Functional renormalization for ultracold quantum gases

Transcription

1 Functional renormalization for ultracold quantum gases Stefan Floerchinger (Heidelberg) S. Floerchinger and C. Wetterich, Phys. Rev. A 77, (2008); S. Floerchinger and C. Wetterich, arxiv: ERG 2008 Heidelberg, Jun. 2008

2 Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ λ(φ φ) 2} x

3 Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ λ(φ φ) 2} x natural units: = k B = 2M = 1

4 Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ λ(φ φ) 2} x natural units: = k B = 2M = 1 µ is chemical potential

5 Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ λ(φ φ) 2} x natural units: = k B = 2M = 1 µ is chemical potential homogeneous space and Matsubara formalism = 1/T x 0 dτ d d x

6 Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ λ(φ φ) 2} x natural units: = k B = 2M = 1 µ is chemical potential homogeneous space and Matsubara formalism = 1/T x 0 consider here d = 3 and d = 2 dτ d d x

7 Symmetries of the microscopic action

8 Symmetries of the microscopic action rotation and translation

9 Symmetries of the microscopic action rotation and translation global U(1) symmetry φ e iα φ, φ e iα φ broken spontaneously in the superfluid phase

10 Symmetries of the microscopic action rotation and translation global U(1) symmetry φ e iα φ, φ e iα φ broken spontaneously in the superfluid phase Galilean invariance with analytic continuation to real time φ(t, x) e i( q2 t q x) φ(t, x 2 qt) φ (t, x) e i( q2 t q x) φ (t, x 2 qt) also broken spontaneously in the superfluid phase.

11 Symmetries of the microscopic action rotation and translation global U(1) symmetry φ e iα φ, φ e iα φ broken spontaneously in the superfluid phase Galilean invariance with analytic continuation to real time φ(t, x) e i( q2 t q x) φ(t, x 2 qt) φ (t, x) e i( q2 t q x) φ (t, x 2 qt) also broken spontaneously in the superfluid phase. with µ = µ(t) semilocal gauge invariance φ(t, x) e iα(t) φ(t, x), φ (t, x) e iα(t) φ (t, x) µ µ + t α.

12 Grand canonical partition function. Z = e βφg = Tre β(h µn) = Dχ e S[χ]

13 Grand canonical partition function. Z = e βφg = Tre β(h µn) = Dχ e S[χ] from effective action Γ[φ] βφ G = Γ[φ eq ] with δγ[φ] δφ = 0 φ=φeq

14 Grand canonical partition function. Z = e βφg = Tre β(h µn) = Dχ e S[χ] from effective action Γ[φ] βφ G = Γ[φ eq ] with δγ[φ] δφ = 0 φ=φeq use flow equation to obtain effective action (Wetterich (1993)) k Γ k = 1 2 Tr(Γ(2) k + R k ) 1 k R k

15 Grand canonical partition function. Z = e βφg = Tre β(h µn) = Dχ e S[χ] from effective action Γ[φ] βφ G = Γ[φ eq ] with δγ[φ] δφ = 0 φ=φeq use flow equation to obtain effective action (Wetterich (1993)) k Γ k = 1 2 Tr(Γ(2) k + R k ) 1 k R k finite size system described by average action Γ kph instead of Γ = Γ k=0

16 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ)

17 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints

18 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency

19 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ

20 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ propagator depends on (µ µ 0 )

21 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ propagator depends on (µ µ 0 ) similar truncations were studied at T = 0 by:

22 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ propagator depends on (µ µ 0 ) similar truncations were studied at T = 0 by: C. Castellani, C. Di Castro, F. Pistolesi, G. C. Strinati, Phys. Rev. Lett. 78, 1612 (1997); Phys. Rev. B 69, (2004).

23 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ propagator depends on (µ µ 0 ) similar truncations were studied at T = 0 by: C. Castellani, C. Di Castro, F. Pistolesi, G. C. Strinati, Phys. Rev. Lett. 78, 1612 (1997); Phys. Rev. B 69, (2004). C. Wetterich, Phys. Rev. B 77, (2008).

24 Truncation of average action Γ k = { φ ( S τ V τ 2 ) φ x } +2V (µ µ 0 )φ ( τ ) φ + U(ρ,µ) motivated by more systematic derivative expansion and analysis of symmetry constraints terms linear and quadratic in Matsubara frequency implicit wavefunction renormalization φ φ = φ Ā φ propagator depends on (µ µ 0 ) similar truncations were studied at T = 0 by: C. Castellani, C. Di Castro, F. Pistolesi, G. C. Strinati, Phys. Rev. Lett. 78, 1612 (1997); Phys. Rev. B 69, (2004). C. Wetterich, Phys. Rev. B 77, (2008). N. Dupuis, K. Sengupta, Europhys. Lett. 80, (2007).

25 Effective Potential, Cutoff U(ρ,µ) = U(ρ 0,µ 0 ) n k (µ µ 0 ) + ( m 2 + α(µ µ 0 ) ) (ρ ρ 0 ) (λ + β(µ µ 0)) (ρ ρ 0 ) 2.

26 Effective Potential, Cutoff U(ρ,µ) = U(ρ 0,µ 0 ) n k (µ µ 0 ) + ( m 2 + α(µ µ 0 ) ) (ρ ρ 0 ) (λ + β(µ µ 0)) (ρ ρ 0 ) 2. minimum ρ 0 is k-dependent ( ρ U)(ρ 0,µ 0 ) = 0 for allk

27 Effective Potential, Cutoff U(ρ,µ) = U(ρ 0,µ 0 ) n k (µ µ 0 ) + ( m 2 + α(µ µ 0 ) ) (ρ ρ 0 ) (λ + β(µ µ 0)) (ρ ρ 0 ) 2. minimum ρ 0 is k-dependent ( ρ U)(ρ 0,µ 0 ) = 0 for allk k-independent µ 0 determines particle number ( µ U)(ρ 0,µ 0 ) = n at k = k ph

28 Effective Potential, Cutoff U(ρ,µ) = U(ρ 0,µ 0 ) n k (µ µ 0 ) + ( m 2 + α(µ µ 0 ) ) (ρ ρ 0 ) (λ + β(µ µ 0)) (ρ ρ 0 ) 2. minimum ρ 0 is k-dependent ( ρ U)(ρ 0,µ 0 ) = 0 for allk k-independent µ 0 determines particle number ( µ U)(ρ 0,µ 0 ) = n at k = k ph use cutoff (Litim (2001)) R k = Ā(k2 p 2 m 2 )θ(k 2 p 2 m 2 )

29 Effective Potential, Cutoff U(ρ,µ) = U(ρ 0,µ 0 ) n k (µ µ 0 ) + ( m 2 + α(µ µ 0 ) ) (ρ ρ 0 ) (λ + β(µ µ 0)) (ρ ρ 0 ) 2. minimum ρ 0 is k-dependent ( ρ U)(ρ 0,µ 0 ) = 0 for allk k-independent µ 0 determines particle number ( µ U)(ρ 0,µ 0 ) = n at k = k ph use cutoff (Litim (2001)) R k = Ā(k2 p 2 m 2 )θ(k 2 p 2 m 2 ) all momentum integrations and Matsubara sums are performed analytically

30 Flow of kinetic coefficients Ā, S, V Ā S 0.6 V ln(k)

31 Flow of kinetic coefficients Ā, S, V Ā S 0.6 V wavefunction renormalization Ā conected to condensate depletion Ā = ρ 0 ρ 0 = n S n C ln(k) (= n n C at T = 0)

32 Flow of kinetic coefficients Ā, S, V Ā S 0.6 V wavefunction renormalization Ā conected to condensate depletion Ā = ρ 0 ρ 0 = n S n C linear frequency term S goes to zero (d = 3 logarithmically, d = 2 linearly) ln(k) (= n n C at T = 0)

33 Flow of kinetic coefficients Ā, S, V Ā S 0.6 V wavefunction renormalization Ā conected to condensate depletion Ā = ρ 0 ρ 0 = n S n C linear frequency term S goes to zero (d = 3 logarithmically, d = 2 linearly) quadratic frequency term V takes over ln(k) (= n n C at T = 0)

34 Vacuum flow of interaction strength λ λ(k) ln(k)

35 Vacuum flow of interaction strength λ λ(k) ln(k) in d = 2 logarithmic running t λ = 1 4π λ2, λ(k) = 1 1 λ Λ + 1 4π ln(λ/k), λ = λ(k ph) 4π ln(λ/k ph )

36 Vacuum flow of interaction strength λ λ(k) ln(k) in d = 2 logarithmic running t λ = 1 4π λ2, λ(k) = 1 1 λ Λ + 1 4π ln(λ/k), λ = λ(k ph) 4π ln(λ/k ph ) in d = 3 linear running t λ = k 6π 2 λ2, λ(k) = 1 1 λ Λ + 1 6π (Λ k), 2 6π2 λ = λ(0) Λ

37 Vacuum flow of interaction strength λ λ(k) ln(k) in d = 2 logarithmic running t λ = 1 4π λ2, λ(k) = 1 1 λ Λ + 1 4π ln(λ/k), λ = λ(k ph) 4π ln(λ/k ph ) in d = 3 linear running t λ = k 6π 2 λ2, λ(k) = 1 1 λ Λ + 1 6π (Λ k), 2 6π2 λ = λ(0) Λ triviality bound on interaction strength!

38 Vacuum flow of interaction strength λ λ(k) 4 λ ln(k) in d = 2 logarithmic running λλ t λ = 1 4π λ2, λ(k) = 1 1 λ Λ + 1 4π ln(λ/k), λ = λ(k ph) 4π ln(λ/k ph ) in d = 3 linear running t λ = k 6π 2 λ2, λ(k) = 1 1 λ Λ + 1 6π (Λ k), 2 6π2 λ = λ(0) Λ triviality bound on interaction strength!

39 Dispersion relation follows from propagator with analytic continuation (T = 0) ( ) p det 2 V ω 2 + U + 2ρU, isω isω, p 2 V ω 2 + U = 0 S 2 ω 2 + ( p 2 V ω 2 + U + 2ρU )( p 2 V ω 2 + U ) = 0

40 Dispersion relation follows from propagator with analytic continuation (T = 0) ( ) p det 2 V ω 2 + U + 2ρU, isω isω, p 2 V ω 2 + U = 0 S 2 ω 2 + ( p 2 V ω 2 + U + 2ρU )( p 2 V ω 2 + U ) = 0 solution for ω has two branches

41 Dispersion relation ln ω±(p) ln ω+(p) ln ω (p) ln(p) follows from propagator with analytic continuation (T = 0) ( ) p det 2 V ω 2 + U + 2ρU, isω isω, p 2 V ω 2 + U = 0 S 2 ω 2 + ( p 2 V ω 2 + U + 2ρU )( p 2 V ω 2 + U ) = 0 solution for ω has two branches

42 Dispersion relation ln ω±(p) ln ω+(p) ln ω (p) ln(p) follows from propagator with analytic continuation (T = 0) ( ) p det 2 V ω 2 + U + 2ρU, isω isω, p 2 V ω 2 + U = 0 S 2 ω 2 + ( p 2 V ω 2 + U + 2ρU )( p 2 V ω 2 + U ) = 0 solution for ω has two branches lower branch describes phase fluctuations (sound mode)

43 Dispersion relation ln ω+(p) ln ω±(p) cs/n 1/ ln ω (p) ln(p) λ follows from propagator with analytic continuation (T = 0) ( ) p det 2 V ω 2 + U + 2ρU, isω isω, p 2 V ω 2 + U = 0 S 2 ω 2 + ( p 2 V ω 2 + U + 2ρU )( p 2 V ω 2 + U ) = 0 solution for ω has two branches lower branch describes phase fluctuations (sound mode) sound velocity deviates from mean field for large λ

44 Sound at T > 0 Landau (1941): For temperatures 0 < T < T c two fluid hydrodynamics: First and Second sound

45 Sound at T > 0 Landau (1941): For temperatures 0 < T < T c two fluid hydrodynamics: First and Second sound can be calculated in grand canonical ensemble

46 Sound at T > 0 Landau (1941): For temperatures 0 < T < T c two fluid hydrodynamics: First and Second sound can be calculated in grand canonical ensemble use flow equation for p = U(ρ 0,µ 0 )

47 Sound at T > 0 Landau (1941): For temperatures 0 < T < T c two fluid hydrodynamics: First and Second sound can be calculated in grand canonical ensemble use flow equation for p = U(ρ 0,µ 0 ) perform various derivatives with respect to T and µ nummerically

48 Sound at T > 0 Landau (1941): For temperatures 0 < T < T c two fluid hydrodynamics: First and Second sound can be calculated in grand canonical ensemble use flow equation for p = U(ρ 0,µ 0 ) perform various derivatives with respect to T and µ nummerically c 2 /n 2/ T/n 2/3

49 Critical temperature in d = 3 Tc/n 2/ an 1/3

50 Critical temperature in d = 3 Tc/n 2/ an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 =

51 Critical temperature in d = 3 Tc/n 2/ an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = for a > 0 shift in T c : T c /T c = κan 1/3

52 Critical temperature in d = 3 Tc/n 2/ an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = for a > 0 shift in T c : we find κ = 2.1 T c /T c = κan 1/3

53 Critical temperature in d = 3 Tc/n 2/ an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = for a > 0 shift in T c : we find κ = 2.1 T c /T c = κan 1/3 Monte-Carlo and RG studies in the high-t limit find κ = 1.3 (including only the n = 0 Matsubara frequency) (Arnold, Moore (2001); Kashurnikov et al. (2001); Baym et al. (1999); Ledowski et al. (2004); Blaizot et al. (2006).)

54 Kosterlitz-Thouless physics in d = 2

55 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem)

56 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem) for small temperatures T < T c the renormalized order parameter ρ = Ā ρ (superfluid density) remains finite

57 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem) for small temperatures T < T c the renormalized order parameter ρ = Ā ρ (superfluid density) remains finite crititcal temperature T c depends on λ, not computable from Kosterlitz-Thouless theory

58 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem) for small temperatures T < T c the renormalized order parameter ρ = Ā ρ (superfluid density) remains finite crititcal temperature T c depends on λ, not computable from Kosterlitz-Thouless theory perturbative treatment (Popov (1983)) gives T c /n = 4π/ln(ζ/λ) and Monte-Carlo (Prokof ev et al. (2001)) determines ζ = 380

59 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem) for small temperatures T < T c the renormalized order parameter ρ = Ā ρ (superfluid density) remains finite crititcal temperature T c depends on λ, not computable from Kosterlitz-Thouless theory perturbative treatment (Popov (1983)) gives T c /n = 4π/ln(ζ/λ) and Monte-Carlo (Prokof ev et al. (2001)) determines ζ = 380 we find that ζ depends on the size of system k 1 ph

60 Kosterlitz-Thouless physics in d = 2 in the infinite size limit (k 0) bare order parameter ρ 0 vanishes (Mermin-Wagner theorem) for small temperatures T < T c the renormalized order parameter ρ = Ā ρ (superfluid density) remains finite crititcal temperature T c depends on λ, not computable from Kosterlitz-Thouless theory perturbative treatment (Popov (1983)) gives T c /n = 4π/ln(ζ/λ) and Monte-Carlo (Prokof ev et al. (2001)) determines ζ = 380 we find that ζ depends on the size of system k 1 ph Tc/n λ circles: k ph = 10 4, ζ = 100 boxes: k ph = 10 6, ζ = 225 diamonds: k ph = 10 8, ζ = 424

61 Conclusions

62 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons.

63 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons. Varios thermodynamic observables in equilibrium have been calculated.

64 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons. Varios thermodynamic observables in equilibrium have been calculated. A simple truncation describes the essential features of superfluidity in d = 3 and d = 2.

65 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons. Varios thermodynamic observables in equilibrium have been calculated. A simple truncation describes the essential features of superfluidity in d = 3 and d = 2. Deviations from Mean-Field results are found for large interaction strength λ and close to T c.

66 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons. Varios thermodynamic observables in equilibrium have been calculated. A simple truncation describes the essential features of superfluidity in d = 3 and d = 2. Deviations from Mean-Field results are found for large interaction strength λ and close to T c. Extentions of the truncation are straight forward and should lead to quantitative improvements for large λ.

67 Conclusions Functional renormalization for the average action yields a unified description of ultracold bosons. Varios thermodynamic observables in equilibrium have been calculated. A simple truncation describes the essential features of superfluidity in d = 3 and d = 2. Deviations from Mean-Field results are found for large interaction strength λ and close to T c. Extentions of the truncation are straight forward and should lead to quantitative improvements for large λ. Thank you for your attention!

68 First and second sound sound velocities c follow are solutions of the equation ( ) (Mc 2 ) 2 p n s Ts 2 n + s (n n S )c v n 2 (Mc 2 ) n n s Ts 2 p + (n n s )c v n 2 n = 0 T use here specific heat per particle c v = T n s T n

69 Order in d = 2 n, ρ 0, ρ ln(k) Flow of the density n (solid), the superfluid density ρ 0 (dashed) and the condensate density ρ 0 (dotted) for temperatures T = 0 (top), T = 2.4 (middle) and T = 2.8 (bottom).

70 Jump in superfluid density ρ 0/n T/n Superfluid fraction of the density ρ 0 /n at different scales k ph. dotted: simple truncation solid: improved truncation