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
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
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
Two dimensional BCS-BEC crossover ✷❉ s❝❛tt❡r✐♥❣ ❛❧✇❛②s ❛❧❧♦✇s ❛ ❜♦✉♥❞ st❛t❡ ❛♥❞ ✐s ❡♥❡r❣② ❞❡♣❡♥❞❡♥t✱ � 2 4 π f ( q ) = + i π , ε B = � 1 / a 2 2D q 2 � ma 2 ln 2d ◆♦ ✉♥✐t❛r② r❡❣✐♠❡ ❜✉t ✐♥t❡r❛❝t✐♦♥s ❝❛♥ ❜❡ ❝❤❛♥❣❡❞ ❢r♦♠ t❤❡ ❇❊❈ ✲ ❇❈❙ s✐❞❡ t❤r♦✉❣❤ η = ln ( k F a 2D ) = − 1 2 ln ( 2 E F /ε B ) ❇❈❙ 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
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
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
2D breathing mode anomaly ❯s✐♥❣ t❤❡ ✷❉ ❡q✉❛t✐♦♥ ♦❢ st❛t❡ ✇❡ ❝❛♥ ❡①♣❧♦r❡ t❤❡ ❜r❡❛t❤✐♥❣ ♠♦❞❡ ❛♥♦♠❛❧②✿ ❉❡❧t❛ ❢✉♥❝t✐♦♥ V 2D ( r − r ′ ) = g 2 D δ ( 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❝❛❧✐♥❣✿ g 2 D → log( k F a 2 D ) ■♥❝❧✉❞✐♥❣ ❛ ❤❛r♠♦♥✐❝ tr❛♣✱ H trap = 1 2 m ω 2 r 2 ✱ ❜r❡❛❦s t❤❡ s❝❛❧❡ ✐♥✈❛r✐❛♥❝❡✱ r → λ r , H trap → λ 2 H trap ❍♦✇❡✈❡r t❤❡r❡ ✐s ❛ ❤✐❞❞❡♥ SO ( 2 , 1 ) s②♠♠❡tr② ❚❤✐s s②♠♠❡tr② ❝❛♥ ❡①❝✐t❡ ❛ ❜r❡❛t❤✐♥❣ ♠♦❞❡✱ ω B = 2 ω → 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
Realisation 0.20 T 0 polytropic fit T 0 BEC 0.15 T 0 BCS T 0.42 T F V ogt et al. 30 0.10 0 0.05 0.00 0.05 10 5 0 5 10 ln k F a 2 D Brendan Mulkerin (Theoretical Condensed Matter Physics Swinburne University ) Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition 8 / 20
���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� 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 − V trap ( r ) ✱ � µ � µ , P ( r ) λ 2 � � , df p ( x ) n ( r ) λ 2 = f n = f p = f n ( x ) k B T k B T k B T dx � ∇ V trap � ∂ P S ( 2 ) = 1 � � � � ∇ V trap � � � ω 2 ρ 0 u − ( ∇ ρ 0 · u ) ( ∇ · u ) 2 · u + 2 ρ 0 · u ( ∇ · u ) − ρ 0 d r ∂ρ 2 M M ¯ s ◆♦ 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
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 − V trap ( r ) ✱ � µ � µ , P ( r ) λ 2 � � , df p ( x ) n ( r ) λ 2 = f n = f p = f n ( x ) k B T k B T k B T dx � ∇ V trap � ∂ P S ( 2 ) = 1 � � � � ∇ V trap � � � ω 2 ρ 0 u − ( ∇ ρ 0 · u ) ( ∇ · u ) 2 · u + 2 ρ 0 · u ( ∇ · u ) − ρ 0 d r ∂ρ 2 M M ¯ s ���� ���� ���� ���� ���� ���� ���� ���� - ���� - ���� - ���� - ���� - ���� - ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ◆♦ 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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