Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables

This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al., Ann. Phys. 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling exponents. In particular, all exponents corresponding to polynomial pure-scaling observables up to the order $q=5$ are found numerically. For the observables for which the percolation mapping is derived, analytical and numerical results are in perfect agreement with each other. The analytical and numerical results unambiguously demonstrate that the generalized parabolicity (i.e., proportionality to eigenvalues of the quadratic Casimir operator) does not hold for the spectrum of generalized-multifractality exponents. This excludes Wess-Zumino-Novikov-Witten models, and, more generally, any theories with local conformal invariance, as candidates for the fixed-point theory of the spin quantum Hall transition. The observable construction developed in this work paves a way to the investigation of generalized multifractality at Anderson localization critical points of various symmetry classes.

Figure 1

Schematic representation of path configurations yielding individual contributions within the percolation mapping for the $q=1$ and $q=2$ correlation functions presented in Secs. 2b1 and 2b2. Paths corresponding to each Green's function are shown by the color corresponding to the line underlining this Green's function. When a path traverses a segment of the loop for the second time, it is shown by a dashed line.

Figure 2

Schematic illustration of $q=3$ percolation probabilities of the type (58), with three spatial arguments ${\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3$ and three hull-length arguments $N_1,N_2,N_3$. The hierarchy of hull lengths $N_1>N_2>N_3$ is assumed, and the factors $(N_i^{-\frac{4}{7}}r{)}^{x_i^{\mathrm{h}}-x_{i-1}^{\mathrm{h}}}$ in Eq. (58) that depend on this hierarchy are indicated, with the hull exponents $x_i^{\mathrm{h}}$ abbreviated as $x_i$. On the right-hand side of the figure, different possible connections between the paths are shown. They yield different percolation probabilities that exhibit the same scaling (58).

Figure 3

Numerical simulations of the percolation probability $p({\mathbf{r}}_1;N)$ entering Eq. (25) for the $q=1$ SQH correlation function ${\mathcal{D}}_{\left(1\right)}({\mathbf{r}}_1;z)$. The data fully confirm the analytical prediction $p({\mathbf{r}}_1;N)\sim N^{-1-\frac{4}{7}{x}_1^{\mathrm{h}}}=N^{-8/7}$, see Eq. (49). In combination with Eq. (25), it yields the exponent $x_{\left(1\right)}=x_1^{\mathrm{h}}=1/4$. For $N>1000$, strong statistical fluctuations are observed, which are related to insufficient averaging.

Figure 4

Numerically determined coefficient functions ${\tilde{\mathcal{D}}}_{(1,1)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2)$ and ${\tilde{\mathcal{D}}}_{\left(2\right)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2)$ in the percolation representation (85) and (86) of the $q=2$ SQH correlation functions. The links ${\mathbf{r}}_1,{\mathbf{r}}_2$ are chosen to be either horizontal or vertical nearest neighbors. (Left) The Hartree coefficient function ${\tilde{\mathcal{D}}}_{(1,1)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is shown by light blue symbols, while the Fock coefficient function ${\tilde{\mathcal{D}}}_{\left(2\right)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2)$ by dark blue symbols. (When the Fock function is negative, its absolute value is shown by red symbols.) The results agree with the analytically predicted scaling ${\tilde{\mathcal{D}}}_{(1,1)}^{\left(N\right)},{\tilde{\mathcal{D}}}_{\left(2\right)}^{\left(N\right)}\propto N^{-1-\frac{4}{7}{x}_1^{\mathrm{h}}}=N^{-\frac{8}{7}}$, see Eq. (50), as shown by light blue and dark blue straight lines. The black symbols show the combination ${\tilde{\mathcal{D}}}_{(1,1)}^{\left(N\right)}-2{\tilde{\mathcal{D}}}_{\left(2\right)}^{\left(N\right)}$ (multiplied by ${10}^{-2}$ for clarity) corresponding to the pure-scaling correlation function ${\mathcal{P}}_{(1,1)}^C$. It is seen that the leading terms $\propto N^{-\frac{8}{7}}$ cancel and the scaling follows the law ${\tilde{\mathcal{D}}}_{(1,1)}^{\left(N\right)}-2{\tilde{\mathcal{D}}}_{\left(2\right)}^{\left(N\right)}\propto N^{-1-\frac{4}{7}{x}_2^{\mathrm{h}}}=N^{-\frac{12}{7}}$ (straight black line), in agreement with analytical prediction. (Right) Same data normalized to the analytically predicted power laws. In this representation, the predicted scaling corresponds to $N$-independence of the plotted functions (horizontal straight lines). To demonstrate violation of the generalized parabolicity, we also included the dashed line corresponding to $x_{(1,1)}=1$ which would hold for generalized parabolicity.

Figure 5

Numerically determined coefficient functions ${\tilde{\mathcal{D}}}_{\lambda }^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ in the percolation representation (88) of the $q=3$ SQH correlation functions. The links ${\mathbf{r}}_1,{\mathbf{r}}_2,\text{and}{\mathbf{r}}_3$ are chosen to be either horizontal or vertical (next) nearest neighbors. (Top) ${\tilde{\mathcal{D}}}_{\left(3\right)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ as given by Eq. (91) (black) and individual probabilities $p^{\left(s\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3;N)$ (blue) and $p_1^{\left(s\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3;N)$ (red) entering this formula. (Middle) ${\tilde{\mathcal{D}}}_{(2,1)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ as given by Eq. (92) (black) and individual terms entering this formula (light blue, dark blue, red). (Bottom) ${\tilde{\mathcal{D}}}_{\left(1^3\right)}^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$, Eq. (93) (black), and the individual terms in this formula given by Eq. (94) (light blue, dark blue, and red). The results for the individual terms agree with the analytically predicted scaling $\propto N^{-1-\frac{4}{7}{x}_1^{\mathrm{h}}}=N^{-\frac{8}{7}}$ as shown by straight lines of the corresponding colors. At the same time, the total expressions for the coefficient functions ${\tilde{\mathcal{D}}}_{\lambda }^{\left(N\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ exhibit the analytically predicted scaling $\propto N^{-1-\frac{4}{7}{x}_2^{\mathrm{h}}}=N^{-\frac{12}{7}}$ (black straight lines). (Right) Same data as in the respective left panels normalized to the analytically predicted power laws. In this representation, the predicted scaling corresponds to $N$-independence of the plotted functions (horizontal straight lines).

Figure 6

Numerical determination of the generalized-multifractality exponents at the SQH transition by SU(2) Chalker-Coddington network (CCN) simulations of pure scaling wave-function observables, Eqs. (151) and (168). Ensembles of systems of linear sizes $L=64,96,...,1024$ with ${10}^4$ disorder configurations for each length $L$ are studied. All observables are averaged over $\sim L^2$ spatial points and over ${10}^4$ disorder realizations. Each of the four panels shows data for all Young diagrams $\lambda$ in a given order $q=\left|\lambda \right|$, including $q=2$ (top left), $q=3$ (top right), $q=4$ (bottom left), and $q=5$ (bottom right). The spatial arguments ${\mathbf{r}}_i$ in the wave-function observable are chosen to be separated by approximately the same distance $r$. The data are scaled by a factor $r^{{\mathrm{\Delta }}_{q_1}+\cdots +{\mathrm{\Delta }}_{q_n}}$, which ensures the collapse of the data with different $r$ onto a single line $\sim {(r/L)}^{{\mathrm{\Delta }}_{\lambda }}$. The data corresponding to the smallest distance $r$ for each combination are highlighted as bold data points. These bold data points are used in the fit to extract the scaling exponent. Solid lines are fits to the highlighted small-$r$ data points, whereas dashed lines are predictions from the percolation theory (whenever available). For $q=4$, the dotted lines represent numerical results obtained earlier in Ref. [5] by an alternative (less general) numerical approach. Scaling exponents obtained numerically in the present work are summarized in Table 2, see the column $x_{\lambda }^{\mathrm{qn}}$. They are also compared there with analytical predictions of the percolation theory, with percolation numerics, and with previous numerical results of Ref. [5] (whenever available).

Figure 7

Schematic representation of path configurations yielding individual contributions within the percolation mapping for the $q=3$ correlation functions presented in Appendix pp1-s1b [Eqs. (A12, A13, A14)]. Paths corresponding to each Green's function are shown by the color corresponding to the line underlining this Green's function. When a path traverses a segment of the loop for the second time, it is shown by a dashed line. Only contributions containing explicitly all three points ${\mathbf{r}}_1, {\mathbf{r}}_2$, and ${\mathbf{r}}_3$ are shown. In addition, there are contributions originating from unity terms in the expansion of some of the Green's functions $G({\mathbf{r}}_i,{\mathbf{r}}_i;z)$, see Eqs. (A13) and (A14).