Higher-order topological superconductors in ${\mathcal P}$-, ${\mathcal T}$-odd quadrupolar Dirac materials
Bitan Roy
cond-mat.mes-hall cond-mat.str-el cond-mat.supr-con




bitan.roy@lehigh.edu
Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA




The presence or absence of certain symmetries in the normal state (NS) also determines the symmetry of the Cooper pairs. Here we show that parity (𝒫) and time-reversal (𝒯) odd Dirac insulators (trivial or topological) or metals, sustain a local or intra-unit cell pairing that supports corner (in d=2) or hinge (in d=3) modes of Majorana fermions and stands as a higher-order topological superconductor (HOTSC), when the NS additionally breaks discrete four-fold (C_4) symmetry. Although these outcomes does not rely on the existence of a Fermi surface, around it (when the system is doped) the HOTSC takes the form of a mixed parity, 𝒯-odd (due to the lack of 𝒫 and 𝒯 in the NS, respectively) p+id pairing, where the p(d)-wave component stems from the Dirac nature of quasiparticles (lack of C_4 symmetry) in the NS. Thus, when strained, magnetically ordered Dirac materials, such as doped magnetic topological insulators (MnBi_2Te_4), can harbor HOTSCs, while the absence of an external strain should be conducive for the axionic p+is pairing.     


Higher-order topological superconductors in 𝒫-, 𝒯-odd quadrupolar Dirac materials
    Bitan Roy
    December 30, 2023
=================================================================================





Introduction. The symmetry of the normal state (NS) plays an important role in understanding the nature of the Cooper wavefunctions inside the paired states at low temperatures <cit.>. For example, (1) the lack of the inversion symmetry in the NS mixes the even- and odd-parity pairings <cit.>, and (2) Dirac materials composed of linearly dispersing massive or massless quasiparticles in the NS, described by the angular momentum ℓ=1 harmonics, harbor a plethora of exotic p-wave pairings <cit.>, analogous to the ones in the B- and polar phases of ^3He <cit.>. Here we investigate the role of time-reversal (𝒯), inversion (𝒫), as well as discrete crystalline four-fold (C_4) symmetry breaking on the paired states in two- and three-dimensional Dirac materials (insulator or metal) and come to the following conclusions.    


In Dirac materials 𝒫 and 𝒯 can be simultaneously broken by a Dirac mass <cit.>. Here we show that both two- and three-dimensional Dirac systems allow one local or inter-unit cell or momentum independent pairing that anticommutes with such a Dirac mass [Tables <ref> and <ref>]. In addition, when it breaks the C_4 symmetry (hereafter referred as the Wilson-Dirac (WD) mass) the boundary modes (with codimension d_c=1) of such paired states suffer dimensional reduction, producing zero energy corner and hinge modes (d_c=2) of neutral Majorana fermions, respectively in two and three dimensions [Figs. <ref>, <ref> and <ref>]. The paired states then represent a higher-order topological superconductor (HOTSC), a topic of immense current interest <cit.>. By contrast, other topological p-wave pairings, which commute with the WD mass, continue to represent first-order TSCs.     


We primarily support these outcomes by numerically diagonalizing the effective single-particle Hamiltonian for various local pairings in trivial Dirac insulators (devoid of any boundary modes) and in the absence of a Fermi surface (thus involving both intra- and inter-band couplings). Therefore, emergent topology can solely be attributed to the Bogoliubov-de Gennes (BdG) quasiparticles <cit.>. In the presence of a Fermi surface (conforming to weak coupling pairing) the intraband component of a HOTSC mimics p+id pairing. Respectively, the parity mixing and 𝒯-breaking arise from the lack of 𝒫 and 𝒯 symmetries, whereas the d-wave component roots into the C_4 symmetry breaking in the NS, as its restoration converts the paired state into a trivial p+is pairing. Therefore, antiferromagnetic topological insulators <cit.>, such as  MnBi_2Te_4 <cit.>, can be the ideal platform to search for HOTSCs, when they are doped (yielding a pairing conducive Fermi surface) and strained (lifting the crystalline symmetry).     

















Two-dimensions. The Hamiltonian for 2D quadrupolar massive Dirac fermions is H^ 2D_ Dir=H_0 + H_1 + H_2 (θ), where
[4]

    H_0    =    t [ Γ_331 S_1 + Γ_302 S_2 ], 
    H_1= Γ_303[ m + t_0 ∑^2_j=1 C_j ], 
    
    H_2 (θ)    =   Δ  g( k)  [ Γ_011cosθ + Γ_021sinθ].

Here S_j=sin(k_j a), C_j=cos(k_j a) with k_js as the components of spatial momenta, a is the lattice spacing, and θ is an arbitrary parameter <cit.>. Eight-dimensional Hermitian matrices Γ_λνρ=η_λσ_ντ_ρ, where λ, ν, ρ=0,⋯, 3, and the Pauli matrices {τ_ν}, {σ_ν} and {η_ν} operator on the parity, spin and particle-hole indices, respectively. The Nambu spinor basis is Ψ^⊤_ Nam ( k)= [ Ψ_ k, σ_2 τ_0 ( Ψ^†_- k )^⊤], where  Ψ^⊤_ k= [ c^+_ k,↑, c^-_ k, ↑, c^+_ k,↓, c^-_ k,↓], and c^τ_ k,σ is fermion annihilation operator with parity τ=±, spin projection σ=↑,↓ and momentum k. When Δ=0, H^ 2D_ Dir describes a quantum spin Hall insulator and a trivial or normal insulator (NI) for |m/t_0|<2 and |m/t_0|>2, respectively. Here we set m/t_0=3. 



Notice that H_2(θ) is odd under 𝒯:  k→ - k and 𝒯=Γ_020𝒦, where 𝒦 is the complex conjugation, and 𝒫:  r→ - r and 𝒫Ψ_ Nam( k) 𝒫^-1 = Γ_003Ψ_ Nam(- k). In addition,  H_2(θ) breaks C_4 symmetry, if g ( k)=C_1-C_2. Then it represents a WD mass, as { H_2(θ), H_1+H_2 }=0, and the excitations are massive quadrupolar Dirac fermions.  


Now we address topology of the paired states in this system, described by the single-particle Hamiltonian 
[4]

    H^ 2D_ pair   =   Δ_1 Γ_j22 + Δ_2 Γ_j32 + Δ_3 Γ_j12 +Δ_4 Γ_j00
       +   Δ_5 Γ_j01 + Δ_6 Γ_j03,

where j=1 or 2, reflecting the U(1) gauge freedom in the choice of superconducting phase, and Δ_l's are the pairing amplitudes. We choose j=1. Two s-wave pairings (Δ_4,Δ_6) are topologically trivial. By contrast, Δ_1 and Δ_3 pairings fully anticommute with the Dirac kinetic energy H_0 and represent fully gapped topological superconductor when |Δ_j|> |m-2 t_0| and Δ=0. They support one-dimensional (1D) edge modes [Figs. <ref> and <ref>]. The remaining two pairings (Δ_2 and Δ_5) break rotational symmetry. Respectively, they produce two Dirac points along the k_x and k_y directions, and support Fermi arcs along the x [Fig. <ref>] and y [Fig. <ref>] directions.         















The effect of the WD mass on the pairings and boundary modes can be anticipated from its (anti)commutation relations with the pairing matrices [Table <ref>]. Here we consider θ=0 and θ=π/2 only. The Δ_3 [Δ_1] pairing anticommutes with H_2(0) [H_2(π/2)]. As H_2(θ) changes sign under four-fold rotation, it acts as a domain wall mass (see Ref. <cit.>) for the one-dimensional counter propagating edge modes of Δ_1 (Δ_3) pairing when θ=0(π/2). Consequently, the boundary modes of Δ_3 pairing undergoes a dimensional reduction, yielding four corner localized Majorana zero modes with d_c=2 when θ=0 [Fig. <ref>]. And we realize a second-order TSC. By contrast, Δ_1, Δ_2 and Δ_5 pairings commute with H_2(0). So, their boundary modes are mildly affected [Figs. <ref>, <ref> and <ref>], and they continue to be first-order TSCs. Similarly, for θ=π/2, only the Δ_1 pairing represents a HOTSC and supports four corner localized zero modes [Fig. <ref>]. 



Based on this observation, we propose a general principle of realizing HOTSCs. If an effective single-particle Hamiltonian (H_ FO) describes a first-order TSC, supporting boundary modes of d_c=1, then the addition of an appropriate discrete-symmetry breaking WD mass (H_ WD) can covert the paired state into a HOTSC, when { H_ FO, H_ WD}=0. Next we show that this mechanism is operative in d=3, yielding 1D hinge modes of d_c=2. 




The Hamiltonian for a collection of three-dimensional quadrupolar massive Dirac fermions is H^ 3D_ Dir=H_0+ H_1+H_2, where H_2 = Γ_020Δ  g( k) is the WD mass, 
[4]

    H_0 = t ∑^3_j=1Γ_31j S_j,  
    H_1= Γ_330[m- t_0 ∑^3_j=1 C_j ],

and Γ_ρλν=η_ρτ_λσ_ν. The spinor basis is Ψ^⊤_ Nam ( k)=[ Ψ_ k, τ_0 σ_2 ( Ψ^†_- k )^⊤], with Ψ^⊤_ k =[c^+_ k,↑, c^+_ k,↓, c^-_ k,↑, c^-_ k,↓]. While H_0 gives rise to massless Dirac fermions, H_1 stands as the symmetry preserving Dirac mass. Under 𝒯: k→ - k and Ψ_ Nam( k) →Γ_002Ψ_ Nam(- k), and the corresponding antiunitary operator is 𝒯=Γ_002𝒦. Under 𝒫:  r→ - r and 𝒫Ψ_ Nam( k) 𝒫^-1 =Γ_330Ψ_ Nam(- k). The WD mass H_2 breaks both 𝒫 and 𝒯 symmetries, but preserves the conjugate 𝒫𝒯 symmetry, and in addition also breaks the discrete C_4 symmetry if g( k)=C_1-C_2.   















For Δ=0, the above model supports a strong (weak) topological insulator for 1<m/t_0<3 (-1<m/t_0<1), and NIs for m/t_0>3 and m/t_0<-1 <cit.>. Here we choose m/t_0=4. So the Dirac insulator is always trivial, and we solely unveil the topology of the paired states.



The single-particle Hamiltonian for local pairings is 
[4]

    H^ 3D_ pair= Δ_ sΓ_j00 + Δ_ psΓ_j10 + Δ_0 Γ_j30 + ∑^3_i=1Δ_i Γ_j2i,
  
where j=1 or 2. In the Dirac language Δ_ s transforms as a scalar, while the odd-parity Δ_ ps transforms as a pseudoscalar (PS). On the other hand, Δ_0 and Δ_i transform as the temporal and three spatial components (i=1,2,3) of vector pairing, respectively <cit.>. The (anti)commutation relations of all pairings with various components of H^ 3D_ Dir are shown in Table <ref>, which determines their topology. First we set Δ=0.















Two s-wave pairings Δ_ s and Δ_ 0 are topologically trivial. The PS pairing is a fully gapped class DIII TSC, supporting gapless Majorana modes on six surfaces of a cubic system [Fig. <ref>], when |Δ_ ps|>|m-3t_0| <cit.>. The spatial components of vector pairing break the rotational symmetry and represent nematic superconductors <cit.>. The Δ_i pairing supports two Dirac points along the k_i direction <cit.>, and Fermi arc surface states (with d_c=1) along the ith direction in the real space [Figs. <ref>- <ref>]. 
     















Now we switch on the WD mass H_2. It breaks the C_4 symmetry and changes sign four times in the xy plane for any z. The PS pairing anticommutes with H_2 [Table <ref>], which then acts as a domain wall mass for the surface Majorana fermions, yielding 1D hinge modes (with d_c=2) along the z direction [Fig. <ref>]. The PS pairing then represents a second-order TSC. For g( k)=C_3-C_1 and C_2-C_3 the hinge modes are aligned along the y and x directions, respectively. By contrast, all vector pairings commute with H_2, and thus the Fermi arcs are mildly affected [Figs. <ref>- <ref>]. These outcomes are in agreement with the general principle of realizing HOTSC.


Finally, we anchor these findings by projecting the HOTSCs onto a Fermi surface when the chemical doping (here measured from the bottom of the conduction band) μ>0. We return to two-dimensional Dirac system and consider its low-energy model for θ=0
[4]

    H^ 2D, low_ Dir=v [ Γ_331 k_1 + Γ_302 k _2 ] + Γ_303 M + Γ_011Δ̅ (k^2_1-k^2_2),

obtained by expanding H^ 2D_ Dir around the Γ=(0,0) point, where M=|m - 2t_0|, v = t a bears the dimension of the Fermi velocity, and Δ̅=-Δ a^2/2. The corresponding band diagonalizing matrix is U= U( k, Δ̅) ⊕ U( k, -Δ̅), where 
[4]

    U( k, Δ̅)=( 
    [    -λ_-/√(2 λλ_-)                 0     λ_+/√(2 λλ_+)                 0;   v k_+/√(2 λλ_-) -g̅( k)/√(2 λλ_-)   v k_+/√(2 λλ_+) -g̅( k)/√(2 λλ_+);                 0     λ_-/√(2 λλ_-)                 0    -λ_+/√(2 λλ_+);  g̅( k)/√(2 λλ_-)   v k_-/√(2 λλ_-)  g̅( k)/√(2 λλ_+)   v k_+/√(2 λλ_+) ]),

with k_± = k_1 ± i k_2, λ=√(v^2 k_+k_- + M^2 + g̅( k)^2), λ_± =λ± M and g̅( k)=Δ̅(k^2_1-k^2_2). In the presence of Δ_3 pairing, the single-particle Hamiltonian in the vicinity of the Fermi surface for large mass (M ≫ v k, Δ) reads 
[4]

    H^ 2D_ FS = ξ_ kΓ_30 + Δ_3/M{ v [ k_x Γ_11-k_y Γ_12] 
    + g̅( k) Γ_20},
 
where ξ_ k=v^2 k^2/(2 M)- μ, and Γ_ρν=η_ρσ_ν. The Pauli matrices {η_ν} ({σ_ν}) operate on the particle-hole (pseudospin) indices. For Δ̅=0, H^ 2D_ FS describes a 𝒯-symmetric fully gapped p-wave pairing. The p-wave BdG quasiparticles solely arises from the Dirac nature of NS fermions. If we set g̅( k)=g_0 (constant), then H^ 2D_ FS describes a trivial p+is pairing. The breaking of 𝒯 and mixing of parity inside the paired state stem from the lack of 𝒯 and 𝒫 symmetries in the NS, respectively. By contrast, when g̅( k)=Δ̅(k^2_1-k^2_2), H^ 2D_ FS describes a HOT p+id pairing, and the appearance of the d-wave component can now solely be attributed to the lack of C_4 symmetry in the NS. Therefore, the absence of each symmetry in the NS plays a crucial role in determining the symmetry and topology of the paired states <cit.>. Note that H^ 2D_ FS assumes the form of Dirac fermions (p-wave pairing), subject to an inverted-band regular Dirac mass (yielding a Fermi surface when μ>0) and 𝒫- and 𝒯-odd, C_4 symmetry breaking WD mass (d-wave pairing); together giving rise to a HOTSC and four corner localized Majorana zero modes [Fig. <ref>]. The same argument is applicable for the Δ_1 pairing when θ=π/2.    


Similarly, the single-particle Hamiltonian for the three-dimensional PS pairing around the Fermi surface takes the form H^ 3D_ FS=ξ_ kΓ_30 + H^ 3D_p+id, with M=|m-3t_0| and  
[4]

    H^ 3D_p+id = Δ_ ps/M{ v [ k_x Γ_11-k_y Γ_12 + k_z Γ_13] 
    + g̅( k) Γ_20}.
 
It takes the form of the 𝒫- and 𝒯-odd p+id (p+is) pairing for g̅( k)=Δ̅(k^2_1-k^2_2) (constant), standing as HOT (axionic <cit.>) superconductor. Therefore, around the Fermi surface the PS pairing is described by Dirac fermions (p-wave pairing) in the presence of both regular (ξ_ k) and 𝒫- and 𝒯-odd, C_4 symmetry breaking (d-wave pairing) Dirac masses, which gives birth to 1D hinge modes [Fig. <ref>] <cit.>. Imprint of each discrete symmetry in the paired state is identical to that for the 2D HOTSC.  



Discussion. To summarize, here we demonstrate a general principle of realizing HOTSCs. In both d=2 and 3, a first-order TSC can be converted into a HOTSC in the presence of a discrete symmetry breaking, 𝒫- and 𝒯-odd anticommuting WD mass [Tables <ref>, <ref>], respectively yielding corner and hinge modes of codimension d_c=2 [Figs. <ref>, <ref>, <ref>]. Therefore, magnetically doped or ordered (due to strong electronic correlations) Dirac insulators (trivial or topological) can accommodate HOTSCs, when they supports a Fermi surface (conducive for weak-coupling pairings) and are subject to external strain (breaking discrete rotational symmetry). Hopefully, the present discussion will stimulate a search for HOTSCs (both theoretically and experimentally) in strained but doped magnetic topological insulators <cit.>, such as MnBi_2Te_4 <cit.>. In the future, it will also be worth investigating possible HOTSCs in candidate HOT insulators, such as Bi <cit.> and strained Luttinger materials <cit.>.        





Acknowledgments. B.R. was supported by the startup grant from Lehigh University. 




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