Figure 1

Energy structure of the BHM for large $U/J$ without disorder. The levels correspond to the energies of the exact many-particle energy eigenstates of the system with interactions. Due to the U(1) symmetry, the energy eigenstates can be chosen to be of well-defined particle number. For commensurate particle number $N$, the ground state (MI) is separated from a band of particle-hole excitations by a charge gap, while for $N+1$ particles, all eigenstates are delocalized, forming a band of hybridized particle states. At $T=0$, the single-particle DOS has finite weight at frequencies corresponding to energy differences $E_i^{(N+1)}-E_0^{(N)}$ of transitions between $N$ and $N+1$ states. At $T>0$, transitions between different excited states also contribute, causing the gap to vanish if the two bands overlap and the respective matrix elements do not vanish.Reuse & Permissions

Figure 2

Compressibility $\kappa$ and local single-particle DOS in an insulator for box disorder at $\mu =1.1U$ at various temperatures (in units of $U^{-1}$). At zero temperature $\kappa$ vanishes in the MI, but not in the BG and may thus be taken as a quantity to distinguish between them. At finite temperature however, $\kappa$ becomes nonzero in the normal phase and the sharp features become rounded off by thermal fluctuations. In this case, the disorder-averaged DOS of local excitations $\overline{\rho }(\omega =0)$, which retains all sharp features at $T>0$, can be used to determine the crossover from a BG to the normal phase, where it remains zero.Reuse & Permissions

Figure 3

Finite-temperature phase diagrams for box disorder with $\Delta =0.5U$ for three temperature regimes: zero (top row), intermediate $T=0.03U$ (middle row), and high $T=0.2U$ (bottom row). The left column shows the expectation value of the self-consistently determined distribution, which is the order parameter for the SF-insulator transition. It vanishes in the MI, BG, and the $T>0$ normal phase. In the $T=0$ diagrams, the MI-BG phase borders from strong coupling theory are indicated by the orange solid lines. The BG phase is marked by white stripes; at finite $T>0$ this is determined by a vanishing local DOS only (i.e., not by strong coupling theory). At finite temperature there is only a crossover between the normal and the BG phase, and the borders determined by the DOS of purely local excitations are marked by the horizontal white lines. The right column shows the compressibility $\kappa$ in units of $U^{-1}$. Whereas this is a suitable quantity to distinguish the MI (incompressible) from the BG at $T=0$, it is nonzero, but exponentially small in $T$ in the MI in the intermediate regime and of order $U^{-1}$ in the high-temperature regime. The white borders indicate the transition to the SF, determined from the respective diagram on the left.Reuse & Permissions

Figure 4

Phase diagram for box disorder at fixed density $n=1$ in the $U/J$-$\Delta /J$ plane at $T=0.03U$ showing the mean order parameter $\overline{\psi }=\sqrt{f_c}$. Re-entrant superfluidity is reflected by the protruding SF lobe (we also find a second, less pronounced lobe at higher $\Delta$), where increasing $\Delta$ can drive the system through a sequence of SF-insulator transitions. The orange line specifies the crossover from the normal to the BG phase (at $T=0$ this line becomes the MI-BG phase border) as determined by by shifted mean-field phase borders at this temperature, giving a better approximation than the strong-coupling approach for three dimensions. The simpler criterion of looking at purely local particle and hole excitations [Eq. (16)] would lead to the MI-BG phase border at $U=\Delta$ and predict a direct MI-SF transition at small $\Delta$. Including collective excitations in the BG, as done here, always leads to an intermediate BG phase between the MI and the SF in the $T=0$ limit.Reuse & Permissions

Figure 5

Typical self-consistent distributions $P(\psi )$ for speckle disorder for $\Delta =1U$, $\mu =1.2U$, $Z=6$, and $\mathit{JZ}=0.3U$ at different temperatures. For low temperatures, the mean-field parameters in the condensed phase are robust against finite-temperature fluctuations, but with increasing temperature the system is eventually driven into an insulating BG phase, as indicated by the distribution at $P(\psi )=\delta (\psi )$ in the at $T=0.35U$.Reuse & Permissions

Figure 6

Disorder-averaged MFP $\overline{\psi }=\int d\psi \psi P(\psi )$ characterizing the SF-insulator transition (left column) and compressibility (right column, in units of $U^{-1}$) in the $\mathit{JZ}/U$-$\mu /U$ plane for speckle disorder. Diagrams are shown for increasing disorder strength ($\Delta =0.1U$, $\Delta =0.3U$, $\Delta =1U$) in the zero- or low-temperature regime ($T=0$, $T=0.05U$, $T=0.05U$ for the top, middle, rows, respectively). In contrast to the behavior for box disorder, the structure of each lobe does not change symmetrically, but rapidly extends in the direction of decreasing $\mu /U$ with increasing disorder strength $\Delta$. Since the MI-normal phases do not exist for speckle disorder, the insulating region (black in the left figures) is always a BG. The dashed white lines denote the SF-insulator phase boundaries, indicating where $\overline{\psi }$ takes on a finite value.Reuse & Permissions

Figure 7

The compressibility, local single-particle DOS and chemical potential as a function of disorder strength $\Delta$ for speckle disorder at constant filling. Comparison of the top two plots shows a diverging compressibility and local DOS for $\Delta \to 0$ for noninteger filling (the system remains SF), while for integer filling it drops to an exponentially small value in $\Delta$ at a value of $\Delta \approx 0.1U$ and vanishes in this limit. The bottom figure shows the same quantities at finite temperature $\mathit{kT}=0.3U$, where thermal fluctuations have totally smeared out the sharp features in the compressibility. However, these persist in the local DOS, although their position is changed due to a temperature-induced shift in $\mu$.Reuse & Permissions

Figure 8

The disorder-averaged single-particle density of states $\rho (\omega =0,\mu )$ at zero frequency for $T=0.05U$ for different speckle-disorder intensities. For $\Delta =0$ this consists of a sequence of $\delta$ peaks at positive integer values of $\mu /U$; however, for any $\Delta >0$ this quantity is nonzero for $\mu ⩾0$ and the system is in the BG phase.Reuse & Permissions

Figure 9

(Main image) Phase diagram in the $T/J$-$U/J$ plane for fixed $\Delta =10J$ at fixed filling $n=1$ in three dimensions. For this specific disorder strength, the SF region is enlarged by the disorder in comparison to the pure case (i.e., disorder-induced condensation; see Sec. 5a). In the lower left corner, the SF exists at sufficiently low $T$ and $\Delta$ and the value of disorder-averaged $\overline{\psi }$ is color coded. Outside the SF region, the system is always in a BG phase but undergoes two crossovers from a regime with exponentially low compressibility $\kappa U$ for intermediately low $0.02\lesssim T/U\lesssim 0.065$ (for $\Delta /J=10$) into a strongly compressible BG regime, in the limits of both high and very low $T/U$. To clarify the quantitative behavior, the compressibility $\kappa$, the chemical potential $\mu$, and the local single-particle DOS $\rho (\omega =0)$, along the dashed line $U/J=100$, are shown in the inset.Reuse & Permissions

Figure 10

Low-temperature $kT=0.03U$ phase diagram showing the disorder-averaged MFP $\overline{\psi }$ for speckle disorder at constant filling $n=1$ in three dimensions. Multiple re-entrant behavior can be seen within a small window of $U/J$. The red line at $\Delta =0$ indicates the presence of MI-normal state; for any $\Delta >0$ the insulator is a BG.Reuse & Permissions

Figure 11

Compressibility $\kappa$ in units of $1/U$ at $kT=0.03U$ for speckle disorder at constant filling $n=1$ in three dimensions (same parameters as in Fig. 10). The incompressible normal phase only exists on the line $\Delta /J=0$, but the region where $\kappa$ is exponentially small in the insulator extends linearly with $U/J$. The white line is the SF-insulator phase border, where $\overline{\psi }$ becomes zero. The compressibility along the blue dashed line can be understood as the low-temperature case and compared to $\kappa$ in the lower two plots of Fig. 7, where the sharp features have been washed out by temperature but are still pronounced peaks.Reuse & Permissions

Figure 12

(Main background image) Same $\overline{\psi }$ phase diagram as in Fig. 10, but in the experimentally relevant high-temperature regime $\mathit{kT}=0.3U$ at $n=1$, where disorder cannot induce condensation. (Insets) Disorder-averaged MFP (a), compressibility (b), and local DOS $\overline{\rho }(\omega =0)$ in the $\mu /U$-$\mathit{JZ}/U$ plane at the same temperature $T=0.3U$ for $\Delta =1U$. In this regime the most interesting structure has been washed out by thermal fluctuations and disorder and an increase in either $U$, $\Delta$, or $T$ always suppresses condensation.Reuse & Permissions

Figure 13

Diagrams showing the effect of hopping disorder at $T=0.05U$ and an on-site speckle-disorder strength $\Delta =U$. (Main three-dimensional figure) Disorder-averaged MFP $\psi$ for speckle on-site and experimentally corresponding hopping disorder. For orientation purposes, the MF-phase boundary for the pure system is plotted as a thin dashed white line. The value $J/U$ refers to the most likely value of the hopping disorder distribution $p_J(J)$, whereas the width of the distribution $p_J(J)$ is constant. (Top left inset) Comparison of the order parameter at fixed $\mu =2.6U$ with and without hopping disorder, clearly demonstrating that the addition of hopping disorder stabilizes the SF phase and leads to a lower critical value of $J$ in this regime. (Top right inset) Subsequent comparison of phase borders in the $\mu /U$-$J/U$ plane.Reuse & Permissions

Figure 14

The condensate fraction as a function of the lattice strength $s$ for various total particle numbers in a harmonic trap, calculated within LDA and SMFT (connected dotted lines). The calculations were performed at fixed temperature $\mathit{kT}=0.3U$ and disorder strength $\Delta =1U$.Reuse & Permissions

Figure 15

Illustration of the multigrid procedure.Reuse & Permissions