Quantum phase diagram of bosons in optical lattices

We work out two different analytical methods for calculating the boundary of the Mott-insulator–superfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from quantum Monte Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.

Figure 1

(Color online) Quantum phase diagram of the first MI-SF lobe at $T=0$ for the three-dimensional case. Dot-dashed blue line is the mean-field result [14], dotted black line is from the third-order strong-coupling expansion [18], and red dots are recent high-precision Monte Carlo data [19].

Figure 2

(Color online) Quantum phase diagram of the first MI-SF lobe $(n=1)$ at $T=0$. Solid green lines are results from our field-theoretic method, dashed purple lines are results from our variational method, dot-dashed blue lines are from mean-field theory [14], dotted black lines are from the third-order strong-coupling expansion [18], and red triangles are the numerical data. For the one-dimensional case the numerical data stem from density-matrix renormalization-group calculations [29], while the data for two and three dimensions are obtained from quantum Monte Carlo simulations [19, 30].

Figure 3

(Color online) Phase diagram of MI-SF lobes at $T=0$ for different dimensions. Solid green lines are results from our field-theoretic method, dashed purple lines are results from our variational method, dot-dashed blue lines are from mean-field theory, and dotted black lines are from the third-order strong coupling-expansion [18]. The nonphysical ten-dimensional case is included to show that our two methods converge to the mean-field theory [14] in the limit $d\to \infty$, as is expected on general grounds [16].