Finite-size scaling analysis of localization transitions in the disordered two-dimensional Bose-Hubbard model within the fluctuation operator expansion method

The disordered Bose-Hubbard model in two dimensions at noninteger filling admits a superfluid to Bose-glass transition at weak disorder. Less understood are the properties of this system at strong disorder and energy densities corresponding to excited states. In this work, we study the Bose-glass transition of the ground state and the related finite energy localization transition, the mobility edge of the quasiparticle spectrum, a critical energy separating extended from localized quasiparticle excitations. To study these the fluctuation operator expansion is used. The level spacing statistics of the quasiparticle excitations, the fractal dimension and decay of the corresponding wave functions are consistent with a many-body mobility edge. The finite-size scaling of the lowest gaps yields a correction to the mean-field prediction of the superfluid to Bose-glass transition. In its vicinity, we discuss spectral properties of the ground state in terms of the dynamic structure factor and the spectral function which also shows distinct behavior above and below the mobility edge.

Figure 1

Characterization of the MF critical point for the 2D BHM with disorder. The Edwards-Anderson parameter $q_{\text{EA}}\left(a\right)$ and the fractal dimension $D_{\varphi }\left(b\right)$ are shown together with their respective numerical derivatives in the insets. Both are given as a function of the disorder $W$ for fixed interaction $U/t=20$ and various system sizes (see legend).

Figure 2

Finite-size scaling and critical points for the MF ground state of the disordered 2D BHM. (a) shows the shift of $D_0$ at the inflection point as function of $L$ for $\alpha =0.44$. Grey colors correspond to $L$ as in Fig. 1 and lines are best fits to Eq. (5) used to determine $D_0$ in the limit $L\to \infty$. Corresponding best infinite system size predictions for $D_c$ (left pointing triangles) and $W_c\left(U\right)$ (right pointing triangles) as determined by the crossing point in the partial collapse [inset (c)] is shown in (b) for fixed values of $\alpha$ and $U/t=20$. The adjusted parameter of convergence of these fits is given in the inset of (b). (c) shows the scaling collapse (4) of $D_{\varphi }$ as a function of the rescaled disorder (unscaled in the inset) for $W_c/t=2.9$ and $U/t=20$. (d) depicts the MF critical disorder $W_c$ determined via the collapse of $D_{\varphi }$ according to Eq. (4) and a corresponding collapse for $U/t=3$ in the inset.

Figure 3

Disorder averaged mean fraction of local fluctuations $\kappa$ quantifying the goodness of the FOE. $\left(a\right)\kappa$ as a function the disorder $W/t$ for fixed $U/t=20$, $N=3$, and various system sizes given in the legend. (b) Contour plot of $\kappa$ as function of $W/t$ and $U/t$ for fixed system size $L=32$ and $N=3$.

Figure 4

Disorder averaged fraction of local fluctuations $\kappa$ for the QP excitations as a function of their disorder average energy ${\omega }_{\gamma }$. The vertical lines mark the inverse variance weighted mean of the ME as determined in Ref. [27] As given in the legend, the lattice average $\overline{\kappa }$ and the disorder average of the (2nd to) maximum (${\kappa }_{2\text{nd}}$) ${\kappa }_{1\text{st}}$ are shown for fixed $U/t=20$ and $W/t=5$. In (a), $N=3$ and $L\in \{10,20,24,32\}$, while $L=20$ and $N\in \{3,4,5\}$ in (b) as specified in the respective legends. For all cases $N_r=25$, except for $L=10$ where $N_r=50$.

Figure 5

Scaling of the lowest QP gaps $\mathrm{\Delta }\omega$ with the grayscale indicating the truncation $N=3,4,5$, respectively, as given in the legend of (d). (a) Exemplary data for different values of $W/t$ given in the legend. Best fit parameters of the scaling ansatz Eq. (29) including one standard deviation in the error of the fit are given in (b) (see legend) and (c). Dashed lines in (b) are guides to the eye. (d) shows the adjusted parameter of convergence ${\tilde{R}}^2$ of the fits. Dashed lines in (c) and (d) correspond to fitting results for fixed $p=2$.

Figure 6

Gap and gap ratio distributions $P\left(s\right)$ [(a) and (c)] and $P\left(r\right)$ [(b) and (d)], respectively, for the QP spectrum of (1) for $U/t=20$ and $W/t=5$. For reference the analytic predictions related to Poisson statistics (dashed lines) and for the GOE (dotted lines) are included in all plots. In (a) and (b), each distribution covers an energy window ${\omega }_{\gamma }$ of width $t$ centered at various ${\omega }_{\gamma }/t\in [2,7]$ (see legend), with $L=40$. For (c) and (d), the central energy is fixed to ${\omega }_{\gamma }/t=5$, while the linear system size $L$ is varied (see legend).

Figure 7

Gap ratio $r$ (left ordinate) and fractal dimension $D$ data (right ordinate) as functions of the QP energy ${\omega }_{\gamma }/t$ for $U/t=20$, $W/t=5$ and $L=32$ averaged over 95 realizations. Black lines are moving averages as a guide to the eye and dashed lines mark $r_P$ and $r_G$. The crossing point (vertical arrow) of the data with the critical $r_c$ (shaded red, narrow) and $D_c$ (shaded blue, wide) mark the ME. (Insets) Exemplary squared QP wave functions $|{\mathbf{v}}_{\ell }^{\left(\gamma \right)}{|}^2$ normalized to the maximum. Reproduced from [27].

Figure 8

Decay behavior of $|{\mathbf{v}}_{\ell }^{\left(\gamma \right)}{|}^2$ for $U/t=20$ in (1) with $L=32$ and $N=3$ if not specified otherwise. For $W/t=7$ examples of $\mathcal{A}\left(r\right)$ and $\mathcal{I}\left(r\right)$ are shown in (a) and (b). All distributions are averaged over bins of 16 levels closest in energy to $\omega /t\in \left[0.75,5.75\right]$ increasing in steps of 0.5 from light gray to black (top to bottom at large distances). Inset in (a) shows a corresponding typical state ${〈{log}_{10}{\left|{\mathbf{v}}_{\ell }^{\left(\gamma \right)}\right|}^2〉}_d$ at $\omega /t=5.5$. Dashed lines in (a) are exponential fits of Eq. (35) and corresponding inverse decay lengths $1/\lambda$ are given in (c) for $W/t=10$ and $W/t\in \left[1,15\right]$ in (d). Data sets in (c) for $L\in [10,20,24,32,40]$ are accompanied by linear fits (solid lines) and circles have $L=32$ and $N=4$ (see legend). Zeros of these fits are given in the inset with solid (dashed) linear fit lines for all $L$ ($L>10$). ${\omega }_c$ in (c) and lines in (d) represent the ME determined in [27] via fractal dimension $D$ or gap ratio $r$ (see legend), while circles (crosses) are derived from the decay length $\lambda$ ($\tilde{\lambda }$) for all sizes ($L>10$).

Figure 9

Finite-size ($L=40$) spectral functions for the QP ground state (see text) of (1) at $U/t=20$ and weak disorder $W/t=1$. Shown are (a) the momentum distribution $n\left(\mathbf{k}\right)$, [(b), inset (a)] the spectral function $\mathcal{A}(\mathbf{k},\omega )$ with separate color axis for the retarded and advanced branches, (c) the static structure factor $SSF\left(\mathbf{k}\right)$, and (d) the dynamic structure factor $DSF(\mathbf{k},\omega )$. All spectral functions are shown along a path of high-symmetry points of the first Brillouin zone of the square-lattice, in units of $\pi /a$: $(0,1)\to (0,0)\to (1,1)\to (1,0)\to (0.5,0.5)$. Dashed lines in (b) and (d) mark the ME as determined in Ref. [27].

Figure 10

Finite-size ($L=40$) spectral functions for the QP ground state (see text) of (1) at $U/t=20$ and moderate disorder $W/t=5$. Shown are (a) the momentum distribution $n\left(\mathbf{k}\right)$, [(b), inset (a)] the spectral function $\mathcal{A}(\mathbf{k},\omega )$ with separate color axis for the retarded and advanced branches, (c) the static structure factor $SSF\left(\mathbf{k}\right)$, and (d) the dynamic structure factor $DSF(\mathbf{k},\omega )$. All spectral functions are shown along a path of high-symmetry points of the first Brillouin zone of the square lattice, in units of $\pi /a$: $(0,1)\to (0,0)\to (1,1)\to (1,0)\to (0.5,0.5)$. Dashed lines in (b) and (d) mark the ME as determined in Ref. [27].