▼❛❥♦r❛♥❛ st❛t❡ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❞✐s♦r❞❡r❡❞ ✸❉ t♦♣♦❧♦❣✐❝❛❧ ✐♥s✉❧❛t♦r P✳ ❆✳ ■♦s❡❧❡✈✐❝❤✱ P✳ ▼✳ ❖str♦✈s❦②✱ ▼✳ ❱✳ ❋❡✐❣❡❧✬♠❛♥ ✶✸ ❙❡♣t❡♠❜❡r ✷✵✶✷ ❚❤❡ ❙❝✐❡♥❝❡ ♦❢ ◆❛♥♦str✉❝t✉r❡s✿ ◆❡✇ ❋r♦♥t✐❡rs ✐♥ t❤❡ P❤②s✐❝s ♦❢ ◗✉❛♥t✉♠ ❉♦ts ❈❤❡r♥♦❣♦❧♦✈❦❛
❖❞❞ ❛♥❞ ❡✈❡♥ ❝❧❛ss❡s ♦❢ ❍ ❛♥❞ t❤❡ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ Majorana level protected by 0 0 BdG symmetry ◮ ❇♦❣♦❧②✉❜♦✈✲❞❡ ●❡♥♥❡s ❤❛♠✐❧t♦♥✐❛♥ ❤❛s ❜✉✐❧t✲✐♥ C ✲s②♠♠❡tr② C ❍ C = − ❍ ✱ ❜r❡❛❦✐♥❣ ❧❡✈❡❧s ✐♥t♦ ❝♦♥❥✉❣❛t❡ ± ❊ ♣❛✐rs ❛♥❞✱ ♣♦ss✐❜❧②✱ ❛ s❡❧❢✲❝♦♥❥✉❣❛t❡ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ ◮ s②♠♠❡tr✐❡s ❧✐❦❡ t✐♠❡✲r❡✈❡rs❛❧ T ♦r s♣✐♥ r♦t❛t✐♦♥ s②♠♠❡tr② ❣✉❛r❛♥t❡❡ ❍ t♦ ❜❡ ❡✈❡♥✳ ◮ ❍ ✇✐t❤ ♦♥❧② C ✲s②♠♠❡tr② ❜❡❧♦♥❣s t♦ t❤❡ ❉✲❝❧❛ss ♦❢ s②♠♠❡tr② ✭♣r♦✈✐❞❡❞ C ✷ = ✶✮✳
❙✉r❢❛❝❡ st❛t❡s ♦❢ ❛ ✸❉ ❚■ ✇✐t❤ T ✲s②♠♠❡tr② ◮ ❙✐♥❣❧❡ ❉✐r❛❝ ❝♦♥❡ ✭ ❇✐ ✷ ❙❡ ✸ , ❇✐ ✷ ❚❡ ✸ ❡t❝✮ ❍ ✵ = ✈ ❢ ( s · ♣ ) , ◮ ❙♣✐♥✲♣♦❧❛r✐③❡❞ ❡❧❡❝tr♦♥ st❛t❡s � � ✶ ❡ ✐ ♣ · r Ψ = ❊ = ± ✈ ❢ | ♣ | ± ♣ ① + ✐♣ ② | ♣ | ◮ T ✲s②♠♠❡tr② ❝♦♥♥❡❝ts st❛t❡s ✇✐t❤ ♦♣♣♦s✐t❡ ♣ ❛♥❞ s ✳
▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ ✐♥ ❛ ✈♦rt❡① ❝♦r❡ m m spinless usual p-wave SC or s-wave SC s-wave on top of TI surface ◮ s✲✇❛✈❡ s✉♣❡r❝♦♥❞✉❝t✐✈✐t② ✐s ✐♥❞✉❝❡❞ ❜② ♣r♦①✐♠✐t② ❡✛❡❝t ◮ ✈♦rt❡① ❜r❡❛❦s T ✲s②♠♠❡tr② ❛♥❞ ♣r♦❞✉❝❡s ❛♥ ♦❞❞ ❤❛♠✐❧t♦♥✐❛♥✿ ❍ = ( ✈ ❢ s · ♣ − µ ) τ ③ + ∆( r )( τ ① ❝♦s θ + τ ② s✐♥ θ ) ◮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❞❡ ●❡♥♥❡s s♣❡❝tr✉♠ ω ✵ ∼ ∆ ✷ / ❊ ❢ , ❊ ♠ = ω ✵ ♠ , ◮ ♠ = ✵ ✐s ❛ ▼❛❥♦r❛♥❛ st❛t❡ ✭❋✉✱ ❑❛♥❡✱ ✷✵✵✽✮
❙❡t✉♣ ❛♥❞ ●♦❛❧ h Tunneling probe 2e Strong disorder changes SC subgap spectrum significantly Majorana level is protected by symmetry and stays at E=0 Topological Insulator ✇❡ ✜♥❞ ◮ ❛✈❡r❛❣❡ ❧♦❝❛❧ ❞❡♥s✐t② ♦❢ st❛t❡s ✭❉♦❙✮ ρ ( r , ❊ ) ✱ ◮ ■ ( ❱ , ❚ ) ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛ t✉♥♥❡❧✐♥❣ ♣r♦❜❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❚■ s✉r❢❛❝❡ ❢♦r ❛♥② ♣❛rt✐❝✉❧❛r ❞✐s♦r❞❡r r❡❛❧✐③❛t✐♦♥ ❛♥❞ t❤❡ ❛✈❡r❛❣❡✳ ◮ s♣❡❝✐❛❧ ❜❡❤❛✈✐♦✉r ❛t ③❡r♦✲❜✐❛s✿ ✷ ❡ ✷ / ❤ ♣❡❛❦ ❢♦r ❇✲❝❧❛ss ✭❡♥s❡♠❜❧❡ ✇✐t❤ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥✮ ❛♥❞ ❞✐♣ t♦ ③❡r♦ ❢♦r ❉✲❝❧❛ss ✭♥♦ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥✮
❍❛♠✐❧t♦♥✐❛♥ ◮ ❚❤❡ ❞✐s♦r❞❡r❡❞ s✉♣❡r❝♦♥❞✉❝t✐♥❣ ❚■ s✉r❢❛❝❡ ✐s ❞❡s❝r✐❜❡❞ ❜② ❍ = ( ✈ ❢ s · ♣ − µ + ❱ ( r )) τ ③ + ∆( r )( τ ① ❝♦s θ + τ ② s✐♥ θ ) ✇✐t❤ ✇❤✐t❡✲♥♦✐s❡ ❞✐s♦r❞❡r ♣♦t❡♥t✐❛❧ ❱ ( r ) ✿ � ❱ ( r ) ❱ ( r ′ ) � = δ ( r − r ′ ) πντ ❲❡ ❝♦♥s✐❞❡r t❤❡ r❡❣✐♠❡ ❊ ❚❤ = ❉ / ❘ ✷ ≪ ∆ ≪ ❊ ❢ ✱ ✐♠♣❧②✐♥❣ � ✵ , r < ❘ , ∆( r ) = ∆ , r ≥ ❘ .
❍❛♠✐❧t♦♥✐❛♥ ◮ ❚❤❡ ❞✐s♦r❞❡r❡❞ s✉♣❡r❝♦♥❞✉❝t✐♥❣ ❚■ s✉r❢❛❝❡ ✐s ❞❡s❝r✐❜❡❞ ❜② ❍ = ( ✈ ❢ s · ♣ − µ + ❱ ( r )) τ ③ + ∆( r )( τ ① ❝♦s θ + τ ② s✐♥ θ ) ✇✐t❤ ✇❤✐t❡✲♥♦✐s❡ ❞✐s♦r❞❡r ♣♦t❡♥t✐❛❧ ❱ ( r ) ✿ � ❱ ( r ) ❱ ( r ′ ) � = δ ( r − r ′ ) πντ ◮ ❲❡ ❝♦♥s✐❞❡r t❤❡ r❡❣✐♠❡ ❊ ❚❤ = ❉ / ❘ ✷ ≪ ∆ ≪ ❊ ❢ ✱ ✐♠♣❧②✐♥❣ � ✵ , r < ❘ , ∆( r ) = ∆ , r ≥ ❘ .
❙✉♣❡rs②♠♠❡tr✐❝ s✐❣♠❛✲♠♦❞❡❧ ❛❝t✐♦♥ ❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ]+ ❙ θ [ ◗ ] ✽ ◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭ τ ✮ ❛♥❞ P❛rt✐❝❧❡✲❍♦❧❡ ✭ σ ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗ ✷ = ✶ ❛♥❞ � σ ① � ✵ ◗ = ❈◗ ❚ ❈ ❚ ✇✐t❤ ❈ = τ ① , ✵ ✐ σ ② ❋❇ t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙ θ [ ◗ ] ✱ ǫ = ❊ + ✐● t δ ( r − r ✵ ) / ✹ πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ ③ τ ③ ✳ ♣❧❛❝❡❤♦❧❞❡r
❙✉♣❡rs②♠♠❡tr✐❝ s✐❣♠❛✲♠♦❞❡❧ ❛❝t✐♦♥ ❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ]+ ❙ θ [ ◗ ] ✽ ◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭ τ ✮ ❛♥❞ P❛rt✐❝❧❡✲❍♦❧❡ ✭ σ ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗ ✷ = ✶ ❛♥❞ � σ ① � ✵ ◗ = ❈◗ ❚ ❈ ❚ ✇✐t❤ ❈ = τ ① , ✵ ✐ σ ② ❋❇ ◮ t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙ θ [ ◗ ] ✱ ǫ = ❊ + ✐● t δ ( r − r ✵ ) / ✹ πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ ③ τ ③ ✳ ♣❧❛❝❡❤♦❧❞❡r
❙✉♣❡rs②♠♠❡tr✐❝ s✐❣♠❛✲♠♦❞❡❧ ❛❝t✐♦♥ ❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ]+ ❙ θ [ ◗ ] ✽ ◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭ τ ✮ ❛♥❞ P❛rt✐❝❧❡✲❍♦❧❡ ✭ σ ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗ ✷ = ✶ ❛♥❞ � σ ① � ✵ ◗ = ❈◗ ❚ ❈ ❚ ✇✐t❤ ❈ = τ ① , ✵ ✐ σ ② ❋❇ ◮ t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙ θ [ ◗ ] ✱ ◮ ǫ = ❊ + ✐● t δ ( r − r ✵ ) / ✹ πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ ③ τ ③ ✳ ♣❧❛❝❡❤♦❧❞❡r
❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ] + ❙ θ [ ◗ ] ✽ ❈♦♥❞✐t✐♦♥s ◗ ✷ = ✶ ❛♥❞ ◗ = ◗ ❧❡❛❞ t♦ t❤❡ str✉❝t✉r❡✿ � ◗ ❋ � ✵ ◗ = ❱ − ✶ ❱ ✵ ◗ ❇ ✇❤❡r❡ ❱ ❝♦♥t❛✐♥s ✽ ❣r❛ss♠❛♥ ✈❛r✐❛❜❧❡s✱ ❛♥❞ ◗ ❋ , ❇ ❛r❡ ♣❛r❛♠❡t❡r✐③❡❞ ❜② ✹ ❛♥❣❧❡s ❡❛❝❤✳ ◗ ❋ , ❇ = ◗ ❋ , ❇ ✳
❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ] + ❙ θ [ ◗ ] ✽ x
❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ] + ❙ θ [ ◗ ] ✽ variation at zero energy: x Usadel equation vortex term at origin
❙ [ ◗ ] = πν � ❞ ✷ r str [ ❉ ( ∇ ◗ ) ✷ + ✹ ( ✐ ǫ Λ − ˆ ∆) ◗ ] + ❙ θ [ ◗ ] ✽ structure of the symmetry class D, U Change in means D- odd manifold consists of two disjoint parts
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