Topological Superconductivity of the Unconventional Type, S = 1, S^z = 0, in a Layer of Adatoms

Abstract

In this paper, we study the appearance of topological p-wave superconductivity of the type $S=1$, $S^z=0$ in a layer of adatoms. This unconventional superconductivity arises due to an anti-symmetric hybridization between the orbitals of the adatoms and those of the atoms in the superconducting BCS substrate. This two-dimensional system is topologically non-trivial only in the absence of a magnetic field and belongs to class DIII of the Altland–Zirnbauer classification. We obtain the Pfaffian that characterizes the topological phases of the system and its phase diagram. We discuss the differences between the two-dimensional case and a chain with the same type of superconductivity.

1. Introduction

Topological superconductivity in two dimensions was formalized by Read and Green in 2010 [1]. Similar to the Kitaev model [2], this concept also involves Majorana fermions; however, instead of manifesting at the edges, the Majorana modes are observed in the vortex cores [3,4,5,6,7,8,9]. Topological materials have garnered significant attention from the scientific community due the their potential technological applications [10,11]. Particularly, topological superconductors have received significant attention from the scientific community, largely due to their strong connection with Majorana fermions, which can be used as qubits in quantum computing [12,13,14,15]. The starting point for understanding these superconductors involves p-wave superconductivity. However, superconductors of this type are difficult to achieve experimentally, a challenge that can be overcome through induced superconductivity [16,17,18,19,20,21,22,23].

In this work, we consider a lattice consisting of atoms deposited on a BCS superconducting substrate as shown in Figure 1. The lattice is formed from adatoms, i.e., atoms strongly adsorbed in the substrate through a hybridization between the orbitals of these atoms and those in the substrate. Experimentally, various phenomena associated with this effect can be observed [24,25]. We previously studied this problem for a single chain of adsorbed atoms considering that the hybridization between the adatoms and substrate involves orbitals with different parities and consequently is anti-symmetric [23]. A necessary condition for implementing our model is to deposit atoms on the substrate whose electronic orbitals have parities different from those of the substrate, at the Fermi level. This is the case of mixing orbitals like s-p and p-d, where the angular momenta differ by an odd number. We showed that in this case of anti-symmetric hybridization, the superconductivity induced in the chain by the substrate is of the type $S=1$ with p-wave symmetry but with $S_z=0$. Notice that this process of inducing superconductivity through hybridization is distinct from the usual proximity effect. In the present study, the hybridization mixes orbitals with different parities, leading to induced unconventional pairing in the chain of the type $S=1$, $S^z=0$.

The present work extends our previous study [23] to the case of a two-dimensional layer. We showed in Ref. [23] that the induced superconducting gap in the adsorbed layer ${\Delta }_{ind}^p\left(k\right)$ is related to that of the BCS substrate ${\Delta }_a^s$ by

$$\begin{array}{c}\hfill {\Delta }_{ind}^p\left(k\right)={\displaystyle \frac{V_k}{\sqrt{{({ϵ}_k^b-{ϵ}_k^a)}^2+4{\left|V_k\right|}^2}}}{\Delta }_a^s,\end{array}$$

(1)

where ${ϵ}_k^a$ and ${ϵ}_k^b$ are the dispersion relations of the electrons in the substrate and layer, respectively, and $V_k$ is an anti-symmetric hybridization between these electrons [23]. Notice that the order parameter in the layer will have the same temperature dependence as that of the substrate, which follows the behavior of a conventional BCS superconductor reaching its maximum value at $T=0$. In this work, we consider only $T=0$, disregarding temperature variations.

In one dimension, the $S=1$, $S^z=0$ superconducting chain, in the presence of a magnetic field, belongs to the BDI class in the Altland–Zirnbauer classification [26]. According to this table, a system in the BDI class does not exhibit a topological phase transition in two dimensions [27,28,29,30,31,32]. This motivated us to study the $S=1$, $S^z=0$ superconductor in two dimensions and explore its eventual topological properties. In this work, we do not consider the effects of disorder. Since superconductivity is induced in the layer, this is robust as long as the substrate remains a superconductor [33].

We find that in 2D, the system is topological only in the absence of an applied magnetic field. In this case, conventional time-reversal symmetry is restored, and both the one- and two-dimensional systems belong to class DIII of the Altland–Zirnbauer table, which admits topological transitions in d = 1 and d = 2 [34].

In 1D, without a magnetic field, we showed that the superconducting $S=1$, $S^z=0$ chain is topological and hosts four Majorana modes, with two at each end. This system is in the DIII class, and the topological phase is characterized by a winding number. In the presence of a magnetic field, these modes disappear and in strong fields, the chain becomes gapless with stable Fermi points exhibiting linear dispersions, unlike the gapped Kitaev chain. In this region of the phase diagram, the system is topological and belongs to class BDI.

In this work, we consider a $S=1$, $S^z=0$ superconductor in a two-dimensional square lattice in the absence of a magnetic field. This model exhibits gap-closing not only at the time-reversal invariant points of the Brillouin zone but also at lines which are symmetry protected. Importantly, this gap-closing behavior distinguishes it from other models in the same class. We show that the model has time-reversal symmetry, characterized by the time-reversal operator T, with $T^2=-1$ [3]. The system belongs to the DIII class and has a ${\mathbb{Z}}_2$ topological invariant, implying the existence of only two possible topological states, which can be characterized using the Pfaffian.

The paper is organized as follows. In Section 2, we present the Hamiltonian of the model and obtain the energy spectra of an infinite lattice. Section 3 is devoted to the calculation of the topological invariant. Finally, we conclude in Section 4.

2. The 2D Model

This section introduces the model of a two-dimensional lattice with induced p-wave superconductivity, which exhibits topological phase transitions. We also discuss its key features, including the phase transition as a function of model parameters such as chemical potential, spin–orbit coupling, hybridization, and the hopping term. This model is schematized in Figure 1.

The Hamiltonian describing a layer of $N^2$ sites and spinful fermions with $S=1$, $S^z=0$, p-wave induced superconductivity is given by

$\mathcal{H}={\mathcal{H}}_{\mathcal{N}}+{\mathcal{H}}_{\mathcal{HS}}$, where

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{\mathcal{N}}& =& -\mu \sum _{j,\sigma }{c}_{j,\sigma }^{†}{c}_{j,\sigma }-\sum _{i,j,\sigma }\left(t{c}_{i,\sigma }^{†}{c}_{j,\sigma }+h.c.\right)+\sum _{i,j,\sigma ,\overline{\sigma }}\left(i\lambda c_{i,\sigma }^{†}{\left({\sigma }^y\right)}_{\sigma \overline{\sigma }}{c}_{j,\overline{\sigma }}+h.c.\right)\hfill \\ \hfill {\mathcal{H}}_{\mathcal{HS}}& =& -{\displaystyle \frac{1}{2}}\sum _{i,j,\sigma }\left(\Delta (c_{i,\sigma }^{†}{c}_{j,-\sigma }^{†}-c_{j,\sigma }^{†}{c}_{i,-\sigma }^{†})+h.c.\right).\hfill \end{array}$$

(2)

The first equation describes the normal layer. The quantity $\mu$ is the chemical potential, t is a nearest neighbor hopping, and $\lambda$ is a spin–orbit coupling. In our strict model, the latter Rashba-like term is essentially an anti-symmetric spin–flip hopping due to the spin–orbit interaction.

The Hamiltonian ${\mathcal{H}}_{\mathcal{HS}}$ represents the induced superconductivity in the layer of adatoms due to its hybridization with the BCS superconducting substrate. Since this hybridization is anti-symmetric, $\Delta$ is an induced anti-symmetric superconducting pairing (${\Delta }_{ij}=-{\Delta }_{ji}=\Delta$) between fermions with anti-parallel spins in neighboring sites. Notably, this anti-symmetric pairing mechanism plays a crucial role in determining the topological properties of the system. The total anti-symmetry of the order parameter with anti-parallel spins is guaranteed by the spatial anti-symmetric wave function of the Cooper pairs. Then, the induced superconductivity is a triplet state with $S=1$ but with the z-component of the total spin of the Cooper pair, $S^z=0$. The term h.c. stands for Hermitian conjugate.

First, we consider an infinite layer with periodic boundary conditions where we can Fourier transform Hamiltonian Equation (2) in momentum space. Choosing the basis ${\mathsf{\Psi }}_{\mathbf{k}}={(c_{\mathbf{k},\sigma }^{†},c_{\mathbf{k},-\sigma }^{†},c_{-\mathbf{k},\sigma },c_{-\mathbf{k},-\sigma })}^T$, with $\sigma =\uparrow$ and $-\sigma =\downarrow$, we obtain

$$\begin{array}{c}\hfill H={\displaystyle \frac{1}{2}}\sum _{\mathbf{k}}{\mathsf{\Psi }}_{\mathbf{k}}^{†}\mathcal{H}\left(\mathbf{k}\right){\mathsf{\Psi }}_{\mathbf{k}}+{\displaystyle \frac{1}{2}}\sum _{\mathbf{k}}[{\epsilon }_{\mathbf{k}\uparrow }+{\epsilon }_{-\mathbf{k}\downarrow }],\end{array}$$

(3)

and the second term of the Hamiltonian can be disregarded since it is a constant. This allows us to rewrite the total Hamiltonian in its matrix form in momentum space:

$$\begin{array}{c}\hfill \mathcal{H}\left(k\right)=\left(\begin{array}{cccc}(E_k-\mu )& {\lambda }_k& 0& {\Delta }_k^{*}\\ {\lambda }_k^{*}& (E_k-\mu )& {\Delta }_k^{*}& 0\\ 0& {\Delta }_k& -(E_k-\mu )& -{\lambda }_{-k}^{*}\\ {\Delta }_k& 0& -{\lambda }_{-k}& -(E_k-\mu )\end{array}\right).\end{array}$$

(4)

In the two-dimensional case, we have $E_k=-2t(cos{k}_x+cos{k}_y)$, ${\lambda }_k=2i\lambda (sin{k}_x+sin{k}_y)$ and ${\Delta }_k=2i\Delta (sin{k}_x+sin{k}_y)$ since ${\lambda }_k\mathrm{and}{\Delta }_k$ are odd and complex functions ${\lambda }_{-k}^{*}={\lambda }_k$, with a similar relation for ${\Delta }_k$.

It is interesting that we may realize a crossover between the one- and two-dimensional problems, considering the different hopping parameters and interactions along the x and y axes of the layer. By doing so, it is possible to reduce these terms to zero in one direction, effectively reducing the system to a set of one-dimensional wires.

This Hamiltonian of the infinite layer can be diagonalized, and the dispersion relations of the quasi-particles are given by

$$\begin{array}{ccc}\hfill {\omega }_1\left(k\right)& =& \sqrt{E_k^2-2\mu E_k+{\mu }^2+{\Delta }_k^2+{\lambda }_k^2-2{\lambda }_k\sqrt{E_k^2-2\mu E_k+{\mu }^2+{\Delta }_k^2}}\hfill \\ \hfill {\omega }_2\left(k\right)& =& \sqrt{E_k^2-2\mu E_k+{\mu }^2+{\Delta }_k^2+{\lambda }_k^2+2{\lambda }_k\sqrt{E_k^2-2\mu E_k+{\mu }^2+{\Delta }_k^2}},\hfill \end{array}$$

(5)

and ${\omega }_3\left(k\right)=-{\omega }_1\left(k\right)\mathrm{and}{\omega }_4\left(k\right)=-{\omega }_2\left(k\right)$. These dispersion relations exhibit gap-closing points at high-symmetry points. For $k_x=k_y=0$, the system shows a gap-closing when the ratio between the hopping term and the chemical potential is $t=-{\displaystyle \frac{\mu }{4}}$. When $t={\displaystyle \frac{\mu }{4}}$, the gap closes at four points: $k_x=k_y=\pm \pi$, $k_x=-\pi$ with $k_y=\pi$, and $k_x=\pi$ with $k_y=-\pi$.

In addition to these points, as shown in Figure 2, for values of $|\mu /4t|<1$, the dispersion relation also shows lines where the gap closes. For example, for $\mu =4\sqrt{{\lambda }_0^2-{\Delta }_0^2}$ and $t=1$, there are gap-closing lines passing through $k_x=k_y=\pm {\displaystyle \frac{\pi }{2}}$, which only cease to exist at the boundaries.

Then, we find a topological phase with zero-energy modes for $|\mu /4t|<1$. We observe that in the two-dimensional system, the critical value for the topological transition differs from the one-dimensional case by a factor of 2, where the topological phase transition parameter is given by $|\mu /2t|=1$ [23].

It is worth highlighting the cases where $\mu =0$, in which we also have gap-closing, as we can see in Figure 3. This differs from the one-dimensional case, where two pairs of magnetic monopoles are formed [23].

When $\mu =0$, two gap-closing lines appear, following the equation $k_y=k_x-\pi$. This gap-closing is due to the anti-symmetric nature of the hybridization and the induced superconductivity, and it is independent of the parameter values. Moreover, when $\Delta =\lambda$ in addition to the lines defined by $k_y=-k_x-\pi$, there is also a gap-closing at $k_y=-k_x-\pi$. Finally, when $t=0$, a gap-closing line given by $k_y=k_x$ is observed, which is due to the anti-symmetric nature of the system, without dependence on the values of $\lambda$ and $\Delta$. Consequently, in these cases, we observe gap-closings along lines that do not correspond to the symmetry-protected edge states as shown in Figure 3. Thus, for $\mu =0$, there are no topological phase transitions.

3. Topological Invariant

In this section, we present the topological invariant for the two-dimensional case and discuss the differences with the one-dimensional case, highlighting the distinctions between the phase transition parameters.

The Hamiltonian (4) satisfies the time-reversal symmetry and particle–hole symmetry such that

$$\begin{array}{cc}\hfill TH\left(k\right)T^{-1}& =H(-k)\mathrm{and}\hfill \\ \hfill PH\left(k\right)P^{-1}& =-H(-k)\hfill \end{array}$$

where the time-reversal operator $T=i{\sigma }_3\otimes {\sigma }_{2}K$ and the particle–hole operator $P={\sigma }_2\otimes {\sigma }_{2}K$ with K as the complex conjugate operator. As a result of the time-reversal symmetry and particle–hole symmetry, the Hamiltonian acquires a chiral symmetry:

$$SH\left(k\right)S^{-1}=-H\left(k\right),$$

(6)

with $S=TP$. This system shows differences compared to the one-dimensional case, representing a BDI class, for which we have $T_{BDI}={\sigma }_0\otimes {\sigma }_{0}K$ and $P={\sigma }_1\otimes {\sigma }_{0}K$. It is important to note that $T^2=-1$. This defines the Hamiltonian in class DIII [3,26]. If we apply the same basis transformation that diagonalizes S, with $US{U}^{-1}=S^{\prime }$, we will have a Hamiltonian written in the form $H^{\prime }\left(k\right)=UH\left(k\right)U^{-1}$, where

$$\begin{array}{c}\hfill {\mathcal{H}}^{\prime }\left(k\right)=\left(\begin{array}{cc}0& A\left(k\right)\\ A{\left(k\right)}^{†}& 0\end{array}\right).\end{array}$$

(7)

Here, it is important to highlight a significant difference. Unlike the BDI case, where the off-diagonal form is written in terms of matrix $B\left(k\right)$ and $B{(-k)}^T$, in the DIII case, the Hamiltonian is expressed in terms of $A\left(k\right)$ and $A{\left(k\right)}^{†}$, where

$$\begin{array}{c}\hfill \begin{array}{cc}& A\left(k\right)=\hfill \\ & \left(\begin{array}{cc}2t(cos{k}_x+cos{k}_y)+2i\Delta (sin{k}_x+sin{k}_y)& 2i\lambda (sin{k}_x+sin{k}_y)\\ -2i\lambda (sin{k}_x+sin{k}_y)& 2t(cos{k}_x+cos{k}_y)-2i\Delta (sin{k}_x+sin{k}_y)\end{array}\right)\hfill \end{array}\end{array}$$

The difference in how the Hamiltonians are written for the one-dimensional and two-dimensional cases is reflected in the calculation of the winding number. In the one-dimensional case, the winding number is of the type $\mathbb{Z}$, while in our case, we are dealing with a ${\mathbb{Z}}_2$ class, which only accommodates two topological values: 1 for a topologically trivial system and $-1$ for the topological phase.

The winding number $\mathcal{M}$ can be defined as a product of the signs of the Pfaffians at the high-symmetry points $k_x=k_y=0$, $k_x=\pi \mathrm{and}{k}_y=0$, $k_x=0\mathrm{and}{k}_y=\pi$ and $k_x=k_y=\pi$. We obtain

$$\mathcal{M}=sgn\left[\mu \right(\mu -4t\left)\right(\mu +4t\left)\right].$$

(8)

The trivial topological phase is characterized by $\mathcal{M}=+1$, while the topological phase is characterized by $\mathcal{M}=-1$. Then, in the case $\mu <4t$, we have a topological superconductor, and for $\mu =4t$, there is a quantum topological phase transition to a trivial superconducting state. The critical value of the ratio between the chemical potential and the hopping, ${(\mu /T)}_c$, for the topological phase transition to occur differs from the one-dimensional model by a factor of two.

4. Conclusions

We have studied in this paper the topological properties of a superconducting a layer of adatoms adsorbed on the surface of a BCS substrate. The induced superconductivity in the layer is unconventional since it arises from an anti-symmetric hybridization between the orbitals of the atoms in the layer and the substrate. The requirement for this type of hybridization is that the mixing orbitals have angular momenta differing from an odd number, like $s-p$ or $d-p$ orbitals. We find the layer presents topological protected superconductivity whenever the condition $|\mu /4t|<1$ is satisfied. The non-trivial topological character of the induced superconductivity has been demonstrated by the calculation of a topological index associated with a Pfaffian.