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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
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
The model and resultsIdea of the proof
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∑
i=1
(−∆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 ay
⟩−−−−−−→|x−y |→∞ρ fixed
const.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
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∑
i=1
(−∆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 ay
⟩−−−−−−→|x−y |→∞ρ fixed
const.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
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∑
i=1
(−∆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 ay
⟩−−−−−−→|x−y |→∞ρ fixed
const.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Set of the problem with functional integrals
The interacting partition function can beformally expressed as a functional integral:
ZΛ
Z 0Λ
=
∫PΛ(dϕ) e−VΛ(ϕ)
I ϕ+x,t = (ϕ−x,t)∗ complex fields (coherent states)
VΛ(ϕ) =λ
2
∫Ω×Ω
ddx ddy
∫ β/2
−β/2
dt |ϕx,t |2 v(x−y) |ϕy,t |2−µ∫
Ω
ddx
∫ β/2
−β/2
dt |ϕx,t |2
P0Λ(dϕ) is a complex Gaussian measure with covariance
S0Λ(x , y) =
⟨a+
x ay
⟩∣∣∣λ=0
=
∫P0
Λ(dϕ)ϕ−x ϕ+y → ρ0 +
1
(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)
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Set of the problem with functional integrals
The interacting partition function can beformally expressed as a functional integral:
ZΛ
Z 0Λ
=
∫PΛ(dϕ) e−VΛ(ϕ)
I ϕ+x,t = (ϕ−x,t)∗ complex fields (coherent states)
VΛ(ϕ) =λ
2
∫Ω×Ω
ddx ddy
∫ β/2
−β/2
dt |ϕx,t |2 v(x−y) |ϕy,t |2−µ∫
Ω
ddx
∫ β/2
−β/2
dt |ϕx,t |2
P0Λ(dϕ) is a complex Gaussian measure with covariance
S0Λ(x , y) =
⟨a+
x ay
⟩∣∣∣λ=0
=
∫P0
Λ(dϕ)ϕ−x ϕ+y → ρ0 +
1
(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)
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Set of the problem with functional integrals
The interacting partition function can beformally expressed as a functional integral:
ZΛ
Z 0Λ
=
∫PΛ(dϕ) e−VΛ(ϕ)
I ϕ+x,t = (ϕ−x,t)∗ complex fields (coherent states)
VΛ(ϕ) =λ
2
∫Ω×Ω
ddx ddy
∫ β/2
−β/2
dt |ϕx,t |2 v(x−y) |ϕy,t |2−µ∫
Ω
ddx
∫ β/2
−β/2
dt |ϕx,t |2
P0Λ(dϕ) is a complex Gaussian measure with covariance
S0Λ(x , y) =
⟨a+
x ay
⟩∣∣∣λ=0
=
∫P0
Λ(dϕ)ϕ−x ϕ+y → ρ0 +
1
(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)
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Set of the problem with functional integrals
The interacting partition function can beformally expressed as a functional integral:
ZΛ
Z 0Λ
=
∫PΛ(dϕ) e−VΛ(ϕ)
I ϕ+x,t = (ϕ−x,t)∗ complex fields (coherent states)
VΛ(ϕ) =λ
2
∫Ω×Ω
ddx ddy
∫ β/2
−β/2
dt |ϕx,t |2 v(x−y) |ϕy,t |2−µ∫
Ω
ddx
∫ β/2
−β/2
dt |ϕx,t |2
P0Λ(dϕ) is a complex Gaussian measure with covariance
S0Λ(x , y) =
⟨a+
x ay
⟩∣∣∣λ=0
=
∫P0
Λ(dϕ)ϕ−x ϕ+y → ρ0 +
1
(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)
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Bogoliubov approximation (1947)
I V(ϕ) = Qξ(ψ) + Vξ(ψ)
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) =
⟨a+
x ay
⟩∣∣Bog
=
∫PB
Λ (dϕ)ϕ−x ϕ+y → ρ0 +
1
(2π)d+1
∫Rd+1
ddk dk0 e−ikx g−+(k)
g−+(k) =c2
B
k20 + c2
Bk2
g free−+(k) =
1
−ik0 + k2
Main goal: to control and compute in a systematic way
the corrections to Bogoliubov theory at weak coupling.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Bogoliubov approximation (1947)
I V(ϕ) = Qξ(ψ) + Vξ(ψ)
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) =
⟨a+
x ay
⟩∣∣Bog
=
∫PB
Λ (dϕ)ϕ−x ϕ+y → ρ0 +
1
(2π)d+1
∫Rd+1
ddk dk0 e−ikx g−+(k)
g−+(k) =c2
B
k20 + c2
Bk2 g free
−+(k) =1
−ik0 + k2
Main goal: to control and compute in a systematic way
the corrections to Bogoliubov theory at weak coupling.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
Bogoliubov approximation (1947)
I V(ϕ) = Qξ(ψ) + Vξ(ψ)
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) =
⟨a+
x ay
⟩∣∣Bog
=
∫PB
Λ (dϕ)ϕ−x ϕ+y → ρ0 +
1
(2π)d+1
∫Rd+1
ddk dk0 e−ikx g−+(k)
g−+(k) =c2
B
k20 + c2
Bk2 g free
−+(k) =1
−ik0 + k2
Main goal: to control and compute in a systematic way
the corrections to Bogoliubov theory at weak coupling.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
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).
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
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) =
1
(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
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
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.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Set of the problemBogoliubov and PTMain result
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.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Rigorous RG scheme
ϕ±x = ξ±+ ψ±x⟨ξ−ξ+
⟩= ρ0 ,
⟨ψ−x ψ
+x
⟩decaying
ZΛ
Z 0Λ
=
∫P0
Λ(dϕ)e−VΛ(ϕ)
y+
y-
( )V y
yl
yt
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
+µh
+γ2hνh
+Zh
+∂0 ∂0
Bh
+∂ ∂
Ah
+∂0
Eh
3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Rigorous RG scheme
ϕ±x = ξ±+ ψ±x⟨ξ−ξ+
⟩= ρ0 ,
⟨ψ−x ψ
+x
⟩decaying
ZΛ
Z 0Λ
=
∫PΛ(dξ)
∫PΛ(dψ)e−Qξ(ψ)−Vξ(ψ)
y+
y-
( )V y
yl
yt
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
+µh
+γ2hνh
+Zh
+∂0 ∂0
Bh
+∂ ∂
Ah
+∂0
Eh
3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Rigorous RG scheme
ϕ±x = ξ±+ ψ±x⟨ξ−ξ+
⟩= ρ0 ,
⟨ψ−x ψ
+x
⟩decaying
ZΛ
Z 0Λ
=
∫PΛ(dξ)
∫P≤h
B (ψ)e−Vh(ψ)
y+
y-
( )V y
yl
yt
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
+µh
+γ2hνh
+Zh
+∂0 ∂0
Bh
+∂ ∂
Ah
+∂0
Eh
3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Rigorous RG scheme
ϕ±x = ξ±+ ψ±x⟨ξ−ξ+
⟩= ρ0 ,
⟨ψ−x ψ
+x
⟩decaying
ZΛ
Z 0Λ
=
∫PΛ(dξ)
∫P≤h
B (ψ)e−Vh(ψ)
y+
y-
( )V y
yl
yt
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
+µh
+γ2hνh
+Zh
+∂0 ∂0
Bh
+∂ ∂
Ah
+∂0
Eh
3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Rigorous RG scheme
ϕ±x = ξ±+ ψ±x⟨ξ−ξ+
⟩= ρ0 ,
⟨ψ−x ψ
+x
⟩decaying
ZΛ
Z 0Λ
=
∫PΛ(dξ)
∫P≤h
B (ψ)e−Vh(ψ)
y+
y-
( )V y
yl
yt
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
+µh
+γ2hνh
+Zh
+∂0 ∂0
Bh
+∂ ∂
Ah
+∂0
Eh
3 Using Gallavotti–Nicolo tree expansion we proved that RVh is well definedwith explicit bounds if the terms in LVh are bounded.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
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.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Result
I The theory is order by order finite in the renormalized coupling constantsprovided that
d = 3 λh −−−−→h→−∞
0
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.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons
The model and resultsIdea of the proof
Rigorous RG schemeWard IdentitiesResult
Perspectives
I interacting bosons on a lattice;
I weak coupling and high density regime;
I critical temperature;
I . . .
I constructive theory.
S. Cenatiempo ICMP12, Aalborg 10 August 2012 BEC for low dimensional interacting bosons