Topological surface-state destruction via trivializing proximity effect: Lattice localization despite continuum criticality

TL;DR

The paper investigates how surface states of 3D class CI topological superconductors respond to disorder and a trivializing proximity effect (TPE), comparing bulk lattice models to continuum Dirac theories. Using multifractal analysis and kernel-polynomial methods, it finds that lattice CI surface states become Anderson localized under disorder when coupled to a trivial 2D flat band (TPE), while uncoupled lattice surfaces retain spectrum-wide quantum criticality. In contrast, the 2D continuum CI surface models show robust SWQC under disorder, and stronger disorder tends to heal the surface, filling spectral gaps; the Bistritzer-MacDonald chiral model exhibits similar behavior, lacking a robust TPE. The results reveal a fundamental mismatch between lattice-scale surface physics and effective continuum field theories for localizable topological phases, with implications for moiré materials and surface engineering in topological superconductors.

Abstract

In a significant conceptual revision to the tenfold classification scheme for topological insulators and superconductors, it was recently demonstrated that most three-dimensional (3D) classes are simultaneously "localizable" in two distinct, but intricately connected ways: (1) There is no obstruction to Wannier localization of all bulk eigenstates, and (2) Almost all surface states can be Anderson localized by arbitrarily weak symmetry-preserving quenched disorder. Here we consider the localizable class CI in 3D, and numerically investigate the stability of surface states. We demonstrate that surface states of a bulk class-CI topological lattice model are fragile in that they can be Anderson localized by the combination of weak quenched randomness and hybridization with an additional trivial 2D band (a trivializing proximity effect, TPE). With the TPE, stronger disorder is more destructive to the surface states of the bulk lattice model. By contrast, without additional bands the surface states remain extended, exhibiting robust spectrum-wide quantum criticality. We also investigate the fragility of surface states in effective 2D class-CI continuum Dirac theories, including the chiral limit of the Bistritzer-MacDonald model for twisted bilayer graphene. Although the continuum models exhibit signs of Anderson localization near gap edges for weak disorder, stronger disorder instead appears to heal the surface, restoring criticality whilst filling in spectral energy gaps. Our results provide further evidence that effective continuum field theories fail to capture key aspects of surface-state physics in localizable topological phases.

Figures (13)

• Figure 1: Contrasting the behavior of lattice and continuum models of class-CI topological surface states, subject to disorder and a trivializing proximity effect (TPE). As a proxy for wave function criticality or Anderson localization, we plot the numerically computed curvature parameter $\theta$ for the multifractal spectrum (see text), averaged over a certain finite energy window, versus the effective disorder strength. Larger values of $\theta$ indicate stronger spatial rarefaction, tending towards Anderson localization. Results are shown for 5 different models: 2D surface states of a 3D class-CI lattice model without and with coupling to a trivial class-CI 2D flat band, the same for states of a 2D continuum Dirac theory of the surface, and the continuum Bistritzer-MacDonald (BM) model for twisted bilayer graphene in the chiral limit, subject to chiral disorder. The BM model is tuned to the first magic angle; the band gaps play a similar role as the coupling to a trivial flat band for the surface states. The lines marked "WZW" and "SQHPT" refer to predictions for zero- and finite-energy critical surface states, in the so-called spectrum-wide quantum criticality (SWQC) scenario Ghorashi18Karcher21UFO24. The main takeaways are (1) Increasing disorder localizes lattice surface states (away from zero energy) with the TPE, but not significantly without it, and (2) In the 2D continuum theories, increasing disorder has either no effect (states remain critical) or suppresses an initial trend towards localization for weak disorder. Disorder strengths are selected relative to each model and can only be qualitatively compared across models; details for these and the selected energy windows are provided in Appendix \ref{['sec:fig1params']}.
• Figure 2: Slab spectrum of the cubic-lattice class-CI topological superconductor model defined by the lattice Hamiltonian in Eq. (\ref{['eq:hamiltonian']}) with winding number $|\nu| = 2$. (a) $z$-cut, $k_y = 0$ line. The surface states form a quadratic touching, analogous to a single valley of Bernal bilayer graphene. (b) $x$-cut, $k_z = 0$ line. The surface states appear as two Dirac cones shifted away from the surface $\Gamma$ point. The parameters for these plots are: $\mu = - 2$, $\Delta_1 = 1$, and $\Delta_2 = 1$. In addition, a surface Chern mass $m_c = 1$ is used to gap out one of the two slab surfaces in each plot, so as to energetically isolate the low-energy surface states on the "active" surface. The slab thickness is $N_{z / x} = 60$.
• Figure 3: Slab energy spectrum versus $k_y$ of the topological CI lattice model (as in Fig. \ref{['fig:ci_surf_spec']}), now coupled to a 2D flat band via Eq. (\ref{['eq:HamTPE']}). (a) $z$-cut. (b) $x$-cut. The model parameters are $\varepsilon_c = 0.2$, $\gamma = 0.35$, $\mu = - 2$, $\Delta_1 = 1$, and $\Delta_2 = 1$. Again a Chern mass of size $m_c = 1$ is employed to gap out one of the two slab surfaces. (a) has $N_x = 100$, $N_z = 10$ and (b) is the same with $x \Leftrightarrow z$.
• Figure 4: Splitting of the quadratic surface-state touching by proximity coupling. We plot the lowest-energy band for the $z$-cut bulk CI topological lattice model hybridized with the trivial 2D flat band as in Fig. \ref{['fig:lattice_flat_spec']}(a), for various hybridization strengths. (a) $\gamma = 0$. (b) $\gamma = 0.2$. (c) $\gamma = 0.6$. Other parameters are $\varepsilon_c = 0.4$, $\mu = - 2$, $\Delta_1 = 1$, $\Delta_2 = 1$, $N_z = 20$.
• Figure 5: Spectrum of the CI 2D continuum surface-Dirac model hybridized with a class-CI flat band, in the absence of disorder. The parameters are $\varepsilon_c = 0.4$ and $\gamma_0 = 0.35$.