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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
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
Review of BEC and BCS-Theory: BCS
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
Sebastian Diehl, Heidelberg University
Review of BEC and BCS-Theory: BCS
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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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.
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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.
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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!).
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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).
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
→ 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.
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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.
Sebastian Diehl, Heidelberg University
Unified Description for the Crossover Problem
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!
Sebastian Diehl, Heidelberg University
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).
Sebastian Diehl, Heidelberg University
Approximation Scheme
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!
Sebastian Diehl, Heidelberg University
Approximation Scheme
• 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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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
Sebastian Diehl, Heidelberg University
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).
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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).
Sebastian Diehl, Heidelberg University
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).
Sebastian Diehl, Heidelberg University
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.)
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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.
Sebastian Diehl, Heidelberg University
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