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

Abstract

Combining the process-chain method with a symbolized evaluation we work out in detail a highorder 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 (EPLT).In order to obtain higher orders than the 10th order, Eckardt et al. developed a computer assisted processchain algorithm (PCA) 19,20 based on Kato's formulation of the perturbation calculus. 21 Using the PCA, highprecision results were obtained for both the ground-state energy and the correlation function within the Mott insulator. 20 Furthermore, by implementing PCA for the effective potential Landau theory (EPLT), 22,23 the critical line of the quantum phase transition from the Mott insulator to the superfluid phase was determined with high precision 20,24 so that even the corresponding critical exponents can be extracted. 25 In order to deal with degenerate states, such as particle-hole excitations, the

Quantum phase transition of the Bose-Hubbard model on cubic lattice with anisotropic hopping

“…Then, we can achieve very high order (e.g. 10th order [26]) by numerically computing each diagram. Meanwhile, the whole process is symbolized in our previous work [26] so that the results can be considered as exact Taylor expansion of the phase boundaries, so it can provide more accurate analysis of the QPT.…”

“…10th order [26]) by numerically computing each diagram. Meanwhile, the whole process is symbolized in our previous work [26] so that the results can be considered as exact Taylor expansion of the phase boundaries, so it can provide more accurate analysis of the QPT. Considering the computational resource consumption is more sensitive to the orders than dimensionality, the HSSCE is very suitable for quantitatively studying the QPT during the dimensional crossover.…”

“…In previous work Ref. [26], we implement HSSCE to isotropic 1D, 2D and 3D Bose-Hubbard model, together with Padé re-summation, we get very accurate phase boundaries, which are non-distinguishable from numerical results, and also the critical exponents obtained from the function of charge gap. Thus, we expect the HSSCE can also provide the high accuracy results of the anisotropic 3D Bose-Hubbard model, which is not only benefit for understanding the QPT and its universality class during the 1D to 3D dimensional crossover, but also used for benchmark of the experiments and numerical simulations.…”

“…In this paper, we use high order symbolic strong coupling expansion method (HSSCE) [24][25][26] to study 2D arrays of coupled 1D tubes of ultra-cold bosonic gas, where the inter-tube coupling is varied from zero to the same value as the intra-tube coupling, so that the system can be detuned from 1D to 3D. Basing on the processchain algorithm [25], strong coupling expansion [24] and symbolization, the HSSCE can give the symbolic series expansion function of the critical line between compressible phase (Mott-insulator) and incompressible one (superfluid) up to very high order for different atom filling number and anisotropy (up to eighth order in this work, see Appendix.A).…”

Quantum Hall States for $$\alpha = 1/3$$ α = 1 / 3 in Optical Lattices

“…In this model the kinetic energy of the bosons competes with the on site interaction and drives a quantum phase transition (QPT) from superfluid (SF) to bosonic Mott insulator (MI) phase [4,21]. Various theoretical methods, such as mean field theory [19], strong coupling expansion [22][23][24][25], quantum Monte Carlo [26], density matrix renormalization group [27] have been used to study the role of quantum fluctuation and short range on site interaction on QPT. The SF phase is compressible with finite SF order parameter and phase coherent; MI phase on the other hand is incompressible with zero order parameter and shows integer commensurate filling per lattice site.…”