Mott transition in bosonic systems: Insights from the variational approach

We study the Mott transition occurring for bosonic Hubbard models in one, two, and three spatial dimensions by means of a variational wave function benchmarked by Green’s function Monte Carlo calculations. We show that a very accurate variational wave function, which is constructed by applying a long-range Jastrow factor to the noninteracting boson ground state, can describe the superfluid-insulator transition in any dimensionality. Moreover, by mapping the quantum averages over such a wave function into the partition function of a classical model, important insights into the insulating phase are uncovered. Finally, the evidence in favor of anomalous scenarios for the Mott transition in two dimensions is reported whenever additional long-range repulsive interactions are added to the Hamiltonian.

Figure 1

(Color online) Accuracy of different variational wave functions as a function of $U/t$ for 60 sites and 60 bosons. $\Delta E=E_0-E_{\text{VMC}}$, where $E_{\text{VMC}}$ is the variational energy and $E_0$ is the ground-state one, which is obtained by GFMC. The state of Eq. (5) is denoted by “Jastrow,” that of Eq. (6) by “$\text{Gutzwiller}+\text{MB}$,” and that of Eq. (8) by “$\text{Jastrow}+\text{MB}$.”

Figure 2

(Color online) Jastrow parameters $v_q$ multiplied by $q^2$ as a function of $\left|q\right|$ for 60 sites and 60 bosons.

Figure 3

(Color online) Density structure factor $N_q$ divided by $\left|q\right|$, which is calculated with variational Monte Carlo (left panel) and GFMC (right panel) for different $U/t$ and $L=60$. From top to bottom, $U/t=1.6$, 1.8, 2, 2.2, 2.4, 2.5, 3, and 4.

Figure 4

(Color online) Variational results for the momentum distribution $n_k$ for $L=60$ (squares), 100 (circles), and 150 (triangles) across the transition ($U/t=2.4$ and 2.5). Inset: Size scaling of the condensate fraction $n_0/L$ for $U/t=2.4$ (upper curve) and $U/t=2.5$ (lower curve).

Figure 5

(Color online) Superfluid stiffness $D_s$ calculated by GFMC as a function of $U/t$ for different sizes in the 1D Bose–Hubbard model. Lower inset: Size scaling of $D_s$ for different $U/t$ (same values as those of the main panel, with $U/t$ increasing from top to bottom). Upper inset: $D_s\times L$ as a function of $U/t$. The point where the different curves cross marks the transition point.

Figure 6

(Color online) Accuracy of different variational wave functions as a function of $U/t$ for the $10\times 10$ cluster and 100 bosons. The symbols are the same as in Fig. 1.

Figure 7

(Color online) Jastrow parameters $v_q$ multiplied by $q^2$ as a function of $\left|q\right|$ [along the (1,0) direction] for $20\times 20$ (circles), $26\times 26$ (squares), and $30\times 30$ (triangles) clusters. Inset: The many-body variational parameter $g_{\text{MB}}$ as a function of $U/t$.

Figure 8

(Color online) Jastrow parameters $v_q$ multiplied by $q^2/\text{log}\left(q/\pi \right)$ as a function of $\left|q\right|$ [along the (1,0) direction] for different $U/t$ and the same sizes as in Fig. 7.

Figure 9

(Color online) Left panel: Density structure factor $N_q$ divided by $\left|q\right|$, which is calculated by the variational Monte Carlo for different $U/t$ and $L=20\times 20$. From top to bottom, $U/t=10$, 10.2, 10.4, 11, and 12. Right panel: The same as for the GFMC on the $L=16\times 16$ cluster. From top to bottom, $U/t=8$, 8.2, 8.4, 8.6, and 8.8.

Figure 10

(Color online) GFMC results for the sound velocity $v_s$, which are obtained through a finite-size scaling of the ground-state energy [see Eq. (19)]. The line is a fit of the numerical results.

Figure 11

(Color online) Variational results for the momentum distribution $n_k$ in two dimensions for $10\times 10$ (squares), $16\times 16$ (circles), $20\times 20$ (upward triangles), and $26\times 26$ (downward triangles) clusters with $U/t=10$ and 12. Inset: Size scaling of the condensate fraction $n_0/L$ for $U/t=10$ (upper curve) and $U/t=12$ (lower curve).

Figure 12

(Color online) Superfluid stiffness $D_s$ as a function of $U/t$ for different sizes. Inset: Size scaling of $D_s$ for different $U/t$ (the same values as those of the main panel are reported, with $U/t$ increasing from top to bottom).

Figure 13

(Color online) Accuracy of different variational wave functions as a function of $U/t$ for the $10\times 10\times 10$ cluster and 1000 bosons. The symbols are the same as in Fig. 1.

Figure 14

(Color online) Jastrow parameters $v_q$ multiplied by ${\left|q\right|}^3$ as a function of $\left|q\right|$ for $8\times 8\times 8$ (circles), $10\times 10\times 10$ (squares), and $12\times 12\times 12$ (triangles) clusters [along the (1,0,0) direction].

Figure 15

(Color online) Upper panel: Variational results for the density structure factor $N_q$ for three dimensions and $U/t=20$. Lower panels: $N_q$ for nonoptimized wave functions with $v_q\sim {\beta }_{3\text{D}}/{\left|q\right|}^3$ for two values of ${\beta }_{3\text{D}}$.

Figure 16

(Color online) Jastrow parameters $v_q$ multiplied by ${\left|q\right|}^{3/2}$ as a function of $\left|q\right|$ for the potential of Eq. (3) with different $V/t$ and $20\times 20$ (circles), $26\times 26$ (squares), and $30\times 30$ (triangles) clusters. Inset: The many-body variational parameter $g_{\text{MB}}$ as a function of $V/t$.

Figure 17

(Color online) Variational results for the density structure factor $N\left(q\right)$ divided by ${\left|q\right|}^{3/2}$ for the potential of Eq. (3) with different $V/t$ and $20\times 20$ (circles), $26\times 26$ (squares), and $30\times 30$ (triangles) clusters.

Figure 18

(Color online) The same as in Fig. 16, for the potential of Eq. (4).

Figure 19

(Color online) The same as in Fig. 17, for the potential of Eq. (4).