Characterization of Mott-insulating and superfluid phases in the one-dimensional Bose-Hubbard model

We use strong-coupling perturbation theory, the variational cluster approach (VCA), and the dynamical density-matrix renormalization group (DDMRG) method to investigate static and dynamical properties of the one-dimensional Bose–Hubbard model in both the Mott-insulating and superfluid phases. From the von Neumann entanglement entropy we determine the central charge and the transition points for the first two Mott lobes. Our DMRG results for the ground-state energy, momentum distribution function, boson correlation function decay, Mott gap, and single-particle spectral function are reproduced very well by the strong-coupling expansion to fifth order, and by VCA with clusters up to 12 sites as long as the ratio between the hopping amplitude and onsite repulsion, $t/U$, is smaller than 0.15 and 0.25, respectively. In addition, in the superfluid phase VCA captures well the ground-state energy and the sound velocity of the linear phonon modes. This comparison provides an authoritative estimate for the range of applicability of these methods. In strong-coupling theory for the Mott phase, the dynamical structure factor is obtained from the solution of an effective single-particle problem with an attractive potential. The resulting resonances show up as double-peak structures close to the Brillouin zone boundary. These high-energy features also appear in the superfluid phase which is characterized by a pronounced phonon mode at small momenta and energies, as predicted by Bogoliubov and field theory. In one dimension, there are no traces of an amplitude mode in the dynamical single-particle and two-particle correlation functions.

Figure 1

Phase diagram of the 1D Bose-Hubbard model showing superfluid (SF) and Mott-insulating (MI) regions as a function of the chemical potential $\mu /U$ and the electron transfer amplitude $t/U$. The boundaries delimiting the first two Mott lobes were determined by DMRG, using system sizes up to $L=128$ and open boundary conditions [20] (see text).

Figure 2

Entanglement entropy difference, $c^{*}$ from Eq. (65), for the 1D Bose-Hubbard model with $\rho =1$ (upper panel) and $\rho =2$ (lower panel). Data are obtained by DMRG for lattices up to $L=64$ with PBC. The closed symbols indicate the maximum value for each system size. An extrapolation of the $t/U$ values at these maxima to the thermodynamic limit provides the Kosterlitz-Thouless transition point (see insets); here, the lines correspond to a polynomial fit. The vertical dashed lines in the main panels mark the Kosterlitz-Thouless transition point.

Figure 3

Ground-state energy $E_0/\left(LU\right)$ as a function of interaction strength $t/U$ for $\rho =1$. Weak- and strong-coupling results are compared with the $L\to \infty$ extrapolated DMRG data obtained from chains up to $L=128$ with OBC. For the VCA calculations (crosses) a cluster with $L_c=12$ ($L_c=4$) sites is used in the MI (SF) phase.

Figure 4

Decay of bosonic correlations in the 1D Bose-Hubbard model within the first Mott lobe ($\rho =1$) for decreasing interaction strengths $x=0.05$ (a), 0.1 (b), 0.15 (c), and 0.2 (d). DMRG results are obtained for a chain with $L=128$ sites and OBC. To minimize the boundary effects we place $j$ and $\ell$ symmetrically around the center of the system. The VCA data were calculated using clusters with $L_c=4$ (green triangles), 8 (blue squares), and 12 (red circles).

Figure 5

Momentum distribution function $n\left(k\right)$ within the first Mott lobe from DMRG with $L=64$ and PBC (symbols), VCA (dashed lines), and third-order strong-coupling expansion (69) (solid lines). The inset in panel (d) shows the dependence of the VCA results on the cluster size $L_c$ for $k\gtrsim 0$ in comparison with the DMRG data.

Figure 6

Single-particle spectral function $A(k,\omega )$ in the $k$-$\omega$ plane (left panels) and density of states $N\left(\omega \right)$ (right panels) for the 1D Bose-Hubbard model with $\rho =1$ and $t/U=0.05$ (upper panels), $t/U=0.1$ (lower panels). We compare DDMRG data for a system with $L=64$ and OBC (squares) with the results of VCA for $L_c=12$ and $\eta =0.03U$ (density plots) and strong-coupling expansions (24) and (25) (lines).

Figure 7

Spectral functions and density of states for intermediate couplings $t/U=0.15$ (upper panels), $t/U=0.2$ (middle panels) and $t/U=0.25$ (lower panels). Notations are the same as in Fig. 6.

Figure 8

Spectral functions in superfluid phase of the 1D Bose-Hubbard model with $t/U=0.35$ (left panel), $t/U=0.4$ (right panel). Compared are DDMRG dispersions with VCA density plots for $L_c=4$ and the dispersion of the condensate excitations $E\left(k\right)$ from Bogoliubov theory (black lines, other symbols are the same as in Fig. 6). See Eq. (3) with (2).

Figure 9

Intensity of the dynamical structure factor $S(q,\omega )$ in the MI phase of the 1D Bose-Hubbard model for different $t/U$ where $\rho =1$. DDMRG data were obtained for an $L=32$ site system with PBC, using $\eta =0.5t$. Red crosses mark the positions of the maximum in each $q=2\pi m_q/L$ sector.

Figure 10

Frequency scans of the dynamical structure factor in the MI state at fixed momenta $q=\pi /2$ (left panels) and $q=\pi$ (right panels). DDMRG data (circles) were obtained for $L=64$, PBC, and $\eta =0.5t$; at $t/U=0.15$ ($q=\pi$) also for $L=128$, PBC, and $\eta =0.2t$ (green crosses). Blue solid (red dashed) lines give the corresponding results of the strong-coupling theory with $\eta =0$ ($\eta =0.5t$). Please note the different scales of the ordinates.

Figure 11

Intensity of the dynamical structure factor $S(q,\omega )$ in the superfluid phase of the Bose-Hubbard model with $\rho =1$. Again we use $L=32$, PBC, and $\eta =0.5t$. The yellow line gives the Bogoliubov result (3).

Figure 12

Frequency dependence of dynamical structure factor $S(q,\omega )$ at $q=\pi /2$ and $q=\pi$. Only DDMRG data are shown. Circles (crosses) mark the results for $L=64$, PBC, and $\eta =0.5t$ ($L=128$, PBC, and $\eta =0.2t$).