bec for low dimensional interacting bosons -...

BEC for low dimensional interacting bosons

Serena Cenatiempo

joint work with A. Giuliani

ICMP12

Aalborg, 10 August 2012

The model and resultsIdea of the proof

Motivations

Since 1995 BEC of ultracold dilute atomic gases has been subject ofintensive studies, driven by always new experimental techniques.

Hansel et al. (2001) Kruger et al. (2007) Billy, Josse, Zuo, Guerin, Aspect, Bouyer (2008)

State of the art:

there is a single model (Dyson, Lieb, Simon (1978)) proving Bosecondensation for homogeneous interacting bosons.

S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons

Motivations

Since 1995 BEC of ultracold dilute atomic gases has been subject ofintensive studies, driven by always new experimental techniques.

Hansel et al. (2001) Kruger et al. (2007) Billy, Josse, Zuo, Guerin, Aspect, Bouyer (2008)

State of the art:

there is a single model (Dyson, Lieb, Simon (1978)) proving Bosecondensation for homogeneous interacting bosons.

Set of the problemBogoliubov and PTMain result

Set of the problem

I N bosons in a periodic box Ω in Rd

I weak repulsive short range potential

HΩ,N =N∑

(−∆x i

− µ)

+ λ∑

1≤i<j≤N

v(x i − x j

Goal: ground state properties

|Ω| → ∞ with ρ =⟨N⟩/|Ω| fixed

BEC for interacting bosons:

S(x , y) =⟨a+

⟩−−−−−−→|x−y |→∞ρ fixed

const.

Set of the problem

HΩ,N =N∑

(−∆x i

− µ)

+ λ∑

1≤i<j≤N

v(x i − x j

S(x , y) =⟨a+

⟩−−−−−−→|x−y |→∞ρ fixed

const.

Set of the problem

HΩ,N =N∑

(−∆x i

− µ)

+ λ∑

1≤i<j≤N

v(x i − x j

S(x , y) =⟨a+

⟩−−−−−−→|x−y |→∞ρ fixed

const.

Set of the problem with functional integrals

The interacting partition function can beformally expressed as a functional integral:

∫PΛ(dϕ) e−VΛ(ϕ)

I ϕ+x,t = (ϕ−x,t)∗ complex fields (coherent states)

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

dt |ϕx,t |2 v(x−y) |ϕy,t |2−µ∫

∫ β/2

−β/2

dt |ϕx,t |2

P0Λ(dϕ) is a complex Gaussian measure with covariance

S0Λ(x , y) =

⟩∣∣∣λ=0

Λ(dϕ)ϕ−x ϕ+y → ρ0 +

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

I ϕ±x = ξ± + ψ±x with⟨ξ−ξ+

⟩= ρ0

I V(ϕ) = Qξ(ψ) + Vξ(ψ)

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

Bogoliubov approximation (1947)

Neglecting Vξ(ψ) the model is exactly solvable and predicts a linear dispersionrelation for low momenta

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

→ Landau argument for superfluidity

Schwinger function for Bogoliubov model:

SBΛ (x , y) =

⟩∣∣Bog

Λ (dϕ)ϕ−x ϕ+y → ρ0 +

(2π)d+1

∫Rd+1

ddk dk0 e−ikx g−+(k)

g−+(k) =c2

k20 + c2

g free−+(k) =

−ik0 + k2

Main goal: to control and compute in a systematic way

the corrections to Bogoliubov theory at weak coupling.

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

SBΛ (x , y) =

⟩∣∣Bog

(2π)d+1

∫Rd+1

g−+(k) =c2

k20 + c2

Bk2 g free

−+(k) =1

−ik0 + k2

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

SBΛ (x , y) =

⟩∣∣Bog

(2π)d+1

∫Rd+1

g−+(k) =c2

k20 + c2

Bk2 g free

−+(k) =1

−ik0 + k2

Perturbation theory

Numerous papers were then devoted to analyze the correctionsto Bogoliubov model: Beliaev (1958), Hugenholtz and Pines(1959), Lee and Yang (1960), Gavoret and Nozieres (1964),Nepomnyashchy and Nepomnyashchy (1978), Popov (1987).More recently Benfatto (1994) and Pistolesi, Castellani, DiCastro, Strinati (1997, 2004).

The effective model

The condensation problem depends only on the long–distance behaviorof the system → effective model with an ultraviolet momentum cutoff:

g≤0−+(x) =

(2π)d+1

∫ddkdk0 χ0(k, k0) e−ikx g−+(k)

χ0(k , k0) is a regularization of the characteristic function of the set

k20 + c2

Bk2 ≤ 1

Main result

For bosons in d = 2, 3, interacting with a weak repulsive short range potential,in the presence of an ultraviolet cutoff and at zero temperature we proved that:

I the interacting theory is well defined at all orders in terms ofseries in an effective parameter related to the intensity of theinteraction, with coefficient of order n bounded by (const.)n n!.

I the correlations do not exhibit anomalous exponents, i.e. themodel is in the same universality class of the exactly solvableBogoliubov model.

→ justification of the validity at all ordersof Bogoliubov theory for d = 2, 3

Remark. 3d case first solved by Benfatto; a more satisfactory method– based on WI’s – allows us also to control the 2d case.

Main result

For bosons in d = 2, 3, interacting with a weak repulsive short range potential,in the presence of an ultraviolet cutoff and at zero temperature we proved that:

I the interacting theory is well defined at all orders in terms ofseries in an effective parameter related to the intensity of theinteraction, with coefficient of order n bounded by (const.)n n!.

I the correlations do not exhibit anomalous exponents, i.e. themodel is in the same universality class of the exactly solvableBogoliubov model.

→ justification of the validity at all ordersof Bogoliubov theory for d = 2, 3

Remark. 3d case first solved by Benfatto; a more satisfactory method– based on WI’s – allows us also to control the 2d case.

Rigorous RG schemeWard IdentitiesResult

Rigorous RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

Λ(dϕ)e−VΛ(ϕ)

( )V y

1 Multiscale decomposition: we integrate iteratively the fields of decreasing

energy scale, k20 + c2

Bk2 ' 2h, h ∈ (−∞, 0]

2 Integration over the fields higher than γh gives Vh(ψ) = LVh(ψ) +RVh(ψ)

LVh =λh

+γ2hνh

+∂0 ∂0

+∂ ∂

3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.

Rigorous RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫PΛ(dψ)e−Qξ(ψ)−Vξ(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

LVh =λh

+γ2hνh

+∂0 ∂0

+∂ ∂

Rigorous RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫P≤h

B (ψ)e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

LVh =λh

+γ2hνh

+∂0 ∂0

+∂ ∂

Rigorous RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫P≤h

B (ψ)e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

LVh =λh

+γ2hνh

+∂0 ∂0

+∂ ∂

Rigorous RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫P≤h

B (ψ)e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

LVh =λh

+γ2hνh

+∂0 ∂0

+∂ ∂

Ward Identities

Idea: the WI’s reduce the number of independent couplings:

to implement the local WI’s within the constructive RG scheme;

a non trivial task since the momentum cutoffs explicitly breakthe local gauge invariance.

→ In low-dimensional systems of interacting fermions ( Luttingerliquids ) the corrections to WI are crucial for establishing theinfrared behavior of the system.

using techniques by ( Benfatto, Falco, Mastropietro, 2009) we havestudied the flow of the corrections (marginal) and proved that givecorrections of higher order in λ to the formal WI’s.

Result

I The theory is order by order finite in the renormalized coupling constantsprovided that

d = 3 λh −−−−→h→−∞

d = 2 λh −−−−→h→−∞

λ∗ = const.

∗ for d = 3 one can prove that λh has an asintotically free flow;

∗ for d = 2 a second order calculation suggests λ∗ to be of order one.

I Bogoliubov linear spectrum is exactly constrained by Ward identities:

gBogoliubov−+ (k) ∝ 1

k20 + c2

B k2 −→ g interacting−+ (k) ∝ 1

k20 + c2(λ) k2

with cB =√

2λρ0v(0) the speed of sound of Bogoliubov model and c(λ) therenormalized speed of sound.

Perspectives

I interacting bosons on a lattice;

I weak coupling and high density regime;

I critical temperature;

I . . .

I constructive theory.

bec for low dimensional interacting bosons -...

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