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Jan 27, 2023 213 likes •445 views

Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition r r rt s ttr Pss

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Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition

❇r❡♥❞❛♥ ▼✉❧❦❡r✐♥

❚❤❡♦r❡t✐❝❛❧ ❈♦♥❞❡♥s❡❞ ▼❛tt❡r P❤②s✐❝s ❙✇✐♥❜✉r♥❡ ❯♥✐✈❡rs✐t②

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 1 / 20

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SLIDE 2

Todays talk

  • ❚❤❛♥❦s ❆✴Pr♦❢s✳ ❳✐❛✲❥✐ ▲✐✉✱ ❈❤r✐s ❱❛❧❡✱ P❛✉❧ ❉②❦❡✱ ❏✐❛ ❲❛♥❣✱ ▲✐❛♥②✐ ❍❡✱

❛♥❞ ❍✉✐ ❍✉ ❢♦r ♦✉r ❝♦❧❧❛❜♦r❛t✐✈❡ ✇♦r❦

  • ❚❤❛♥❦s t♦ ♦✉r P❤❉ st✉❞❡♥ts ❯♠❜❡rt♦ ❚♦♥✐♦❧♦✱ ❙❡❜❛st✐❛♥ ❙❝❤❛✛❡r✱

❳✐❛♦✲▲♦♥❣ ❈❤❡♥✱ ❛♥❞ ❈❤r✐st♦♣❤❡r ❍♦❡❣❛❛r❞ ❢♦r ❛❧❧ t❤❡✐r ❤❛r❞ ✇♦r❦

  • ❇❊❈✲❇❈❙ ❝r♦ss♦✈❡r ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s
  • ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ st❛t❡✿ ❇r❡❛t❤✐♥❣ ♠♦❞❡ ✐♥ ✷❉ ❣❛s❡s
  • ❇❑❚ tr❛♥s✐t✐♦♥

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 2 / 20

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Why strongly interacting Fermi gases?

❙tr♦♥❣❧② ✐♥t❡r❛❝t✐♥❣ ❋❡r♠✐ ❣❛s❡s ✇✐t❤ ❜❛❧❛♥❝❡❞ ♣♦♣✉❧❛t✐♦♥s ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡

  • ❙tr♦♥❣❧② ❝♦rr❡❧❛t❡❞ ❋❡r♠✐ s②st❡♠s ❛r❡ ❛ ♣❧❛②❣r♦✉♥❞ ❢♦r ♠❛♥②✲❜♦❞② ♣❤②s✐❝s
  • ❚❤❡② ❛r❡ st❛❜❧❡ ♦♥ ❧♦♥❣ t✐♠❡s❝❛❧❡s ❛♥❞ ❢♦r str♦♥❣ ✐♥t❡r❛❝t✐♦♥s

Figure: ❳✐❛✲❏✐ ▲✐✉ P❤②s✐❝s ❘❡♣♦rts ✺✷✹ ✭✷✮✱ ✸✼✲✽✸✳

  • ❚❤❡② ♣❧❛② ❛ ❢✉♥❞❛♠❡♥t❛❧ r♦❧❡ ✐♥ ✈❡r② ❞✐✛❡r❡♥t ❛r❡❛s ♦r ♣❤②s✐❝s
  • ▲♦✇❡r ❞✐♠❡♥s✐♦♥s ✐♥❝r❡❛s❡ t❤❡ ✢✉❝t✉❛t✐♦♥s✱ q✉❛♥t✉♠ ❡✛❡❝ts ❛r❡ ❧❛r❣❡r

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 3 / 20

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Two dimensional BCS-BEC crossover

✷❉ s❝❛tt❡r✐♥❣ ❛❧✇❛②s ❛❧❧♦✇s ❛ ❜♦✉♥❞ st❛t❡ ❛♥❞ ✐s ❡♥❡r❣② ❞❡♣❡♥❞❡♥t✱ f(q) = 4π ln

  • 1/a2

2Dq2

+ iπ , εB = 2 ma2

2d

◆♦ ✉♥✐t❛r② r❡❣✐♠❡ ❜✉t ✐♥t❡r❛❝t✐♦♥s ❝❛♥ ❜❡ ❝❤❛♥❣❡❞ ❢r♦♠ t❤❡ ❇❊❈ ✲ ❇❈❙ s✐❞❡ t❤r♦✉❣❤ η = ln (kFa2D) = − 1

2 ln (2EF/εB)

❇❈❙ s✐❞❡✿ ✇❡❛❦❧② ✐♥t❡r❛❝t✐♥❣ ♣❛✐rs ❇❊❈ s✐❞❡✿ ❚✐❣❤t❧② ❜♦✉♥❞ ❜♦s♦♥✐❝ ♠♦❧❡❝✉❧❡s

Fluctuations in 2D are larger:

❚❤✐s ♣r❡✈❡♥ts ❧♦♥❣✲r❛♥❣❡ ♦r❞❡r ❬▼❡r♠✐♥✲❲❛❣♥❡r✲❍♦❤❡♥❜❡r❣❪

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 4 / 20

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Experimental progress

❊①♣❡r✐♠❡♥t❛❧✐sts ❝❛♥ ❞✐r❡❝t❧② ♠❡❛s✉r❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ✭❊✳❖✳❙✮ ❛♥❞ t❤❡r♠♦❞②♥❛♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❣❛s

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 5 / 20

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Equation of state

❚❤❡ ❡q✉❛t✐♦♥ ♦❢ st❛t❡ s❤♦✇s t❤❡ ♥♦♥✲tr✐✈✐❛❧ ❊✳❖✳❙✳ ❡✈❡♥ ✐♥ t❤❡ ♥♦r♠❛❧ st❛t❡

Figure: ❋❡♥❡❝❤ ❡t ❛❧ P❘▲ ✶✶✻ ✵✹✺✸✵✷ ✭✷✵✶✻✮ ✭❚♦♣✮ ❛♥❞ ❇♦❡tt❡❝❤❡r ❡t ❛❧ P❘▲ ✶✶✻ ✵✹✺✸✵✸ ✭✷✵✶✻✮ ✭❇♦tt♦♠✮✳

❲❡ ❝❛♥ ✉s❡ t❤❡ ❊✳❖✳❙✳ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ❛♥♦♠❛❧②

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 6 / 20

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2D breathing mode anomaly

❯s✐♥❣ t❤❡ ✷❉ ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ✇❡ ❝❛♥ ❡①♣❧♦r❡ t❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ❛♥♦♠❛❧②✿ ❉❡❧t❛ ❢✉♥❝t✐♦♥ V2D(r − r′) = g2Dδ(r − r′) ✐♥t❡r❛❝t✐♦♥ ✐s t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✐♥t❡r❛❝t✐♦♥ ✐♥ ❛ t✇♦✲❝♦♠♣♦♥❡♥t ❋❡r♠✐ ❣❛s s❝❛❧❡s ❛s λ−2 ✐♥ ✷❉✱ r❡❣✉❧❛r✐s❛t✐♦♥ ❞❡str♦②s t❤✐s s❝❛❧✐♥❣✿ g2D → log(kFa2D) ■♥❝❧✉❞✐♥❣ ❛ ❤❛r♠♦♥✐❝ tr❛♣✱ Htrap = 1

2mω2r2✱ ❜r❡❛❦s t❤❡ s❝❛❧❡ ✐♥✈❛r✐❛♥❝❡✱

r → λr, Htrap → λ2Htrap ❍♦✇❡✈❡r t❤❡r❡ ✐s ❛ ❤✐❞❞❡♥ SO(2, 1) s②♠♠❡tr② ❚❤✐s s②♠♠❡tr② ❝❛♥ ❡①❝✐t❡ ❛ ❜r❡❛t❤✐♥❣ ♠♦❞❡✱ ωB = 2ω → t❤❡ q✉❛♥t✉♠ ❛♥♦♠❛❧② ✇✐❧❧ ❜r❡❛❦ t❤✐s ❤✐❞❞❡♥ s②♠♠❡tr②

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 7 / 20

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Realisation

T 0 polytropic fit T 0 BEC T 0 BCS T 0.42TF V

  • gt et al. 30

10 5 5 10 0.05 0.00 0.05 0.10 0.15 0.20 ln kFa2D

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 8 / 20

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Theoretical results

❯s✐♥❣ t❤❡ ❤②❞r♦❞②♥❛♠✐❝ ❢♦r♠❛❧✐s♠ ❛♥❞ ❡q✉❛t✐♦♥ ♦❢ st❛t❡✿ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ❛♥❞ t❤❡ ❧♦❝❛❧ ❞❡♥s✐t② ❛♣♣r♦①✐♠❛t✐♦♥✱ µ(r) = µ − Vtrap(r)✱ n(r)λ2 = fn µ kBT

  • , P(r)λ2

kBT = fp µ kBT

  • , dfp(x)

dx = fn(x) S(2) = 1 2

  • dr
  • ω2ρ0u − (∇ρ0 · u)

∇Vtrap M · u

  • + 2
  • ρ0

∇Vtrap M · u

  • (∇ · u) − ρ0

∂P ∂ρ

  • ¯

s

(∇ · u)2

  • ◆♦ s✐❣♥✐✜❝❛♥t r❡s✉❧t ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ → ✇❡ ❞♦ s❡❡ ❞❛♠♣✐♥❣

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 9 / 20

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Theoretical results

❯s✐♥❣ t❤❡ ❤②❞r♦❞②♥❛♠✐❝ ❢♦r♠❛❧✐s♠ ❛♥❞ ❡q✉❛t✐♦♥ ♦❢ st❛t❡✿ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ❛♥❞ t❤❡ ❧♦❝❛❧ ❞❡♥s✐t② ❛♣♣r♦①✐♠❛t✐♦♥✱ µ(r) = µ − Vtrap(r)✱ n(r)λ2 = fn µ kBT

  • , P(r)λ2

kBT = fp µ kBT

  • , dfp(x)

dx = fn(x) S(2) = 1 2

  • dr
  • ω2ρ0u − (∇ρ0 · u)

∇Vtrap M · u

  • + 2
  • ρ0

∇Vtrap M · u

  • (∇ · u) − ρ0

∂P ∂ρ

  • ¯

s

(∇ · u)2

  • ◆♦ s✐❣♥✐✜❝❛♥t r❡s✉❧t ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ → ✇❡ ❞♦ s❡❡ ❞❛♠♣✐♥❣

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 9 / 20

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SLIDE 11

Theoretical results

❯s✐♥❣ t❤❡ ❤②❞r♦❞②♥❛♠✐❝ ❢♦r♠❛❧✐s♠ ❛♥❞ ❡q✉❛t✐♦♥ ♦❢ st❛t❡✿ ❊q✉❛t✐♦♥ ♦❢ st❛t❡ ❛♥❞ t❤❡ ❧♦❝❛❧ ❞❡♥s✐t② ❛♣♣r♦①✐♠❛t✐♦♥✱ µ(r) = µ − Vtrap(r)✱ n(r)λ2 = fn µ kBT

  • , P(r)λ2

kBT = fp µ kBT

  • , dfp(x)

dx = fn(x) S(2) = 1 2

  • dr
  • ω2ρ0u − (∇ρ0 · u)

∇Vtrap M · u

  • + 2
  • ρ0

∇Vtrap M · u

  • (∇ · u) − ρ0

∂P ∂ρ

  • ¯

s

(∇ · u)2

  • ❚❤❡r❡ ✐s ❛ str❛♥❣❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ✐♥ t❤❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡ r❡❣✐♠❡

Figure: ❚❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ❛♥♦♠❛❧② ❢♦r T/TF = 0.8

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 10 / 20

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SLIDE 12

Path integral 1/3

❙tr♦♥❣❧② ✐♥t❡r❛❝t✐♥❣ ❋❡r♠✐ ❣❛s❡s ✇✐t❤ ❜❛❧❛♥❝❡❞ ♣♦♣✉❧❛t✐♦♥s✱ ✇❡❧❧ st✉❞✐❡❞✱ ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡✳ ❚♦ ❜❡❣✐♥ ✇✐t❤ ✇❡ ❤❛✈❡ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ♣♦t❡♥t✐❛❧ ❢♦✉♥❞ t❤r♦✉❣❤ t❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ Ω = −β−1 ln Z, ✇❤❡r❡ t❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❛❝t✐♦♥ ❛r❡ ❣✐✈❡♥ ❜②✱ Z =

  • D
  • ψ, ¯

ψ

  • e−S[ψ, ¯

ψ] ❛♥❞ S =

β dτ

  • dr
  • σ

¯ ψσ(x)∂τψσ(x) + H

  • ,

❛♥❞ t❤❡ ❛❝t✐♦♥ ❞❡✜♥❡❞ ❜② ❛ ❍❛♠✐❧t♦♥✐❛♥ ✐s S = β dτ

  • dr
  • σ

¯ ψσ(x)∂τψσ(x) + H

  • ,

❉❡❝♦✉♣❧❡ t❤r♦✉❣❤ t❤❡ ❍✉❜❜❛r❞✲❙tr❛t♦♥♦✈✐❝❤ tr❛♥s❢♦r♠❛t✐♦♥✿ Seff [∆, ∆∗] =

  • dx

|∆(x)|2 U0 − Tr ln [−G−1]

  • .

❚❤✐s ✐s tr✉❡ ❢♦r ❣❡♥❡r❛❧ ❞✐♠❡♥s✐♦♥✱ ✇❤❡r❡

  • dx =
  • ddrdτ ❛♥❞ U0 ✐s r❡❣✉❧❛r✐s❡❞ ❛♣♣r♦♣r✐❛t❡❧②✳

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 11 / 20

slide-13
SLIDE 13

Path integral 2/3

❙tr♦♥❣❧② ✐♥t❡r❛❝t✐♥❣ ❋❡r♠✐ ❣❛s❡s ✇✐t❤ ❜❛❧❛♥❝❡❞ ♣♦♣✉❧❛t✐♦♥s✱ ✇❡❧❧ st✉❞✐❡❞✱ ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡✳ ❊①♣❛♥❞ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ♣♦t❡♥t✐❛❧ ❜② t❛❦✐♥❣ t❤❡ ❇♦s❡ ✜❡❧❞ ∆(r, t) ❛❜♦✉t ✐ts s❛❞❞❧❡ ♣♦✐♥t ∆0✱ ∆(r, t) = ∆0 + ϕ(r, t), ❚❤❡ ❛❝t✐♦♥ ✐s ❡①♣❛♥❞❡❞ ✐♥ ♦r❞❡r ♦❢ ∆0 ❛♥❞ t❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ♣♦t❡♥t✐❛❧ ✐s Ω = ΩMF + ΩGF. ❊①t❡♥❞ t♦ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ❝♦♥❞❡♥s❡❞ ♣❛✐rs ✢♦✇ ✇✐t❤ ❛ ✇❛✈❡✈❡❝t♦r Q✿ ∆eiQ·r ■♥ t❤✐s ❝❛s❡✱ t❤❡ ♠❡❛♥✲✜❡❧❞ t❤❡r♠♦❞②♥❛♠✐❝ ♣♦t❡♥t✐❛❧ ✐s ❣✐✈❡♥ ❜② ΩMF (Q) = ∆2 U +

  • k
  • ˜

ξk − Ek − 2 β ln

  • 1 + eβE+

k

  • ,

❛♥❞ t❤❡ ♠❡❛♥✲✜❡❧❞ ❣❛♣ ❡q✉❛t✐♦♥✱

  • k

  1 − 2f

  • E+

k

  • 2Ek

− 1 2k2/M + εB   = 0.

❍✳ ❍✉✱ ❳✳✲❏✳ ▲✐✉✱ P✳ ❉✳ ❉r✉♠♠♦♥❞✱ ❊✉r♦♣❤②s✳ ▲❡tt✳ ✼✹✱ ✺✼✹ ✭✷✵✵✻✮ ❘✳ ❇✳ ❉✐❡♥❡r✱ ❘✳ ❙❡♥s❛r♠❛✱ ❛♥❞ ▼✳ ❘❛♥❞❡r✐❛✱ P❤②s✳ ❘❡✈✳ ❆ ✼✼✱ ✵✷✸✻✷✻ ✭✷✵✵✽✮ ❚❛②❧♦r✱ ❆✳ ●r✐✣♥✱ ◆✳ ❋✉❦✉s❤✐♠❛✱ ❛♥❞ ❨✳ ❖❤❛s❤✐✱ P❤②s✳ ❘❡✈✳ ❆ ✼✹✱ ✵✻✸✻✷✻ ✭✷✵✵✻✮

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 12 / 20

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SLIDE 14

Path integral 3/3

❚❤❡ t❤❡r♠♦❞②♥❛♠✐❝ ♣♦t❡♥t✐❛❧ ❢♦r ❣❛✉ss✐❛♥ ♣❛✐r ✢✉❝t✉❛t✐♦♥s ✭●P❋✮ ✐s✿ ΩGF (Q) = kBT

  • Q≡(q,iνl)

S (Q) eiνl0+, S (Q) = 1 2 ln

  • 1 −

M2

12 (Q)

M11 (Q) M11 (−Q)

  • + ln M11 (Q) ,

❙❡❡ ♦✉r r❡❝❡♥t ♣❛♣❡r P❘❆ ✾✻ ✵✺✸✻✵✽ ✭✷✵✶✼✮ ❢♦r t❤❡ ❧♦♥❣ ❞❡✜♥✐t✐♦♥s ♦❢ M11 ❛♥❞ M12 ❚❤✐s ✐s ❞✐✣❝✉❧t t♦ s♦❧✈❡ ❜❡❧♦✇ Tc✱ t❤❡ ▼❛ts✉❜❛r❛ s✉♠♠❛t✐♦♥ ✐s tr✐❝❦②✱ ✐♥st❡❛❞ ✇❡ ✉s❡✿ 1 β

  • |l|>l0

Sη (q, iνl) = − 1 π +∞

−∞

dω ImSη (q, ω + iγ) eβω + 1 ✇❤❡r❡ Sη(q, iνl) ≡ S(q, iνl)eiνlη ❛♥❞ γ = (2l0 + 1)π/β ❢♦r ❛r❜✐tr❛r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r l0

▲✳ ❍❡✱ ❍✳ ▲ü✱ ●✳ ❈❛♦✱ ❍✳ ❍✉✱ ❛♥❞ ❳✳✲❏✳ ▲✐✉✱ P❤②s✳❘❡✈✳ ❆ ✾✷✱ ✵✷✸✻✷✵ ✭✷✵✶✺✮ ❏✳ ❚❡♠♣❡r❡✱ ❙✳ ◆✳ ❑❧✐♠✐♥✱ ❏✳ ❚✳ ❉❡✈r❡❡s❡✱ ❛♥❞ ❱✳ ❱✳ ▼♦s❤❝❤❛❧❦♦✈✱ P❤②s✳ ❘❡✈✳ ❇ ✼✼✱ ✶✸✹✺✵✷ ✭✷✵✵✽✮

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 13 / 20

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SLIDE 15

Two-dimensional Fermi gas

❚♦ ✐❧❧✉str❛t❡ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ♦✉r ❢✉❧❧ tr❡❛t♠❡♥t ♦❢ t❤❡ ●P❋ ✇❡ ✜♥❞ t❤❡ ❊✳❖✳❙✳

Figure: ❈♦♠♣❛r✐♥❣ t❤❡ r❡s✉❧ts t♦ ❡①♣❡r✐♠❡♥t ❛♥❞ ▲✉tt✐♥❣❡r✲❲❛r❞ T✲♠❛tr✐① t❤❡♦r② P❘❆ ✾✻ ✵✺✸✻✵✽ ✭✷✵✶✼✮

❚❤❡ ●P❋ ✉♥❞❡rst✐♠❛t❡s t❤❡ ♣r❡ss✉r❡ ❜✉t ❤❛s ❛ s✉♣❡r✢✉✐❞ ♦r❞❡r ♣❛r❛♠❡t❡r

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 14 / 20

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SLIDE 16

Equation of state and superfluid density in 2D

❚❤❡ s✉♣❡r✢✉✐❞ ❞❡♥s✐t② ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ❛❞❞✐♥❣ ❛ t✇✐st✱ ❣✐✈✐♥❣ t❤❡ ❞❡♥s✐t② t♦ ❜❡ ns = 4m 2 ∂2Ω(Q) ∂Q2

  • Q=0

❲❡ ❝❛♥ ♥♦✇ ❧♦♦❦ ❛t t❤❡ str♦♥❣❧② ❝♦rr❡❧❛t❡❞ ❇❈❙ s✐❞❡ ✐♥ ✷❉

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0

∆MF ∆GPF ns,GPF ∆/εF and ns/n T/TF

Figure: ❚❤❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ❛♥❞ s✉♣❡r✢✉✐❞ ❢r❛❝t✐♦♥ ✐♥ ✷❉ ❢♦r εB/εF = 0.1

❚❤❡r❡ ✐s ❛ r❡❣✐♦♥ ✇❤❡r❡ ns = 0 ❛♥❞ ∆GPF > 0

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 15 / 20

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SLIDE 17

Superfluid density in 2D

❚❤❡ ❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss ❝r✐t❡r✐♦♥ ❞❡✜♥❡s t❤❡ ❇❑❚ tr❛♥s✐t✐♦♥ t❡♠♣❡r❛t✉r❡✿ kBTBKT = π 2 2 4mns(T)

Figure: ❚❤❡ ♦r❞❡r ♣❛r❛♠❡t❡r ❛♥❞ s✉♣❡r✢✉✐❞ ❢r❛❝t✐♦♥ ✐♥ ✷❉ ❢♦r εB/εF = 0.1 ❛♥❞ ♦t❤❡r t❤❡♦r❡t✐❝❛❧ ❛tt❡♠♣ts

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 16 / 20

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SLIDE 18

Superfluid density in 2D

  • 5
5 10 15 20 0.0 0.5 1.0 1.5 2.0 2.5 5 10 15 20 0.0 0.5 1.0 s n n normal n s (GPF) 8/ln[1+exp( )] n s (MF) n s /n B = 0.5 superfluid

Figure: ❚❤❡ s✉♣❡r✢✉✐❞ ❢r❛❝t✐♦♥ ♥♦r♠❛❧✐s❡❞ ❜② ❛♥ ✐❞❡❛❧ ❣❛s

normal ( c + B /2) B superfluid

❈r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡r❛❝t✐♦♥ str❡♥❣t❤ ❢♦r ❙✇✐♥❜✉r♥❡ ✭sq✉❛r❡s✮ ❛♥❞ ❍❡✐❞❡❧❜❡r❣ ✭❝✐r❝❧❡s✮

❲❡ ❝❛♥ ❞❡✜♥❡ ❛ ❝r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ♠❡❛s✉r❡❞ ❞✐r❡❝t❧② ✐♥ ❡①♣❡r✐♠❡♥t ❈r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ ❝r✐t✐❝❛❧ r❛❞✐✉s ✿

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 17 / 20

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SLIDE 19

Superfluid density in 2D

  • 5
5 10 15 20 0.0 0.5 1.0 1.5 2.0 2.5 5 10 15 20 0.0 0.5 1.0 s n n normal n s (GPF) 8/ln[1+exp( )] n s (MF) n s /n B = 0.5 superfluid

Figure: ❚❤❡ s✉♣❡r✢✉✐❞ ❢r❛❝t✐♦♥ ♥♦r♠❛❧✐s❡❞ ❜② ❛♥ ✐❞❡❛❧ ❣❛s

normal ( c + B /2) B superfluid

Figure: ❈r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡r❛❝t✐♦♥ str❡♥❣t❤ ❢♦r ❙✇✐♥❜✉r♥❡ ✭sq✉❛r❡s✮ ❛♥❞ ❍❡✐❞❡❧❜❡r❣ ✭❝✐r❝❧❡s✮

❲❡ ❝❛♥ ❞❡✜♥❡ ❛ ❝r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ♠❡❛s✉r❡❞ ❞✐r❡❝t❧② ✐♥ ❡①♣❡r✐♠❡♥t ❈r✐t✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ → ❝r✐t✐❝❛❧ r❛❞✐✉s ✿ µc = µ − V(rc)

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 17 / 20

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SLIDE 20

Unambiguously find the BKT transition

❆♥ ✉♥❛♠❜✐❣✉♦✉s ♠❡t❤♦❞ ✜♥❞ t❤❡ ❇❑❚ tr❛♥s✐t✐♦♥✿ ✉s❡ t❤❡ ▲❉❆ µg = µ − V(r)

  • 5
5 10 15 20 0.0 0.5 1.0 1.5 0.0 0.1 0.2 0.3 0.4
  • 1.58
  • 1.57
  • 1.56
normal v c /v T B = 0.5 superfluid (Q)/| | (Q/k T ) 2 Q c

Figure: ❚❤❡ ❝r✐t✐❝❛❧ ✈❡❧♦❝✐t② vc = Q/(2m) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❞✐♠❡♥s✐♦♥❧❡ss ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 18 / 20

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SLIDE 21

Summary

  • ❚❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ✐s ❞❛♠♣❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❡♠♣❡r❛t✉r❡ ❛♥❞ ✐s

s✐❣♥✐✜❝❛♥t ✐♥ t❤❡ ❤✐❣❤ t❡♠♣❡r❛t✉r❡ r❡❣✐♠❡

  • ❲❡ ❤❛✈❡ ❡①♣❧✐❝✐t❧② ✐♥❝❧✉❞❡❞ ♣❛✐r✐♥❣ ✢✉❝t✉❛t✐♦♥s ✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡

s✉♣❡r✢✉✐❞ ❞❡♥s✐t②

  • ❚❤r♦✉❣❤ st✐r✐♥❣ t❤❡ ❣❛s ✇❡ ❝❛♥ ✜♥❞ ❛♥ ✉♥❛♠❜✐❣✉♦s ♠❡t❤♦❞ t♦ ♠❡❛s✉r❡

t❤❡ ❇❑❚ tr❛♥s✐t✐♦♥

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 19 / 20

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SLIDE 22

Thank you

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥ t♦❞❛②

Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 20 / 20