Strong-coupling expansion for the momentum distribution of the Bose-Hubbard model with benchmarking against exact numerical results

A strong-coupling expansion for the Green’s functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator–to–superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.

Figure 1

(Color online) Strong-coupling diagrams for the single-particle Green’s functions up to second order in $\mathfrak{t}$. The horizontal directed dashed lines indicate the hopping matrix element $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$ evolving between the respective imaginary times. The ellipse (yellow online) at multiply visited sites denotes the appearance of connected or cumulant $n$-particle Green’s functions.

Figure 2

(Color online) Strong-coupling diagrams for the single-particle Green’s functions corresponding to third order in $\mathfrak{t}$. The horizontal directed dashed lines indicate the hopping matrix element $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$ evolving between the respective imaginary times. The ellipses (yellow online) at multiply visited sites denote the appearance of connected or cumulant $n$-particle Green’s functions.

Figure 3

(Color online) Strong-coupling diagrams for the correlation functions up to second order in $\mathfrak{t}$. The horizontal directed dashed lines indicate $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$. The ellipses (red online) at multiply visited sites denote the appearance of connected or cumulant Green’s functions. Initial- and intermediate-state labels for the different possible imaginary time orderings shown are also indicated.

Figure 4

Strong-coupling diagrams for the correlation functions $C^{\left(3,0\right)}$. The horizontal directed dashed lines indicate the hopping matrix $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$. There are no multiply visited sites in these diagrams. Initial- and intermediate-state labels for the different possible imaginary time orderings shown are also indicated.

Figure 5

(Color online) Strong-coupling diagrams for the correlation functions $C^{\left(3,1\right)}$. The horizontal directed dashed lines indicate the hopping matrix $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$. The one ellipse (red online); at multiply visited sites denotes the appearance of connected or cumulant Green’s functions. Initial- and intermediate-state labels for the different possible imaginary time orderings shown are also indicated.

Figure 6

(Color online) Strong-coupling diagrams for the correlation functions $C^{\left(3,2\right)}$. The horizontal directed dashed lines indicate the hopping matrix element $\mathfrak{t}$ between the sites labeled, and the vertical lines indicate single-site Green’s functions $\mathcal{G}$. The two ellipses (red online) at multiply visited sites denote the appearance of connected or cumulant Green’s functions. Initial- and intermediate-state labels for the different possible imaginary time orderings shown are also indicated.

Figure 7

(Color online) Momentum distribution function for the three-dimensional case with $x=0.0625$ as a function of the band energy ${ϵ}_{\mathbf{k}}$. Note how the QMC data agree better with the scaling-theory results than it does with the strong-coupling results or the scaled RPA, although deviations can be seen in the data. Both the scaling theory and the scaled RPA use the QMC critical point as an input.

Figure 8

(Color online) Momentum distribution function for the three-dimensional case with $x=0.09$ as a function of the band energy ${ϵ}_{\mathbf{k}}$. Once again, the QMC data agree better with the scaling theory than it does with the strong-coupling results or the scaled RPA, although deviations can still be seen in the data. Note that the scaled RPA works better than the truncated strong-coupling expansion. Both the scaling theory and the scaled RPA use the QMC critical point as an input.

Figure 9

(Color online) Phase diagram of the three-dimensional Bose-Hubbard model. Note how the truncated strong-coupling expansion does not agree so well with the QMC data [17], but the scaled results nearly fit the Mott-phase lobe perfectly.

Figure 10

(Color online) Momentum distribution function in two dimensions with $x=0.05$. We plot the strong-coupling expansion against the scaling-theory results, the scaled RPA, and QMC simulations. Note how the QMC results agree much better with the scaled results and do not show the change in curvature near ${\xi }_{\mathbf{k}}=1$. In addition, the scaled RPA does not work as well here as it did in three dimensions. Both the scaling theory and the scaled RPA use the QMC critical point as an input.

Figure 11

(Color online) Momentum distribution function in two dimensions with $x=0.1$, which is close to the critical point. We plot the strong-coupling expansion against the scaling-theory results, the scaled RPA, and the QMC simulations. Note how the QMC results agree much better with the scaling-theory results. Both the scaling theory and the scaled RPA use the QMC critical point as an input.

Figure 12

(Color online) Phase diagram of the two-dimensional Bose-Hubbard model. Note how the truncated strong-coupling expansion [8, 9] does not agree so well with the QMC data [16], but the scaled results nearly fit the Mott-phase lobe perfectly.

Figure 13

(Color online) Momentum distribution function in one dimension with $x=0.1$, which is far from the critical point. We plot the strong-coupling expansion against the scaling-theory results, the scaled RPA, and the DMRG calculations. Note how the DMRG results agree much better with the scaling-theory results than the truncated expansion or the scaled RPA. Both the scaling theory and the scaled RPA use the DMRG critical point as an input.

Figure 14

(Color online) Momentum distribution function in one dimension with $x=0.2$, which is two-thirds of the way to the critical point. We plot the strong-coupling expansion against the scaling-theory results, the scaled RPA, and the DMRG calculations. Note how the DMRG results agree much better with the scaling-theory results than the truncated expansion or the scaled RPA, but one can see that the scaling approach is beginning to fail. Both the scaling theory and the scaled RPA use the DMRG critical point as an input.