High-order symbolic strong-coupling expansion for the Bose-Hubbard model

Combining the process-chain method with a symbolized evaluation, we work out in detail a high-order symbolic strong-coupling expansion (HSSCE) for determining the quantum phase boundaries between the Mott insulator and the superfluid phase of the Bose-Hubbard model for different fillings in hypercubic lattices of different dimensions. With a subsequent Padé approximation we achieve for the quantum phase boundaries a high accuracy, which is comparable to high-precision quantum Monte-Carlo simulations, and show that all the Mott lobes can be rescaled to a single one. As a further cross-check, we find that the HSSCE results coincide with a hopping expansion of the quantum phase boundaries, which follow from the effective potential Landau theory.

Figure 1

Second-order arrow diagrams for (a) the ground state of a Mott insulator of filling one, and also for (b) open and (c) closed arrow diagrams for the case of adding one particle. As all arrow diagrams irrelevant to site $k$, indicated by black dashed arrows, cancel each other, only the arrow diagrams relevant to site $k$, highlighted by red solid arrows, need to be considered.

Figure 2

Quantum phase diagrams of Bose-Hubbard chain for filling $n=1$ (black solid line), $n=2$ (blue dashed line), and $n=3$ (orange dot-dashed line). The red dots are DMRG results for filling $n=1$ and $n=2$ [34].

Figure 3

Quantum phase diagram of 2D bosonic lattice system for $n=1$ (black solid line), $n=2$ (blue dashed line), and $n=3$ (orange dot-dashed line). For $n=1$ we also show QMC simulation results (red dot) [37].

Figure 4

Quantum phase diagram of 3D bosonic lattice system for $n=1$ (black solid line), $n=2$ (blue dashed line), and $n=3$ (orange dot-dashed line). For $n=1$ we also show QMC simulation results (red dots) [38].

Figure 5

Critical hopping amplitudes $t_c$ for filling numbers $n$ from 1 to 100 (red dots) compared with the value of $f_t\left(d\right)$ times the rescaling function $g_t\left(n\right)$ (black line) in two and three dimensions.

Figure 6

Scaled Mott lobe obtained from different fillings $n$ (red dots) compared with infinite filling quantum phase diagram (black line). For both dimensions $d=2$ and $d=3,$ the scaled Mott lobes deduced from filling number $n=1,2,3,4,10,100,10000$ are almost on top of each other.

Figure 7

Comparison of quantum phase diagrams for $d=2$ via HSSCE (black line) and HEPLT (red dots) [20] in tenth order with QMC results (green dots) [37] at $n=1$.

Figure 8

Arrow diagrams of fourth order in the square lattice, which are topologically equivalent

Figure 9

One sixth-order arrow diagram labeled with $\left(rdurlu\right)$ is shown in the left panel, its corresponding tree graph is shown in the right panel. Red solid arrows in the right panel are paths, which are passed due to the algorithm, and the dashed arrows represent other possible paths. The arrows (2,3,6,7) indicate trace back process, so the smallest path is (1,4,5,8,9,10). Thus, the smallest diagram representation is provided by $\left(rrlduu\right)$.

Figure 10

One process chain of energy correction Kato-list with arrow order 1 4 2 3 matching the Kato-list $\left({\alpha }_1^{\prime }0{\alpha }_2^{\prime }\right)$.

Figure 11

One process chain of SC energy Kato-list with arrow order 1 4 2 3 matches Kato-list $\left(00{\alpha }_1^{\prime }\right)$.