetry. At $t∕U=0.034$, the sound velocity is $c∕t=6.3\pm 0.4$.

Figure 7

(Color online). Energy per particle at $t∕U=0.005$, unity filling factor and linear system size $L=20$. Solid circles are prime data (error bars within symbol size), the solid line is the analytical prediction from Eq. (7), where, at each temperature, the chemical potential has been fixed by imposing equal number of particle and hole excitations. The inset shows the total number of excitations present in the system.

Figure 8

(Color online) Energy (top), specific heat (middle), and entropy (bottom) per particle at $t∕U=0.025$ (MI ground state) and unity filling factor. On the top, solid circles refer to prime data (error bars within symbol size). the data were taken for linear system sizes $L=10$ and 20. Within the error bars, we are not able to resolve any finite size effect. Solid lines in all plots are obtained from spline-interpolated data for energy, with subsequent analytic differentiation/integration of the interpolation curve.

Figure 9

(Color online) Entropy per particle at $t∕U=0.005$, unity filling factor and linear system size $L=20$. The dashed line is the entropy of the uniform WIBG. The solid line is the result of analytical derivation/integration of numerical data for energy and the dashed-dotted line (barely visible) is the analytical prediction (12), where, at each temperature, the chemical potential has been fixed by the condition of having equal number of particle and hole excitations. If the system was initially cooled down to $T^{\mathrm{in}}=0.25T_c$, the final temperature is $T^{\mathrm{fin}}=22t$ and nearly a hundred of thermally activated particle-hole excitations are present in the final state (see inset in Fig. 7).

Figure 10

(Color online) Energy per particle at $t∕U=0.005$, $\mu =116.5t$, and trap frequency $\omega =2\pi 60\mathrm{Hz}$. Solid circles are numerical data, the solid line is the energy of particle and hole excitations in MI deduced from Eq. (7). The inset shows a zoom of the low temperature range. At $T\sim 2.3t$ the contribution of MI excitations to energy is $\sim 10\%$ only.

Figure 11

(Color online) Entropy per particle for the same system as in Fig. 10. The dashed line is the entropy of the trapped WIBG, before the optical lattice is loaded; the solid line is the final entropy. If the system was initially cooled down to $T^{\left(\mathrm{in}\right)}=0.25T_c$, the final temperature is $T^{\left(\mathrm{fin}\right)}=(2.35\pm 0.3)t$ and the liquid at the perimeter of the trap is normal (see text).

Figure 12

(Color online) Energy (top), specific heat (middle), and entropy (bottom) per particle at $t∕U=0.0385$ (SF ground state) and unity filling factor. On the top, solid circles refer to prime data (error bars within symbol size). Data were taken for system size $L=10$ and 20. Within error bars, we are not able to resolve any finite size effect. Solid lines in all plots are obtained from spline-interpolated data for energy, with subsequent analytic differentiation/integration of the interpolation curve.

Figure 13

(Color online) Finite size scaling of the superfluid stiffness. $n_{s}L$ vs $T∕t$ for system size $L=5$ (circles), $L=10$ (squares), $L=20$ (triangles). We estimate the critical temperature to be $T_c=3.25\left(1\right)t$, already nearly half the noninteracting gas value.