Quasiperiodic Bose-Hubbard model and localization in one-dimensional cold atomic gases

We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasiperiodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wavelength is incommensurate with the main lattice wavelength. After discussing the conditions under which the model can be realized experimentally, the study of the density vs the chemical potential curves for a nontrapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density-wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth $V_2$ in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott-insulating phase is found at finite $V_2$. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The latter feature can be investigated by looking at the expansion of the cloud after releasing the trap.

Figure 1

(Color online) Phase diagrams of the bichromatic Bose-Hubbard model for densities $n=1$, $r$ (the ratio of the potential wavelengths), and $n=0.5$. The diagrams are shown as a function of the interaction strength $U$ and the bichromatic potential strength $V_2$, both normalized by the hopping $J$ (lines are guides to the eyes). SF stands for the superfluid phase, MI for the Mott-insulating phase, BG for the “Bose-glass” phase (meaning localized but with zero one-particle gap), and ICDW for incommensurate charge-density-wave phase. The $U=V_2$ line on the phase diagram with $n=1$ indicates the $J=0$ limit for which the gap of the one-particle excitation vanishes. Black error bars are deduced from calculations averaging over the phase shift $\varphi$ (cf. Sec. 1A) and finite-size scaling (see Figs. 12, 13 for details on the $n=1$ phase diagram). Gray error bars are roughly evaluated from calculations on systems with $L=35$ and fixed $\varphi =0$ (see Figs. 11, 14 for details). In the phase diagram with density $n=1$, the darker (violet) region in the BG phase is localized but could have a small gap which cannot be resolved numerically.

Figure 2

(Color online) The bichromatic potential (full line) with the same parameters as in the experiment of Ref. 23. Dashed lines show the two beating standing waves from which the bichromatic potential originates. We observe that not only the depths of the potential wells fluctuate but so do their positions and their width. The energetic landscape displays wells of typical width $1∕(1-r)$. The plot on the right-hand side shows the bounded distribution which behaves as $1∕\sqrt{x(V_2-x)}$ and is thus peaked around 0 and $V_2$.

Figure 3

(Color online) Comparison of $n\left(\mu \right)$ for a random and a bichromatic potential (irrational $r$) for hard-core bosons. Plateaus open for the bichromatic potential, the main ones being at $n=r$ and $1-r$.

Figure 4

(Color online) Opening of plateaus in the $n\left(\mu \right)$ curves with $V_2$ for hard-core bosons ($V_2$ is increased by steps of $J∕4$ and is in units of $J$, as $\mu$). Above the critical value $V_2^c=4$, the curves acquire a devil’s staircaselike behavior: plateaus become dense.

Figure 5

(Color online) $n\left(\mu \right)$ for soft-core bosons with a large interaction parameter $U=16$ at small and large $V_2$ computed on a system with $L=35$ and a fixed phase shift $\varphi =0$. Inset: curves at low density show the comparison between hard-core bosons $(U=\infty )$ and soft-core ones, which is good up to finite-size effects.

Figure 6

(Color online) Plateaus with a large $V_2\sim U$. Some of the largest plateaus found do not correspond to the hard-core boson limit. The two sizes $L=35$ and 70 give an idea of the (weak) finite-size effects.

Figure 7

(Color online) Ground-state wave function of the lattice model (3) with a smooth trap but no interaction for increasing perturbing potential $V_2$. There is a crossover from a Gaussian wave function (logarithm scale) to an exponential one. Note that the maximum of the wave function in the presence of strong $V_2$ is not centered at the middle of the trap.

Figure 8

(Color online) The averaged inverse participation ratio ${⟨I\left({\psi }_0\right)⟩}_{\varphi }$ as a function of $V_2$ for a bichromatic lattice and a random box distribution. There is a sharp transition at $V_2^c=4$ in the thermodynamical limit ($L=420$ for $\omega =0$). A similar sharp transition is also found for the system with a smooth trap.

Figure 9

(Color online) The Luttinger parameter $K$ for hard-core bosons as a function of $V_2$ and the density $n$ on a finite system with $L=175$ for an irrational $r=0.7714\dots$. Strictly speaking, $K$ should be equal to 1 in all superfluid phases and 0 in the gapped phases. Up to finite-size effects, vertical bands reveal the successive openings of gaps as $V_2$ is increased.

Figure 10

(Color online) Superfluid–Bose-glass localization transition. The parameters $U=2$ and $n=0.63$ are chosen such that the system is in the superfluid phase at $V_2=0$. (a) The inverse correlation length scales roughly as $\sqrt{V_2-V_2^c}$, where $V_2^c=6.9$ is much larger than the noninteracting critical value $V_2^c=4$. Below $V_2^c$, we have the scaling $\xi \sim L$ typical of a superfluid state. (b) ${\rho }_s$ provides an independent determination of the transition. (c) Evolution of the Luttinger exponent $K$ as a function of $V_2$. The dashed line displays the RBD result $K_c=3∕2$. (d) Averaged one-particle density matrix $G\left(x\right)$ for increasing $V_2$ showing the transition from an algebraic to an exponential decay.

Figure 11

(Color online) Phase diagram for $n=1$. Observables are computed with DMRG for a system with $L=35$ and fixed phase shift $\varphi =0$ as a function of $U$ and $V_2$. The $V_2=0$ line shows the Mott transition at $U_c=3.3$ while the $U=0$ line shows the free boson localization transition around $V_2^c=4$. The Mott-insulating phase gets qualitatively delocalized as $V_2$ increases for $U$ not too large. Increasing $U$ delocalizes the BG phase if $V_2$ is not too large.

Figure 12

(Color online) Superfluid–Mott-insulator transition for $n=1$. Cuts along the $U$ axis for $V_2=0$ and 2. Left: data giving similar results to those of Ref. 48 (the vertical bar being $U_c\simeq 3.33\pm 0.1$). Data are computed with $N^{\mathrm{bos}}=4$ and the Luttinger exponent $K$ is determined using either ${\rho }_s$ and $\kappa$ or $G\left(x\right)$ (see Sec. 3A). Scaling of ${\rho }_s$, $\xi ∕L$, and the criteria $K_c=2$ gives the same critical point within error bars. The scaling of the condensate fraction $f_c$ (not shown) is not simple at the transition and the HCB scaling $f_c\propto \sqrt{N}$ does not hold. Right: the same observables for $V_2=2$ and $N^{\mathrm{bos}}=3$ (this cutoff induces a nonphysical decrease of the superfluidity at small $U$, but does not affect the transition as seen, for instance, from the behavior of the Luttinger parameter $K$ for $N^{\mathrm{bos}}=4$). The insets show scaling behavior for ${\rho }_s$ and $\xi ∕L$ after averaging over several different $\varphi$. A critical point $U_c\simeq 3.6\pm 0.1$ is found which corresponds to $K_c\simeq 2$. Furthermore, the one-particle gap (not averaged over $\varphi$) is best fitted by a Kosterlitz-Thouless opening (we fixed $U_c=3.6$ for the fit). Up to numerical precision, we infer from these results that there is a direct transition between the SF and the MI phases, with no intervening BG phase.

Figure 13

(Color online) Reentrances of the SF phase with increasing interactions. Cut at $V_2=10$ in the phase diagram of Fig. 11. Error bars are related to the average over the phase shift $\varphi$ (about 20 samples). The scalings of the correlation length (a) and of the superfluid density (b) suggest two reentrances of the SF phase as $U$ is increased. In between, a localized phase is found and a MI phase is obtained at large $U$ according to the gap (c). In the intermediate BG phase (for $U\simeq 6–9$), a small gap is found, but for the chosen sizes $L=22$, 35, and 70, the scaling $\xi \sim 1∕\Delta$ shown in (d) seems to be satisfied only in the MI phase, meaning that strong finite-size effects are present.

Figure 14

(Color online) Phase diagram for $n\simeq r$. Observables are computed on a system with fixed size $L=35$ and fixed phase shift $\varphi =0$. A large ICDW phase emerges at large $U$. A finite $V_2$ is required to stabilize it. The SF phase extends along the $V_2=0$ contrary to the phase diagram with $n=1$.

Figure 15

(Color online) Delocalization by increasing the number of particles. Observables for increasing density when $V_2\simeq U$ (same parameters as in Fig. 6). There is a delocalization transition with increasing density. A few ICDW phases can be seen at intermediate fillings (dashed brown vertical lines in the right-hand figure showing the one-particle gap ${\Delta }_c$). Gray areas denote the localized regions (either BG or ICDW). The weakening of the superfluidity at large density might be an artifact of the cutoff in the number of bosons kept per site which is here $N^{\mathrm{bos}}=4$.

Figure 16

Typical $n\left(k\right)$ in the Bose-glass phase of Fig. 10 in a system without a trap. Satellite peaks at wave vector $2\pi (1-r)$ are visible if the disorder is not too strong. The width of the middle peak typically gives the inverse correlation length [a Lorentzian of width $\xi$ computed from Eq. (14) is given in dashed lines for a qualitative comparison].

Figure 17

(Color online) Expansion of HCB condensates. At $t=0$, the system is in the ground state of the Hamiltonian with the trap plus “disorder” potentials using a fixed chemical potential $\mu =0$ (about 51 particles) and a trap frequency $\omega =0.03$. At $t>0$, the trap confinement is switched off abruptly. Figures show the evolution of the local density profile as a function of time. An expansion is observed for systems with $V_2=0$ (without disorder) and $V_2=2$, but not for a random potential and a bichromatic potential with $V_2=6>V_2^c$. For the RBD, $V_2$ defines the width of the box distribution. When there is an expansion, reflections on the boundaries of the box in which the condensate expands can be seen.

Figure 18

(Color online) Dynamical critical point—width of the atomic distribution $W=\sqrt{\overline{n_j{(j-j_0)}^2}}$ of a HCB condensate as a function of $V_2$ for several increasing times (in units if the hopping). The width $W_0$ at $t=0$ has been subtracted for clarity. Parameters are the same as in Fig. 17. Up to finite trap frequency effects (see the text), the dynamical critical point is identical to the equilibrium one (at least for HCB).

Figure 19

(Color online) Effective dispersion relations for HCB—the Fourier transform of the single-particle wave functions ${\mid {\psi }_k\mid }^2$ is plotted as a function of the single-particle energy and the pseudomomentum $k$. At the right of each dispersion curve is the initial occupation numbers (from the state prepared using the trap confinement) vs the same single-particle energies for two different chemical potentials $\mu =-1.5,0$. This shows which states participate to the expansion. Other parameters are the same as for the expansions shown in Fig. 17.

Figure 20

(Color online) A typical example of fitting the averaged data from DMRG with conformal field theory results to extract the Luttinger exponent $K$. System size is $L=128$.