Universality in the BEC-BCS Crossoverklein/MariaLaach/SebastianDiehl.pdf · Universality in the BEC-BCS Crossover Sebastian Diehl Institute for Theoretical Physics, Heidelberg University

Universality in the BEC-BCS Crossover

Sebastian Diehl

Institute for Theoretical Physics, Heidelberg University

in collaboration with

Prof. Christof Wetterich

Sebastian Diehl, Heidelberg University

Outline of the talk

• Introduction

• Tunable Interactions: Feshbach Resonance

• Review of Bose-Einstein Condensation (BEC) and BCS-Theory

• Unified Description for the Crossover Problem

• Approximation Scheme

• Results: Macroscopic Properties and Universality

• Conclusions and Outlook


Introduction

• BEC: bosons populate a single quantum state macroscopically.

• BCS: attractively interacting fermions form bosonic “Cooper

pairs” and successively condense.

• There are fermionic systems in which the interaction can be

manipulated to smoothly connect these simple pictures.

• Universality:

→ unified description of the condensation phenomenon.

→ Macrophysics characterized by three dimensionless parameters

(temperature, interaction strength, Yukawa coupling), no further

reference to microphysics.


Tunable Interactions: Feshbach Resonance

B

scattering length a and binding energy εM

a(B)

εM (B)

→ Physical origin: resonant hyperfine interaction between two

electron spin interaction channels.

⇒ a(B) ≈ abg∆B−B0

, εM (B) = Ma(B)2 ≈

(

B−B0

abg∆

)2

→ a(B) ∝ λR parameterizes the interaction strength for the fermions.

a→ 0+ : microscopic bosonic bound states → BEC of molecules

a→ 0− : weakly attractive interaction → BCS type superfluid


Review of BEC and BCS-Theory: BEC

Classical euclidean action for free, nonrelativistic bosons

SB =

∫

x

φ∗(x)(∂τ − 42MB

− µB)φ(x)

with

• µB chemical potential, “source” for bosonic density.

• φ∗, φ: complex scalar bosonic fields.

• The classical action has a global U(1) - symmetry.


Review of BEC and BCS-Theory: BCS

• Conserved charge for the full quantum theory

(ZB =∫

Dφ exp(−SB [φ])) particle number N or, in a

homogeneous setting, the particle density

n = 〈φ∗φ〉c + 〈φ∗〉〈φ〉

with a thermal part nB = 〈φ∗φ〉c and a condensate part

nC = φ∗φ = 〈φ∗〉〈φ〉 which is nonzero in the low temperature

regime only.

• BEC is a purely statistical effect.

• NB: the macroscopic observable φ breaks the U(1) - symmetry;

this spontaneous symmetry breaking (SSB) gives rise to

phenomena like superfluidity, the existence of vortices etc.



Classical euclidean action for interacting, nonrelativistic fermions

SF =

∫

x

ψ†(x)(∂τ − 42MB

− µ)ψ(x)

+1

2

∫

x,y,z,w

λ(x, y, z, w)ψ†(x)ψ(y)ψ†(z)ψ(w)

with

• µ chemical potential, “source” for fermionic density.

• ψ†, ψ: complex Grassmann-valued 2-spinors (hyperfine spin

states of the atoms).

• Again: global U(1) - symmetry (gauge degrees of freedom

“integrated out”).

• Again: particle number (homogeneous: density) is conserved



charge for the full quantum theory ZF =∫

Dψ exp−SF [ψ]:

nF = 〈ψ†ψ〉c + 〈ψ†〉〈ψ〉

with a thermal part nF = 〈ψ†ψ〉c and 〈ψ†〉 = 〈ψ〉 = 0 due to

Fermi statistics.

• With an attractive interaction, however, possibly

〈ψ†ψ†〉, 〈ψψ〉 6= 0.

BCS theory for weak coupling: transition temperature

Tc ∝ e−κ/|a|, κ > 0 → out of experimental reach.

• NB: these correlation functions also break the U(1) symmetry.

The similar symmetry properties for BEC and BCS cases already

suggest a close connection of the condensation phenomena.


Unified description for the Crossover Problem

• Macroscopic variables for BEC:

nB = 〈φ∗φ〉c, φ = 〈φ〉.

• Macroscopic variables for BCS:

nF = 〈ψ†ψ〉, 〈ψψ〉.

• Unified description of the phase transition by SSB of U(1)

symmetry.

• Elementary constituents of the ensemble are interacting fermions;

their interaction is usually described by a microscopic 4-fermion

vertex.

=⇒ Interpret the fermionic bilinears ψψ ∝ φ, ψ†ψ† ∝ φ∗ as

“molecular fields” and introduce them explicitly into the fermionic

path integral: Partial bosonization of the 4-fermion vertex.


Unified Description for the Crossover Problem

Implementation in the path integral formalism:

1. Write (Gaussian integral) (integration over both φ and φ∗ understood,

m2φ > 0, assume momentum independent (i.e. local) interaction for simplicity)

1 = const.×∫

Dφ exp(

−∫

x

m2φ(φ∗ − hφ

2m2φ

ψ†εψ∗)(φ+hφ

2m2φ

ψT εψ))

.

2. Insert this factor of unity in the original fermionic path integral

and multiply out,∫

DψDφ exp(

−∫

x

ψ†(∂τ − 42MB

− µ)ψ + φ∗m2φφ

+1

2hφ

(

φ∗ψT εψ − φψ†εψ∗)

+1

2

(

λ+h2

φ

m2φ

)

(ψ†ψ)2)

.

3. Impose U = − h2φ

m2φ

⇒ fermionic theory mapped on a Yukawa model.



More generally, one can also deal with nonlocal, i.e. momentum

dependent 4-fermion interactions.

λ(Q) = −h2

φ

P clφ (Q)

where hφ is the atom-molecule Yukawa coupling and

P clφ (Q) = iωm +

q2

4M+ ν − 2µ

is the bare molecule propagator. Its form is determined by symmetry

considerations.



Some remarks:

• Partial bosonization is not unique. Several bosonic “channels”

(e.g. σ ∝ ψ†ψ, vector bosons...) can be included.

• It cannot reproduce the most general momentum structure of the

4-fermion vertex. More channels allow for more complex

momentum structures.

• Close to the Feshbach resonance, it should be very good!

We include also σ, but omit a momentum dependence.



This results in the following classical action (position space)

SFM =

∫

dx[

ψ†(

∂τ − 42M

− σ)

ψ + φ∗(

∂τ − 44M

+ ν − 2σ)

φ

−hφ

2

(

φ∗ψT εψ − φψ†εψ∗)

+m2

2σ2

]

.

The full quantum theory can be extracted from

WFM [J ] = log

∫

DψDφ exp

− SFM [ψ, φ]

+

∫

dx[

J(x)(

ψ†(x)ψ(x) + 2φ∗(x)φ(x))

]

.

→ The local source generalizes the chemical potential µ,

J(x) = µ− Vl(x) (Vl(x): trapping potential!).



Exact properties:

• The conserved charge is (homogeneous situation)

n = 〈ψ†ψ〉c + 2〈φ∗φ〉c + 2〈φ∗〉〈φ〉 = nF + 2nM + nC .

• For constant φ and vanishing source for φ

φ =hφ

2ν − 4σ〈ψT εψ〉

⇒ formalism does not differentiate between “condensation of

molecules” φ and a “condensate of atom pairs” 〈ψT εψ〉!

Up to now the theory is defined as a functional of the external field

J(x).



→ Switch to the effective action Γ via a Legendre transform with

respect to the field expectation value σ = δWFM/δJ ,

Γ = −WFM +∫

jσ ( j = m2J).

⇒ Formulation in terms of the field expectation values or “classical

fields”

φ = 〈φ〉, σ = 〈σ〉, Γ = Γ[σ, φ].

Now concentrate on the homogeneous case → characterized by the

effective potential U (Γ =∫

xU = V U).

Extract the field equations by variation wrt the classical fields,

1. n!= −∂U(σ, φ)

∂σ= nF + 2nM + 2φ∗φ.

• ⇒ the above exact identity is implemented in the formalism!

• n given ⇒ σ plays the role of an effective chemical potential.



By symmetry, the effective potential can depend on the combination

ρ = φ∗φ only. Hence the “field equation” for φ has the form

2. 0!=∂U(σ, ρ)

∂φ∗= α(σ, ρ)φ.

The phase structure is characterized by

Symmetric phase φ(T > Tc) = 0

Phase transition φ(T = Tc) = 0 and α(T = Tc) = 0

Broken phase φ(T < Tc) > 0 and α(T < Tc) = 0

→ Solve these field equations to extract macrophysics.



α also enters the bosonic 2 × 2 mass matrix. Choosing a real basis in

field space φ1, φ2 with φ = (φ1 + iφ2)/√

2 and choosing φ to be real,

it has the form

m2φ =

∂2U

∂φa∂φb=

α(σ, ρ) + 2ρβ(σ, ρ) 0

0 α(σ, ρ)

.

The vanishing of α in the broken phase is associated to the massless

“Goldstone mode”, responsible for “long range” effects like

superfluidity.

Therefore, α must be determined accurately!


Approximation Scheme

• Here: treat σ classical field or effective chemical potential from

the outset, σ = σ.

• Treat hφ as a free parameter.

The effective action to one loop order has the form

φ∗(ν − 2σ)φ+ U(F )1 (σ, ρ) + U

(B)1 (σ, ρ).

→ Classical contribution from the condensate field.

→ Contributions from the one loop fermion and boson fluctuations

U(F )1 = − 1

2Tr logPF

U(B)1 = 1

2Tr logPφ

→ PF : inverse fermion propagator (φ - dependent 4 × 4 matrix for

the discrete indices if computed from the classical action SFM ),

→ Pφ: inverse boson propagator (diagonal degenerate 2× 2 matrix in

the classical approximation).



Strategy: first integrate out the fermions on the Gaussian level,

WFM = log

∫

Dφ exp(−S[φ]),

S[φ] =

∫

x

φ∗(

∂τ − 44M

+ ν − 2σ)

φ+ U(F )1 [σ, φ∗φ].

U(F )1 is still a functional of fluctuating bosonic fields!



• Expand around the classical field φ(K) = φ+ δφ(K):

U(F )1 [φ∗φ] = U

(F )1 [φ∗φ] +

∫

K

δφ∗(−K)∆P(F )φ (K)δφ(K) + O((δφ∗δφ)2)

Then

U = φ∗(ν − 2σ)φ+ U(F )1 (σ, φ∗φ) +

1

2Tr log(P cl

φ + ∆P(F )φ )

→ inconsistent mass determination!

→ The phase transition is of first order.

• Schwinger-Dyson equations for the bosonic theory, particularly

the mass. ⇒ Set of (partially) coupled self-consistent equations.

As the bare mass and coupling parameters, take those from 2.(fermions integrated out to give effective bosonic theory)

→ m2φ ≥ 0 and consistently determined. Additionally, IR

divergences impose an infrared free theory for the bosons in the

broken phase.


Relevant Parameters, Momentum and Energy Scales

• Explicit calculations require UV renormalization.

→ Zero-point shifts to the densities and effective potential.

→ Linear divergence of the fermionic loop contribution to the

boson mass ⇒ renormalization of the bare “detuning” ν → νR.

• characteristic scales: n = k3F /(3π

2) → interparticle spacing

k−1F = (3π2n)1/3, Fermi energy εF = k2

F /(2M).

Define dim.less parameters, e.g. T = T/εF , hφ = 2Mhφ/k1/2F .

⇒ Effective low energy formulation free of cutoff Λ and M .

• Relate the effective scattering length aR (∝ λR) to our

calculational parameters. For the dimensionless “concentration”

c = aRkF we find1

c= −8π(ν − 2σ)

h2φ

⇒ Everything formulated in the dim.less parameters T , c, hφ.


Kinetic part of Pφ: Wave function renormalization

0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.02

0.04

0.06

0.08

0.1

0.12

q

q3NM (q)

(a)- 3 - 2 - 1 0 1 2 3 4

2

4

6

8

c−1

BCS BEC

•

Zφ(0)

(b)

Figure 1: (a) Bosonic distribution function with classical WFR

Zφ = 1/4M (dashed-dotted), with Zφ(q) (solid) and Zφ(0).

(b) WFR for various Yukawa couplings hφ = 0.1, 1, 10, 100 (left to

right).

Zφ(0) =1

2+ h2

φ(Integral); supression nM ∝ Z−3/2φ


Chemical Potential and mass contributions

- 2 - 1 0 1 2

- 4

- 3

- 2

- 1

0

1

c−1

σ

T = Tc

(a)- 2 - 1 0 1 2

- 1

- 0.5

0

0.5

1

c−1

contributions to bosonic mass

T = Tc

(b)

Figure 2: (a) Effective chemical potential for vari-

ous Yukawa couplings hφ = 0.1, 1, 10, 100 (down to up)

(b) Fluctuation contributions to the full bosonic mass at the

critical temperature, m2φ(T = Tc) = 0: “classical” (solid), fermionic

(dashed) and bosonic (dashed-dotted) (hφ = 10).


Crossover Phase Diagram

- 1 0 1 2 3 4

0.1

0.2

0.3

0.4

0.5

c−1

Tc

BCS BEC

Figure 3: Phase diagram for the crossover system for various Yukawa

couplings (hφ = 0.1, 1, 10, 100 (downmost to uppermost). The expec-

tations for the limiting cases are reproduced.


Density Fractions at Tc

- 3 - 2 - 1 0 1 20

0.2

0.4

0.6

0.8

1

c−1

ΩF = nF /n, ΩM = 2nM/n

ΩF

BCS

T = Tc

ΩM

BEC

Figure 4: Crossover from fermion to boson domination at the criti-

cal temperature for various Yukawa couplings. Fermionic density de-

creases slower for larger Yukawa couplings (still ΩC = 0).


Density Fractions at T = 0

- 3 - 2 - 1 0 1 2 30

0.2

0.4

0.6

0.8

1

c−1

ΩF

T = 0

ΩC

Figure 5: Crossover from fermion to condensate domination at T = 0

for various Yukawa couplings. Fermionic density decreases slower for

larger Yukawa couplings (ΩM = 0 at zero temperature).


Density Fractions in the broken phase

0.05 0.1 0.15 0.2

0.2

0.4

0.6

0.8

1

T

ΩF ,ΩM ,ΩC

hφ = 1

0.05 0.1 0.15 0.2 0.25

0.2

0.4

0.6

0.8

1

T

ΩF ,ΩM ,ΩC

hφ = 10

Figure 6: Second order phase transition: Density fractions ΩF (solid),

ΩM (dashed), ΩC (dashed-dotted) in the crossover regime (c−1 =

−0.70 for the left, c−1 = 0.46 for the right plot.)


Further aspects of present work

• Establish contact to microphysics (scattering properties).

Effective action contains all information about the (1PI) vertices

⇒ scattering physics obtained in the limit of infinite interparticle

spacing kF → 0, but fixed dimensionless temperature

T = T/TF = 2MT/k2F . Reproduces several aspects of quantum

mechanical calculations.

• Implementation of a local trapping potential

Use position dependent local source terms J(x) = µ− Vl(x)

explicitly. Allows e.g. for a description of vortices.


Conclusions and Outlook

• Unified description of the condensation phenomenon for all

coupling and temperature regimes in terms of the effective

action. The condensation is signalled by the spontaneous

symmetry breaking of the U(1) symmetry.

• All observables can be expressed in only 3 dimensionless

parameters c−1, hφ, T = T/TF → universal description of the

ensemble with no further reference to the microphysics.

• The extreme BEC and BCS regimes reveal a particularly high

degree of universality.

• The infrared properties of the bosons are particularly simple (IR

free theory in the broken phase). The phase transition is of

second order.


Conclusions and Outlook

• extension to the inhomogeneous case to properly account for the

trapping potential.

• More sophisticated analysis: functional renormalization group

equations.

→ Gain control over the true Yukawa coupling hφ. Further

aspect of universality possible.

→ Characterize critical region more accurately.

• Finally: direct quantitative comparison to current experiments.


Documents

Universality in the BEC-BCS Crossoverklein/MariaLaach/SebastianDiehl.pdf · Universality in the BEC-BCS Crossover Sebastian Diehl Institute for Theoretical Physics, Heidelberg University