Helical Hinge Majoranas in Iron-Based Superconductors
Rui-Xing Zhang William S. Cole S. Das Sarma


















































    †↑↓⟨⟩ruixing@umd.eduwcole1@umd.eduCondensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
            Motivated by recent experiments on FeTe_1-xSe_x, we construct an explicit minimal model of an iron-based superconductor with band inversion at the Z point and non-topological bulk s_± pairing. While there has been considerable interest in Majorana zero modes localized at vortices in such systems, we find that our model -  without any vortices - intrinsically supports 1D helical Majorana modes localized at the hinges between (001) and (100) or (010) surfaces, suggesting that this is a viable platform for observing “higher-order” topological superconductivity. We provide a general theory for these hinge modes and discuss their stability and experimental
            manifestation. Our work indicates the possible experimental observability of hinge Majoranas in iron-based topological superconductors.
            Helical Hinge Majoranas in Iron-Based Superconductors
                S. Das Sarma
                December 30, 2023
            =====================================================
            
             Introduction - In the decade since their initial discovery, iron-based superconductors (FeSCs) have been studied vigorously <cit.>. A substantial diversity of materials has been realized in this class, all sharing characteristically high critical temperatures and rich phase diagrams with nearby magnetic phases, as a function of chemical substitution. More recently it has been appreciated that several members of this family also have topologically nontrivial normal-state band structures <cit.>, and therefore may be a natural platform for the realization of Majorana zero modes (MZM) <cit.> and effective topological superconductivity (TSC) <cit.>. As in the original Fu-Kane proposal <cit.>, the pairing symmetry is taken to be conventional; then, through a “self-proximity effect” induced by the bulk superconductor, the topological surface states or vortex lines are imbued with pair correlations from the bulk, leading to lower-dimensional surface or vortex TSC.
            
            
            This indeed appears to be the case with the vortex core MZM recently observed in the unconventional iron-based superconductor FeTe_1-xSe_x (x=0.45) (FTS) <cit.>. Above the superconducting transition at T_c= 14.5K, this system exhibits non-trivial band topology with a band inversion along the Γ-Z line in the Brillouin zone. On the (001) surface, angle-resolved photoemission spectroscopy (ARPES) measurements clearly see a helical Dirac surface state <cit.>, the hallmark of three dimensional (3D) time-reversal invariant (TRI) topological insulators (TI) <cit.>. Below T_c, the surface state develops a gap, as expected. STM measurements of the same system in a weak magnetic field reveal robust zero-bias peaks inside vortex cores, a strong indication of zero energy subgap states and associated MZM physics <cit.>.
            
            
            In this letter, we present a completely different mechanism for Majorana states in FeSCs in the absence of vortices. Specifically, we demonstrate the emergence of higher-order topology <cit.> and corresponding helical Majorana hinge states in an explicit minimal lattice model based on two key ingredients <cit.>: (i) the band inversion along Γ-Z<cit.> and (ii) extended s-wave (s_±) pairing <cit.>, with a sign change of the pairing potential between the Γ and M points. We expect this to be a reasonable effective model that captures the topological properties of FTS <cit.>, Li_0.84Fe_0.16OHFeSe <cit.>, LiFeAs <cit.>, and other FeSCs. We explain the origin of these Majorana hinge states with an analytic theory for the projection of the bulk extended s-wave pairing onto the Dirac cone of an arbitrary surface termination, and numerically demonstrate their stability at nonzero chemical potential and in the presence of chemical potential disorder.
            
             Model Hamiltonian - Our minimal BdG model describes a 3D TI with s_± pairing on a cubic lattice:
            
            H(k) = [      H_0(k)-μ        -iD(k);         iD(k) μ - H_0^*(-k) ].
            
            The normal state Hamiltonian is H_0(k)=v(sin k_xΓ_1 + sin k_yΓ_2+sin k_z Γ_3)+ m(k)Γ_5 with m(k)=m_0-m_1(cos k_x + cos k_y)-m_2cos k_z<cit.>. The 4 × 4 matrices Γ_i are chosen to be
            Γ_1    =   σ_x ⊗ s_x, Γ_2 = σ_x ⊗ s_y, Γ_3 = σ_x ⊗ s_z, 
            Γ_4    =   σ_y ⊗ s_0, Γ_5 = σ_z ⊗ s_0,
            
            where σ and s are Pauli matrices for the orbital and spin degrees of freedom, respectively. The time-reversal (Θ) and the parity (P) symmetry operations are given by
            Θ = iσ_0⊗ s_y K = -iΓ_13 K, P=σ_z⊗ s_0 = Γ_5
             
            where Γ_ij=[Γ_i,Γ_j]/2i. The band topology of H_0 is easily read off using the Fu-Kane criterion <cit.>. 
            
            For our purposes, we choose v=1,m_0=-4,m_1=-2,m_2=1 which locates the band inversion at Z[The band inversion at Z happens when m_0<0, m_1<0, m_2>0, 2m_1-m_2<m_0<min{-m_2,-|2m_1+m_2|}.]. H_0 then describes a strong TI phase with helical Dirac surface states.
            
            The s_± pairing function is D(k) = Δ(k) Γ_13, with
            Δ(k) = Δ_0 + Δ_1 (cos k_x + cos k_y)
            
            We first restrict to μ=0 for simplicity, which is also justified by the small chemical potential observed in the ARPES experiments. As we adiabatically turn on the pairing, the bulk remains a topologically trivial superconductor. However, unlike the case of uniform s-wave pairing, the s_± pairing will gap out the TI surface states in an anisotropic way. This is the basic principle that enables the realization of new surface topological phenomena beyond the conventional bulk topology.
            
            
            
            
            
            
            
            
            
            
            
             Theory of Surface States - To visualize the anisotropy of surface state pairing, we first notice that any surface termination of a crystal can be described as a tangent plane of the unit sphere <cit.>, as shown in Fig. <ref>(a). In particular, an arbitrary surface Σ(ϕ,θ) can be uniquely labeled by its unit normal vector defined as n_Σ=(sinθcosϕ, sinθsinϕ, cosθ)^T. In the long-wavelength limit, we expand H_0(k) around Z to O(k^2) and obtain H_0^Z( k)=v(k_xΓ_1+k_yΓ_2-k_zΓ_3) +[m̃_0+m_1/2(k_x^2+k_y^2)-m_2/2k_z^2]Γ_5, where m̃_0=m_0-2m_1+m_2. To solve for the effective surface theory for Σ(ϕ,θ), we apply the coordinate rotation R(ϕ,θ)=R_Y(-θ)R_Z(-ϕ) to k=(k_x,k_y,k_z)^T and arrive at
             k'   =   (k_1,k_2,k_3)^T = R(ϕ,θ) k,
            
            
            
            
            
            
            where R_Y(θ) and R_Z(ϕ) are Euler rotations around y-axis and z-axis, respectively. It is easy to check that k_3 =  n_Σ· k and any surface state on Σ(θ,ϕ) can now be obtained by imposing open boundary conditions on H_0^Z( k') along k_3 direction.
            
            
            
            Generally, H_0^Z( k') has a complicated form in k'. However, we notice that there always exists the unitary transformation U(ϕ,θ)=e^iΓ_13θ/2e^iΓ_12ϕ/2 such that H̃_0^Z=U(ϕ,θ) H_0^Z U(ϕ,θ)^†=h̃_0 + h̃_1 has a simple form:
            h̃_0    =    -vk_3Γ_3 + (m̃_0 - m̃_2k_3^2) Γ_5 
            h̃_1    =    v(k_1Γ_1+k_2Γ_2)+ (m̃_13k_1k_3+m̃_1k_1^2+m_1/2k_2^2)Γ_5. 
                  
            We have defined m̃_1=(m_1cos^2θ-m_2sin^2θ)/2, m̃_2=(m_2cos^2θ-m_1sin^2θ)/2 and m̃_13=(m_1+m_2)sin 2θ/2. Now we solve for the zero mode equation of h̃_0(k_1=k_2=0) and consider h̃_1 as a perturbation to extract the surface state dispersion of Σ(ϕ,θ). 
            
            Replacing k_3 → -i ∂_x_3 yields the zero mode equation
            
            [iv∂_x_3Γ_3 + (m̃_0+m̃_2∂_x_3^2)Γ_5]ψ̃(x_3) = 0
            
            With the boundary conditions ψ̃(x_3=0)=ψ̃(x_3=-∞)=0, we find two solutions ψ̃_1,2 that are exponentially localized near x_3=0:
            ψ̃_i =  Nsin (β x_3) e^α x_3 e^i(k_1x_1+k_2x_2)ξ_i,  i=1,2.
            
            where we have defined
            α = v/2m̃_2, β = √(4m̃_0m̃_2-v^2)/2m̃_̃2̃,
            
            and the normalization factor N =√(4α(α^2+β^2)/β^2). The spinor part of ψ̃_i are the eigenstates of Γ_35 with eigenvalue -1: 
            ξ_1 = (0,i,0,1)^T, ξ_2 = (-i,0,1,0)^T.
            
            Treating h̃_1 as a perturbation, the low energy description of the surface state is given by
            
            h_ss = k_1ς_2+k_2ς_1,
            
            where ς are pseudo-spin Pauli matrices in the space spanned by ξ_1 and ξ_2. 
            
             Surface State Pairing - The effective pairing h_Δ(θ,ϕ) on a surface Σ(ϕ,θ) is solved by treating the s_± SC pairing Δ(k) τ_y ⊗Γ_13 as a perturbation (τ are the Pauli matrices for the particle-hole basis). The particle-hole redundancy requires the hole counterparts of Eq. <ref>, and the spinor part of the zero modes in the BdG basis are
            χ_1    =    (1,0)^T⊗ U^†ξ_1, χ_3 = (0,1)^T⊗ (U^†ξ_1)^*, 
            χ_2    =    (1,0)^T⊗ U^†ξ_2, χ_4 = (0,1)^T⊗ (U^†ξ_2)^*,
            
            where we have undone the unitary transformation U(ϕ,θ) of ξ_i. By rotating Δ( k) to Δ( k') and expanding it around Z, the effective pairing at the surface Dirac point with k_1=k_2=0 becomes
            Δ(0,0,-i∂_x_3) = Δ_0 + 2Δ_1 + Δ_1 sin^2θ/2∂_x_3^2.
            
            We expect that h_Δ(ϕ,θ)=Δ_eff(θ)Λ(ϕ,θ), where Δ_eff and Λ come from the zero mode projection of the scalar momentum part Δ(k) and the matrix part τ_y⊗Γ_13 respectively.
            Interestingly, the matrix part Λ is independent of the surface Σ(ϕ,θ):
            
            (Λ)_i,j =  χ_i^†τ_y⊗Γ_13χ_j = (τ_y⊗ς_y)_i,j.
            
            The spatial anisotropy of h_Δ(ϕ,θ) arises completely from Δ_eff(θ). Our main analytic result is an approximate formula for the effective surface pairing,
            Δ_eff   =    N^2∫_-∞^0 dx_3 sin (β x_3) e^α x_3Δ(-i∂_x_3)sin (β x_3) e^α x_3
               =   Δ_0 + 2Δ_1 - Δ_1 m_0-2m_1+m_2/m_2cos^2θ - m_1sin^2 θsin^2 θ.
            	
            The effective surface SC theory for any facet Σ(ϕ,θ) is finally given by,
            
            H_Σ(k_1,k_2)= [  k_1ς_2 + k_2ς_1   -iς_y Δ_eff(θ);    iς_y Δ_eff(θ) -k_1ς_2 + k_2ς_1 ].
             Helical hinge Majoranas - The angular anisotropy of Δ_eff(θ) enables a nontrivial situation where the pairing gap flips its sign while crossing the hinge between two facets Σ(ϕ,θ_1) and Σ(ϕ,θ_2). Mathematically,
            Δ_eff(θ_1)Δ_eff(θ_2) < 0.
            
            With Eq. <ref>, the hinge physics manifests itself as a TRI topological domain wall problem of 2D BdG Dirac fermions. As a result, the sign reversal of the pairing gaps necessarily binds the hinge with a pair of 1D helical Majoranas. 
            
            Furthermore, the hinge Majorana condition in Eq. <ref> requires the existence of a topological critical angle θ_c, where Δ_eff(θ_c)=0. This critical angle θ_c is given by
            ^2θ_c = m_1/m_2+Δ_1/Δ_0+2Δ_1( m_0-2m_1/m_2+1 ).
            
            Helical hinge Majoranas will appear as long as θ_c exists, i.e., when ^2 θ_c > 0. With our choice of model parameters, the condition is
            
            -2Δ_1<Δ_0<-3/2Δ_1.
            
            In Fig. <ref> (b), we schematically plot the surface gap distribution for all facets, for parameters with a non-zero θ_c, where the blue and red regions denote positive and negative surface gaps, respectively [Notice that spatial inversion symmetry will enforce Δ_eff(θ) = Δ_eff(π-θ) in our system.].
            
            
            
            We now confirm this continuum theoretical analysis by carrying out a direct numerical solution of the full lattice model, with results shown in Fig. <ref>. The gap is obtained in a slab geometry on the (001) surface with θ=0 and the (010) surface with θ=π/2. The numerical results shown in Fig. <ref> (a) agree well with the analytic result from Eq. <ref>. As Δ_0 is tuned from zero, a surface topological phase transition first occurs at Δ_0= -3/2Δ_1 with vanishing surface gap on the (010) surface. The gap then reverses sign between the (001) and (010) surfaces, and the hinge Majorana condition is satisfied. When Δ_0=-2Δ_1, a second surface topological phase transition occurs, trivializing the surface topology and eliminating the hinge Majoranas. In Fig. <ref> (b) and (c), we show the clear surface gaps at Δ_0=-1.75Δ_1 for translation invariant (001) and (010) surfaces. In Fig. <ref> (d), we calculate the band dispersion along k_x for a wire geometry with periodic boundary conditions along x̂ and open boundary conditions along ŷ and ẑ. Correspondingly, we observe four pairs of helical Majoranas states that appear inside the surface gap. In Fig. <ref> (e), we plot the spatial profile for these states. Each pair of helical Majoranas is exponentially localized around each corner of the y-z cross section (i.e., the edge between (001) and (010) surfaces). For a sample with a cubic geometry, and open boundaries in all directions (as shown in Fig. <ref> (a)) the hinge Majoranas circulate the edges between the top/bottom surfaces and the side surfaces. In Fig. <ref> (b), we plot the intensity |ψ(r)|^2 for the lowest-energy eigenstate (summed over σ, s, τ) of the lattice model on this geometry, showing that the support is exponentially localized to the hinges, as expected.
            
            The appearance of helical hinge Majoranas has a topological origin. In the supplementary materials <cit.>, we study the Wannier bands <cit.> of H(k) in different slab geometries and find “helical" surface Wannier bands that characterize the time-reversal-symmetric ℤ_2 pump of BdG quasi-particles on the surfaces. Such ℤ_2 pumps also occur in 2D TRI TIs/TSCs and offer a topological picture for the helical edge physics <cit.>. Similarly, the helical surface Wannier band spectrum here unambiguously signals the helical hinge Majoranas. 
            
            We emphasize that the helical surface Wannier bands should not be interpreted as a surface topological index for any single isolated surface, e.g. the (001) surface. Importantly, any surface Hamiltonian H_Σ in Eq. <ref> is topologically trivial by itself, and the helical hinge Majoranas only arise when a domain wall is formed between neighboring facets. Therefore, the higher order topology here is a combined effect with contributions from all surfaces in a crystal.
            
             Stability - Having demonstrated the existence of hinge modes under ideal circumstances, we now show that they persist even when the chemical potential is moved from the Dirac point. The analytic theory can no longer be easily applied for finite chemical potential, but we can characterize the phase hosting hinge modes through either its surface spectrum or its Wannier band spectrum. As the surface gap closes and reopens, the helical (gapped) surface Wannier bands will simultaneously develop a gap (become helical), indicating a surface topological phase transition. By checking both surface energy band spectra and Wannier band spectra, we numerically obtain the phase diagram of our model as a function of the isotropic pairing Δ_0 (in units of Δ_1) and the chemical potential μ, shown in Fig. <ref>. We find that the non-trivial phase with helical hinge Majoranas persists, over a decreasing range of Δ_0, until the chemical potential reaches the bulk conduction band. However, this is also the scale at which we expect our effective model to no longer accurately represent the topological bands of realistic FeSCs.
            
            
            We have also checked the stability against onsite potential variations V(r) τ_z ⊗σ_0 ⊗ s_0, taking < V(r) > = 0 and < V(r) V(r') > = W^2 δ_rr'. In a real-space calculation using the kernel polynomial method <cit.>, we find the hinge states persist for small W (say, half the bulk gap), but defer a full investigation of disorder effects to future work.
            
            
             Conclusion and Discussion - Motivated by experimental results on the band topology of several FeSCs, we have constructed an explicit model of higher-order topological superconductivity supporting helical Majorana states at the hinges between different facets, depending on their orientations. The main ingredients are the band inversion at the Z point and s_± pairing. Unlike some related constructions for 2D higher-order TSC in heterostructures <cit.>, our proposal applies intrinsically for a single FeSC sample, avoiding all the complications of an interface.
            
            Some practical observations are in order. First, the helical hinge Majoranas can be probed by the same STM techniques already used to identify vortex bound states <cit.>; i.e., the surfaces should be locally gapped everywhere away from the hinges, while the hinge LDOS should be finite and approximately constant at low energies, indicating massless dispersing modes localized to the hinges. In a terraced sample, one expects the two pairs of narrowly separated Majoranas that occur at a terrace edge to hybridize. A terrace edge could still host a massive localized mode below the surface gap, but a more detailed modeling would be required to answer this question fully. Finally, the experimental identification of s_± pairing remains controversial. While vortex bound MZMs are agnostic to whether the pairing is nodeless or s_±, the appearence of hinge states requires the latter. Therefore, the experimental observation of hinge Majoranas would also be a simultaneous demonstration of s_± pairing.
            
            In addition to more realistic future numerical modeling of specific FeSCs to search for this physics, we also foresee some interesting generalizations of our theoretical construction. In our model, and in the 2D models of Refs. <cit.>, a first-order TI is proximitized by a nodal pairing function resulting in fully gapped but spatially anisotropic superconductivity; a sign change of the gap between different surfaces creates a domain wall that binds helical Majoranas. We conjecture that a second-order 3D TI with s_± pairing could realize, under the right conditions, a third-order gapped TSC with corner-localized, 0D Majorana modes.  Acknowledgement - R.-X.Z is indebted to Fan Zhang, Fengcheng Wu, Sheng-Jie Huang and Lun-Hui Hu for helpful discussions. R.-X.Z is supported by a JQI Postdoctoral Fellowship. W.S.C. is supported by LPS-MPO-CMTC. This work is also additionally supported by Microsoft Q.
            
            
            
            apsrev4-1
            
             §.§  Supplemental Material for “Helical Hinge Majoranas in Iron-Based Superconductors"
            
            
            § TOPOLOGICAL CHARACTERIZATION OF HELICAL HINGE MAJORANAS
            
            
            In this appendix, we discuss the topological characterization of helical hinge Majoranas in our model using the concept of Wannier bands. Consider a slab geometry where x̂ and ŷ directions are periodic while ẑ direction is open, the corresponding slab Hamiltonian H^(z)(k_x,k_y) gives rise to a gapped (001) surface dispersion, as shown in Fig. <ref> (b) in the main text. The Wannier center P^(z)_x(k_y) at k_y can be calculated by constructing the Wilson loop operator of all occupied BdG bands of H^(z)(k_x,k_y) along k_x. 
            
            As shown in Fig. <ref> (a), the evolution of Wannier centers along k_y forms a set of Wannier bands with interesting features: (i) Wannier bands from bulk states are gapped; (ii) Wannier bands from (001) surface states (for both top and bottom surfaces) are gapless. In particular, the “helical" surface Wannier band of P^(z)_x(k_y) characterizes a ℤ_2 pump of BdG quasi-particles along y direction. Similar ℤ_2 pump physics has been observed in 2D TRI topological insulators/superconductors and accounts for their heliacl edge physics. Therefore, the helical surface Wannier band is the topological origin of the helical hinge Majorana that circulates the top/bottom surface.
            
            On the other hand, we also plot the Wannier bands of P^(x)_z(k_y) and P^(x)_y(k_z) in Fig. <ref> (b) and (c) with open boundary conditions along x̂: (i) P^(x)_z(k_y) has the same in-gap helical surface Wannier band spectrum as that of P^(z)_x(k_y); (ii) P^(x)_y(k_z) is gapped. Therefore, there is no non-trivial BdG quasi-particle pumping process along ẑ direction, which implies the absence of any in-gap Majorana physics on the edge between (100) and (010) surfaces. This is consistent with our analytical analysis and the numerical results in Fig. <ref> (b). 
            
            However, it should be emphasized that the nontrivial ℤ_2 pump (helical surface Wannier bands) should not be interpreted as the topological invariant of any single isolated surface system. This is because the effective surface Hamiltonian in Eq. <ref> is topologically trivial for any surface Σ. To prove the above argument numerically, we use iterative Green function method to calculate the boundary Green function G^(z)(ω, k_x,k_y) of the (001) surface with a semi-infinite slab geometry along ẑ direction. Unlike the slab Hamiltonian method, the boundary Green function is only concerned with the (001) surface states and does not know anything about the side surfaces. Following Ref. <cit.>, we define an effective boundary Hamiltonian H_Green^(z)(k_x,k_y) as the inverse of G^(z)(ω, k_x,k_y) at zero frequency ω=0. We further denote P̃^(z)_x(k_y) as the Wannier center of H_Green^(z)(k_x,k_y). As shown in Fig. <ref> (d), we find the Wannier bands of P̃^(z)_x(k_y) is gapped, which is in contrast to the gapless Wannier bands of H^(z)(k_x,k_y) in Fig. <ref> (a). Similar gapped Wannier bands are also observed for P̃^(x)_z(k_y) and P̃^(x)_y(k_z). This confirms the topological triviality of the effective surface Hamiltonians.
