Extended self-energy functional approach for strongly correlated lattice bosons in the superfluid phase

Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper, we show how to overcome this shortcoming. To this end, we identify an appropriate quantity, which we term $D$, that represents the correlation correction of the condensate order parameter, as it does the self-energy for Green's function. An appropriate functional is derived, which is stationary at the exact physical realization of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and, therefore, applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the “pseudoparticle” approach proposed in Phys. Rev. B 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which shows a remarkable agreement with those of quantum-Monte-Carlo calculations.

Figure 1

(Color online) (a) Superfluid density ${\rho }_s$ evaluated for constant hopping strength $t/U=0.02$ as a function of the chemical potential $\mu /U$. VCA results for reference systems of size $L=1\times 1$, $L=2\times 1$, and $L=2\times 2$ and for essentially infinitely large physical systems are compared to QMC results for physical systems of size $32\times 32$ and inverse temperature $U/T=128$. (b) Comparison of the superfluid density ${\rho }_s$ and condensed density ${\rho }_c$ for reference systems of size $L=1\times 1$ and essentially infinitely large physical systems (cf. Ref. 10).

Figure 2

(a) Superfluid density ${\rho }_s$ and (b) superfluid fraction ${\rho }_s/n$ ranging deep in the superfluid phase evaluated for constant chemical potential $\mu /U=0.4$ as a function of the hopping strength $t/U$. Results obtained by means of VCA for reference systems of size $L=1\times 1$ and essentially infinitely large physical systems are compared to QMC results for physical systems of size $32\times 32$ and inverse temperature $U/T=128$.

Figure 3

(Color online) Particle density $n$ (left), condensate density ${\rho }_c$ (middle), and superfluid density ${\rho }_s$ (right) evaluated around the quantum critical region close to the tip of the first Mott lobe. Comparison of the data obtained by means of VCA (for essentially infinitely large physical systems and reference systems as stated in the legends), QMC (for physical systems of size $32\times 32$ and inverse temperatures $U/T=128$), and mean field. (a.1)–(a.3) The first row shows results for a fixed chemical potential $\mu /U=0.4$ as a function of the hopping strength $t/U$, whereas (b.1)–(b.3) the second row shows results for the fixed hopping strength $t/U=0.05$ as a function of the chemical potential $\mu /U$.