Noncrystalline topological superconductors

Sourav Manna, Sanjib Kumar Das, and Bitan Roy

Phys. Rev. B 109, 174512 – Published 6 May 2024

Abstract

Topological insulators, featuring bulk-boundary correspondence, have been realized on a large number of noncrystalline materials, among which amorphous network, quasicrystals, and fractal lattices are the most prominent ones. By contrast, topological superconductors beyond the realm of quantum crystals are yet to be harnessed, as their nucleation takes place around a well-defined Fermi surface with a Fermi momentum, the existence of which rests on the underlying translational symmetry. Here we identify a family of noncrystalline Dirac materials, devoid of time-reversal ( $T$ ) and translational symmetries, on which a suitable local or on-site pairing yields topological superconductors. We showcase this outcome on all the abovementioned noncrystalline platforms embedded in a two-dimensional flat space. The resulting noncrystalline topological superconductors possess quantized topological invariants (Bott index and local Chern marker) and harbor robust one-dimensional Majorana edge modes, analogs of $T$ -odd $p + i p$ pairing in noncrystalline materials.

Received 18 March 2024
Accepted 22 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.174512

Physics Subject Headings (PhySH)

Superconductivity Topological phases of matter

Amorphous materials Fractals Quasicrystals

Lattice models in condensed matter Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sourav Manna ¹, Sanjib Kumar Das ², and Bitan Roy ²

¹Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
²Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 109, Iss. 17 — 1 May 2024

Reuse & Permissions

Access Options

Article part of CHORUS

Accepted manuscript will be available starting 6 May 2025.

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Cuts of the global phase diagram of NCTSCs on (a) the $(M, Δ)$ plane for zero chemical doping ( $μ = 0$ ) and (b) the ( $M, μ$ ) plane for a fixed pairing amplitude $Δ = 1$ [Eq. (4)], and for $t = t_{0} = 1$ and $r_{0} = 1$ [Eq. (3)]. We find identical phase diagrams on a square lattice and an amorphous network both containing 400 lattice sites, Penrose and Ammann-Beenker quasicrystals with 481 and 357 lattice sites, respectively, and third generation Sierpiński carpet and fifth generation glued Sierpiński triangle fractals, containing 512 and 454 lattice sites, respectively, with open boundary conditions. This is so because the hopping elements in the normal state are sufficiently long ranged, washing out microscopic details of the underlying noncrystalline order. We find NCTSCs with the topological Bott indices $B = - 1$ and $- 2$ [Eq. (5)], both of which support one-dimensional Majorana edge modes (Figs. 2 and 3), besides the trivial one with $B = 0$ .
Reuse & Permissions
Figure 2
Energy spectra of $H_{pair}^{real}$ [Eq. (4)] on (a) a square lattice containing 400 sites, where $r_{0}$ is the nearest-neighbor (NN) distance; (b) an amorphous network with 600 sites with $r_{0} = 0.05 \times$ linear dimension of the system in each direction; (c) Penrose (481 sites) and (d) Ammann-Beenker (617 sites) quasicrystals, where $r_{0}$ is the arm length; (e) third generation Sierpiński carpet (512 sites); and (f) fifth generation glued Sierpiński triangle (454 sites) fractal lattices, where $r_{0}$ is the NN distance, with periodic (black) and open (red) boundary conditions, for $M = 2, t = t_{0} = 1, Δ = 0.5$ , and $μ = 0$ . We then realize a NCTSC in all these systems with the Bott index $B = - 1$ (Fig. 1). Local density of states (LDOS) for two closest to zero energy states (normalized by its maximum value) with open boundary conditions on (g) square lattice, (i) Penrose and (j) Ammann-Beenker quasicrystals, (k) Sierpiński carpet, and (l) glued Sierpiński triangle fractals, showing sharp edge localization. In panel (h) we show the total LDOS (normalized by its maximum value) for all the near-zero energy states that can only be found with open boundary conditions, displaying sharp edge localization. On two fractal lattices the spectra remain qualitatively unchanged with periodic and open geometries due to the inner edges. The LDOS of two closest to zero energy states in open (periodic) fractal lattices shows sharp localization dominantly around the outer (inner) edges. See also Fig. 3. We arrive at qualitatively similar results for NCTSCs with $B = - 2$ (not shown explicitly). The trivial paired states with $B = 0$ are devoid of any near-zero energy edge modes.
Reuse & Permissions
Figure 3
The LDOS for two closest to zero energy states on a (a) Sierpiński carpet and (b) glued Sierpiński triangle fractal lattices with periodic boundary conditions, displaying their localization dominantly around the inner edges. Compare with Figs. 2 and 2, respectively.
Reuse & Permissions
Figure 4
Distribution of the local Chern marker $C_{L} (r)$ [Eq. (7)] in noncrystalline topological superconducting phases on a square lattice, amorphous system, Pensore and Amman-Beenker quasicrystals, Sierpiński carpet, and glued Sierpiński triangle fractals (from left to right) for $μ = 0, M = 6$ , and $Δ = 2$ (top row); $μ = 1, M = 6$ , and $Δ = 1$ (second row); $μ = 0, M = 2$ , and $Δ = 2$ (third row); and $μ = 1, M = 2$ , and $Δ = 1$ (bottom row). For these parameter values the Bott index takes the values $B = - 1$ (first two rows) and $B = - 2$ (last two rows) as shown in Fig. 1. The system size and the other parameter values are the same as those in Fig. 1. In each system and row, the values of the local Chern marker on the gray sites fall outside the window specified in the corresponding color bar.
Reuse & Permissions
Figure 5
Inverse participation ratio (IPR), denoted by $P$ (in units of $10^{- 3}$ ) [Eq. (8)] on (a) square lattice, (b) amorphous network, (c) Penrose and (d) Ammann-Beenker quasicrystals, (e) Sierpiński carpet, and (f) glued Sierpiński triangle fractal lattices, containing $N$ sites (see legends). In each subfigure, panel (1) shows the IPR of all the states, panel (2) shows the scaling of the IPR for the closest to zero energy state (edge mode) as a function of $\sqrt{N}$ , and panel (3) shows the scaling of a large energy state (bulk mode) as a function of $N$ . Except on two fractal lattices, IPR in panel (2) scales linearly with $\sqrt{N}$ , confirming sharp outer-edge localization of the near-zero energy mode (Fig. 2). Absence of such a linear scaling on fractal lattices stems from the existence of the inner boundaries, which host near-zero energy modes with periodic geometry (Fig. 3).
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics