Low dimensional interacting bosons

Transcription

1 Low dimensional interacting bosons Serena Cenatiempo PhD defence Supervisors: E. Marinari and A. Giuliani Physics Department Sapienza, Università di Roma February 7, 2013

2 It was only in 1995 Anderson et al., BEC in a vapor of rubidium 87 atoms A separation is effected; one part condenses, the rest remains a saturated ideal gas. Einstein, 1925 ρβ,ω = hni hn0 i 1 X 1 = β(k 2 µω,β ) Ω Ω Ω e 1 k6=0 S. Cenatiempo Rome, BEC occurs when lim Ω hn0 i Ω Low dimensional interacting bosons = (const.)

3 BEC nowadays is an ultralow temperature laboratory for quantum optics, atomic physics and condensed matter physics Hänsel et al., Atom chip (2001) Krüger et al. (2007) Billy et al., Guided atom laser (2008) Nature Physics 1 (2005) Klaers et al., BEC of phonons (2010)

4 BEC nowadays is an ultralow temperature laboratory for quantum optics, atomic physics and condensed matter physics State of the art Hänsel et al., Atom chip (2001) Billy et al., Guided atom laser (2008) Krüger et al. (2007) There is a single model in which we can prove BEC for homogeneous interacting bosons (Dyson, Lieb and Simon, 1978). More recent results on trapped bosons (Lieb, Seiringer and Yngvason, 2002). Nature Physics 1 (2005) Klaers et al., BEC of phonons (2010)

5 Bogoliubov and PT N bosons in a periodic box Ω in R d weak repulsive short range potential N ( H Ω,N = x i µ ) λ v ( ) x i x j i=1 1 i<j N Goal: ground state properties Ω with ρ = N / Ω fixed

6 Bogoliubov and PT N bosons in a periodic box Ω in R d weak repulsive short range potential N ( H Ω,N = x i µ ) λ v ( ) x i x j i=1 1 i<j N Goal: ground state properties Ω with ρ = N / Ω fixed BEC for interacting bosons: S(x, y) = a x a y β x y ρ fixed const.

7 Bogoliubov and PT with functional integrals The interacting partition function can be formally expressed as a functional integral: Z Λ Z 0 Λ = P 0 Λ(dϕ) e V Λ(ϕ) ϕ x,t = (ϕ x,t) complex fields (coherent states)

8 Bogoliubov and PT with functional integrals The interacting partition function can be formally expressed as a functional integral: Z Λ Z 0 Λ = P 0 Λ(dϕ) e V Λ(ϕ) ϕ x,t = (ϕ x,t) complex fields (coherent states) V Λ (ϕ) = λ 2 Ω Ω d d x d d y β/2 β/2 β/2 dt ϕ x,t 2 v(x y) ϕ y,t 2 µ d d x dt ϕ x,t 2 Ω β/2

9 Bogoliubov and PT with functional integrals The interacting partition function can be formally expressed as a functional integral: Z Λ Z 0 Λ = P 0 Λ(dϕ) e V Λ(ϕ) ϕ x,t = (ϕ x,t) complex fields (coherent states) V Λ (ϕ) = λ 2 Ω Ω d d x d d y β/2 β/2 β/2 dt ϕ x,t 2 v(x y) ϕ y,t 2 µ d d x dt ϕ x,t 2 Ω β/2 P 0 Λ(dϕ) is a complex Gaussian measure with covariance SΛ(x, 0 y) = a x a y λ=0 = PΛ(dϕ)ϕ 0 x ϕ y = Ω,β ρ0 1 Rd1 d d e ikx k dk (2π) d1 0 ik 0 k 2

10 Bogoliubov and PT with functional integrals The interacting partition function can be formally expressed as a functional integral: Z Λ Z 0 Λ = P 0 Λ(dϕ) e V Λ(ϕ) ϕ x,t = (ϕ x,t) complex fields (coherent states) V Λ (ϕ) = λ 2 Ω Ω d d x d d y β/2 β/2 β/2 dt ϕ x,t 2 v(x y) ϕ y,t 2 µ d d x dt ϕ x,t 2 Ω β/2 P 0 Λ(dϕ) is a complex Gaussian measure with covariance SΛ(x, 0 y) = a x a y λ=0 = PΛ(dϕ)ϕ 0 x ϕ y = Ω,β ρ0 1 Rd1 d d e ikx k dk (2π) d1 0 ik 0 k 2 ϕ ± x = ξ ± ψ ± x with ξ ξ = ρ 0

11 Bogoliubov and PT with functional integrals The interacting partition function can be formally expressed as a functional integral: Z Λ Z 0 Λ = P 0 Λ(dϕ) e V Λ(ϕ) ϕ x,t = (ϕ x,t) complex fields (coherent states) V Λ (ϕ) = λ 2 Ω Ω d d x d d y β/2 β/2 β/2 dt ϕ x,t 2 v(x y) ϕ y,t 2 µ d d x dt ϕ x,t 2 Ω β/2 P 0 Λ(dϕ) is a complex Gaussian measure with covariance SΛ(x, 0 y) = a x a y λ=0 = PΛ(dϕ)ϕ 0 x ϕ y = Ω,β ρ0 1 Rd1 d d e ikx k dk (2π) d1 0 ik 0 k 2 ϕ ± x = ξ ± ψ ± x with ξ ξ = ρ 0 V(ϕ) = Q ξ (ψ) V ξ (ψ)

12 Bogoliubov and PT Bogoliubov approximation (1947) V(ϕ) = Q ξ (ψ) V ξ (ψ) Neglecting V ξ (ψ) the model is exactly solvable and predicts a linear dispersion relation for low momenta E(k) = k 4 2 λρ 0ˆv(k)k 2 2λρ0ˆv(0) k = c B k k 0 Landau argument for superfluidity

13 Bogoliubov and PT Bogoliubov approximation (1947) V(ϕ) = Q ξ (ψ) V ξ (ψ) Neglecting V ξ (ψ) the model is exactly solvable and predicts a linear dispersion relation for low momenta E(k) = k 4 2 λρ 0ˆv(k)k 2 2λρ0ˆv(0) k = c B k k 0 Landau argument for superfluidity Schwinger function for Bogoliubov model: SΛ B (x, y) = a x a Bog y = PΛ B (dϕ)ϕ x ϕ 1 y ρ 0 d d k dk (2π) d1 0 e ikx g (k) R d1

14 Bogoliubov and PT Bogoliubov approximation (1947) V(ϕ) = Q ξ (ψ) V ξ (ψ) Neglecting V ξ (ψ) the model is exactly solvable and predicts a linear dispersion relation for low momenta E(k) = k 4 2 λρ 0ˆv(k)k 2 2λρ0ˆv(0) k = c B k k 0 Landau argument for superfluidity Schwinger function for Bogoliubov model: SΛ B (x, y) = a x a Bog y = PΛ B (dϕ)ϕ x ϕ 1 y ρ 0 d d k dk (2π) d1 0 e ikx g (k) R d1 g (k) c 2 B k 2 0 c2 B k 2 ik 0 = ±c B k g (k) free 1 = ik 0 k 2

15 Bogoliubov and PT Bogoliubov approximation (1947) V(ϕ) = Q ξ (ψ) V ξ (ψ) Neglecting V ξ (ψ) the model is exactly solvable and predicts a linear dispersion relation for low momenta E(k) = k 4 2 λρ 0ˆv(k)k 2 2λρ0ˆv(0) k = c B k k 0 Landau argument for superfluidity Schwinger function for Bogoliubov model: SΛ B (x, y) = a x a Bog y = PΛ B (dϕ)ϕ x ϕ 1 y ρ 0 d d k dk (2π) d1 0 e ikx g (k) R d1 g (k) c 2 B k 2 0 c2 B k 2 ik 0 = ±c B k Main goal: to control and compute in a systematic way the corrections to Bogoliubov theory at weak coupling. g (k) free 1 = ik 0 k 2

16 Bogoliubov and PT Perturbation theory Numerous papers were then devoted to analyze the corrections to Bogoliubov model: Beliaev (1958), Hugenholtz and Pines (1959), Lee and Yang (1960), Gavoret and Nozières (1964), Nepomnyashchy and Nepomnyashchy (1978), Popov (1987).

17 Bogoliubov and PT Perturbation theory Numerous papers were then devoted to analyze the corrections to Bogoliubov model: Beliaev (1958), Hugenholtz and Pines (1959), Lee and Yang (1960), Gavoret and Nozières (1964), Nepomnyashchy and Nepomnyashchy (1978), Popov (1987). Benfatto (1994, 1997): first systematic study of the infrared divergences for the 3d theory at T = 0 justification of BEC at all orders

18 Bogoliubov and PT Perturbation theory Numerous papers were then devoted to analyze the corrections to Bogoliubov model: Beliaev (1958), Hugenholtz and Pines (1959), Lee and Yang (1960), Gavoret and Nozières (1964), Nepomnyashchy and Nepomnyashchy (1978), Popov (1987). Benfatto (1994, 1997): first systematic study of the infrared divergences for the 3d theory at T = 0 justification of BEC at all orders Pistolesi, Castellani, Di Castro, Strinati (1997, 2004): 3d and 2d bosons at T = 0 by using Ward Identities (RG scheme based on a dimensional regularization)

19 Bogoliubov and PT Perturbation theory Numerous papers were then devoted to analyze the corrections to Bogoliubov model: Beliaev (1958), Hugenholtz and Pines (1959), Lee and Yang (1960), Gavoret and Nozières (1964), Nepomnyashchy and Nepomnyashchy (1978), Popov (1987). Benfatto (1994, 1997): first systematic study of the infrared divergences for the 3d theory at T = 0 justification of BEC at all orders Pistolesi, Castellani, Di Castro, Strinati (1997, 2004): 3d and 2d bosons at T = 0 by using Ward Identities (RG scheme based on a dimensional regularization) Exact RG approach Explicit bounds at all orders Complete control of all the diagrams (irrelevant terms included) Momentum cutoff regularization (in perspective, not perturbative construction)

20 Bogoliubov and PT The effective model The condensation problem depends only on the long distance behavior of the system effective model with an ultraviolet momentum cutoff: g (x) 0 1 = (2π) d1 d d kdk 0 χ 0 (k, k 0 ) e ikx g (k) χ 0 (k, k 0 ) is a regularization of the characteristic function of the set k 2 0 c 2 Bk 2 1

21 Bogoliubov and PT For bosons in d = 2, interacting with a weak repulsive short range potential, in the presence of an ultraviolet cutoff and at zero temperature we proved that:

22 Bogoliubov and PT For bosons in d = 2, interacting with a weak repulsive short range potential, in the presence of an ultraviolet cutoff and at zero temperature we proved that: the interacting theory is well defined at all orders in terms of two effective parameters (three and two particles effective interactions) with coefficient of order n bounded by (const.) n n!.

23 Bogoliubov and PT For bosons in d = 2, interacting with a weak repulsive short range potential, in the presence of an ultraviolet cutoff and at zero temperature we proved that: the interacting theory is well defined at all orders in terms of two effective parameters (three and two particles effective interactions) with coefficient of order n bounded by (const.) n n!. the correlations do not exhibit anomalous exponents: same universality class of the exactly solvable Bogoliubov model. g Bogoliubov (k) 1 k 2 0 c2 B k2 g interacting (k) 1 k 2 0 c2 (λ) k 2

24 Bogoliubov and PT For bosons in d = 2, interacting with a weak repulsive short range potential, in the presence of an ultraviolet cutoff and at zero temperature we proved that: the interacting theory is well defined at all orders in terms of two effective parameters (three and two particles effective interactions) with coefficient of order n bounded by (const.) n n!. the correlations do not exhibit anomalous exponents: same universality class of the exactly solvable Bogoliubov model. g Bogoliubov (k) 1 k 2 0 c2 B k2 g interacting (k) 1 k 2 0 c2 (λ) k 2 Key points 1. three new effective coupling constants 2. to use local WIs within a RG scheme based on a momentum regularization

25 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x ξ ξ = ρ 0, ψx ψ x decaying Z Λ Z 0 Λ = P 0 Λ(dϕ)e V Λ(ϕ)

26 Ward Identities Flow equations ϕ ± x = ξ ± ψ x ± ξ ξ = ρ 0, ψx ψ x decaying Z Λ = P ZΛ 0 Λ (dξ) P Λ (dψ)e Q ξ(ψ) V ξ (ψ)

27 Ward Identities Flow equations ϕ ± x Z Λ Z 0 Λ = ξ ± ψ x ± = P Λ (dξ) ξ ξ = ρ 0, ψx ψ x decaying P Λ (dψ)e Q ξ(ψ) V ξ (ψ) } {{ } e Λ W Λ (ξ) V( ) t l

28 Ward Identities Flow equations ϕ ± x = ξ ± ψ x ± ξ ξ = ρ 0, ψx ψ x decaying e W Λ(ρ 0 ) = P Λ (dψ)e Qρ 0 (ψ) Vρ 0 (ψ) V( ) t l

29 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x ξ ξ = ρ 0, ψx ψ x decaying e W Λ(ρ 0 ) = P Λ (dψ)e Qρ 0 (ψ) Vρ 0 (ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l

30 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x e W Λ(ρ 0 ) = ξ ξ = ρ 0, ψx ψ x decaying P h Λ (dψ) e V h(ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l

31 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x e W Λ(ρ 0 ) = ξ ξ = ρ 0, ψx ψ x decaying P h Λ (dψ) e V h(ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l 2 Integration over the fields higher than 2 h : V h (ψ) = LV h (ψ) RV h (ψ) LV h = λ 6 h γ h λ h γ h 2 µ h γ 2h ν h Z h 0 0 B h x x A h E h 0

32 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x e W Λ(ρ 0 ) = ξ ξ = ρ 0, ψx ψ x decaying P h Λ (dψ) e V h(ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l 2 Integration over the fields higher than 2 h : V h (ψ) = LV h (ψ) RV h (ψ) LV h = λ 6 h γ h λ h γ h 2 µ h γ 2h ν h Z h 0 0 B h x x A h E h 0 3 Using Gallavotti Nicoló tree expansion we prove that RV h is well defined with explicit bounds if the terms in LV h are bounded.

33 Ward Identities Flow equations Gallavotti Nicolò tree expansion The h th step of the iterative integration can be graphically represented as a sum of trees over h scale labels. The number n of endpoints represents the order in perturbation theory. τ = 1 2 Γ = Gallavotti Nicolò trees are a synthetic and convenient way to isolate the divergent terms, avoiding the problem of overlapping divergences.

34 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x e W Λ(ρ 0 ) = ξ ξ = ρ 0, ψx ψ x decaying P h Λ (dψ) e V h(ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l 2 Integration over the fields higher than γ h : V h (ψ) = LV h (ψ) RV h (ψ) LV h = λ 6 h γ h λ h γ h 2 µ h γ 2h ν h Z h 0 0 B h x x A h E h 0 3 Using Gallavotti Nicólo tree expansion we prove that RV h is well defined with explicit bounds if the terms in LV h are bounded.

35 Ward Identities Flow equations ϕ ± x = ξ ± ψ ± x e W Λ(ρ 0 ) = ξ ξ = ρ 0, ψx ψ x decaying P h Λ (dψ) e V h(ψ) V( ) 1 Multiscale decomposition: we integrate iteratively the fields of decreasing energy scale, k 2 0 c 2 Bk 2 2 h, h (, 0] t l 2 Integration over the fields higher than γ h : V h (ψ) = LV h (ψ) RV h (ψ) LV h = λ 6 h γ h λ h γ h 2 µ h γ 2h ν h ω h Z h 0 0 B h x x A h E h 0 λ h µ h 3 Using Gallavotti Nicólo tree expansion we prove that RV h is well defined with explicit bounds if the terms in LV h are bounded.

36 Ward Identities Flow equations Ward Identities to reduce the number of independent couplings 2 3 Global WIs λ h λ 6 h 3 Local WIs Task: the momentum cutoffs break the local gauge invariance In low-dimensional systems of interacting fermions ( Luttinger liquids ) the corrections to WIs are crucial for establishing the infrared behavior

37 Ward Identities Flow equations Ward Identities to reduce the number of independent couplings 2 3 Global WIs λ h λ 6 h 3 Local WIs Task: the momentum cutoffs break the local gauge invariance In low-dimensional systems of interacting fermions ( Luttinger liquids ) the corrections to WIs are crucial for establishing the infrared behavior Result: using techniques by ( Benfatto, Falco, Mastropietro, 2009) we studied the flow of the corrections (marginal and relevant) and proved that * they do not change the way WIs are used to control the flow; * they are possibly observable in the relations among thermodynamic and response functions.

38 Ward Identities Flow equations Two independent flow equations coupled among them at all orders =

39 Ward Identities Flow equations Two independent flow equations coupled among them at all orders = h Numerical solutions to the leading order flows for λλ h and λ 6,h /(λλ 2 h )

40 Ward Identities Flow equations Two independent flow equations coupled among them at all orders = h Remark.The behavior of the propagator only depends on the existence of the fixed points for λ h and λ 6,h. Numerical solutions to the leading order flows for λλ h and λ 6,h /(λλ 2 h )

41 Ward Identities Flow equations Outlook renormalizability of the UV region (work in progress!) interacting bosons on a lattice weak coupling and high density regime critical temperature... constructive theory

42 Ward Identities Flow equations Formal local WIs 0 = E h 0 0 = B h = A h J 0 J 0 J 1 µ J0 h E J0 h E J1 h 0 E h Z h h λ 1 B h h λ 1 A h h 1 o (1)

43 Ward Identities Flow equations Corrections to local WIs 0 E h = J 0 µ J0 h J = B h J 0 E J0 h 0 J = A h J 1 E J1 h J 1 2