Spectral weight redistribution in strongly correlated bosons in optical lattices

We calculate the single-particle spectral function for the one-band Bose-Hubbard model within the random-phase approximation (RPA). In the strongly correlated superfluid, in addition to the gapless phonon excitations, we find extra gapped modes, which become particularly relevant near the superfluid-Mott quantum phase transition (QPT). The strength in one of the gapped modes, a precursor of the Mott phase, grows as the QPT is approached and evolves into a hole (particle) excitation in the Mott insulator depending on whether the chemical potential $\mu$ is above (below) the tip of the lobe. The sound velocity $c$ of the Goldstone modes remains finite when the transition is approached at constant density; otherwise, it vanishes at the transition. It agrees well with Bogoliubov theory except close to the transition. We also calculate the spatial correlations for bosons in an inhomogeneous trapping potential creating alternating shells of Mott insulator and superfluid. Finally, we discuss the capability of the RPA to correctly account for quantum fluctuations in the vicinity of the QPT.

Figure 1

(Color online) Mean-field phase diagram in the $\mu /U$ vs $t/U$ plane. The blue line shows the Mott insulating lobe at density $n=1$. On this diagram, we indicate two points where the QPT happens, which we discuss in this paper: a generic one at $\mu /U=0.5$ and $t/U\approx 0.167$ (black) and the tip of the lobe at $\mu /U=\sqrt{2}-1$ and $t/U\approx 0.1716$ (red), where the QPT takes place at constant density.

Figure 2

(Color online) Spectral function $\mathcal{A}\left(q,\omega \right)$ as a function of $\omega$ for various $q$. Results obtained by RPA (black) and Bogoliubov theory (green). (a) Weakly interacting SF: $t/U=10$ with $\approx 10$ bosons per site; the RPA calculation agrees extremely well with Bogoliubov theory; note indications of additional modes at higher $\omega$. (b) $t/U=1$ with $\approx 1.8$ bosons per site. (c) Strongly interacting SF: $t/U=0.25$ with $\approx 1.1$ bosons per site; stronger deviations from Bogoliubov theory are present especially at larger $q$. Additional modes are clearly visible in the spectrum. (d) Mott insulating phase for $t/U=0.16$ and 1 boson per site. In all these figures, $\mu /U=0.5$.

Figure 3

(Color online) [(a) and (b)] Dispersion and [(c) and (d)] strength of excitation modes at $\mu /U=0.5$: (a) and (c) are in the Mott regime $t/U=0.1$; (b) and (d) are in the SF regime $t/U=0.25$. For clarity, in (b) the fourth pair of resonances at $\omega \approx \pm 18.4$ is not shown and in (d) only the strength of the four modes at lower energy is shown. Note that modes at positive (negative) energy have positive (negative) strength.

Figure 4

(Color online) (a) Energy and (b) strength of the modes at low $q=\pi /50$ as a function of $t/U$ for $\mu /U=0.5$. Note the presence of both phonons and gapped modes in the strongly interacting SF and their evolution into two gapped modes into the MI. One of the gapped modes in the MI evolves from the phonon mode in the SF and the other one from a gapped mode in the SF. The thin vertical line at $t/U\approx 0.167$ indicates the QPT.

Figure 5

(Color online) (a) Energy and (b) strength of the modes at low $q=\pi /50$ as a function of $t/U$ at $\mu /U=\sqrt{2}-1$ corresponding to the tip of the lobe. The frequency of the lowest four modes (two phononic and two gapped) in the SF vanish at the QPT. The thin vertical line at $t/U\approx 0.1716$ indicates the QPT. In the inset, the zoom around the QPT in panel (b) is shown.

Figure 6

(Color online) Density of states for several values of $t/U$. (a) $t/U=0.25$, SF regime; (b) $t/U=0.17$, SF regime, very close to the QPT; (c) $t/U=0.16$, MI regime, very close to the QPT; (d) $t/U=0.1$, MI regime. All these results are for $\mu /U=0.5$.

Figure 7

(Color online) Sound velocity $c$ (blue solid line) extracted from the slope of the dispersion of the phonon modes and from Eq. (14); order parameter $\phi$ (red dashed line). Also shown for comparison are the sound velocity and the order parameter obtained from Bogoliubov theory (thin lines). In the inset $d\rho /d\mu$ is shown. (a) $\mu /U=0.5$; (b) $\mu /U=\sqrt{2}-1$, corresponding to the tip of the lobe.

Figure 8

(Color online) Momentum distribution $n\left(q\right)$ for a 2D system for $\mu /U=0.5$ and (a) $t/U=0.05$ deep in the MI; (b) $t/U=0.15$ in the MI but closer to the QPT; (c) $t/U=0.175$ in the SF phase close to the QPT; (d) $t/U=0.25$ in the SF phase but further from the QPT.

Figure 9

(Color online) Density matrix $\rho \left(r,r^{\prime }\right)$ as a function of the relative distance $r-r^{\prime }$ for a 2D system (black line for a cut in the center of the trap and blue line along the diagonal). (a) $t/U=0.05$ deep in the MI showing only nearest-neighbor correlations; (b) $t/U=0.15$ in the MI but closer to the QPT showing an increase in the scale of the short-range correlations; (c) $t/U=0.175$ in the SF phase close to the QPT showing long-range order; (d) $t/U=0.25$ in the SF phase but farther from the QPT. Note that the asymptotic value for large $r-r^{\prime }$ has been subtracted.

Figure 10

(Color online) Condensate fraction obtained from the strength of the poles of the spectral function, as in Eq. (19), for the phonon mode at positive (upper green line) and negative (lower red line) frequencies. As $q\to 0$ both functions approach the condensate fraction $n_0$ obtained with MFT (dashed blue line). In this figure $\mu /U=0.5$ and $t/U=0.5$.

Figure 11

(Color online) Inhomogeneous system: (a) order parameter in the trap. It is zero in the central MI core and finite and large in the surrounding SF ring. The other panels show $G\left(\mathbf{r},{\mathbf{r}}^{\prime }\right)$ for a given value of the energy as a function of ${\mathbf{r}}^{\prime }$ for a fixed $\mathbf{r}$ (black dot): (b) $\mathbf{r}=\left(-6,0\right)$ in the SF ring; (c) $\mathbf{r}=\left(-3,0\right)$ at the interface between the MI core and the SF ring; (d) $\mathbf{r}=\left(0,0\right)$ in the center of the MI core.

Figure 12

(Color online) Normalization of $n\left(q\right)$ for $\mu /U=0.5$ as a function of $t/U$ in one (blue dashed-dotted line), two (green dashed line), and three (red solid line) dimensions. The thin vertical line indicates the phase transition.