BEC for low dimensional interacting bosons€¦ · Milano, 11 December 2012. Motivations The model...

BEC for low dimensional interacting bosons

Serena Cenatiempo

joint work with A. Giuliani

Dipartimento di Matematica “F. Enriques”

Milano, 11 December 2012

MotivationsThe model and results

Idea of the proof

Bose–Einstein condensationfor non interacting bosons (1925)

The equilibrium properties of the ideal Bose gas at β−1 temperature areencoded in the grand–canonical partition function:

ZΩ,β =∑N≥0

e−β(HΩ−µΩ,βN) HΩ = − ~2m

N∑i=1

∆x i

When we calculate the density of the system

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1'

βk2 ≤ ρcriticalβ

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

lim|Ω|→∞〈N0〉|Ω| = (const.)

S. Cenatiempo Milano, 11.12.2012 BEC for low dimensional interacting bosons

Idea of the proof

ZΩ,β =∑N≥0

N∑i=1

∆x i

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1'

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

Idea of the proof

ZΩ,β =∑N≥0

N∑i=1

∆x i

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1

'k→0

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

Idea of the proof

ZΩ,β =∑N≥0

N∑i=1

∆x i

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1'

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

Idea of the proof

ZΩ,β =∑N≥0

N∑i=1

∆x i

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1'

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

Idea of the proof

ZΩ,β =∑N≥0

N∑i=1

∆x i

ρΩ,β =〈N〉|Ω| =

|Ω|∑

eβ(k2−µΩ,β) − 1k = 2π

Ln n ∈ Zd

lim|Ω|→+∞

ρΩ,β =1

(2π)d

eβ(k2−µΩ,β ) − 1'

ρβ,Ω =〈N〉|Ω| =

〈N0〉|Ω| +

|Ω|∑k 6=0

eβ(k2−µΩ,β ) − 1

BEC occurrs when

Idea of the proof

It was only in 1995 The ground state wave function is simply aproduct of single particle wave–functions:

Ψ0(x1, . . . , xN ) =∏N

i ψ0(xi )

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

Idea of the proof

It was only in 1995 The ground state wave function is simply aproduct of single particle wave–functions:

Ψ0(x1, . . . , xN ) =∏N

i ψ0(xi )

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

Idea of the proof

BEC nowadays is an ultralow–temperature laboratory for

quantum optics, atomic physics and condensed matter physics

Hansel et al., “Atom chip” (2001)Kruger et al. (2007)

Billy et al., Guided atom laser (2008)

Nature Physics 1 (2005)Klaers et al., BEC of phonons (2010)

Idea of the proof

BEC nowadays is an ultralow–temperature laboratory for

quantum optics, atomic physics and condensed matter physics

Hansel et al., “Atom chip” (2001)Kruger et al. (2007)

Billy et al., Guided atom laser (2008)

Nature Physics 1 (2005)Klaers et al., BEC of phonons (2010)

State of the art

There is a single model in which we can prove BEC for homogeneousinteracting bosons (Dyson, Lieb and Simon, 1978). More recent resultson trapped bosons (Lieb, Seiringer and Yngvason, 2002).

Idea 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∑

(−∆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+

⟩β−−−−−−→|x−y |→∞ρ fixed

const.

Idea of the proof

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+

⟩β−−−−−−→|x−y |→∞ρ fixed

const.

Idea of the proof

Set of the problem with functional integrals

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

Λ(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 =|Ω|,β→∞

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)

Idea of the proof

Λ(dϕ) e−VΛ(ϕ)

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0=

ρ0 +1

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

Idea of the proof

Λ(dϕ) e−VΛ(ϕ)

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0=

ρ0 +1

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

Idea of the proof

Λ(dϕ) e−VΛ(ϕ)

VΛ(ϕ) =λ

∫Ω×Ω

ddx ddy

∫ β/2

−β/2

∫ β/2

−β/2

dt |ϕx,t |2

S0Λ(x , y) =

⟩∣∣∣λ=0=

ρ0 +1

(2π)d+1

∫Rd+1

ddk dk0e−ikx

−ik0 + k2

⟩= ρ0

Idea of the proof

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) ' c2B

k20 + c2

Bk2 → ik0 = ±cB |k|

g free−+(k) =

−ik0 + k2

Main goal: to control and compute in a systematic way

the corrections to Bogoliubov theory at weak coupling.

Idea of the proof

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

SBΛ (x , y) =

⟩∣∣Bog

(2π)d+1

∫Rd+1

g−+(k) ' c2B

k20 + c2

Bk2 → ik0 = ±cB |k|

g free−+(k) =

−ik0 + k2

Idea of the proof

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

SBΛ (x , y) =

⟩∣∣Bog

(2π)d+1

∫Rd+1

g−+(k) ' c2B

k20 + c2

Bk2 → ik0 = ±cB |k|

g free−+(k) =

−ik0 + k2

Idea of the proof

E(k) =√

k4 + 2λρ0v(k)k2 '|k|→0

√2λρ0v(0) |k| = cB |k|

SBΛ (x , y) =

⟩∣∣Bog

(2π)d+1

∫Rd+1

g−+(k) ' c2B

k20 + c2

Bk2 → ik0 = ±cB |k|

g free−+(k) =

−ik0 + k2

Idea of the proof

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).

Benfatto (1994, 1997): first systematic study of the infrared diver-

gences 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 (non rigorous RG scheme)

“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)

Idea of the proof

Perturbation theory

Idea of the proof

Perturbation theory

Idea of the proof

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

Idea of the proof

Main result

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

I the interacting theory is well defined at all orders in terms of twoeffective parameters (three and two particles effective interactions)

with coefficient of order n bounded by (const.)n n!.

I the correlations do not exhibit anomalous exponents: same universalityclass of the exactly solvable Bogoliubov model.

gBogoliubov−+ (k) ∝ 1

k20 + c2

B k2 −→ g interacting−+ (k) ∝ 1

k20 + c2(λ) k2

Key points1. three new effective coupling constants

2. to use local WIs within a RG schemebased on a momentum regularization

Idea of the proof

Main result

k20 + c2

k20 + c2(λ) k2

Idea of the proof

Main result

k20 + c2

k20 + c2(λ) k2

Idea of the proof

Main result

k20 + c2

k20 + c2(λ) k2

Idea of the proof

Exact RG schemeWard IdentitiesFlow equations

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

Λ(dϕ)e−VΛ(ϕ)

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 2h : Vh(ψ) = LVh(ψ) +RVh(ψ)

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

3 Using Gallavotti–Nicolo tree expansion we prove that RVh

is well defined with explicit bounds if the terms in LVh are bounded.

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫PΛ(dψ)e−Qξ(ψ)−Vξ(ψ)

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

∫PΛ(dξ)

∫PΛ(dψ)e−Qξ(ψ)−Vξ(ψ)︸︷︷︸

e−|Λ|WΛ(ξ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫PΛ(dψ)e−Qρ0

(ψ)−Vρ0(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫PΛ(dψ)e−Qρ0

(ψ)−Vρ0(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫P≤h

Λ (dψ) e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫P≤h

Λ (dψ) e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫P≤h

Λ (dψ) e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Gallavotti–Nicolo tree expansion

The |h|–th step of the iterative integration can be graphically representedas a sum of trees over |h| scale labels. The number n of endpointsrepresents the order in perturbation theory.

−2 −1 0 1

⇐ Γ =

Gallavotti–Nicolo trees are a syntetic and convenient way to isolate thedivergent terms, avoiding “overlaps in divergences”.

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫P≤h

Λ (dψ) e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

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

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

Idea of the proof

Exact RG scheme

ϕ±x = ξ±+ ψ±x⟨ξ−ξ+

⟩= ρ0 ,

⟨ψ−x ψ

⟩decaying

e−WΛ(ρ0) =

∫P≤h

Λ (dψ) e−Vh(ψ)

( )V y

Bk2 ' 2h, h ∈ (−∞, 0]

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

γhλh

γh2 µh

+γ2hνh

+∂0 ∂0

+∂x ∂x

λ′h µ′

Idea of the proof

Ward Identities to reduce the number of independent couplings

I 2 +3 Global WIsλh

I 3 Local WIs

Task: the momentum cutoffsbreak the local gauge invariance

→ In low-dimensional systems of interacting fermions( Luttinger liquids ) the corrections to WIs arecrucial for establishing the infrared behavior

Result: using techniques by ( Benfatto, Falco, Mastropietro, 2009)we studied the flow of the corrections (marginal and relevant) andproved that

* they do not change the way WIs are used to control the flow;

* but they are possibly observable in the relationsamong thermodynamic and response functions.

Idea of the proof

Ward Identities to reduce the number of independent couplings

I 2 +3 Global WIsλh

I 3 Local WIs

Task: the momentum cutoffsbreak the local gauge invariance

→ In low-dimensional systems of interacting fermions( Luttinger liquids ) the corrections to WIs arecrucial for establishing the infrared behavior

Result: using techniques by ( Benfatto, Falco, Mastropietro, 2009)we studied the flow of the corrections (marginal and relevant) andproved that

* they do not change the way WIs are used to control the flow;

* but they are possibly observable in the relationsamong thermodynamic and response functions.

Idea of the proof

Two independent flow equations coupled among them at all orders

= + + +

+ + + +

5 10 15 20 h¤

Numerical solutions to the leading order

flows for λλh and λ6,h/(λλ2h)

I The two effective parametershave non trivial flows

The behavior of the propagatoronly depends on the

existence of the fixed pointsfor λh and λ6,h.

Idea of the proof

= + + +

+ + + +

5 10 15 20 h¤

Idea of the proof

= + + +

+ + + +

5 10 15 20 h¤

Idea of the proof

Outlook

I renormalizability of the UV region;

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€¦ · Milano, 11 December 2012. Motivations The model...

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