Theoretical description

The studied amorphous topological superconductor is comprised of randomly distributed magnetic moments on a superconducting surface with a Rashba spin-orbit coupling. The moments can arise from magnetic atoms, molecules or nanoparticles. Regular 1D structures of this type have been predicted to host Majorana states10,11,12,13,14,15,16,17,18 with supporting experimental evidence19,20,21. More recently, ferromagnetic 2D lattices have emerged as a promising platform for chiral superconductivity22, 23 with a rich topological phase diagram24,25. Classical magnetic moments embedded in a gapped s-wave superconductor give rise to Yu–Shiba–Rusinov (YSR) subgap states26, localised subgap states which decay algebraically for distances smaller than the superconducting coherence length. In 2D superconductors, such as layered systems, thin films and surfaces, the decay of the wavefunctions from the deep-lying impurity has a functional form \({\mathrm{e}}^{ - r/\xi }{\mathrm{/}}\sqrt {k_{\mathrm{F}}r}\), where ξ and k F are the superconducting coherence length and the Fermi wave vector of the underlying bulk. The Shiba glass results from a hybridisation of randomly distributed YSR states. To model the system, we consider deep-lying YSR states with energies ε 0 located in the vicinity of the gap centre \(\varepsilon _0{\mathrm{/\Delta }} \ll 1\), where Δ is the pairing gap in the bulk. The energy of a single YSR state is given by \(\varepsilon _0 = {\mathrm{\Delta }}\frac{{1 - \alpha ^2}}{{1 + \alpha ^2}}\), where \(\alpha = \pi JS{\cal N}\) is a dimensionless impurity strength, J is the magnetic coupling, S is the magnitude of the magnetic moment and \({\cal N}\) is the spin-averaged density of states at the Fermi level. The deep-impurity assumption translates to \(\left| {1 - \alpha } \right| \ll 1\) and the energy of an impurity state is given by ε 0 ≈ Δ(1 − α). As outlined in the Methods section, the low-energy properties of the coupled impurity moments are modelled by a tight-binding Bogoliubov-de Gennes Hamiltonian24

$$\begin{array}{*{20}{c}} {H_{mn} = \left( {\begin{array}{*{20}{c}} {h_{mn}} & {{\mathrm{\Delta }}_{mn}} \\ {\left( {{\mathrm{\Delta }}_{nm}} \right)^ \ast } & { - h_{mn}^ \ast } \end{array}} \right),} \end{array}$$ (1)

which describes a long-range hopping between YSR states centred at random positions r n . The entries h mn , Δ mn for arbitrary configuration of magnetic moments is lengthy and given in Supplementary Note 1. Physical intuition can be obtained by considering the special case of fully out-of-plane ferromagnetic spins, where the model reduces to

$$\begin{array}{l}h_{mn} = \left\{ {\begin{array}{*{20}{c}} {\varepsilon _0} & {m = n} \\ {\frac{{\alpha {\mathrm{\Delta }}}}{2}\left[ {I_1^ - (r_{mn}) + I_1^ + (r_{mn})} \right]} & {m

e n} \end{array}} \right.,\\ {\mathrm{\Delta }}_{mn} = \left\{ {\begin{array}{*{20}{c}} 0 & {m = n} \\ {\frac{{\alpha {\mathrm{\Delta }}}}{2}\left[ {I_4^ + (r_{mn}) - I_4^ - (r_{mn})} \right]\frac{{x_{mn} - {\mathrm {i}}y_{mn}}}{{r_{mn}}}} & {m

e n.} \end{array}} \right.\end{array}$$ (2)

In the above expression \(r_{mn} = \left| {\mathbf r}_{m} - {\mathbf r}_{n} \right|\), and x mn and y mn are components of r m − r n ≡ (x mn , y mn ). The hopping elements are expressed in terms of the functions

$$\begin{array}{l}I_4^ \pm (r) = \frac{{{\cal N}_ \pm }}{{\cal N}}{\kern 1pt} \Re \left[ {{\mathrm{i}}J_1\left( {k_{\mathrm{F}}^ \pm r + {\mathrm{i}}r{\mathrm{/}}\xi } \right) + H_{ - 1}\left( {k_{\mathrm{F}}^ \pm r + {\mathrm{i}}r{\mathrm{/}}\xi } \right)} \right],\\ I_1^ \pm (r) = \frac{{{\cal N}_ \pm }}{{\cal N}}\Re \left[ {J_0\left( {k_{\mathrm{F}}^ \pm r + {\mathrm{i}}r{\mathrm{/}}\xi } \right) + {\mathrm{i}}H_0\left( {k_{\mathrm{F}}^ \pm r + {\mathrm{i}}r{\mathrm{/}}\xi } \right)} \right],\end{array}$$ (3)

where J n and H n are Bessel and Struve functions of order n. The Rashba spin-orbit coupling induces two helical Fermi surfaces with density of states \({\cal N}_\pm\) = \({\cal N}\left( {1 \mp {\mathit \lambda} /\sqrt {1 + \lambda ^2} } \right)\) and Fermi wavenumber \(k_{\mathrm{F}}^ \pm\) = \(k_{\mathrm{F}}\left( {\sqrt {1 + {\mathit \lambda} ^2} \mp {\mathit \lambda} } \right)\), where λ = α R /(ħv F ) is the dimensionless Rashba coupling and k F ,v F the Fermi wavenumber and velocity in the absence of spin-orbit coupling. The Rashba coupling also slightly modifies the superconducting coherence length \(\xi = (\hbar v_{\mathrm{F}}{\mathrm{/\Delta }})\sqrt {1 + {\mathit \lambda} ^2}\). For ferromagnetic textures, the pairing term Δ ij vanishes with vanishing Rashba coupling α R = 0. The low-energy Hamiltonian (1) describes an odd-parity pairing Δ mn = −Δ nm , which is a long-range hopping variant of a p x + ip y superconductivity. In Eq. (1) the hopping and pairing functions decay as \(f(r) \propto \frac{{{\mathrm{e}}^{ - r/\xi }}}{{r^{1/2}}}\) and display oscillations at wave vectors \(k_{\mathrm{F}}^ \pm\).

Physical properties of the Shiba glass

The spectrum and the topological phase diagram of a finite system can be calculated by diagonalising the effective Hamiltonian (1) for spatially uncorrelated random positions of magnetic moments. After deriving the finite-size properties, we discuss the extrapolation to the thermodynamic limit. For 2D time-reversal breaking topological superconductors, the relevant topological index classifying the state is the Chern number. We will evaluate Chern numbers by employing the real-space approach of Eq. (5).

By evaluating the Chern number, we uncover the topological phase diagram of finite Shiba glass systems which can be seen in Fig. 2a. For sufficiently high densities, a ferromagnetically ordered system is generally in a topological phase with Chern number \(\left| {\cal C} \right| = 1\). For the employed parameters, the critical density ρ c corresponds to the characteristic length scale \(\bar r_{\mathrm{c}} = \rho _{\mathrm{c}}^{ - 1/2} \approx k_{\mathrm{F}}^{ - 1}\). For lower densities \(\left( {\bar r \gg k_{\mathrm{F}}^{ - 1}} \right)\), the system is in general topologically trivial and gapless; rare configurations can manage to enter a topological phase but do not survive disorder averaging. The pattern persists even when the directions of the local spins deviate from the perfect ferromagnetic configuration; in Fig. 2b we plot the phase diagram for spin configurations drawn from a thermal distribution where the angles θ j between the moments and the surface normal are determined by the Boltzmann weights \({\mathrm{e}}^{ - \beta E_{\mathrm{Z}}{\kern 1pt} {\mathrm{cos}}\, \theta _j}\). This situation corresponds to an ensemble of decoupled spins at Zeeman field E Z polarising the moments perpendicular to the plane and disordered by thermal fluctuations at inverse temperature β. Alternatively, the situation can be regarded as a magnetic disorder where the disorder is parametrised by the thermal distribution and βE Z instead of some other random distribution. For βE Z = 10, as indicated by Fig. 2b, the phase diagram remains qualitatively unchanged when compared to that for the completely polarised case. The robustness to moment disorder is not an artefact of the thermal distribution, and we discover qualitatively similar results for other disorder averages exhibiting comparable polarisation.

Fig. 2 Topological superconductivity in the Shiba glass. a Topological phase diagram for a ferromagnetic Shiba glass as a function of the single-moment bound-state energy ε 0 and the characteristic length between the moments \(\bar r = \rho ^{ - \frac{1}{2}}\), where ρ is the moment density per unit area. The colour bar indicates the value of the Chern number. The adatom number is held fixed at 600, with \(k_{\mathrm{F}}\xi = \frac{{4\pi }}{5}\) and λ = 0.2. The displayed diagram is an average over 10 configurations. Inset: Line along ε 0 = 0.1, averaged over 500 configurations. b Same as in a, but for magnetic moment directions drawn from the Boltzmann distribution with βE Z = 10 and averaged over 30 configurations, and with the number of moments fixed at 900. The deviation from the quantised values and the width of the transition region diminish as the system size is increased. c Local density of states (LDOS) for a 12.5ξ × 12.5ξ square Shiba glass system comprising 2500 randomly distributed sites, integrated over subgap energies \(\left| E \right| < 0.1{\mathrm{\Delta }}\). Parameters used same as in a, with onsite energy ε 0 = 0. The areas of the orange discs correspond to the magnitude of the LDOS; each site is additionally represented by a grey point which is visible when the LDOS is negligible. d The thermal conductance (in units of \(\frac{{\pi k_{\mathrm{B}}^2T}}{{3\hbar }}\)) along the line ε 0 = 0 for the same system parameters as in the previous figures, but with 2500 adatoms. The vertical width of the conduction plateau (yellow) corresponds to the mobility gap of the system, and can be seen to close as the system approaches the transition to the trivial gapless phase Full size image

The physical consequences of the topological nature of the Shiba glass are illustrated in Fig. 2c, d. The first one shows that the local density of states (LDOS) is concentrated on the sample edges. This is a consequence of a topological edge mode enclosing a finite system and is directly observable as discussed below. In Fig. 2d we have plotted the thermal conductance of finite systems coupled to external leads, as detailed in Supplementary Note 2. In the topological phase, the system exhibits a quantised thermal conductance which is a direct consequence of the nontrivial topology. The quantised conductance is effected by the edge modes despite the system being highly irregular in real space. In finite-size systems, for parameters close to the phase boundary, the quantised conductance plateau is destroyed and the conductance assumes continuous values. The non-quantised conductance in the trivial phase indicates that the low-energy states there extend over the sample.

The behaviour and exact phase transition point depends on the system parameters, though the overall trend of a topological phase at high densities remains. In Fig. 2 we have used parameters with high Rashba splitting λ and low value of k F ξ as appropriate for a proximity-superconducting 2D semiconductor; a phase diagram for parameters more appropriate for metals are presented in Supplementary Fig. 2b, also indicating a transition to a topological phase at sufficiently high densities.

Now we turn to discuss the features seen when increasing the system size. First of all, in the thermodynamic limit the Shiba glass phase is gapless. While this is a generic feature of a superconductor with magnetic impurities25, a qualitatively new mechanism for low-energy excitations arises in the topological phase. These emerge from rare fluctuations that leave a substantial area where magnetic moments are sparse. As depicted in Fig. 1, these empty antipuddles give rise to low-energy modes which are reminiscent of the gapless edge states circulating around a hole punched in a gapped topological phase. While the probability of formation of antipuddles is exponentially suppressed as a function of their size and their effect is relatively unimportant in finite systems with high density, in infinite systems antipuddles give rise to a tail down to zero energy in the DOS. The antipuddle mechanism provides a simple physical argument why the energy gap must scale to zero in the thermodynamic limit. The second important notion is that, in the thermodynamic limit, the system has well-defined topological nature despite being gapless. The low-energy modes, as we have argued above, are localised perturbations and the states with non-localised wavefunctions have a finite energy threshold. Thus, instead of an energy gap, the system exhibits a mobility gap protecting the topological state. This behaviour is analogous to the integer quantum Hall effect where the extended states carrying Chern numbers are separated by localised states in the Landau level gap27. In Supplementary Fig. 2b we have calculated the thermal conductance for an antipuddle configuration, which shows that for isolated antipuddles, the system has a vanishing energy gap but a finite well-defined mobility gap within which the heat conductance is quantised. In the topological phase the antipuddles are rare and effectively decoupled, thus they cannot destroy the conductance quantisation.