Majorana corner modes in a high-temperature platform

TL;DR

This paper proposes a high-temperature platform for Majorana Kramers pairs using topological insulators coupled with high-$T_c$ superconductors, where pairing symmetry induces corner-localized Majorana zero modes.

Contribution

It introduces a novel approach to realize Majorana zero modes at high temperatures using topological insulators and high-$T_c$ superconductors, with specific criteria and candidate materials.

Findings

Proposes a topologically trivial superconductor-based method for Majorana modes.

Identifies pairing symmetry as a key factor for corner Majorana modes.

Suggests candidate materials for experimental realization.

Abstract

We introduce two-dimensional topological insulators in proximity to high-temperature cuprate or iron-based superconductors as high-temperature platforms of Majorana Kramers pairs of zero modes. The proximity-induced pairing at the helical edge state of the topological insulator serves as a Dirac mass, whose sign changes at the sample corner because of the pairing symmetry of high-$T_c$ superconductors. This sign changing naturally creates at each corner a pair of Majorana zero modes protected by time-reversal symmetry. Conceptually, this is a topologically trivial superconductor-based approach for Majorana zero modes. We provide quantitative criteria and suggest candidate materials for this proposal.

Equations

$$H(\mathbf{k})=$$

$$\Delta (\mathbf{k})={\Delta }_0+{\Delta }_x\cos k_x+{\Delta }_y\cos k_y,$$

$${\Delta }_0=0,{\Delta }_x=-{\Delta }_y\equiv {\Delta }_d.$$

$$H(\mathbf{k})$$

$$H_0(-i{\partial }_x,k_y)$$

$$H_p(-i{\partial }_x,k_y)$$

$${\psi }_{\alpha }(x)={\mathcal{N}}_x\sin ({\kappa }_{1}x)e^{-{\kappa }_{2}x}{e}^{i{k}_{y}y}{\chi }_{\alpha },$$

$${\chi }_1=|{\sigma }_y=-1⟩\otimes |\uparrow ⟩\otimes |{\tau }_z=+1⟩,$$

$${\chi }_2=|{\sigma }_y=+1⟩\otimes |\downarrow ⟩\otimes |{\tau }_z=+1⟩,$$

$${\chi }_3=|{\sigma }_y=+1⟩\otimes |\uparrow ⟩\otimes |{\tau }_z=-1⟩,$$

$${\chi }_4=|{\sigma }_y=-1⟩\otimes |\downarrow ⟩\otimes |{\tau }_z=-1⟩,$$

$$H_{\mathrm{I},\alpha \beta }(k_y)={\int }_0^{+\infty }dx{\psi }_{\alpha }^{\ast }(x)H_p(-i{\partial }_x,k_y){\psi }_{\beta }(x),$$

$$H_{\mathrm{I}}(k_y)=-A_y{k}_y{s}_z+M_{\mathrm{I}}{s}_y{\tau }_y,$$

$$M_{\mathrm{I}}=({\Delta }_x/2){\int }_0^{+\infty }dx{\psi }_{\alpha }^{\ast }(x){\partial }_x^2{\psi }_{\alpha }(x)={\Delta }_{x}m/t_x.$$

$$H_{\mathrm{II}}(k_x)$$

$$H_{\mathrm{III}}(k_y)$$

$$H_{\mathrm{IV}}(k_x)$$

$$H_{\mathrm{edge}}=-iA(l)s_z{\partial }_l+M(l)s_y{\tau }_y.$$

$$|{\psi }_{\text{MKP}}^{\pm }(l)⟩\propto e^{-{\int }^{l}d{l}^{\prime }M(l^{\prime })/A(l^{\prime })}|s_x={\tau }_y=\pm 1⟩.$$

$$\Delta (\mathbf{k})={\Delta }_0-{\Delta }_1(\cos k_x+\cos k_y),$$

$$H(\mathbf{k})$$

$$H_0(-i{\partial }_x,k_y)$$

$$H_p(-i{\partial }_x,k_y)$$

$$H_{\text{I}}(k_y)$$

$$({\bar{\Delta }}_0+{\Delta }_{1}m/t_x)({\bar{\Delta }}_0+{\Delta }_{1}m/t_y)<0.$$

$$(R_s-R_x)(R_s-R_y)<0,$$

$$H(\mathbf{k})$$

$$H_0(-i{\partial }_x,k_y)$$

$$H_p(-i{\partial }_x,k_y)$$

$${\psi }_{\alpha }(x)={\mathcal{N}}_x\sin ({\kappa }_{1}x)e^{{\kappa }_{2}x}{e}^{i{k}_{y}y}{\tilde{\chi }}_{\alpha }$$

$${\kappa }_1=\sqrt{-\frac{2m}{t_x}-\frac{A_x^2}{t_x^2}},{\kappa }_2=\frac{A_x}{t_x}.$$

$${\tilde{\chi }}_1$$

$${\tilde{\chi }}_2$$

$${\tilde{\chi }}_3$$

$${\tilde{\chi }}_4$$

$$H_{\mathrm{III},\alpha \beta }(k_y)={\int }_{-\infty }^{0}dx{\psi }_{\alpha }^{\ast }(x)H_p(-i{\partial }_x,k_y){\psi }_{\beta }(x).$$

Full text

Majorana zero modes (MZMs)Read and Green (2000); Volovik (1999); Kitaev (2001) have been actively pursued in recent years as building blocks of topological quantum computationsAlicea (2012); Beenakker (2013); Stanescu and Tewari (2013); Leijnse and Flensberg (2012); Elliott and Franz (2015); Sarma et al. (2015); Sato and Fujimoto (2016); Aguado (2017). These emergent excitations can generate robust ground-state degeneracy, supporting storage of nonlocal qubits robust to local decoherenceNayak et al. (2008). Moreover, quantum gates can be implemented by their braiding operationsMoore and Read (1991); Wen (1991); Ivanov (2001); Nayak and Wilczek (1996); Das Sarma et al. (2005). As platforms of MZMs, a variety of realizations of topological superconductors have been proposed, including topological insulators in proximity to conventional superconductorsFu and Kane (2008); Qi et al. (2010a); Akhmerov et al. (2009); Law et al. (2009); Chung et al. (2011), semiconductor heterostructuresSau et al. (2010); Alicea (2010); Xu et al. (2014), cold-atom systemsJiang et al. (2011); Diehl et al. (2011); Sato et al. (2009); Zhang et al. (2008); Tewari et al. (2007); Liu et al. (2014a), quantum wiresOreg et al. (2010); Lutchyn et al. (2010); Vazifeh and Franz (2013); Cook and Franz (2011), to name a few; meanwhile, remarkable experimental progress has been witnessedMourik et al. (2012); Nadj-Perge et al. (2014); Rokhinson et al. (2012); Deng et al. (2012); Das et al. (2012); Finck et al. (2013); Churchill et al. (2013); Lv et al. (2016); Wang et al. (2012, 2016); Albrecht et al. (2016); Deng et al. (2016); Pawlak et al. (2016); Xu et al. (2015); Sun et al. (2016); He et al. (2016); Zhang et al. (2017, 2018); Wang et al. (2017).

A single MZM entails breaking the time-reversal symmetry (TRS); in contrast, time-reversal-invariant (TRI) topological superconductorsQi et al. (2009, 2010b); Zhang et al. (2013a); Wong and Law (2012); Sato (2010); Zhang et al. (2013b); Fu and Berg (2010); Haim et al. (2016) host Majorana Kramers pairs (MKPs) of zero modes, which are robust in the presence of TRS, and have interesting consequences such as TRS-protected non-Abelian statisticsLiu et al. (2014b); Zhang and Kane (2014a); Gao et al. (2016), TRS as local supersymmetryQi et al. (2009), and novel Kondo effectsBao and Zhang (2017) and Josephson effects Zhang and Kane (2014b); Orth et al. (2015); Schrade et al. (2015); Schrade and Fu (2018); Camjayi et al. (2017), indicating their potentials in qubit storage or manipulation and other applications. In addition, MKPs can be used as tunable generators of MZMs by breaking the TRSZhang et al. (2013a); Wong and Law (2012). There have been a few interesting proposals for realizing TRI topological superconductors and MKPs Zhang et al. (2013a); Wong and Law (2012); Keselman et al. (2013); Nakosai et al. (2013, 2012); Wang et al. (2014); Haim et al. (2014); Klinovaja et al. (2014); Yang et al. (2015); Trani et al. (2016); Li et al. (2016); Wu and Martin (2017); Reeg et al. (2017); Wang (2016); Hu et al. (2017), though experimental realizations are yet to come.

In this Letter, we show that simple structures of two-dimensional topological insulators (2D TIs) (also known as quantum spin Hall insulators) in proximity to high-temperature superconductors naturally generate MKPs (Fig.1). Since 2D TIs have been experimentally realized at temperatures as high as 100 KelvinWu et al. (2018); Qian et al. (2014), this setup can be a high-temperature platform of MKPs. The physical picture can be readily described as follows. The helical edge states of TI, described as 1D massless Dirac fermions, are gapped out by the induced superconducting gap, which introduces a Dirac mass. Due to the nature of pairing symmetry (say $d$-wave), the induced Dirac mass changes sign at the corner, which generates a MKP as domain-wall excitations.

It is interesting to note that we do not propose here any realization of TRI topological superconductor. In fact, the helical Majorana edge states of $\mathbb{Z}_{2}$-nontrivial superconductors cannot be gapped out without breaking TRS. In our setup, the 2D TI with a proximity-induced pairing has gapped edges, therefore it is a $\mathbb{Z}_{2}$-trivial superconductor. In fact, it has recently been suggested that, as defect modesTeo and Kane (2010), robust MZMs can be realized in certain topologically trivial superconductorsYan et al. (2017); Chan et al. (2017) (Particularly, MZMs can in principle be created as corner modes in judiciously designed trivial-superconductor junctionsYan et al. (2017)). Conceptually, the present work generalizes the trivial-superconductor-based approach to MKPs, for which ideal candidate materials are available.

*$d$-wave pairing.–*As explained above, the key observation comes from the edge states. For concreteness, however, let us start from a lattice model of 2D TI, in which the proximity-induced pairing is added. The Bogoliubov-de Gennes Hamiltonian is $\hat{H}=\sum_{{\bf k}}\Psi_{{\bf k}}^{{\dagger}}H({\bf k})\Psi_{{\bf k}}$, with $\Psi_{{\bf k}}=(c_{a,{\bf k}\uparrow},c_{b,{\bf k}\uparrow},c_{a,{\bf k}\downarrow},c_{b,{\bf k}\downarrow},c_{a,-{\bf k}\uparrow}^{{\dagger}},c_{b,-{\bf k}\uparrow}^{{\dagger}},c_{a,-{\bf k}\downarrow}^{{\dagger}},c_{b,-{\bf k}\downarrow}^{{\dagger}})^{T}$ and

[TABLE]

where $s_{i}$, $\sigma_{i}$ and $\tau_{i}$ are Pauli matrices in the spin ($\uparrow,\downarrow$), orbital ($a,b$), and particle-hole space, respectively, $M({\bf k})=m_{0}-t_{x}\cos k_{x}-t_{y}\cos k_{y}$ and $A_{x,y}$ measure the kinetic energy, $\Delta$ is the pairing, and $\mu$ is the chemical potential. In the following we will take

[TABLE]

which is sufficiently general to model $d$ wave and $s_{\pm}$ wave. Throughout this paper, $t_{x,y}$, $A_{x,y}$ are taken to be positive. If the pairing is removed, the Hamiltonian becomes the paradigmatic BHZ model of 2D TIsB. A. Bernevig et al. (2006); Qian et al. (2014); Wu et al. (2018). The Hamiltonian has TRS $\mathcal{T}H({\bf k})\mathcal{T}^{-1}=H(-{\bf k})$ with $\mathcal{T}=is_{y}\mathcal{K}$ (where $\mathcal{K}$ is the complex conjugation), and particle-hole symmetry $\mathcal{C}H({\bf k})\mathcal{C}^{-1}=-H(-{\bf k})$ with $\mathcal{C}=\tau_{x}\mathcal{K}$.

We first consider the $d$-wave pairing that is relevant to cuprate superconductors, which is

[TABLE]

The spectra on a cylinder geometry are shown in Fig.2(a), indicating that the helical edge states of TI are gapped out by $d$-wave pairing. From the numerical results for a square geometry [Fig.2(b)], it is clear that each corner hosts a MKP, whose energy is pinned to zero.

It is interesting to note that, unlike the more familiar vortex or end modes, the Majorana modes here are corner modes. As such, they may be viewed in the framework of recently proposed higher-order topological insulatorsBenalcazar et al. (2017a); Schindler et al. (2017); Zhang et al. (2013c); Benalcazar et al. (2017b); Song et al. (2017); Langbehn et al. (2017); Peng et al. (2017); Imhof et al. (2017); Serra-Garcia et al. (2017); Schindler et al. (2018); Ezawa (2018); Peterson et al. (2017) and superconductorsGeier et al. (2018); Zhu (2018); Khalaf (2018); Shapourian et al. (2018), for which crystal symmetries have been highlighted; the present scheme does not rely sensitively on the crystal symmetries.

We also mention that the bulk of $d$-wave superconductor is gapless and the MKPs may hybridize with these gapless modes. Nevertheless, the MKPs remain observable in scanning tunneling microscopy (STM). Near a MKP, the tunneling conductance displays a zero-bias peak (though broadened by hybridization), which is absent in the usual $c$-axis tunneling conductanceFischer et al. (2007) (In other directions there is zero bias peak Kashiwaya et al. (1995); Sinha and Ng (1998); Tanaka and Kashiwaya (1995); Ryu and Hatsugai (2002); Shigeta et al. (2002); Giubileo et al. (2002); Kao et al. (2015), which is irrelevant in our setup). Later, we also study the $s_{\pm}$-wave case with an entirely gapped setup, in which case MKPs are the only low-energy modes.

*Edge theory.–*To gain intuitive understandings, we study the edge theory. To simplify the picture, we take $\mu=0$ and focus on the continuum model by expanding the lattice Hamiltonian in Eq.(1) to second order around ${\bf k}=(0,0)$:

[TABLE]

where $\Delta_{x}+\Delta_{y}=0$ has been used for the $d$ wave, and $m=m_{0}-t_{x}-t_{y}<0$ is assumed to ensure that the 2D insulator without pairing is in the topologically nontrivial regime. We label the four edges of a square as (I), (II), (III), and (IV) [Fig.2(b)], and we focus on the edge (I) first. We can replace $k_{x}\rightarrow-i\partial_{x}$ and decompose the Hamiltonian as $H=H_{0}+H_{p}$, in which

[TABLE]

where the insignificant $k_{y}^{2}$ term has been omitted. The purpose of this decomposition is to solve $H_{0}$ first, and then treat $H_{p}$ as a perturbation, which is justified when the pairing is relatively small (This is the case in real samples).

Solving the eigenvalue equation $H_{0}\psi_{\alpha}(x)=E_{\alpha}\psi_{\alpha}(x)$ under the boundary condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$, we find four zero-energy solutions, whose forms are

[TABLE]

with normalization given by $|\mathcal{N}_{x}|^{2}=4|\kappa_{2}(\kappa_{1}^{2}+\kappa_{2}^{2})/\kappa_{1}^{2}|$ (Here, $\kappa_{1}=\sqrt{|\frac{2m}{t_{x}}|-\frac{A_{x}^{2}}{t_{x}^{2}}},\,\kappa_{2}=\frac{A_{x}}{t_{x}}$. The result remains valid even when $\kappa_{1}$ is imaginary). The eigenvectors $\chi_{\alpha}$ satisfy $\sigma_{y}s_{z}\tau_{z}\chi_{\alpha}=-\chi_{\alpha}$. We can explicitly choose them as

[TABLE]

then the matrix elements of the perturbation $H_{p}$ in this basis are

[TABLE]

therefore, the final form of the effective Hamiltonian is

[TABLE]

where

[TABLE]

Similarly, the low-energy effective Hamiltonians for the other three edges are

[TABLE]

with $M_{\rm II}=M_{\rm IV}=\Delta_{y}m/t_{y}$, and $M_{\rm III}=M_{\rm I}$. To be more transparent, let us take an “edge coordinate” $l$, which grows in the anticlockwise direction (apparently, $l$ is defined mod $2(L_{x}+L_{y})$), then the low-energy edge theory becomes

[TABLE]

The kinetic-energy coefficient $A(l)$ and the Dirac mass $M(l)$ are step functions: $A(l)=A_{y},A_{x},A_{y},A_{x}$ and $M(l)=\Delta_{d}m/t_{x},-\Delta_{d}m/t_{y},\Delta_{d}m/t_{x},-\Delta_{d}m/t_{y}$ for (I), (II), (III), and (IV), respectively. At each corner, the $A_{x,y}$ coefficient does not change sign, while the Dirac mass does, which is due to the sign changing in the $d$-wave pairing: $\Delta_{x}=-\Delta_{y}$. Consequently, there is a MKP at each corner (analogous to the Jackiw-Rebbi zero modesJackiw and Rebbi (1976); Lee et al. (2007)). For example, at the corner between (I) and (II), we have

[TABLE]

TRS ensures that these two modes cannot be coupled to generate an energy gap. In essence, the edge theory above can be regarded as two copies of that of Ref.Yan et al. (2017), with TRS as the key additional input.

By a similar calculation, one can find that the sign changing in $M(l)$ occurs at a corner when one of the edges has a polar angle within $[-\pi/4,\pi/4]$ and the other within $[\pi/4,3\pi/4]$ (the gap-maximum direction is taken as the zero polar angle). In Fig.3(a), the lower corner has a sign changing while the right corner does not, and the existence or absence of MKP is consistent with the edge-theory prediction. If one of the edges lies in the $\pi/4$ direction, the edge states become gapless, which also manifests in the numerical spectrum in Fig.3(b).

Finally, we mention that cuprate superconductors in proximity to 3D topological insulators have been experimentally studied for the purpose of creating vortex (instead of corner) MZMsWang et al. (2013); Zareapour et al. (2012); Li et al. (2015); Ortiz et al. (2018). In these setups, the 2D topological surface states (instead of the 1D edge states) are the key ingredients.

*$s_{\pm}$-wave pairing.–*Now we consider fully gapped $s_{\pm}$-wave superconductors with sign changing in the pairing. A host of candidates can be found in high $T_{c}$ iron-based superconductorStewart (2011); Hirschfeld et al. (2011), whose pairing at the Fermi surfaces near the Brillouin zone center and the Brillouin zone boundary have both $s$-wave nature but with opposite signs. The Fermi surfaces do not cross the pairing nodal rings, therefore, the superconductor is fully gapped. A simplest form of $s_{\pm}$-wave pairing is:

[TABLE]

with $0<\Delta_{0}<2\Delta_{1}$. The pairing node is $\cos k_{x}+\cos k_{y}=\Delta_{0}/\Delta_{1}$.

Let us first study the edge theory of TI. Expanding the Hamiltonian near ${\bf k}=(0,0)$ and taking $\mu=0$, we have

[TABLE]

Following a similar approach as the previous section, for the edge (I), we decompose the Hamiltonian as $H=H_{0}+H_{p}$, where

[TABLE]

Similar to the previous section, four zero-energy solutions of $H_{0}$ can be found, and $H_{p}$ takes the following form within this four-dimensional low-energy subspace:

[TABLE]

with $M_{\text{I}}=\int_{0}^{+\infty}dx\psi^{*}_{\alpha}(x)[\Delta_{0}-2\Delta_{1}-(\Delta_{1}/2)\partial_{x}^{2}]\psi_{\alpha}(x)=\Delta_{0}-2\Delta_{1}-\Delta_{1}m/t_{x}$. The low-energy effective Hamiltonians for the other three edges take the same forms as in Eq.(11), with Dirac masses $M_{\text{III}}=\Delta_{0}-2\Delta_{1}-\Delta_{1}m/t_{x}=M_{\text{I}}$, and $M_{\text{II}}=M_{\text{IV}}=\int_{0}^{+\infty}dy\psi^{*}_{\alpha}(y)[\Delta_{0}-2\Delta_{1}-(\Delta_{1}/2)\partial_{y}^{2}]\psi_{\alpha}(y)=\Delta_{0}-2\Delta_{1}-\Delta_{1}m/t_{y}$. Using the edge coordinate $l$, the effective edge Hamiltonian is the same as Eq.(12) with the same $A(l)$ but different $M(l)$, namely, $M(l)=-\bar{\Delta}_{0}-\Delta_{1}m/t_{x},-\bar{\Delta}_{0}-\Delta_{1}m/t_{y},-\bar{\Delta}_{0}-\Delta_{1}m/t_{x},-\bar{\Delta}_{0}-\Delta_{1}m/t_{y}$ for (I), (II), (III), and (IV), respectively, where we have defined $\bar{\Delta}_{0}=2\Delta_{1}-\Delta_{0}$.

To have MKP at each corner, the sign of Dirac mass $M(l)$ must change from an edge to its adjacent, which leads to the following criterion:

[TABLE]

Let us define $R_{s}\equiv\sqrt{2\bar{\Delta}_{0}/\Delta_{1}}$, whose physical meaning is the radius of the ring of the pairing node, across which the pairing changes sign, and $R_{x}\equiv\sqrt{-2m/t_{x}}$ and $R_{y}\equiv\sqrt{-2m/t_{y}}$, whose meanings are the two semi-axes of the ellipse determined by $m+\frac{t_{x}}{2}k_{x}^{2}+\frac{t_{y}}{2}k_{y}^{2}=0$ (i.e., the “band-inversion ring” of TI, where the sign of the $\sigma_{z}$ term changes). The mode existence criterion in Eq.(18) (for $\mu=0$) becomes

[TABLE]

which means that the band-inversion ring has to cross the pairing nodal ring [Fig.4(a)]. Although derived from the continuum model, Eq.(19) is quite accurate according to our lattice-model numerical resultssup . Intuitively, the low-energy edge modes come mainly from states near the band inversion (where the bulk states have the lowest energies), and inherit the sign of pairing there. When the two rings cross, the pairing sign at band inversion is opposite at two adjacent edges, which supports MKPs. We emphasize that the TI has to be anisotropic in the $x,y$ directions to satisfy Eq.(19) ($R_{x}\neq R_{y}$), which is the case for the high-transition-temperature TI WTe2Wu et al. (2018). In Fig.4(b), one finds the existence of MKP when Eq.(19) is satisfied. Including a modest chemical potential with a Fermi surface is innocuous [Fig.4(c)], as the Fermi surface can be gapped out by the induced pairing as long as it does not cross the pairing node. A $(m,\Delta_{0}/\Delta_{1},\mu)$ phase diagram is shown in Fig.4(d).

So far, we have not discussed disorders. We have numerically confirmed that usual disorders such as on-site random potential does not destroy the MKPssup . In addition, our proposal does not require atomically precise edges. Modest edge imperfections do not affect the MKPs because they are pinned to zero energy by particle-hole symmetry; “big” edge imperfections just create new corners that host their own MKPs, which offer more opportunity to observe MKPssup .

Finally, it is useful to mention that, in both the $d$-wave and $s_{\pm}$-wave cases, a single MZM can be created from the MKP at the corner by killing one mode in the pair. Apparently, TRS must be broken. For example, it can be achieved by adding an in-plane magnetic field with an appropriate magnitude. The physical picture is most transparent in the edge theory (see the Supplemental Material for detailssup ).

Experimental estimations.–For concreteness, let us focus on the high-temperature $s_{\pm}$-wave iron-based superconductors. As emphasized above, in the $s_{\pm}$-wave case the TI band structure is required to be anisotropic in the $x$ and $y$ directions [due to Eq.(19)]. Notably, the monolayer WTe2, which has recently been confirmed as a high-temperature TI in experimentsWu et al. (2018)(up to 100 Kelvin), has the desired band structureQian et al. (2014). According to the ${\bf k}\cdot{\bf p}$ model in Ref.Qian et al. (2014), we fit the parameters to be $R_{x}=0.41$Å*-1*, $R_{y}=0.15$Å*-1* (details are given in the Supplemental Materialsup ). The reciprocal lattice vectors of WTe2 along the $x$ and $y$ directions are $G_{x}\simeq 1.0$Å*-1* and $G_{y}\simeq 1.8$Å*-1*. Thus, the band-inversion ring reaches close to the Brillouin zone boundary in the $x$ direction, while it stays close to the zone center in the $y$ direction, resembling the advantageous shape of the band-inversion ring in Fig.4(a). Although an accurate estimation of the magnitude of the induced pairing gap is not available, we note that cuprate superconductors can induce a gap of tens of meV at the surface states of topological insulatorsWang et al. (2013); Zareapour et al. (2012); presumably similar order of magnitude can be expected in the present setup. Therefore, among other options, a setup composed of a WTe2 monolayer in proximity to a high-$T_{c}$ iron-based superconductor is promising for the present proposal. A WTe2 monolayer in proximity to cuprate superconductors is also promising.

*Conclusions.–*We have shown that a 2D TI with proximity-induced $d$-wave or $s_{\pm}$-wave pairing, though being topologically trivial as a TRI superconductor, is a promising candidate of high-temperature platform for realizing robust Majorana corner modes. We provide quantitative criteria for this proposal. This Letter may also stimulate further studies of topologically-trivial-superconductor-based Majorana modes.

*Acknowledgements.—*We would like to thank Wei Li for helpful discussions. This work is supported by NSFC (No. 11674189). Z. Y. is supported in part by the China Postdoctoral Science Foundation (2016M590082).

Note added: Recently, there appeared a related preprintWang et al. (2018) that focuses on the $s_{\pm}$-wave case.

I I. Edge theory of the II, III, IV edges for the $d$-wave pairing

We start from the low-energy bulk Hamiltonian around ${\bf k}=(0,0)$. Having $d$-wave pairing in mind, we take $\Delta_{0}=0$. The low-energy Hamiltonian at $\mu=0$ reads (not imposing any constraint on $\Delta_{x,y}$ at this stage):

[TABLE]

Note that $m<0$. For the edge III, the $k_{y}^{2}$ terms can be neglected and the Hamiltonian is decomposed as $H=H_{0}+H_{p}$, with

[TABLE]

When solving the eigenvalue equation $H_{0}\psi_{\alpha}(x)=E_{\alpha}\psi_{\alpha}(x)$, the boundary condition is $\psi_{\alpha}(0)=\psi_{\alpha}(-\infty)=0$. A straightforward calculation gives four solutions with $E_{\alpha}=0$, whose forms are

[TABLE]

with the normalization constant $\mathcal{N}_{x}=2\sqrt{\kappa_{2}(\kappa_{1}^{2}+\kappa_{2}^{2})/\kappa_{1}^{2}}$ and the two parameters $\kappa_{1}$ and $\kappa_{2}$ given by

[TABLE]

$\tilde{\chi}_{\alpha}$ are eigenvectors satisfying $\sigma_{y}s_{z}\tau_{z}\tilde{\chi}_{\alpha}=\tilde{\chi}_{\alpha}$. Here we choose

[TABLE]

Then the matrix elements of the perturbation $H_{p}$ in this basis are

[TABLE]

In terms of the Pauli matrices, the final form of the effective Hamiltonian is

[TABLE]

where

[TABLE]

Similarly, for the edge II, we also decompose the Hamiltonian into two parts, discarding terms of the order of $k_{x}^{2}$:

[TABLE]

By solving the eigenvalue equation $H_{0}\psi_{\alpha}(y)=E_{\alpha}\psi_{\alpha}(y)$ with the boundary condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$, we find that there are four solutions with $E_{\alpha}=0$, whose forms are

[TABLE]

with the normalization constant $\mathcal{N}_{y}=2\sqrt{\gamma_{2}(\gamma_{1}^{2}+\gamma_{2}^{2})/\gamma_{1}^{2}}$ and the two parameters $\gamma_{1}$ and $\gamma_{2}$ given by

[TABLE]

$\xi_{\alpha}$ are the eigenvectors satisfying $\sigma_{x}\xi_{\alpha}=\xi_{\alpha}$. Here we choose

[TABLE]

In this basis, the matrix elements of the perturbation $H_{p}$ are

[TABLE]

In terms of the Pauli matrices, the final form of the effective Hamiltonian is

[TABLE]

where

[TABLE]

Similarly, for the edge IV, the effective Hamiltonian is

[TABLE]

and $M_{\rm IV}=M_{\rm II}$.

For the $d$-wave pairing with amplitude satisfying $\Delta_{x}=-\Delta_{y}$, the $\Delta_{x}+\Delta_{y}$ term appearing in the mass term vanishes. Let $\Delta_{x}=-\Delta_{y}\equiv\Delta_{d}$, then the effective Hamiltonian of the four edges are

[TABLE]

where $M_{\rm I}=M_{\rm III}=\Delta_{d}m/t_{x}$, $M_{\rm II}=M_{\rm IV}=-\Delta_{d}m/t_{y}$. It is immediately clear that the mass terms on two neighboring edges always have opposite signs.

II II. Edge theory for the $s_{\pm}$-wave pairing via solving

the lattice model

We start from the Bogoliubov-de Gennes Hamiltonian $H=\sum_{k}\Psi_{k}^{{\dagger}}H({\bf k})\Psi_{k}$ with $\Psi_{k}=(c_{a,k\uparrow},c_{b,k\uparrow},c_{a,k\downarrow},c_{b,k\downarrow},c_{a,-k\uparrow}^{{\dagger}},c_{b,-k\uparrow}^{{\dagger}},c_{a,-k\downarrow}^{{\dagger}},c_{b,-k\downarrow}^{{\dagger}})^{T}$ and

[TABLE]

where $\Gamma_{1}=\sigma_{z}\tau_{z}$, $\Gamma_{2}=\sigma_{x}s_{z}$, $\Gamma_{3}=\sigma_{y}\tau_{z}$ and $\Gamma_{4}=s_{y}\tau_{y}$. Here we have taken the chemical potential $\mu=0$.

We first investigate the edge I. As the system takes open boundary condition in the $x$ direction and periodic boundary condition in the $y$ direction, we do a partial Fourier transformation of the Hamiltonian, which gives

[TABLE]

where $x$ is the integer-valued coordinate (the lattice constant $a=1$) taking values from $1$ to $L$. In the basis $(\Psi_{1,k_{y}},\Psi_{2,k_{y}},...)$, the Hamiltonian can be expressed in a matrix form $H(k_{y})=H_{0}(k_{y})+H_{1}(k_{y})+H_{2}(k_{y})$ with

[TABLE]

where $D_{0}=(m_{0}-t_{y}\cos k_{y}),T_{0}=(-t_{x}\Gamma_{1}+iA_{x}\Gamma_{2})/2$,

[TABLE]

where $D_{1}=(\Delta_{0}-\Delta_{1}\cos k_{y})\Gamma_{4},T_{1}=-\Delta_{1}\Gamma_{4}/2$, and

[TABLE]

To simply the calculation, we take $k_{y}$ to be close to [math] and the pairing amplitude to be small, so that $H_{1}(k_{y})$ and $H_{2}(k_{y})$ can be treated as perturbations. We first solve for the eigenstates of $H_{0}(k_{y}=0)$. Their forms can be written as $\psi=(\psi_{1},\psi_{2}\cdots)^{T}$, which satisfies the following iteration relation,

[TABLE]

here for convenience we have renamed $m_{0}-t_{y}$, $t_{x}$, $A_{x}$ as $m$, $t$, $A$, respectively (Do not confuse it with the $m$ in the main text). Because of the anticommutation relation between $\Gamma_{1}$ and $\Gamma_{2}$, the eigenvalues of $i\Gamma_{1}\Gamma_{2}$ are $\pm 1$. To solve this equation, we choose a trial solution $\psi_{\alpha;n}=\lambda^{n}\phi_{\alpha}$, where $\phi_{\alpha}$ satisfies $-i\Gamma_{1}\Gamma_{2}\phi_{\alpha}=s\phi_{\alpha}$ ($s=\pm 1$). Note that $-i\Gamma_{1}\Gamma_{2}$ is an $8\times 8$ matrix, therefore, there are four $\phi_{\alpha}$’s satisfying this equation. Substituting this trial solution into Eq.(42), it is readily found that

[TABLE]

It can be shown that $|\lambda_{\pm}|<1$ when $s=\text{sgn}(t/A)$ and $m^{2}<t^{2}$. If we consider a semi-infinite geometry with $L\rightarrow\infty$, the wave function should satisfy the two boundary conditions: $\psi_{\alpha;0}=0$ and $\psi_{\alpha;+\infty}=0$ (For convenience, we have added an artificial site $x=0$, on which the wavefunction is zero). The solution is of the form \psi_{\alpha;n}=N(\lambda_{+}^{n}-\lambda_{-}^{n})$$\phi_{\alpha}, where $N$ is the normalization constant:

[TABLE]

Now the effective Hamiltonian for the edge states is obtained from perturbation theory,

[TABLE]

The mass term of the effective Hamiltonian comes solely from $H_{1}(k_{y})$. Ignoring all terms of orders higher than $k_{y}$, we get

[TABLE]

After straightforward calculations, we find

[TABLE]

Similar calculations lead to the mass terms for the other three edges, II, III, IV, which are $M_{\rm III}=M_{\rm I}$ and

[TABLE]

To create corner Majorana Kramers pairs, the mass term must change sign at the corner, which requires

[TABLE]

This criterion is the same as the one obtained from continuum model in the main text.

III III. Majorana Kramers pair is absent when the pairing nodal ring does not cross the

band-inversion ring

In the main text, we have shown when the pairing nodal ring crosses the band-inversion ring, Majorana Kramers pairs are created at the corner of TI. To display the opposite situation, we tune the pairing nodal ring of Fig.4(a) in the main text to be around the $(\pi,\pi)$ point while keeping the band-inversion ring unchanged, so that the two rings no longer cross each other, as shown in Fig.5(a). In this regime, the numerical results demonstrate that there is no zero energy state (see Fig.5(b)), indicating the absence of Majorana Kramers pair.

IV IV. Details of experimental estimations

In this section, we give an experimental estimation based on the band structures of WTe2, which has recently been found as a 2D TI at temperature as high as 100 KelvinWu et al. (2018). In Ref.Qian et al. (2014), a ${\bf k}\cdot{\bf p}$ model has been obtained to fit the band structure near the $\Gamma$ point (the band-inverted region), which is

[TABLE]

where $v_{1}=3.87\times 10^{5}$ m/s, $v_{2}=0.46\times 10^{5}$ m/s, $\delta=-0.33$ eV, $m_{x}^{p}=0.50m_{e}$, $m_{y}^{p}=0.16m_{e}$, $m_{x}^{d}=2.48m_{e}$, $m_{y}^{d}=0.37m_{e}$, $m_{e}$ being the free electron mass. The lattice constant of WTe2 is $a=6.25$Å, $b=3.48$Å. To simply the calculation, we make an approximation that $m_{x}=\sqrt{m_{x}^{p}m_{x}^{d}}=1.11m_{e}$ and $m_{y}=\sqrt{m_{y}^{p}m_{y}^{d}}=0.24m_{e}$, and then transform the ${\bf k}\cdot{\bf p}$ model to the lattice form, which is

[TABLE]

where $t_{x}=\frac{\hbar^{2}}{m_{x}a^{2}}=0.18$ eV, $t_{y}=\frac{\hbar^{2}}{m_{y}b^{2}}=2.64$ eV, $m_{0}=-\delta-t_{x}-t_{y}=-2.49$ eV. $A_{x}=\hbar v_{1}/a=0.41$ eV, $A_{y}=\hbar v_{2}/b=0.09$ eV. The band-inversion ring intersects the $k_{x}$ axis at $R_{x}=\arccos[-(m_{0}+t_{y})/t_{x}]/a=0.41$Å*-1*, and the $y$ axis at $R_{y}=\arccos[-(m_{0}+t_{x})/t_{y}]/b=0.15$Å*-1*. The energy gap is $0.087$ eV (at $(k_{x},k_{y})=(0,\pm R_{y})$). It is notable that $R_{y}$ and the energy gap thus obtained agree excellently with the results ($0.146$Å*-1*, about $0.08$ eV) based on the DFT calculationQian et al. (2014), indicating this lattice model gives an accurate description of the relevant band structure. In addition, the reciprocal lattice vectors are $G_{x}=2\pi/a=1.00$Å*-1*, $G_{y}=2\pi/b=1.80$Å*-1*, and it is straightforward to find that $R_{x}/(G_{x}/2)=0.82$, $R_{y}/(G_{y}/2)=0.16$, indicating that the band-inversion ring reaches close to the Brillouin-zone boundary in the $x$ direction, while stays close to the zone center in the $y$ direction.

In the presence of pairing, the Hamiltonian becomes

[TABLE]

where $\Delta({\bf k})=\Delta_{0}+\Delta_{x}\cos(k_{x}a)+\Delta_{y}\cos(k_{y}b)$. The only difference between this Hamiltonian and the one in Eq.(1) of the main text is in the basis choices.

In high-temperature cuprate superconductor, the reduced gap $2\Delta/k_{B}T$ can be much larger than the expected BCS value $4.3$ for $d$-wave pairingWon and Maki (1994), indicating that the pairing amplitude can be quite large. e.g., it was found that $\Delta$ can be as high as $60$ meV in superconductor Bi2Sr2Ca2Cu3O10+δ ($T_{c}=109$ K)Kugler et al. (2006). In high-temperature iron-based superconductor, the pairing amplitude can also be higher than $10$ meV, e.g., $\Delta=15$ meV in superconductor Ba1-xKxFe2As2 ($T_{c}=37$ K)Hoffman (2011).

In Fig.6, we use the model parameters of WTe2 extracted from the ${\bf k}\cdot{\bf p}$ model, and take the $s_{\pm}$-wave pairing with an amplitude $\sim 100$ meV (which has been exaggerated, yet the result is qualitatively unchanged; similarly, the value of $A_{y}$ has also been increased from $0.09$ eV to $0.2$ eV). Fig.6 demonstrates that Majorana Kramers pairs are created at the corners when the pairing nodal ring crosses the band-inversion ring.

V V. Realizing single Majorana zero mode at the corner

To realize a single Majorana zero mode instead of a Majorana Kramers pair per corner, time-reversal symmetry must be broken. We find that it can be achieved by adding an appropriate in-plane magnetic field in either the $d$-wave or the $s_{\pm}$-wave case, provided that the system is anisotropic. For concreteness, suppose that the magnetic field is added in the $x$ direction, then it introduces a Zeeman energy $V_{x}s_{x}\tau_{z}$. The low-energy effective theory becomes

[TABLE]

Now we can split this $4\times 4$ Hamiltonian into two decoupled blocks. In fact, in the two-dimensional $\tau_{x}=\pm s_{z}$ subspace, we have $V_{x}s_{x}\tau_{z}=-iV_{x}s_{x}\tau_{x}\tau_{y}=\mp iV_{x}s_{x}s_{z}\tau_{y}=\mp V_{x}s_{y}\tau_{y}$, therefore, the two decoupled blocks have Dirac mass $M(l)-V_{x}$ and $M(l)+V_{x}$, respectively, and single MZM appears when one of them has a sign changing, while the other does not. This is satisfied, for example, when $|\Delta_{d}m/t_{y}|<V_{x}<|\Delta_{d}m/t_{x}|$ for the $d$-wave case.

VI VI. Stability of the Majorana corner modes against disorders

In this section, we investigate the stability of the Majorana Kramers pairs against random disorders. For concreteness, we consider random on-site potentials. The real-space Hamiltonian reads

[TABLE]

where $s_{x,y,z}$ and $\sigma_{x,y,z}$ are Pauli matrices acting on spin and orbital degree of freedom, respectively, $s_{0}$ and $\sigma_{0}$ are the $2\times 2$ unit matrix. The last term represents the random on-site potential, which is taken to obey a Gaussian distribution, namely $\langle V_{ij}\rangle=0$, $\langle V_{ij}V_{nm}\rangle=D_{0}^{2}\delta_{in}\delta_{jm}$ with $D_{0}$ characterizing the strength of the randomness.

Fig.7 shows several profiles of Gaussian disorders and the spectra close to zero energy. It is apparent that the Majorana Kramers pairs are robust against disorders for a quite broad range of disorder strength.

VII VII. Effects of edge imperfections

In real experiments, the edges are usually not atomically precise, and the edge orientations can vary in space. It is useful to study the stability of Majorana Kramers pairs in the presence of these imperfections. Thanks to the protection of particle-hole symmetry, a Majorana Kramers pair of zero modes remains robust in the presence of small or modest imperfections, because shifting the energy of a spatially isolated Majorana Kramers pair away from zero is inconsistent with the inherent particle-hole symmetry of BdG equation.

As a representative example of edge imperfections, we calculate the energies and eigenstates on a non-ideal square lattice with a small square removed at a corner (see Fig.8). In Fig.8(b), additional in-gap modes are created with energies far away from zero. Such nonzero-energy modes do not affect detection of the original Majorana Kramers pairs, which remain at zero energy. When the size of the removed square is increased [Fig.8(c)], the energies of in-gap modes are lowered, and finally, their energies come to zero, namely, additional Majorana Kramers pairs are created [Fig.8(d)(e)]. From this example, we can see that lattice imperfections generally do not destroy the Majorana Kramers pairs; instead, they provide more opportunity to detect them: The imperfections provide additional corners built in the samples, hosting additional Majorana Kramers pairs.

VIII VIII. Effects of chemical potential

In this section, we provide more details about the effects of a nonzero chemical potential. We will see that including the chemical potential does not change the physics qualitatively compared to the case of vanishing chemical potential.

First, we study the effect of chemical in the low-energy effective edge theory. At the edge I, nonzero $\mu$ simply generates the following term in the effective edge Hamiltonian:

[TABLE]

Now the edge effective Hamiltonian reads

[TABLE]

This Hamiltonian can be decomposed as the sum of two independent $2\times 2$ Hamiltonians

[TABLE]

where $H_{\rm edge,-}$ acts in the $\tau_{z}=s_{z}$ subspace spanned by $\{|s_{z}=1,\tau_{z}=1\rangle,|s_{z}=-1,\tau_{z}=-1\rangle\}$, and $H_{\rm edge,+}$ acts in the $\tau_{z}=-s_{z}$ subspace spanned by $\{|s_{z}=1,\tau_{z}=-1\rangle,|s_{z}=-1,\tau_{z}=1\rangle\}$. Now the zero modes can be found as

[TABLE]

Here, $\mu$ only adds trivial phase factors to the wavefunctions without modifying their profiles. It indicates that a small or modest $\mu$ (to which the low-energy theory is applicable) does not qualitatively change the physics.

Now we go beyond the low-energy continuum Hamiltonian to the lattice Hamiltonian. As we increase the chemical potential $\mu$ away from [math], there are two possible scenarios that can kill the Majorana Kramers pairs: (i) The bulk gap closes; (ii) the edge gap closes. First, let us consider the first scenario. The BdG Hamiltonian of the bulk is

[TABLE]

whose energy spectra are

[TABLE]

Without the pairing, the gap closing condition is $\sqrt{M^{2}({\bf k})+A_{x}^{2}\sin^{2}k_{x}+A_{y}^{2}\sin^{2}k_{y}}=|\mu|$, which determines the bulk Fermi surface of the doped TI. Without losing generality, we only focus on the $\mu>0$ case. The Fermi surface appears when $\mu>\text{min}\{\sqrt{M^{2}({\bf k})+A_{x}^{2}\sin^{2}k_{x}+A_{y}^{2}\sin^{2}k_{y}}\}$. When the Fermi surface determined by

[TABLE]

and the pairing nodal ring determined by $\Delta({\bf k})=0$ cross each other, the energy gap closes. For the pairing we considered in the main text, $\Delta({\bf k})=\Delta_{0}-\Delta_{1}(\cos k_{x}+\cos k_{y})$ with $0<\Delta_{0}<2\Delta_{1}$, we find the gapless region is

[TABLE]

where $\mu_{1}=\text{min}\{\sqrt{(m_{0}-t_{x}\frac{(\Delta_{0}-\Delta_{1})}{\Delta_{1}}-t_{y})^{2}+A_{x}^{2}(1-\frac{(\Delta_{0}-\Delta_{1})^{2}}{\Delta_{1}^{2}})}$, $\sqrt{(m_{0}-t_{x}-t_{y}\frac{(\Delta_{0}-\Delta_{1})}{\Delta_{1}})^{2}+A_{y}^{2}(1-\frac{(\Delta_{0}-\Delta_{1})^{2}}{\Delta_{1}^{2}})}\}$, and $\mu_{2}=\text{max}\{\sqrt{(m_{0}-t_{x}\frac{(\Delta_{0}-\Delta_{1})}{\Delta_{1}}-t_{y})^{2}+A_{x}^{2}(1-\frac{(\Delta_{0}-\Delta_{1})^{2}}{\Delta_{1}^{2}})},$ $\sqrt{(m_{0}-t_{x}-t_{y}\frac{(\Delta_{0}-\Delta_{1})}{\Delta_{1}})^{2}+A_{y}^{2}(1-\frac{(\Delta_{0}-\Delta_{1})^{2}}{\Delta_{1}^{2}})}\}$.

The scenario (ii) mentioned above, namely closing the edge gap to kill the Majorana Kramers pairs, is not as convenient to quantify. The reason is that the edge states of TI do not exist as a 1D system by itself, and the edge cannot be separately studied beyond the low-energy theory. Nevertheless, we can numerically find the critical chemical potential $\mu_{c}$ above which the Majorana Kramers pairs disappear. When the Cooper pairing is small, we find that $\mu_{c}<\mu_{1}$ in general, which indicates that the gap closing appear at the edge, i.e., the scenario (ii).

A 3D phase diagram with varied $(m_{0},\Delta_{0}/\Delta_{1},\mu)$ is numerically calculated and shown in Fig.4(d) in the main article. In this supplemental material, let us focus on the the $\Delta_{0}/\Delta_{1}=1$ cross section of this 3D phase diagram, which is shown in Fig.9(a). For the chosen parameters, the paring nodal ring and the band-inversion ring cross each other when $0.4<m_{0}<1.3$. From Fig.9(b)(c), it is clear that zero modes are indeed found when $0.4\lesssim m_{0}\lesssim 1.3$, in accordance with the criterion Eq.(19) of the main article. Along the path $\gamma$, Majorana corner mdoes exist for $\mu\lesssim 0.38$ (Fig.9(d)). Note that the bulk gap closes at $\mu_{1}=\sqrt{(m_{0}-1.3)^{2}+0.4^{2}}=0.5$, therefore, the disappearance of Majorana corner modes is due to the gap closing at the edges.