Theory of the Anderson transition in three-dimensional chiral symmetry classes: Connection to type-II superconductors

TL;DR

The work establishes that the 3D Anderson transition in chiral symmetry classes is driven by vortex-loop condensation in the nonlinear sigma model, extending the 2D vortex mechanism to three dimensions. It derives a dual representation in which the chiral NLSM maps to a $\mathrm{U}(N)$ type-II superconductor, with the metal corresponding to a Maxwell normal phase and localization to a Meissner phase, while the 1D weak topology $\bm{\chi}$ becomes an external magnetic field that induces a quasi-localized phase along the topological direction. A variational mean-field analysis on the dual lattice yields a second-order transition at a critical fugacity $t_c$ and reveals an intermediate quasi-localized phase when $\bm{\chi}$ is finite, corresponding to an array of conducting channels aligned with the topology. Overall, the results connect localization physics in 3D chiral classes to the well-understood phases of 3D type-II superconductors under magnetic fields and elucidate how topological weak indices shape anisotropic transport.

Abstract

Phase transitions governed by topological defects constitute a cornerstone of modern physics. Two-dimensional (2D) Anderson transitions in chiral symmetry classes are driven by the proliferation of vortex-antivortex pairs -- a mechanism analogous to the Berezinskii-Kosterlitz-Thouless (BKT) transition in the 2D XY model. In this work, we extend this paradigm to three-dimensional (3D) chiral symmetry classes, where vortex loops emerge as the key topological defects governing the Anderson transition. By deriving the dual representation of the 3D nonlinear sigma model for the chiral unitary class, we develop a mean-field theory of its Anderson transition and elucidate the role of 1D weak band topology in the Anderson transition. Strikingly, our dual representation of the 3D NLSM in the chiral symmetry class uncovers its connection to the magnetostatics of 3D type-II superconductors. The metal-to-quasilocalized and quasilocalized-to-insulating transitions in 3D chiral symmetry class share a unified theoretical framework with the normal-to-mixed and mixed-to-superconducting transitions in 3D type-II superconductors under an external magnetic field, respectively.

Figures (3)

• Figure 1: Schematic pictures of vortex loops in 3D. (a) Unpolarized vortex loops that have finite projections to a plane perpendicular to $\bm{\chi}$, and (b) polarized vortex loops that are confined within a plane parallel to $\bm{\chi}$. Black-colored loops on three 2D planes are projections of the vortex loops onto the three coordinate planes. The projections of the polarized vortex loops onto the plane perpendicular to $\bm{\chi}$ enclose zero area. The figures show only circular loops, while vortex loops can take arbitrary shapes. In the presence of the weak topological term $\bm{\chi}$, unpolarized vortex loops are suppressed by the destructive interference effect, while polarized vortex loops are free from the interference effect, dominating the partition function near the mobility edge.
• Figure 2: An example of closed-loop graphs with direction that have time-reversal pairs on the same links. For this closed-loop graph, the RHS of Eq. (\ref{['eq:10c']}) gives a weight $t^{18}/(2! 2! 2!)$, while the LHS of Eq. (\ref{['eq:10c']}) gives a weight $t^{18}$. The additional weight can be regarded as an additional fugacity for the higher vorticity.
• Figure 3: Variational free energy density $f/\Lambda^3$ as a function of the Higgs mass $m/\sqrt{\Lambda}$ at (a) $\sigma/\Lambda=c/\Lambda=1$, and (b) $\sigma/\Lambda=c/\Lambda=80$ (for the $\chi=0$ case). $f$ has a single minimum at $m=0$ for $t<t_*$, two local minima at $m=0$ and at $m\ne 0$ for $t_*<t$. The two minima of $f$ are degenerate at $t=t_c$, while for $t_c<t$, the global minimum of $f$ is at $m\ne 0$. The critical fugacities for $\sigma/\Lambda=c/\Lambda=1$, and $\sigma/\Lambda=c/\Lambda=80$ are $t_c=1.15 \times 10^{-4}$, and $1.37\times 10^{-2}$, respectively. (c) Critical fugacity $t_{\text{c}}$ for different conductivity $\sigma/\Lambda$ and Gade constant $c/\Lambda$. In these figures, we choose $\Lambda^{-1}=1$ for simplicity.