Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases

We demonstrate many-body multifractality of the Bose-Hubbard Hamiltonian’s ground state in Fock space, for arbitrary values of the interparticle interaction. Generalized fractal dimensions unambiguously signal, even for small system sizes, the emergence of a Mott insulator that cannot, however, be naively identified with a localized phase in Fock space. We show that the scaling of the derivative of any generalized fractal dimension with respect to the interaction strength encodes the critical point of the superfluid to the Mott insulator transition, and provides an efficient way to accurately estimate its position. We further establish that the transition can be quantitatively characterized by one single wave function amplitude from the exponentially large Fock space.

Figure 1

Finite-size fractal dimensions of the BHH ground state versus $J/U$ and filling factor $\nu$ (abscissa axis in both plots). Upper panel: Density plot of ${\tilde{D}}_1$ for $L=6$ after linear interpolation of the numerically calculated points indicated by the black grid. White crosses indicate the position of the SF to MI transition [45, 46]. Lower panel: ${\tilde{D}}_2$ for $J/U=1$ (open symbols), $J/U={10}^{-2}$ (filled symbols) and $L=6$ (black), 7 (blue), 8 (green), 9 (orange).

Figure 2

Intensities $|{\psi }_{\mathit{n}}{|}^2$ in the Fock basis of the BHH ground state versus $\eta$ for $L=10$, $\nu =1$. Solid lines highlight the maximum and minimum intensities on a Fock state with a certain number of particle-hole ($p\text{−}h$) excitations on top of the homogeneous state $|\mathit{\nu }⟩$. The values of $\eta$ considered are highlighted by symbols only for the maximum intensity. Dashed lines indicate the intensity value of the first two $p\text{−}h$ manifolds for $\eta =\infty$ [see Eq. (4)].

Figure 3

Finite-size GFDs ${\tilde{D}}_q$ ($q=1$, 2, $\infty$) of the BHH ground state versus $\eta$ for $\nu =1$. Solid lines in main panel are numerical results ($L=16$ only for $q=\infty$). Horizontal dashed lines mark the $D_q$ values for $\eta =\infty$. The inset shows numerical (symbols) and analytical results from perturbation theory (solid lines) for $L=8$.

Figure 4

Extrapolation of ${\tilde{D}}_{\infty }$ as $L\to \infty$ for $\eta =1/7$, $\nu =1$. Symbols are numerical data. The solid line is the best fit to Eq. (5) (chi square $\simeq 10$ with 13 degrees of freedom). The horizontal dashed line and the shaded area mark, respectively, the $D_{\infty }$ value and its 95% confidence interval. The secondary abscissa axis indicates the size of Fock space for each $L$.

Figure 5

Dimension ${\tilde{D}}_{\infty }$ (left vertical axis) and its derivative (right vertical axis) versus $\eta$ for the BHH ground state and $L=\{5–10,12,14,16,18,20,25,30\}$, $\nu =1$. Symbols indicate numerical data (errors within symbol size), dashed lines correspond to the best Padé fits and solid lines are their respective derivatives. For clarity, symbols are shown only for $L=\{5,7,9,12,18,30\}$.

Figure 6

Position ${\eta }_{*}(L)$ of the maximum of ${\tilde{D}}_{\infty }^{\prime }(\eta )$ and ${\tilde{D}}_2^{\prime }(\eta )$ for $\nu =1$. Solid lines are best fits to Eq. (6), with $b=1.92\pm 0.18(1.85\pm 0.27)$, ${\ell }_q=0.025\pm 0.009(0.032\pm 0.018)$ for ${\tilde{D}}_{\infty }$ (${\tilde{D}}_2$) data. Dashed lines and shaded regions mark, respectively, the estimated ${\eta }_c$ and its 95% confidence interval: ${\eta }_c\in [0.284,0.308]$ from ${\tilde{D}}_{\infty }$, and ${\eta }_c\in [0.270,0.312]$ from ${\tilde{D}}_2$. The inset shows the maximum value of the derivatives versus $L$.