Elliptic mirror of the quantum Hall effect

Abstract

Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a {holomorphic modular symmetry}. By exploiting a quantum equivalence called \emph{mirror symmetry}, these models are mapped to tractable mirror models (also elliptic), in which topological protection is provided by more familiar winding numbers. Phase diagrams and scaling properties of elliptic models are compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One conspicuous model %(arguably the simplest and most natural one) has a critical delocalization exponent $ν_{\rm tor} = 18 \ln 2 /(π^2 G^4) = 2.6051\dots$ ($G$ is Gauss' constant) that is in excellent agreement with the value $ν_{\rm num} = 2.607\pm\,.004$ calculated in the numerical Chalker-Coddington model, suggesting that these models are in the same universality class. The real delocalization exponent may be disentangled from other scaling exponents in finite size scaling experiments, giving an experimental value $ν_{\rm exp} = 2.3\pm 0.2$. The modular model suggests how these theoretical and experimental results may be reconciled, but in order to determine if these theoretical models really are in the quantum Hall universality class, improved finite size scaling experiments are urgently needed.

Figures (12)

• Figure 1: Cartoon illustrating some of the remarkable properties of Calabi-Yau (CY) spaces (compact complex manifolds with vanishing first Chern class) that were discovered in string theory. In general a CY manifold is mirrored in a topologically distinct manifold $\widetilde{\rm CY}$, but in one (complex) dimension the only CY space is a torus, so our target space $\rm T^2$ and its mirror $\widetilde{\rm T}^2$ are topologically equivalent. Heuristically, the mirror image of a stack of $D_{2}$-branes wrapped around $\rm T^2$ (top) (a flat holomorphic vector bundle in the geometric limit) is a $D_1$-brane wrapping $\widetilde{\rm T}^2$ (bottom) (a gauge invariant Wilson-loop in the geometric limit).
• Figure 2: An arbitrary flat torus is spanned by basis vectors $\omega_1$ and $\omega_2$. Multiplying by a complex number $\lambda$ (here: $\vert\lambda\vert <1$) rotates and rescales (here: shrinks) the torus. A second rescaling by $\lambda$ is also displayed. Since the orientation of these tori in the complex plane is of no physical or mathematical interest, they should be rotated into "standard position" in the first quadrant, i.e. so that the rotated $\omega_1$ is real. It is then apparent that the rescalings have not changed the shape (complex structure $\tau$) of the torus. What has changed is the size (area) $a = \Im\sigma = \vert \omega_1\vert^2 \Im\tau$ of the torus.
• Figure 3: Topological modes in the mirror model $\widetilde{\rm T}^2$ with shape (complex structure) $\tilde{\tau} = \sigma = \exp(\pi i/3)$ and size (Kähler structure) $\tilde{\sigma} = \tau = i$. (a) The field theory limit of a rational $D_1$-brane on a torus is a straight Wilson loop with three real moduli $(\tilde{\mu} = m/n, u, v)$, which winds $n$ times around one cycle, and $m$ times around the other cycle. The centre of mass coordinate $u$ and the value $v$ of the Wilson loop parametrizes a torus that is isomorphic to the target space, $u + \sigma v \simeq \widetilde{\phi}\in\widetilde{\rm T}^2$. The two parallel closed loops shown here (in purple and green) have the same winding numbers ($n_1 = n_2 = 1,\, m_1 = m_2 = 2$) and slope ($\tilde{\mu}_1 = \tilde{\mu}_2 = 2$), but since they have distinct moduli ($u_1 = 0 \neq 1/4 = u_2$) they are not the same mode. (b) A state has finite energy iff its charge vector $(n,m)$ belongs to this charge lattice, i.e., iff the slope $m/n$ of the charge vector is rational. For a given rational slope, only the point nearest the origin (red), whose charges are coprime, is stable. The rest of the points (blue), whose charges are not coprime, are unstable.
• Figure 4: Energy landscapes (top row) and minimum energy landscapes (middle row) [$\ln E_{n,m}(\sigma)\propto \ln L_{n,m}(\sigma)$], together with phase diagrams in the moduli space of complex structures $\tilde{\tau} = \sigma$ (bottom row), for the mirror model $\widetilde{\rm T}^2$ without spin structure (left), and the mirror model $\widetilde{\rm T}^2_{\rm T}$ with spin structure (right). Saddle points (${\otimes}$) on the phase boundaries (black) are candidate quantum critical points. (The iconography is explained in the main text.) The red lacework showing level curves of unit height [$L_{n,m} (\sigma) = 1$], are Apollonian (left) and sub-Apollonian (right) gaskets.
• Figure 5: The top inset is a magnification of the small black rectangle in the main modular diagram, which has $\Gamma_{\rm T}$-symmetry, shown here in order to facilitate comparison with other spin polarized experiments that probe different parts of the phase diagram, and other symmetries (cf. Fig. \ref{['fig:TRSgrid']}). The inset contains a comparison of the experimental location of seven critical points (solid red icons), reconstructed from Ref. LVXPTsuiPW:2009, with the modular prediction (blue ${{\hbox{${\color{blue}{{\otimes}}}$}}}$). (The iconography is explained in the main text.) The size of the red plot-marker is a rough estimate of the experimental uncertainty in the data (no error-analysis is provided in LVXPTsuiPW:2009). Purple curves are temperature driven flow lines, obtained by quadratic interpolation of points (black bullets) sampled at different temperatures for a dozen arbitrary fixed values of the magnetic field.