Multifractality at the Quantum Hall Transition: Beyond the Parabolic Paradigm

We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents ${\Delta }_q$ characterizing anomalous scaling of wave function moments $⟨|\psi {|}^{2q}⟩$ at the quantum Hall transition. The result reads ${\Delta }_q=2q(1-q)[b_0+b_1(q-1/2{)}^2+\cdots ]$, with $b_0=0.1291\pm 0.0002$ and $b_1=0.0029\pm 0.0003$. The central finding is that the spectrum is not exactly parabolic: $b_1\ne 0$. This rules out a class of theories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.

Figure 1

Upper row: Ratio $r_q$ of inverse participation numbers $P_q$ and $P_{1-q}$ [Eq. (9)] at $q=0.6$, 1, and 1.5 (from left to right). The flat asymptotics indicates the validity of the reciprocity relation ${\Delta }_q={\Delta }_{1-q}$. Small deviations from the constant behavior at the largest system sizes are due to residual statistical noise. Lower row: Analogous plots for the logarithmic derivative $s_q=(\ln r_q{)}^{\prime }$ defined in Eq. (10).

Figure 2

Family of curves $F_q(N)=P_q{N}^{q-1-\delta (q-1)q}$ for $q=0.6$, 1.1, and 1.5 (top: left, center, and right) and $q=0.4$, $-0.1$, and $-0.5$ (bottom: left, center, and right). Each data set is labeled by a parameter $\delta$, which increases from a minimum to a maximum value (given in the upper plots) in steps of 0.0004. The value of $\delta$ for which $F_q(N)$ is flat determines the anomalous exponent ${\Delta }_q=d\delta q(1-q)$. Such saturating data sets are marked with solid symbols (left: ▵; center:⋄; right: □); typical error in the corresponding value of $\delta$ does not exceed 0.001. The change in symbols (i.e., in $\delta$) for saturating data sets illustrates a $q$ dependence of ${\gamma }_q$ and thus gives direct, unprocessed evidence of nonvanishing quartic terms in ${\Delta }_q$.

Figure 3

The exponents ${\gamma }_q={\Delta }_q/q(1-q)$ (○) and ${\tilde{\gamma }}_q={\Delta }_q^{\prime }/(1-2q)$ (⋄) obtained from Fig. 2 and analogous analysis for other values of $q$. The curvature in ${\gamma }_q$ and ${\tilde{\gamma }}_q$ implies that the multifractal spectrum ${\Delta }_q$ is not parabolic. Also shown are results of the earlier work [9] (solid line). Dotted horizontal lines indicate earlier error bars in the regime $0\le q\le 1$. Dashed lines represent parabolic fits with $b_0=0.1291\pm 0.0002$, $b_1=0.0029\pm 0.0003$ (left) and $a_0=0.1282\pm 0.0001$, $a_1=0.0063\pm 0.0005$ (right). Combination of these data yields a rough estimate of the quartic term $b_2=-0.001\pm 0.001$.

Figure 4

The surface exponents ${\gamma }_q^s={\Delta }_q^s/q(1-q)$. Data are presented in the form analogous to the bulk plot (Fig. 3); the curvature is even more pronounced for the surface exponents. Dashed lines indicate parabolic fits with $b_0^s=0.1855\pm 0.0005$, $b_1^s=0.022\pm 0.002$ (left) and $a_0^s=0.1805\pm 0.001$, $a_1^s=0.048\pm 0.003$ (right). The resulting estimate for $b_2^s$ is $b_2^s=-0.008\pm 0.01$.