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▼❛❥♦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 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

Ψ =

• ✶

± ♣①+✐♣②

|♣|

• ❡✐♣·r

❊ = ±✈❢ |♣|

◮ T ✲s②♠♠❡tr② ❝♦♥♥❡❝ts st❛t❡s ✇✐t❤ ♦♣♣♦s✐t❡ ♣ ❛♥❞ s✳

▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ ✐♥ ❛ ✈♦rt❡① ❝♦r❡

m m

usual s-wave SC

spinless p-wave SC or s-wave on top

• f 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)τ③ + ∆(r)(τ① ❝♦s θ + τ② s✐♥ θ)

◮ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ❞❡ ●❡♥♥❡s s♣❡❝tr✉♠

❊♠ = ω✵♠, ω✵ ∼ ∆✷/❊❢ ,

◮ ♠ = ✵ ✐s ❛ ▼❛❥♦r❛♥❛ st❛t❡ ✭❋✉✱ ❑❛♥❡✱ ✷✵✵✽✮

❙❡t✉♣ ❛♥❞ ●♦❛❧

Topological Insulator SC

h 2e

Tunneling probe

Strong disorder changes subgap spectrum significantly Majorana level is protected by symmetry and stays at E=0

✇❡ ✜♥❞

◮ ❛✈❡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))τ③ + ∆(r)(τ① ❝♦s θ + τ② 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))τ③ + ∆(r)(τ① ❝♦s θ + τ② 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 ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗]+❙θ[◗]

◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭τ✮ ❛♥❞

P❛rt✐❝❧❡✲❍♦❧❡ ✭σ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗✷ = ✶ ❛♥❞ ◗ = ❈◗❚❈ ❚ ✇✐t❤ ❈ = τ① σ① ✵ ✵ ✐σ②

• ❋❇

, t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙θ[◗]✱ ǫ = ❊ + ✐●tδ(r − r✵)/✹πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ③τ③✳ ♣❧❛❝❡❤♦❧❞❡r

❙✉♣❡rs②♠♠❡tr✐❝ s✐❣♠❛✲♠♦❞❡❧ ❛❝t✐♦♥

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗]+❙θ[◗]

◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭τ✮ ❛♥❞

P❛rt✐❝❧❡✲❍♦❧❡ ✭σ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗✷ = ✶ ❛♥❞ ◗ = ❈◗❚❈ ❚ ✇✐t❤ ❈ = τ① σ① ✵ ✵ ✐σ②

• ❋❇

,

◮ t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙θ[◗]✱

ǫ = ❊ + ✐●tδ(r − r✵)/✹πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ③τ③✳ ♣❧❛❝❡❤♦❧❞❡r

❙✉♣❡rs②♠♠❡tr✐❝ s✐❣♠❛✲♠♦❞❡❧ ❛❝t✐♦♥

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗]+❙θ[◗]

◮ ◗ ✐s ❛ ✽ × ✽ s✉♣❡r♠❛tr✐① ✐♥ ◆❛♠❜✉✲●♦r✬❦♦✈ ✭τ✮ ❛♥❞

P❛rt✐❝❧❡✲❍♦❧❡ ✭σ✮ s♣❛❝❡✱ ♦❜❡②✐♥❣ ◗✷ = ✶ ❛♥❞ ◗ = ❈◗❚❈ ❚ ✇✐t❤ ❈ = τ① σ① ✵ ✵ ✐σ②

• ❋❇

,

◮ t❤❡ ❉✐r❛❝ s♣❡❝tr✉♠ ♣r♦❞✉❝❡s ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠ ❙θ[◗]✱ ◮ ǫ = ❊ + ✐●tδ(r − r✵)/✹πν ✇✐t❤ ❊ ❜❡✐♥❣ t❤❡ ❡♥❡r❣② ❛♥❞ t❤❡

s❡❝♦♥❞ t❡r♠ ❞❡s❝r✐❜✐♥❣ t✉♥♥❡❧✐♥❣ t♦ t❤❡ ♣r♦❜❡❀ Λ = σ③τ③✳ ♣❧❛❝❡❤♦❧❞❡r

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗] + ❙θ[◗] ❈♦♥❞✐t✐♦♥s ◗✷ = ✶ ❛♥❞ ◗ = ◗ ❧❡❛❞ t♦ t❤❡ str✉❝t✉r❡✿ ◗ = ❱ −✶ ◗❋ ✵ ✵ ◗❇

• ❱

✇❤❡r❡ ❱ ❝♦♥t❛✐♥s ✽ ❣r❛ss♠❛♥ ✈❛r✐❛❜❧❡s✱ ❛♥❞ ◗❋,❇ ❛r❡ ♣❛r❛♠❡t❡r✐③❡❞ ❜② ✹ ❛♥❣❧❡s ❡❛❝❤✳ ◗❋,❇ = ◗❋,❇✳

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗] + ❙θ[◗]

x

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗] + ❙θ[◗]

x

variation at zero energy: Usadel equation

vortex term at origin

❙[◗] = πν ✽

• ❞✷ rstr [❉(∇◗)✷ + ✹(✐ǫΛ − ˆ

∆)◗] + ❙θ[◗]

U

structure of the symmetry class D, Change in means D-odd

manifold consists of two disjoint parts

❉❡♥s✐t② ♦❢ st❛t❡s ❛t ❧♦✇ ❡♥❡r❣✐❡s

ρ(r, ❊) = ν ✽ ❘❡

• ❉◗ ❙tr[ˆ

❦Λ◗(r)] ❡−❙[◗]. ❚✇♦ ♠❛♥✐❢♦❧❞ ♣❛rts ❝♦♥tr✐❜✉t❡ t♦ t❤✐s ✐♥t❡❣r❛❧ ✇✐t❤ ♦♣♣♦s✐t❡ s✐❣♥s✱ ✇❤✐❝❤ ❞✐st✐♥❣✉✐s❤❡s t❤❡ ❉✲♦❞❞ ✭❇✮ ❝❧❛ss ❢r♦♠ ❉✲❡✈❡♥✳ ❲❡ ✜♥❞ ❢♦r ❊ ≪ ❊❚❤ ρ(r, ❊) = ν ♥(r)❢ (❊/ω✵), ♥(r) = ❝♦s θ✶(r) = ❘✷ − r✷ ❘✷ + r✷ , ❢ (①) = γ π(①✷ + γ✷) + ✶ − s✐♥(✷π①) ✷π① . ✇❤❡r❡ γ = ●t♥(r✵)/✷π ≪ ✶ ✇✐t❤ r✵ ❜❡✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣r♦❜❡✱ ❛♥❞ t❤❡ ❧❡✈❡❧✲s♣❛❝✐♥❣ ω−✶

✵

= ✷ν

• ❞✷r ❝♦s θ✶(r) = ✷π(❧♦❣ ✹ − ✶)ν❘✷

❉❡♥s✐t② ♦❢ st❛t❡s ❛t ❧♦✇ ❡♥❡r❣✐❡s

ρ(r, ❊) = ν ✽ ❘❡

• ❉◗ ❙tr[ˆ

❦Λ◗(r)] ❡−❙[◗]. ❚✇♦ ♠❛♥✐❢♦❧❞ ♣❛rts ❝♦♥tr✐❜✉t❡ t♦ t❤✐s ✐♥t❡❣r❛❧ ✇✐t❤ ♦♣♣♦s✐t❡ s✐❣♥s✱ ✇❤✐❝❤ ❞✐st✐♥❣✉✐s❤❡s t❤❡ ❉✲♦❞❞ ✭❇✮ ❝❧❛ss ❢r♦♠ ❉✲❡✈❡♥✳ ❲❡ ✜♥❞ ❢♦r ❊ ≪ ❊❚❤ ρ(r, ❊) = ν ♥(r)❢ (❊/ω✵), ♥(r) = ❝♦s θ✶(r) = ❘✷ − r✷ ❘✷ + r✷ , ❢ (①) = γ π(①✷ + γ✷) + ✶ − s✐♥(✷π①) ✷π① . ✇❤❡r❡ γ = ●t♥(r✵)/✷π ≪ ✶ ✇✐t❤ r✵ ❜❡✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣r♦❜❡✱ ❛♥❞ t❤❡ ❧❡✈❡❧✲s♣❛❝✐♥❣ ω−✶

✵

= ✷ν

• ❞✷r ❝♦s θ✶(r) = ✷π(❧♦❣ ✹ − ✶)ν❘✷

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0

νD=10

E, ETh N(E), πνR

2

■♥t❡❣r❛t✐♥❣ ρ(r, ❊) ♦✈❡r ❞✷r ✇❡ ♦❜t❛✐♥ ◆(❊) = γ ✷π(❊ ✷/ω✵ + γ✷ω✵) + ✶ ✷ω✵

• ✶ − s✐♥(✷π❊/ω✵)

✷π❊/ω✵

• ❲❤❡♥ ●t = ✵ ✭♥♦ ♣r♦❜❡✮ t❤❡ ✜rst t❡r♠ ❜❡❝♦♠❡s δ(❊)/✷✱

❝❤❛r❛❝t❡r✐st✐❝ ❢♦r t❤❡ ❇✲❝❧❛ss ✭❉✳ ❆✳ ■✈❛♥♦✈✱ ✷✵✵✷✮✳

❚✉♥♥❡❧✐♥❣ ❝✉rr❡♥t

❚❤❡ ❝✉rr❡♥t ✐♥ ❛ t✉♥♥❡❧✐♥❣ ❡①♣❡r✐♠❡♥t ✐s ■ = ❡●t ❤ν

• ρ(❊, r✵)
• ❢ (❊ − ❡❱ ) − ❢ (❊)
• ❞❊

✇✐t❤ t❤❡ ❋❡r♠✐ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ❢ (❊)✳ ❆t ❡❱ ≪ ω✵ ✇❡ ❣❡t ❞■ ❞❱ =        ✷❡✷γ✷ ❤

• γ✷ + (❡❱ /ω✵)✷,

❚ ≪ γω✵, π❡✷γω✵ ✷❤❚ ❝♦s❤✷(❡❱ /✷❚) , γω✵ ≪ ❚ ≪ ω✵. ❆t s♠❛❧❧ ❚ t❤❡r❡ ✐s ❛ ▲♦r❡♥t③ ♣❡❛❦ ❛t ❊ = ✵ ✇✐t❤ ❛ ✇✐❞t❤ ∼ ●tω✵ ❛♥❞ ❛ ✉♥✐✈❡rs❛❧ ❤❡✐❣❤t ❡✷/π ♠❡❛♥✐♥❣ ♣❡r❢❡❝t ❆♥❞r❡❡✈ r❡✢❡❝t✐♦♥ ✭▲❛✇✱ ▲❡❡✱ ◆❣✱ ✷✵✵✾✮✳

1 1 eV Ω0 dIdV e2Π Gt T0 TΓ 1

◮ ❆❧❧ ❧❡✈❡❧s ♣r♦❞✉❝❡ ▲♦r❡♥t③✲s❤❛♣❡❞ r❡s♦♥❛♥❝❡s❀ ◮ ❍❡✐❣❤ts ❡q✉❛❧ ✷❡✷ ❤ ψ†Cψ ψ†ψ ✱ ✇✐❞t❤s ❡q✉❛❧ ●t ψ†ψ ν ❀ ◮ ❋❡r♠✐♦♥✐❝ ❧❡✈❡❧s ♣r♦❞✉❝❡ t✇♦ ♣❡❛❦s ❛t ±❊✱ ▼❛❥♦r❛♥❛ ❧❡✈❡❧

❝r❡❛t❡s ♦♥❡ ♣❡❛❦ ❛t ❊ = ✵✳ ❋♦r ❉✲❡✈❡♥ ■(✵) ≡ ✵✳

③❡r♦✲❜✐❛s ❝♦♥❞✉❝t❛♥❝❡ ❞✐♣ ✐♥ ❝❧❛ss ❉

Symmetric levels interfere destructively, at E=0 Andreev reflection is absent Andreev reflection probability At small voltages Andreev current is suppressed, dI/dV is not proportional to DoS any more DoS dI/dV E eV

◮ ❊①❛❝t❧② ❛t ❊ = ✵ ❝♦♥❞✉❝t❛♥❝❡ ✐s ❡✐t❤❡r ✵ ♦r ✷❡✷/✳ ❚❤✐s

❢♦❧❧♦✇s ❢r♦♠ ❇❞●✲s②♠♠❡tr② ❛♥❞ ✉♥✐t❛r✐t② ❢♦r ❛ s✐♥❣❧❡✲❝❤❛♥♥❡❧ ❙✲♠❛tr✐①✳

◮ ✐♥ ❉✲❝❧❛ss ✭♥♦ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥✮ ❞■/❞❱ = ✵✳ Pr♦❝❡ss❡s

✐♥✈♦❧✈✐♥❣ s②♠♠❡tr✐❝ ❆♥❞r❡❡✈ ❧❡✈❡❧s ±❊❥ ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ♦✉t

◮ ✐♥ ❇✲❝❧❛ss ✭♦♥❡ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥✮ ❞■/❞❱ = ✷❡✷/ ✇❤✐❝❤ ✐s

t❤❡ ▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ ❝♦♥tr✐❜✉t✐♦♥

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s ❬P❘❇✱ ✽✻✱ ✵✸✺✹✹✶ ✭✷✵✶✷✮❪

◮ ❚✉♥♥❡❧✐♥❣ ❝✉rr❡♥t ✐♥ ❛ ✈♦rt❡① ❝♦r❡ ♦♥ ❛ ❚■ s✉r❢❛❝❡ ❤❛s ❜❡❡♥

st✉❞✐❡❞ ✐♥ t❤❡ str♦♥❣ ❞✐s♦r❞❡r ❧✐♠✐t

◮ ❚❤❡ ❛✈❡r❛❣❡ ❉♦❙ ❤❛s ❜❡❡♥ ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛

t✉♥♥❡❧✐♥❣ ♣r♦❜❡

◮ ❚❤❡ ❝✉rr❡♥t ✐♥ ❛ t✉♥♥❡❧✐♥❣ ❡①♣❡r✐♠❡♥t ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❢♦r

❛r❜✐tr❛r② ❞✐s♦r❞❡r r❡❛❧✐③❛t✐♦♥s

◮ ❚❤❡ ❛♣♣r♦❛❝❤ ✐s ✉♥✐✈❡rs❛❧ ❛♥❞ ❛♣♣❧✐❡s ❡✳❣✳ t♦ ▼❛❥♦r❛♥❛

❢❡r♠✐♦♥s ✐♥ ♥❛♥♦✇✐r❡s ✇❤✐❝❤ ❛❧s♦ ❢❛❧❧ ✐♥t♦ t❤❡ ❇ − ❉ ❝❧❛ss ✭❬❇❛❣r❡ts✱ ❆❧t❧❛♥❞ ✷✵✶✷❪✮

◮ ❉♦❙ ❞♦❡s ♥♦t ❞✐r❡❝t❧② tr❛♥s❧❛t❡ ✐♥t♦ ❝♦♥❞✉❝t❛♥❝❡ ✕ ③❡r♦✲❜✐❛s

❝♦♥❞✉❝t❛♥❝❡ ✐s q✉❛♥t✐③❡❞✱ ❧❡❛❞✐♥❣ t♦ ❛ ❞✐♣ ✐♥ ❞■/❞❱ ❛t ❱ = ✵ ✐♥ ❝❧❛ss ❉

❉❡♥s✐t② ♦❢ st❛t❡s ❛t ❤✐❣❤ ❡♥❡r❣✐❡s

❆t ❊ ≫ ❊❚❤ ✇❡ ✉s❡ ❛ ♠❡❛♥✲✜❡❧❞ ❢♦r♠✉❧❛ ρ(r, ❊) = ν ❘❡ ❝♦s θ(r, ❊) ■t ②✐❡❧❞s✱ ✐♥ ♣❛rt✐❝✉❧❛r ◆ = πν❘✷

• ✶ − (✷ −

√ ✷)

• ❊❚❤

❊

• ,

♣❧♦tt❡❞ ✐♥ ❞❛s❤❡❞ ❜❧✉❡ ♦♥ t❤❡ ❧❡❢t ✜❣✉r❡✳ ❚❤❡ r❡❞ ❝✉r✈❡ ✐s ♥✉♠❡r✐❝❛❧✳

❚♦♣♦❧♦❣✐❝❛❧ t❡r♠

❋♦r ♦✉r ♣❛r❛♠❡t❡r✐③❛t✐♦♥✱ ✐t ❡①♣❧✐❝✐t❧② r❡❛❞s ❙θ[◗] = ✐ ✹

• ❞✷r
• s✐♥ θ❢
• ∇θ❢ × ∇φ❢
• + s✐♥ ❦❢
• ∇❦❢ × ∇χ❢
• ■t ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥✈❛r✐❛♥t❧② ❛s ❛ ❲❡ss✲❩✉♠✐♥♦✲❲✐tt❡♠ t❡r♠

❙θ[◗] = ✐ǫ❛❜❝ ✷✹π ✶

✵

❞t

• ❞✷r str
• Q−✶(∇❛Q)Q−✶(∇❜Q)Q−✶(∇❝Q)
• ✇❤❡r❡ ◗ ✐s ❡①t❡♥❞❡❞ ♦♥t♦ ❛ t❤✐r❞ ❞✐♠❡♥s✐♦♥ t✳ ❚❤❡ ✈❛r✐❛t✐♦♥

δ❙θ[◗] ♦♥❧② ❞❡♣❡♥❞s ♦♥ ◗ = Q|t=✶✳ ■❢ ◗✷ = ✶✱ t❤❡♥ δ❙θ[◗] ≡ ✵ ❡q✉❛❧s ❡①❛❝t❧② ③❡r♦✱ s♦ t❤❛t ❙θ[◗] = ❝♦♥st ♦✈❡r ❛♥② ❝♦♥♥❡❝t❡❞ ♣❛rt ♦❢ t❤❡ ♠❛♥✐❢♦❧❞✱ t❤✉s ♣❧❛②✐♥❣ t❤❡ r♦❧❡ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ t❡r♠✳

❚❆ =

• ✶ +

❊ ✷ − ❊ ✷

✵

✷❊Γ −✶ ❋♦r Γ ≪ ω✵ ✇❡ ❣❡t ❛t ③❡r♦ t❡♠♣❡r❛t✉r❡ ❞■ ❞❱ ∝ ρ(❡❱ )

∞

• ✵

❚❆❞❊✵ ✭✶✮ t❤✐s ✐s ❡❛s✐❧② ❣❡♥❡r❛❧✐③❡❞ t♦ ✜♥✐t❡ t❡♠♣❡r❛t✉r❡s

▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥

◮ s❡❧❢✲❝♦♥❥✉❣❛t❡ ✭❤❡r♠✐t✐❛♥✮ ♣❛rt✐❝❧❡

γ† = γ,

◮ ❡♥❡r❣② ❡q✉❛❧ t♦ ❡①❛❝t❧② ③❡r♦✱ s✐♥❝❡

[❍, γ†] = ❊γ† ❛♥❞ [❍, γ] = −❊γ

◮ ❤❛❧❢ ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ❢❡r♠✐♦♥

❝† = γ✶ + ✐γ✷ ❝ = γ✶ − ✐γ✷

◮ ✐❢ γ ✐s ♠❛❞❡ ♦✉t ♦❢ ❡❧❡❝tr♦♥✐❝ ♦♣❡r❛t♦rs ψ, ψ† t❤❡♥

γ = λ∗ψ† + λψ

▼❛❥♦r❛♥❛ ❢❡r♠✐♦♥ ❛♥❞ s✉♣❡r❝♦♥❞✉❝t✐✈✐t②

◮ ❡❧❡❝tr♦♥✲❤♦❧❡ ♠✐①✐♥❣ ✐s ♥❡❝❡ss❛r② ❢♦r γ✱ ❛♥❞ ✐s ♣r♦✈✐❞❡❞ ❜②

s✉♣❡r❝♦♥❞✉❝t✐✈✐t② ❍ = ❍✵ ∆ ∆∗ −Θ−✶❍✵Θ

• ✇✐t❤ t✐♠❡✲r❡✈❡rs❛❧ Θ = s②❑ ✭s② ❛❝ts ♦♥ s♣✐♥✱ ❑ ✐s ❝✳❝✳✮✱

◮ ❍ ❤❛s ❛ ❜✉✐❧t✲✐♥ C✲s②♠♠❡tr②

C❍C = −❍ C = τ②Θ ✐s ❝❤❛r❣❡ ❝♦♥❥✉❣❛t✐♦♥ ✭τ② ❛❝ts ✐♥ ◆❛♠❜✉ s♣❛❝❡✮✳ C✷ = ✶✱ ❡♥❡r❣② ❧❡✈❡❧s s♣❧✐t ✐♥t♦ −❊, ❊ ♣❛✐rs Ψ−❊ = CΨ❊ γ = Cγ

◮ ❇❞● ❞♦✉❜❧❡✲❝♦✉♥t✐♥❣ ♠❡❛♥s ❛ s✐♥❣❧❡ γ✲❡✐❣❡♥st❛t❡ ❝♦✉♥ts ❛s

❤❛❧❢ ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ❢❡r♠✐♦♥