Figure 1

Schematic plot of the multifractal spectrum $f\left(\alpha \right)$. A metal is represented by a “needle,” i.e., $f\left(\alpha \right)$ having zero width, at $\alpha =d$. At criticality $f\left(\alpha \right)$ acquires a finite width and the apex shifts to ${\alpha }_0>d$. The negative parts of $f\left(\alpha \right)$ (gray area) correspond to rare events—values of the wave-function amplitude that typically do not occur in a single sample.Reuse & Permissions

Figure 2

Symmetry of the multifractal spectrum. (a) Multifractal exponents ${\Delta }_q$ for the PRBM model with $b=4$, 1, 0.3, and 0.1. The symmetry (2.37) with respect to the point $q=1∕2$ is evident. A small difference between ${\Delta }_q$ (full line) and ${\Delta }_{1-q}$ (dashed line) is due to numerical errors. (b) Same data now in terms of the singularity spectrum $f\left(\alpha \right)$. Dashed lines represent $f(2-\alpha )+\alpha -1$, demonstrating the validity of Eq. (2.38) From 220.Reuse & Permissions

Figure 3

IPR distribution at the Anderson transition (GOE): (a) 3D (system sizes $L=8$, 11, 16, 22, 32, 44, 64, and 80) and (b) 4D ($L=8$, 10, 12, 14, and 16) From 217.Reuse & Permissions

Figure 4

The rms deviation ${\sigma }_q$ of $\ln P_q$ extrapolated to $L\to \infty$ in 3D $(\times )$ and 4D $(*)$. The dotted line is the analytical result (2.49) for $ϵ=0.2$; the full lines represent Eqs. (2.49, 2.50) with $ϵ=1$. Inset: Evolution of ${\sigma }_q$ with $L$ in 3D for values of $q=0.5$, 1.5, 2, 2.5, 3, 4, 5, and 6. The leading finite size correction of all data has the form $L^{-y}$ with $y=0.25–0.5$ for 3D and $y=0.1–0.4$ in 4D. From 217.Reuse & Permissions

Figure 5

Dimensionality dependence of the multifractal spectra. (a) Singularity spectrum $f\left(\alpha \right)$ in 3D (dashed line) and 4D (full line). To illustrate the evolution of the spectrum from $d=2$ to $d=4$, analytical results for $d=2+ϵ$ are shown for $ϵ=0.2$ (dotted) and $ϵ=0.01$ (dot-dashed). Inset: Comparison between $f\left(\alpha \right)$ for 3D and the one-loop result of the $2+ϵ$ expansion with $ϵ=1$ (solid). (b) Fractal dimensions $D_q$ in 3D (dashed line) and 4D (full line). Analytical results for $d=2+ϵ$ with $ϵ=0.2$ (dotted line) and $ϵ=0.01$ (dot-dashed) are also shown. From 217.Reuse & Permissions

Figure 6

Possible behavior of the singularity spectrum $f\left(\alpha \right)$ at $\alpha \to 0$: (a) no singularity, (b) termination, and (c) freezing. The corresponding behavior of ${\tau }_q$ is shown as well.Reuse & Permissions

Figure 7

Surface and bulk multifractal spectra ${\tau }_q$ and $f\left(\alpha \right)$ for a 2D metal with $\gamma =0.01$. For details see text. From 307.Reuse & Permissions

Figure 8

Critical level spacing distribution of 2D symplectic systems for periodic (PBC) and Dirichlet (DBC) boundary conditions and various system sizes $L$. PBC: $L=20$ $(◇)$; DBC: $L=40,80,120$ $(+,◻,\times )$. Adapted from 290; see also 291.Reuse & Permissions

Figure 9

Crossover from the quasimetallic $(b⪢1)$ to the quasi-insulating $(b⪡1)$ behavior in the PRBM ensemble: fractal dimension $D_2$ vs parameter $b$ $(◻)$. Data points are numerical simulations, while the dashed lines represent $b⪢1$ and $b⪡1$ analytical asymptotics, $D_2=1-1∕\pi b$ [Eq. (3.7)] and $D_2=2b$ [Eq. (3.20)]. Also shown is the spectral compressibility $\chi$ $(○)$. The dotted lines indicate the analytical results for $b⪢1$ and $b⪡1$, Eqs. (3.28, 3.30). From 225.Reuse & Permissions

Figure 10

Multifractal spectrum $f\left(\alpha \right)$ for $b=1$ $(◻)$ and $b=4$ $(○)$. Lines indicate parabolic approximation (2.36). Inset: exponents ${\tau }_q$ $(◻)$, ${\tau }_q^{\mathrm{typ}}$ $(◼)$ for $b=1$. From 225.Reuse & Permissions

Figure 11

Scale invariance of the IPR distribution in the PRBM model. (a) Evolution of the distribution $\mathcal{P}\left(\ln P_2\right)$ for $b=1$ with the system size $L$ (from right to left: $L=256$, 512, 1024, 2048, and 4096). The scale invariance of the IPR distribution is clearly seen. (b) Distribution flow of $\ln P_2$ calculated from the kinetic equation (3.17) at $t=b\ln r=1.2\cdots 1.7$ (from right to left). Oscillations near $\ln P_2=-1.5$ are numerical artifacts due to rounding errors. From 225.Reuse & Permissions

Figure 12

Distribution function $\mathcal{P}\left({\tilde{P}}_q\right)$ at $q=2$ $(○)$, 4 $(◻)$, and 6 $(◇)$ at $b=4$ for systems of size $L=4096$. The solid line represents the analytical result Eq. (3.9). Inset: Asymptotic of $\mathcal{P}\left({\tilde{P}}_4\right)$. Dashed line indicates power law with exponent $x_4=1.7$. From 225.Reuse & Permissions

Figure 13

Strong multifractality in the PRBM model. (a) The function $T\left(q\right)$ characterizing exponents $\tau \left(q\right)$ via $\tau \left(q\right)=2bT\left(q\right)$ at $b⪡1$. Dashed line indicates pole position. Inset: Legendre transform $F\left(A\right)$ describing multifractal spectrum via $f\left(\alpha \right)=2bF(\alpha ∕2b)$. (b) $f\left(\alpha \right)$ for $b=0.25$ $(◇)$ and $b=0.1$ $(△)$. Inset: Exponent $\tau \left(q\right)$. Dashed and dotted lines indicate analytical results, Eqs. (3.23, 3.20). From 225.Reuse & Permissions

Figure 14

Level statistics in the PRBM model. (a) Two-level correlation function $R_2\left(s\right)$ for two system sizes $L=256$ $(○)$ and $L=512$ $(◻)$ at $b=0.1$. The solid line indicates the theoretical result (3.29). (b) Variance of the number of levels $⟨\delta N^2⟩$ as a function of the energy width for the interval parametrized by the mean level number $⟨N⟩$. Traces correspond to $b=1$ (open $\circ$: $L=4096$, closed: $L=2048$), $b=0.25$ (open $◻$: $L=4096$, closed: $L=2048$) and $b=0.05$ (open $◇$: $L=4096$, $◇$ with dot: $L=1024$, closed: $L=512$). The dashed line indicates the analytical prediction Eq. (3.30). From 225.Reuse & Permissions

Figure 15

The ratio $R_p={\Delta }_q^{\mathrm{s}}(b,p)∕{\Delta }_q^{\mathrm{b}}\left(b\right)$ of the surface and bulk anomalous exponents for large $b$, as a function of the reflection parameter $p$. Diamonds represent the results of the $\sigma$-model analysis with a numerical solution of the corresponding classical evolution equation (3.35). Circles represent a direct computer simulation of the PRBM model with $b=8$. The ratio $R_p$ has been evaluated for the range $0<q<1$, where numerical accuracy of the anomalous exponents is the best. Within this interval the obtained $R_p$ is $q$ independent (within numerical errors) in agreement with Eq. (3.34). From 220.Reuse & Permissions

Figure 16

Surface vs bulk criticality in the PRBM model: evolution from the quasimetallic $(b⪢1)$ to the quasi-insulating $(b⪡1)$ regime. Upper panel: Anomalous exponent ${\Delta }_2\equiv D_2-1$ as a function of $b$ from numerical simulations in the bulk and at the boundary for the reflection parameter $p=0$ and $1$. Inset: Data for $p=3$ compared to the $p=0$ bulk values. Lower panel: Surface and bulk data for $p=0$ compared with analytical results for small and large $b$ (using $R_0=2.78$), Eqs. (3.7, 3.34, 3.20, 3.38). From 220.Reuse & Permissions

Figure 17

Surface vs bulk multifractality in the PRBM model for large $b$. Upper panel: Boundary and bulk multifractal spectra ${\Delta }_q^{\mathrm{s}}$ and ${\Delta }_q^{\mathrm{b}}$ at $b=2$, $4$, and $8$ for the reflection parameter $p=1$. In accordance with Eq. (3.37), the surface multifractality spectrum is enhanced by a factor close to 2 compared to the bulk. Middle panel: Surface spectrum divided by the analytical $b⪢1$ result, Eq. (3.34). With increasing $b$, the numerical data converge towards the analytical result (dashed line). Lower panel: Analogous plot for the bulk spectrum. From 220.Reuse & Permissions

Figure 18

Numerically determined boundary and bulk anomalous dimensions ${\Delta }_q$ at $b=0.1$ for $p=0$, $1$, and $3$. As expected, the bulk anomalous dimension is independent of $p$. In accordance with Eq. (3.39), for $p=3$ surface and bulk dimensions have the same values. Inset: The $p=0$ data compared to the analytical results, surface [solid line, Eq. (3.38)] and bulk [dashed line, Eq. (3.20)]. Analytical data have been calculated for $q\ge 0.6$ and mirrored for $q\le 0.4$ using the symmetry relation ${\Delta }_q={\Delta }_{1-q}$. In the vicinity of $q=1∕2$, at $\mid q-1∕2\mid \lesssim 1∕\ln b^{-1}$, the analytical result (3.20) breaks down. From Subramaniam, 220.Reuse & Permissions

Figure 19

Multifractal spectrum ${\delta }_q$ for the Ando model (dashed, $W_c=5.84$) and the SU(2) model (solid, $W_c=5.953$). To highlight deviations from parabolicity, reduced anomalous dimensions ${\delta }_q={\Delta }_q∕q(1-q)$ are plotted. Anomalous dimensions ${\delta }_q^{\mathrm{typ}}$ obtained from typical IPR are also shown $(◇)$. Dashed lines indicate the estimated error $\left(2\sigma \right)$ in ${\delta }_0$. Inset: Blow up of the solid line behavior near $q=\frac{1}{2}$ is represented by empty circles $(○)$. Filled symbols $(●)$ show original trace after reflection at $q=\frac{1}{2}$. Dot-dashed line is a fit (offset: ${10}^{-3}$) ${\delta }_q=0.1705+0.0043(q-\frac{1}{2}{)}^2$. From 216.Reuse & Permissions

Figure 20

Two-parameter flow diagram of the Pruisken $\sigma$ model, as first proposed by 175.Reuse & Permissions

Figure 21

Chalker-Coddington network model of the IQH transition (64). The symbols $\pm c$, $\pm s$ denote the components $\pm \cos \theta$, $\pm \sin \theta$ of scattering matrices at the nodes of the network.Reuse & Permissions

Figure 22

Conductance distribution at IQH (solid) and $2d$ symplectic (dashed) critical points for periodic boundary conditions and a square sample geometry. Adapted from 255.Reuse & Permissions

Figure 23

Multifractal spectrum at the IQH transition. Thick solid line: Numerical results for $f\left(\alpha \right)$ obtained from scaling of average IPR after extrapolation to $L\to \infty$. Data for several finite values of $L=16$, 128, and 1024 are also shown. Thick dashed line: Parabolic approximation, Eq. (6.9). Inset: Data points from typical IPR for $L=\mathrm{16,128}$, and 1024 (open symbols) and extrapolation to infinite system size (closed circles); data from average IPR are shown by a solid line. From 99.Reuse & Permissions

Figure 24

Anomalous multifractal dimensions ${\Delta }_q$ [divided by $q(1-q)$] at the IQHE critical point, as obtained from extrapolation of the average IPR. In the case of exact parabolicity, the plotted quantity should be constant. The dotted lines indicate the error bars obtained for ${\alpha }_0-2$. The circles give ${\Delta }_q^{\mathrm{typ}}∕q(1-q)$ as obtained from the typical IPR. As explained in Sec. 2C5, ${\Delta }_q^{\mathrm{typ}}={\Delta }_q$ for $q<q_{+}$; for IQHE with parabolic spectrum (6.9, 6.10) we find $q_{+}=[2∕({\alpha }_0-2){]}^{1∕2}\simeq 2.76$. Thus the last two data points for ${\Delta }_q^{\mathrm{typ}}∕q(1-q)$ are in the range $q>q_{+}$, which explains their downward deviations from ${\Delta }_q∕q(1-q)$. Based on data of 99.Reuse & Permissions

Figure 25

Possible configurations of paths passing four times through a network node. The symbols $c$ and $\pm s$ denote the elements $\cos \theta$, $\pm \sin \theta$ of the $S$-matrix at the node. Statement (2) in Sec. 6D2 allows one to remove the quantum interference contribution (iii) and to associate with contributions (i) and (ii) weights ${\cos }^2\theta$ and ${\sin }^2\theta$, respectively, yielding a mapping to the classical percolation.Reuse & Permissions

Figure 26

SQHE criticality. (a) Scaling plot of the DOS for system sizes $L=16(◇)$, $32(◻)$, and $96(○)$. Dashed and dotted lines indicate power laws (dashed: $E^{1∕7}$, dotted: $E^2$), $\delta =1∕2\pi L^{7∕4}$ denotes the level spacing at $E=0$. Inset: Same data on a linear scale and the RMT result (solid curve). From 100. (b) Scaling of the two-point conductance with distance $r$ between the contacts: average value (empty symbols), $⟨g⟩$, and typical value (filled symbols), $g_{\mathrm{typ}}=\exp ⟨\ln g⟩$, for $L=128(◻)$ and $L=196(○)$. Also shown is scaling of the two-point Green’s function $⟨\mid G{\mid }^2⟩$ and $\mid G{\mid }_{\mathrm{typ}}^2=\exp ⟨\ln \mid G{\mid }^2⟩$ [$L=128(△)$, $L=196(◇)$]. The lines correspond to the $r^{-1∕2}$ (dotted) and $r^{-3∕4}$ (dashed) power laws. Deviations from power-law scaling at large $r$ are due to the finite system size. From 226.Reuse & Permissions

Figure 27

SQHE multifractality. Upper panel: Anomalous dimensions ${\Delta }_q$ (solid line) and ${\Delta }_q^{\mathrm{typ}}$ (circles) at the SQHE critical point. Numerical results agree well with analytical findings ${\Delta }_2=-1∕4$ and ${\Delta }_3=-3∕4$. Data are presented in the form ${\Delta }_q∕q(1-q)$ emphasizing deviations from exact parabolicity (dashed line). Lower panel: Singularity spectrum $f\left(\alpha \right)$ (solid) and the parabolic approximation (dashed). Inset: Magnified view of the apex region. From 226.Reuse & Permissions

Figure 28

Phase diagram of the Cho-Fisher model as obtained by 65 from transfer-matrix calculations. The plane is spanned by the parameters ${\sin }^2\theta$, the interplaquette tunneling probability, and $p$, the concentration of vortex disorder: these control the short-distance values of the conductivity components ${\sigma }_{xy}$ and ${\sigma }_{xx}$, respectively.Reuse & Permissions

Figure 29

DOS in the thermal metal phase of the Cho-Fisher model. (a) Low-energy DOS in metallic phase. Parameters (upper curves): $p=0.5$, $\theta =\pi ∕4$, system sizes $L=128$ (squares) and $L=256$ (full circles). The straight dashed line represents logarithmic asymptotics. For lowest energies the RMT oscillations are clearly visible; they collapse onto a single curve as shown in the right plot. For comparison, results for $p=0.15$ and $0.1$ are also shown; the latter point is close to the expected boundary of the metallic phase, see Fig. 28. It can be seen that when the system approaches the phase boundary, the logarithmic increase of the DOS disappears and the RMT oscillations get damped. (b) Renormalized DOS at maximal disorder $p=0.5$ and on the symmetry line ${\sin }^2\theta =1∕2$ for different system sizes vs energy in units of the level spacing ${\delta }_L$. The RMT result, Eq. (6.53), plotted as a dashed line for comparison. Inset: Logarithmic dependence of $1∕L^2{\delta }_L$ on the system size $L$, consistent with the data of the left plot. From 218.Reuse & Permissions

Figure 30

DOS near $E=0$ for disorder values $p=0.13$, 0.08, 0.04, and 0.02 and for two system sizes $L=64$ and $128$ at fixed interplaquette coupling ${\sin }^2\theta =0.579$. The DOS diverges logarithmically as $E\to 0$ in the metallic phase $(p=0.13)$ and remains finite in the localized phase (other values of $p$). Results for the lowest impurity concentration, $p=0.02$, show an oscillatory feature induced by the band structure of the clean system, as well as strong scatter in the data at the lowest energies, which is due to insufficient ensemble averaging. Upper inset: Low-energy peak at $p=0.13$; its amplitude increases with $L$, in agreement with Sec. 3A. Lower inset: Low-energy DOS at $p=0.08$. No peak at $E\to 0$ is detected; $\rho (E\to 0)$ is a constant independent on $L$, indicating that the system is in the insulating phase. Statistical noise in the lower inset is more pronounced than in the upper one due to the smallness of the DOS. From 218.Reuse & Permissions

Figure 31

DOS at low energy on the self-dual line ${\sin }^2\theta =0.5$ for disorder concentrations $p=0.2$ ($○$, $◇$), 0.1 ($◻$, x), and 0.05 ($△$, +), where in each case the first symbol is for $L=128$ and the second is for $L=256$. Inset: $\rho \left(E\right)∕\mid E\mid$ at $p=0.05$ on a log-linear scale. The logarithmic correction is clearly observed, in agreement with Eq. (6.54). From 218.Reuse & Permissions