Functional renormalization for ultracold quantum gases

Functional renormalization for ultracold quantum gases Stefan Floerchinger (Heidelberg) S. Floerchinger and C. Wetterich, Phys. Rev. A 77, 053603 (2008); S. Floerchinger and C. Wetterich, arxiv:0805.2571. ERG 2008 Heidelberg, 01. - 06. Jun. 2008

Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ + 1 2 λ(φ φ) 2} x

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

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

Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ + 1 2 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

Microscopic action for nonrelativistic Bose gas S[φ] = {φ ( τ µ)φ + 1 2 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

Symmetries of the microscopic action

Symmetries of the microscopic action rotation and translation

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

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.

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

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

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

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

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

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, 024513 (2004).

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 ) + 1 2 (λ + β(µ µ 0)) (ρ ρ 0 ) 2.

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 ) + 1 2 (λ + β(µ µ 0)) (ρ ρ 0 ) 2. minimum ρ 0 is k-dependent ( ρ U)(ρ 0,µ 0 ) = 0 for allk

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 ) + 1 2 (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 )

Flow of kinetic coefficients Ā, S, V Ā 1.0 0.8 S 0.6 V 0.4 0.2 10 8 6 4 2 0 2 ln(k)

Flow of kinetic coefficients Ā, S, V Ā 1.0 0.8 S 0.6 V 0.4 0.2 10 8 6 4 2 0 2 wavefunction renormalization Ā conected to condensate depletion Ā = ρ 0 ρ 0 = n S n C ln(k) (= n n C at T = 0)

Flow of kinetic coefficients Ā, S, V Ā 1.0 0.8 S 0.6 V 0.4 0.2 10 8 6 4 2 0 2 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)

Vacuum flow of interaction strength λ λ(k) 4 3 2 1 4 3 2 1 0 1 2 ln(k)

Vacuum flow of interaction strength λ λ(k) 4 3 2 1 4 3 2 1 0 1 2 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 )

Vacuum flow of interaction strength λ λ(k) 4 3 2 1 4 3 2 1 0 1 2 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

Vacuum flow of interaction strength λ λ(k) 4 λ 3 1.5 2 1.0 1 0.5 4 3 2 1 0 1 2 ln(k) in d = 2 logarithmic running 5 10 15 20 25 30 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!

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

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

Dispersion relation ln ω±(p) 20 15 10 ln ω+(p) 5 0 5 ln ω (p) 10 10 8 6 4 2 0 2 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

Dispersion relation ln ω+(p) ln ω±(p) 20 15 10 5 0 cs/n 1/2 4 3 2 ln ω (p) 5 10 1 10 8 6 4 2 0 2 ln(p) 0.5 1.0 1.5 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

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

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

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 )

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

Critical temperature in d = 3 Tc/n 2/3 6.64 6.635 6.63 6.625 0.0005 0.001 an 1/3

Critical temperature in d = 3 Tc/n 2/3 6.64 6.635 6.63 6.625 0.0005 0.001 an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = 6.6250

Critical temperature in d = 3 Tc/n 2/3 6.64 6.635 6.63 6.625 0.0005 0.001 an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = 6.6250 for a > 0 shift in T c : T c /T c = κan 1/3

Critical temperature in d = 3 Tc/n 2/3 6.64 6.635 6.63 6.625 0.0005 0.001 an 1/3 for a = λ 8π = 0 free Bose gas: T c/(n 2/3 ) = 4π/ζ(3/2) 2/3 = 6.6250 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).)

Kosterlitz-Thouless physics in d = 2

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

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

Conclusions

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

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

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.

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

Order in d = 2 n, ρ 0, ρ 0 2.0 1.5 1.0 0.5 10 8 6 4 2 0 2 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).

Jump in superfluid density ρ 0/n 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T/n Superfluid fraction of the density ρ 0 /n at different scales k ph. dotted: simple truncation solid: improved truncation