Mott glass to superfluid transition for random bosons in two dimensions

We study the zero-temperature superfluid-insulator transition for a two-dimensional model of interacting, lattice bosons in the presence of quenched disorder and particle-hole symmetry. We follow the approach of a recent series of papers by Altman, Kafri, Polkovnikov, and Refael, in which the strong disorder renormalization group is used to study disordered bosons in one dimension. Adapting this method to two dimensions, we study several different species of disorder and uncover universal features of the superfluid-insulator transition. In particular, we locate an unstable finite disorder fixed point that governs the transition between the superfluid and a gapless, glassy insulator. We present numerical evidence that this glassy phase is the incompressible Mott glass and that the transition from this phase to the superfluid is driven by a percolation-type process. Finally, we provide estimates of the critical exponents governing this transition.

Figure 1

A schematization of the universal features of the proposed flow diagram. The $x$ axis gives the ratio of the mean of the renormalized Josephson coupling distribution to the mean of the renormalized charging energy distribution. The $y$ axis gives the ratio of the standard deviation of the Josephson coupling distribution to its mean. In this context, the Josephson coupling distribution only includes the dominant $2\tilde{N}$ couplings in the renormalized $J$ distribution, where $\tilde{N}$ is the number of sites remaining in an effective, renormalized lattice. See the text of Sec. 3 for the reasoning behind the exclusion of weaker Josephson couplings from statistics.

Figure 2

The site decimation RG step: The site marked with the $X$ has the dominant charging energy and is decimated away, generating bonds between neighboring sites $j$ and $k$ with the effective coupling given in Eq. (21). The new local structure of the lattice is shown to the right.

Figure 3

The link-decimation RG step: The crossed link has the dominant Josephson coupling. The two sites it joins are merged into one cluster, resulting in the effective lattice structure shown to the right. The cluster $C$ has an effective charging energy given in Eq. (26).

Figure 4

Site decimation with the sum rule: The site with the dominant charging energy (marked with an $X$) is coupled to two sites ($j$ and $k$) that are already coupled to one another. After site decimation, the effective Josephson coupling between sites $j$ and $k$ is the sum of the old coupling and the effective coupling generated through decimation of site $X$ [see Eq. (32)].

Figure 5

Link decimation with the sum rule: The two sites connected by the dominant Josephson coupling (sites $j$ and $k$) are both coupled to a third site $m$. Following link decimation, the effective Josephson coupling between site $m$ and the cluster $C$ is the sum of the two preexisting couplings between site $m$ and sites $j$ and $k$ [see Eq. (33)], up to corrections coming from Debye-Waller factors.

Figure 6

The projection, in the $\Delta J/\overline{J}$ vs $\overline{J}/\overline{U}$ plane, of the flows of the coupling distributions at different stages of the RG. The initial distributions $P_i\left(U\right)$ and $P_i\left(J\right)$ are both truncated Gaussians, and $J_0$ (the center of the initial $J$ distribution) is used as the tuning parameter. Each flow corresponds to a different choice of the tuning parameter. The flows start at the bottom of the figure and go up and to the left or up and to the right. A smaller value of the thresholding parameter is used near criticality as indicated by the legend. All runs were done on $L=100$ lattices.

Figure 7

Same as Fig. 6, except $P_i\left(U\right)$ is Gaussian and $P_i\left(J\right)\propto J^{-1.6}$, with cutoffs chosen to make the latter distribution very broad. The parameter $U_0$ is used to tune through the transition. The flows begin near the center of the figure. To the left of the figure, flows initially propagate toward the bottom left but eventually turn around and propagate toward the top left. To the right of the figure, flows initially propagate toward the bottom left but eventually turn around and propagate toward the bottom right. All runs were done on $L=300$ lattices.

Figure 8

Same as Fig. 6, except $P_i\left(U\right)\propto U^5$ and $P_i\left(J\right)\propto J^{-3}$. The tuning parameter is $J_{\text{min}}$, the lower cutoff of $P_i\left(J\right)$. All runs were done on $L=150$ lattices.

Figure 9

In these numerical flow diagrams, $P_i\left(U\right)$ is bimodal and $P_i\left(J\right)\propto J^{-1.6}$, with cutoffs chosen to make the latter distribution very broad. In panel (a), we show a few sample flows that start near the top of the figure and initially propagate toward the lower left-hand corner. The complex features of these flows reflect the structural details of the bimodal $U$ distribution. In panel (b), we exhibit the flows at late RG times, when the procedure has had an opportunity to renormalize away the microscopic details of the initial distributions. Then, the flows are, at least qualitatively, more similar to those seen in Fig. 7. All runs were done on $L=200$ lattices.

Figure 10

Log-log plots of initial and renormalized Josephson coupling distributions for near-critical flows. All runs were done on $L=100$ lattices with $\alpha =5\times {10}^{-6}$. Each plot shows the initial distribution and the distribution when the effective lattice has $1\%$ of the original number of sites. The initial distributions have four different forms, but the distributions after renormalization show a universal power law. Note that the plots of initial distributions in these plots were not constructed from actual data (i.e., actual numerical sampling of the distributions) but were instead constructed by hand.

Figure 11

The distributions from Fig. 10, with the Josephson coupling strength scaled by the mean RG scale $\Omega$ at the corresponding stage of the RG. The near collapse of the distributions reveals the universal power-law form of the Josephson coupling distribution near the unstable fixed point. A refined version of this plot, showing data for larger lattices (but also different choices of initial distributions), can be found in Fig. 13b.

Figure 12

Log-log plots of initial and renormalized charging energy distributions for near-critical flows. All runs were done on $L=200$ lattices with $\alpha =5\times {10}^{-6}$. Each plot shows the initial distribution and the distribution when the effective lattice has $0.5\%$ of the original number of sites. The initial charging energy distributions have four different forms, but the distributions after renormalization show a universal form. Note that the plots of initial distributions in these plots were not constructed from actual data (i.e., actual numerical sampling of the distributions) but were instead constructed by hand.

Figure 13

In panel (a), the renormalized distributions from Fig. 12 are plotted together, with the charging energies $U$ scaled by the mean RG scale $\Omega$. In panel (b), we provide a similar plot for the renormalized Josephson coupling distributions produced by the runs in Fig. 12.

Figure 14

Four characteristic behaviors of $s_{\text{max}}$, $s_2$, ${\Delta }_{\text{min}}$, and $\chi$ as a function of system size. Here, $P_i\left(U\right)$ and $P_i\left(J\right)$ are both bimodal, and $J_h$, the center of the higher peak of the Josephson coupling distribution, is used as the tuning parameter. All quantities have been normalized by their value for $L=10$, and data are shown for $L=10$ to $100$. In panel (a), the quantities reflect the purely local physics of the Mott insulator. In panel (b), $s_{\text{max}}$ and $s_2$ grow subextensively with system size, with what appears to be a power law. The gap also decays with a slow power, and the susceptibility remains constant. In panel (c), all quantities show power-law behavior out to $L=100$. The reference line shows $L^2$ growth. Panel (d) reflects the macroscopic clusters of the superfluid phase. The cluster size $s_{\text{max}}$ is parallel to the $L^2$ reference line, and the susceptibility $\chi$ is parallel to the $L^4$ reference line for large $L$.

Figure 15

Four characteristic behaviors of $s_{\text{max}}$, $s_2$, ${\Delta }_{\text{min}}$, and $\chi$ as a function of system size. The initial distributions are those described in Appendix C. All quantities have been normalized by their value for $L=25$, and data are shown for $L=25$ to $300$. In the four panels, $U_0=400$, $9.2$, $8.97$, and $8.8$, respectively. Panel (a) reflects the purely local physics of the Mott insulator. Panel (b) shows a glassy regime characterized by rare-regions clusters that grow subextensively in system size. The reference line shows the power law that $s_{\text{max}}$ obeys near criticality. This nearly critical regime is shown in panel (c). The reference line here shows $L^2$ growth. Finally, panel (d) shows the superfluid regime, in which $s_{\text{max}}$ is asymptotically parallel to the $L^2$ reference line. The susceptibility $\chi$ is expected to grow as $L^4$ for very large $L$, but it does not quite reach this behavior (indicated with a reference line) for $L\le 300$.

Figure 16

Panel (a) shows the scaling collapse of $s_{\text{max}}$ as a function of tuning parameter, and panel (b) shows a similar collapse of $s_2$. Each line corresponds to a different value of the lattice size. We show data for $L=25$, 50, 75, 100, 150, 200, and 300. The insets show magnified views of the vicinity of the critical point for the four largest lattice sizes. The error bars, which are difficult to see in the larger plots, are clearly visible in the insets. To see the values of the exponents $\nu$ and $d_f$ used in each panel, consult Eqs. (71), (72), (76), and (77).

Figure 17

The number per unit area of clusters with local gap near $\Delta$ for a choice of parameters in the glassy phase. The initial distributions are those described in Appendix C, with $P_i\left(U\right)$ Gaussian and $P_i\left(J\right)\propto J^{-1.6}$. The value of the tuning parameter is $U_0=8.8$, and this puts the system on the glassy side of the transition at $U_0\approx 8.97$. Data are shown for $L=75$, 150, and 300 lattices. The density decays faster than a power law at small $\Delta$.

Figure 18

In panel (a), the density (number per unit area) of clusters with a given gap $\Delta$ for a choice of parameters in the superfluid phase. The initial distributions are those described in Appendix C, with $P_i\left(U\right)$ Gaussian and $P_i\left(J\right)\propto J^{-1.6}$. The value of the tuning parameter is $U_0=9.2$, and this puts the system on the superfluid side of the transition at $U_0\approx 8.97$. Data are shown for $L=75$, 150, and 300 lattices. The density profiles exhibit two peaks. The broad peak that is visible in panel (a) remains invariant with $L$. To expose the second peak, we provide a magnified view of the low-$\Delta$ part of the density in panel (b). This peak simultaneously shrinks and propagates toward $\Delta =0$ as the system size is raised.

Figure 19

Sweeps of $\rho \left(s\right)$ as the system is tuned through the superfluid-insulator transition on L = 300 lattices. The initial distributions are those described in Appendix C, with $P_i\left(U\right)$ Gaussian and $P_i\left(J\right)\propto J^{-1.6}$. The tuning parameter is $U_0$, the center of $P_i\left(U\right)$. Panel (a) shows the sweep from deep in the glassy phase ($U_0=20$) to the transition ($U_0=8.97$). The closest data set to the transition is indicated by the black line with large data point markers. This line is reproduced in panel (b), which shows the sweep from the transition into the superfluid phase (up to $U_0=6.5$). In the superfluid phase, the density plot is characterized by a peak at large $s$ and a remnant decay at low $s$.

Figure 20

Scaling collapse of the $\rho \left(s\right)$ curves from Fig. 19. Small cluster sizes ($s<30$) need to be discarded because they are nonuniversal. Large cluster sizes ($s>100$) need to be discarded because they are noisy. Then, the remaining $\rho \left(s\right)$ curves, taken for different values of the tuning parameter, collapse onto a universal curve.

Figure 21

The behavior of the mean $b$ factor for clusters of size $s$ as a function of $s$. In panel (a), we show data for the glassy regime and for the nonmacroscopic clusters of the superfluid regime. The initial distributions are those described in Appendix C, and data are shown for seven values of the tuning parameter $U_0$ on $L=300$ lattices. The log-log plot shows power-law behavior of $\overline{b}\left(s\right)$ vs $s$. In panel (b), we show data for the macroscopic clusters when $U_0=8.8$. The plot indicates that $\overline{b}\left(s\right)\sim s$ for macroscopic clusters. In both plots, the error bars indicate the error on the mean $\overline{b}\left(s\right)$ over all clusters of size $s$. For some of the largest and smallest values of $s$ in each plot, the absence of an error bar indicates that only one cluster of that size was generated in all of the samples.

Figure 22

A test of the convergence of $s_{\text{max}}$ in the thresholding parameter $\alpha$. The variable $\upsilon$ is the ratio of the estimate of $s_{\text{max}}$ for ${\alpha }_h$, the less conservative value of the thresholding parameter to ${\alpha }_{\ell }$, the more conservative value. We plot $\upsilon$ against the tuning parameter $U_0$. The closest data point to criticality ($U_0=8.97$) is indicated with a diamond. Note that $\upsilon >1$ typically indicates convergence since less conservative thresholding (higher $\alpha$) corresponds to throwing away more bonds and, therefore, biases the system away from cluster formation. Smaller values of ${\alpha }_h$ and ${\alpha }_{\ell }$ are used in the vicinity of the transition. See the text of Appendix C for details.

Figure 23

In panel (a), a sweep of renormalized $J$ distribution through the glass and into the superfluid. The initial distributions are those described in Appendix C: $P_i\left(U\right)$ is Gaussian and $P_i\left(J\right)\propto J^{-1.6}$. All data are taken for $L=300$ lattices, and the renormalized distribution is computed when 300 sites remain in the effective lattice. The values of $U_0$ shown are 18, 12, $9.6$, $8.97$ (near critical), $8.4$, and $7.5$. Panel (b) shows the corresponding sweep of the renormalized $U$ distribution at the same stage of the RG.

Figure 24

The graph structure for the tests reported in Figs. 25 and 26. The links $J_{13}$, $J_{14}$, and $J_{25}$ are assumed to be perturbatively weak. The charging energies $U_1$ and $U_2$ and the Josephson coupling $J_{12}$ are varied to explore potentially troublesome scenarios.

Figure 25

In this test, $J=J_{12}$ is assumed to be the strongest coupling in the system, but $U=U_1=U_2$ may be of the same order. An effective coupling between sites 3 and 5 is calculated using two methods. One is the usual RG scheme used in this paper. Another is a hybrid exact diagonalization and RG scheme: the two-site problem of sites 1 and 2 is exactly diagonalized. Then, the resulting cluster is decimated away, and perturbation theory is used to calculate an effective coupling between sites 3 and 5. The two candidate values for the effective coupling $J_{35}$ are compared in the plot, as a function of $\frac{J}{U}$.

Figure 26

In this test, $J_{12}=0.05$ and $U_2=1$. Hence, $J_{12}$ is relatively quite weak. However, we vary $U_1$ such that it passes through a regime where $|U_1-U_2|<J_{12}$, where there may be a danger of resonance effects. We calculate the effective couplings $J_{34}$ and $J_{35}$ using two schemes. One is the usual RG scheme used in this paper. Another is a hybrid exact diagonalization and RG scheme: the two-site problem of sites 1 and 2 is exactly diagonalized. Then, the resulting cluster is decimated away, and perturbation theory is used to calculate an effective coupling between sites 3 and 4 and between sites 3 and 5. The effective couplings predicted by the two methods are compared in the plot, as a function of $\frac{U_1}{U_2}$. No resonance effects are observed.

Figure 27

A sample-by-sample comparison of the particle-number variance predictions from exact diagonalization and from the renormalization procedure. The disordered couplings are drawn from three different choices of the distributions, with 100 samples per distribution type. See the text of Appendix D for details on the distribution choices GG, GP, and FP. Also pictured is the coincidence line $y=x$, along which the points would ideally fall for full quantitative agreement.

Figure 28

Same as Fig. 27, but the quantity being compared is the charging gap ${\Delta }_{\text{min}}$.