Dimensional crossover of class D real-space topological invariants

TL;DR

The paper addresses how topological properties evolve when dimensionality is varied in symmetry class D systems, particularly for Shiba lattices where 2D circular islands can crossover to 1D chains. It introduces a real-space spectral localizer framework that yields local markers $|C_{\boldsymbol{\lambda}}^\textrm{L}|$ and $z_{2,x}$, along with a localizer gap $\mu_{\boldsymbol{\lambda}}$, to track topology across dimensions without relying on translational symmetry. Applied to Shiba lattices, the study maps a 2D island to a 1D chain, showing topological transitions signaled by changes in local markers and gaps, including coexistence regimes and disorder-induced robustness of 1D topology. The framework is general, robust to disorder, and extensible to other symmetry classes and higher dimensions, offering a computationally efficient route to guide materials design (e.g., slab thickness or nanowire geometry) while preserving topological properties.

Abstract

The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological invariant. Since topological invariants are typically defined for a fixed spatial dimension, there is no existing framework to understand the effects of changing spatial dimensions via invariants. Here, we introduce a framework to study topological phase transitions as a system's dimensionality is altered using real-space topological markers. Specifically, we consider Shiba lattices, which are class D materials formed by magnetic atoms on the surface of a conventional superconductor, and characterize the evolution of their topology when an initial circular island is deformed into a chain. We also provide a measure of the corresponding protection against disorder. Our framework is generalizable to any symmetry class and spatial dimension, potentially guiding the design of materials by identifying, for example, the minimum thickness of a slab required to maintain three-dimensional topological properties.

Figures (9)

• Figure 1: Topological phenomena in 1D and 2D Shiba lattices.a,b Magnetic adatoms (filled circles) arranged in a circular island (a) and a chain (b) geometry, embedded in a two-dimensional superconductor whose sites are represented by empty circles. c,d Zero-energy local density of states (LDOS) for the circular Shiba island and chain, respectively, showing a Majorana mode at their boundary. e,f LDOS as a function of energy at a bulk site (dashed black line) and at a site on the island's and chain's edges (solid orange line). The latter displays zero-energy peaks. The parameters used for the island and chain are $N=20$, where $N$ is the size of the superconductor's lattice, chemical potential $\mu=-4t$, spin-orbit coupling strength $\alpha=0.45t/l$, superconducting s-wave pairing $\Delta=2.0t$, and magnetic moment times exchange coupling $JS=4.8t$.
• Figure 2: Application of the spectral localizer framework to the Shiba island.a,b Chern number $|C_{\boldsymbol{\lambda}}^\textrm{L}|$ (a) and 2D localizer gap (b) as a function of position for the 2D island geometry shown in Fig. \ref{['fig:fig1']}a, for the same parameters and at $E/t=0$. The spectral localizer calculations use $\kappa = 0.05(t/l)$.
• Figure 3: Dimensional crossover topological phase diagram in the Shiba lattice. The phase diagram is shown as a function of the chemical potential $\mu/t$ and the product of the magnetic moment and exchange coupling $JS/t$, for $N=20$, and spin-orbit coupling strength $\alpha=0.45t/l$. The phase diagrams contain $31 \times 31$ points. a-c Phase diagram (a), 1D spectral localizer gap for the chain (b), and 2D spectral localizer gap for the chain (c) for $\Delta=1.2 t$ at $E=0$. The color code indicates the 2D local marker $|C_{\boldsymbol{\lambda}}^\textrm{L}|$ computed at the center of a circular island, and the 1D local marker $z_{2,x}$ computed at the center of a chain using the same parameters and only changing the geometry. The localizer gaps are similarly computed at the center of the 2D island and 1D chain. d-f Similar to a-c, except for $\Delta=0.4t$. Topological transitions not captured in the diagram can occur as the geometry changes from 2D to 1D. The spectral localizer calculations use $\kappa = 0.05 (t/l)$.
• Figure 4: Topological phase transitions due to dimensional crossover between the Shiba island and the Shiba chain. The parameters used are the same as in Fig. \ref{['fig:fig1']}, $\Delta =2.0t$, $\mu = -4t$, $\alpha = 0.45t/l$, and $JS = 4.8t$, such that both the initial Shiba island and final Shiba chain exhibit non-trivial topology. a-e Arrangements of magnetic adatoms (filled circles) and their absence (empty circles) embedded in a two-dimensional superconductor interpolating from the two-dimensional circular island to the one-dimensional chain geometries. f-j 1D (black solid line) and 2D (dashed orange lines) localizer spectrum $\Sigma(L_{\boldsymbol{\lambda}})$ as a function of position $x$ for $y = N/2$ and $E=0$. k-o Local Chern number $|C_{\boldsymbol{\lambda}}^\textrm{L}|$ (orange squares) and 2D localizer gap $\text{min}\{| \Sigma(L^{(2D)}_{\boldsymbol{\lambda}})| \}$ (solid gray line) as a function of position $x$ for $y = N/2$ and $E=0$. p-t Local $z_{2,x}$ invariant (black circles) and 1D localizer gap $\text{min}\{| \Sigma(L^{(1D)}_{\boldsymbol{\lambda}}) | \}$ (solid gray line) as a function of position $x$ for $E=0$. The spectral localizer calculations use $\kappa = 0.05(t/l)$.
• Figure 5: Topological phase transitions due to dimensional crossover between an imperfect Shiba island and Shiba chain. The parameters used are $\mu=-4t$, $\alpha=0.45t/l$, $\Delta=2.0t$, and $JS = 4.8t$, as in Fig. \ref{['fig:fig1']}. a-f Arrangements of magnetic adatoms (filled circles) and their absence (empty circles) embedded in a two-dimensional superconductor interpolating from a two-dimensional island with no spatial symmetries to a 1D chain with imperfections. g-l 1D (black solid line) and 2D (dashed orange lines) localizer spectrum $\Sigma(L_{\boldsymbol{\lambda}})$ as a function of position $x$ for $y = N/2$ and $E=0$. m-r Local Chern number $|C_{\boldsymbol{\lambda}}^\textrm{L}|$ (orange squares) and 2D localizer gap $\text{min}\{| \Sigma(L^{(2D)}_{\boldsymbol{\lambda}})| \}$ (solid gray line) as a function of position $x$ for $y = N/2$ and $E=0$. s-x Local $z_{2,x}$ invariant (black circles) and 1D localizer gap $\text{min}\{| \Sigma(L^{(1D)}_{\boldsymbol{\lambda}}) | \}$ (solid gray line) as a function of position $x$ for $E=0$. The spectral localizer calculations use parameter $\kappa = 0.05(t/l)$.