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Correlations in the Low-Density Fermi Gas:Fermi-Liquid state, BCS Pairing, and Dimerization
With Hsuan-Hao FanDepartment of Physics, University at Buffalo SUNY
and Robert ZillichInstitute for Theoretical Physics, JKU Linz, Austria.
Barcelona, June 7, 2017
Outline
1 GenericaThe equation of state
2 MethodsVariational wave functions and optimizationVerification: Bulk quantum fluids
3 Structure CalculationsWhat we expect and what we getThe normal Fermi Liquid
4 Pairing with strong correlationsPairing interactionMany-Body effectsResultsQuo vadis ?
5 A visitor6 Summary7 Acknowledgements
The equation of stateEasy questions, hard (?) answers...
A truly microscopic many-bodytheory should:
Be robust against the choice ofinteractions;
V(r
)
(s
ome
units
)
r (some units)
ε
σ
Molecules/AtomsHard spheresCoulomb
Generica The equation of state
The equation of stateEasy questions, hard (?) answers...
A truly microscopic many-bodytheory should:
Be robust against the choice ofinteractions;
Be able to deal with self-boundsystems;
V(r
)
(s
ome
units
)
r (some units)
ε
σ
Molecules/AtomsHard spheresCoulomb
ρ (some units)
E/N
(som
e un
its)
bosons
fermions
Generica The equation of state
The equation of stateEasy questions, hard (?) answers...
A truly microscopic many-bodytheory should:
Be robust against the choice ofinteractions;
Be able to deal with self-boundsystems;
Have no answers if “mothernature” does not have them;
V(r
)
(s
ome
units
)
r (some units)
ε
σ
Molecules/AtomsHard spheresCoulomb
ρ (some units)
E/N
(som
e un
its)
bosons
fermions
Generica The equation of state
The equation of stateEasy questions, hard (?) answers...
A truly microscopic many-bodytheory should:
Be robust against the choice ofinteractions;
Be able to deal with self-boundsystems;
Have no answers if “mothernature” does not have them;
Technically: Sum at least the“parquet” diagrams.
V(r
)
(s
ome
units
)
r (some units)
ε
σ
Molecules/AtomsHard spheresCoulomb
ρ (some units)
E/N
(som
e un
its)
bosons
fermions
Generica The equation of state
What physics tells us:
Binding and saturation =⇒ short-ranged structure:“bending” of the wave function at small interparticle distances;“local screening” or “local field corrections” in electron systems.
Generica The equation of state
What physics tells us:
Binding and saturation =⇒ short-ranged structure:“bending” of the wave function at small interparticle distances;“local screening” or “local field corrections” in electron systems.
“No answers when mother nature does not have them”:Show correct instabilities;Deal properly with long-ranged correlations;
Generica The equation of state
What physics tells us:
Binding and saturation =⇒ short-ranged structure:“bending” of the wave function at small interparticle distances;“local screening” or “local field corrections” in electron systems.
“No answers when mother nature does not have them”:Show correct instabilities;Deal properly with long-ranged correlations;
Translate this into the language of perturbation theory:
Short-ranged structure ⇒ Ladder diagramsLong-ranged structure ⇒ Ring diagramsConsistency ⇒ parquet diagrams
Generica The equation of state
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
An intuitive way to include
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Density profiles of 4Hefilms
0.00
0.01
0.02
0.03
0.04
0.05
0 5 10 15 20 25 30
0.068 Å-2
0.137 Å-2
z (Å)
ρ(z)
(Å
-3)
An intuitive way to include inhomogeneity,
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
]
exp(u2(r))
r (Å)
An intuitive way to include inhomogeneity,core exclusion
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
]
exp(u2(r))
r (Å)
An intuitive way to include inhomogeneity,core exclusion and statistics;
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
] g(r)
exp(
u(r)
)
r (Å)
An intuitive way to include inhomogeneity,core exclusion and statistics;Diagram summation methods fromclassical statistics (HNC);
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
]
exp(u2(r))
r (Å)
An intuitive way to include inhomogeneity,core exclusion and statistics;Diagram summation methods fromclassical statistics (HNC);Optimization δE/δun = 0 makescorrelations unique;
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
] g(r)
exp(
u(r)
)
r (Å)
An intuitive way to include inhomogeneity,core exclusion and statistics;Diagram summation methods fromclassical statistics (HNC);Optimization δE/δun = 0 makescorrelations unique;Express everything in terms of physicalobservables;
Methods Variational wave functions and optimization
Correlated wave functions: Putting things together
What looked like a “simple quick and dirty” method:
Ψ0(1, . . . ,N) = exp12
[∑
i
u1(ri) +∑
i<j
u2(ri , rj) + . . .
]
Φ0(1, . . . ,N)
≡ F (1, . . . ,N)Φ0(1, . . . ,N)
Φ0(1, . . . ,N) “Model wave function” (Slater determinant)
Correlation- anddistribution functions
−15
−10
−5
0
5
10
15
0.0
0.5
1.0
1.5
0 2 4 6 8 10
V(r
)
[K
] g(r)
exp(
u(r)
)
r (Å)
An intuitive way to include inhomogeneity,core exclusion and statistics;Diagram summation methods fromclassical statistics (HNC);Optimization δE/δun = 0 makescorrelations unique;Express everything in terms of physicalobservables;
Same as summing localized parquetdiagrams.
Methods Variational wave functions and optimization
Microscopic ground state calculationsAn overkill at low densities ?
Assume an attractive square-well potential
Vsc(r) = ǫθ(σ − r)
or a Lennard-Jones potential
VJL(r) = 4ǫ[(σ
r
)12−
(σ
r
)6] −4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
Methods Verification: Bulk quantum fluids
Microscopic ground state calculationsAn overkill at low densities ?
Assume an attractive square-well potential
Vsc(r) = ǫθ(σ − r)
or a Lennard-Jones potential
VJL(r) = 4ǫ[(σ
r
)12−
(σ
r
)6]
Adjust potential depth ǫ to produce thedesired scattering length,
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
Methods Verification: Bulk quantum fluids
Microscopic ground state calculationsAn overkill at low densities ?
Assume an attractive square-well potential
Vsc(r) = ǫθ(σ − r)
or a Lennard-Jones potential
VJL(r) = 4ǫ[(σ
r
)12−
(σ
r
)6]
Adjust potential depth ǫ to produce thedesired scattering length,
Microscopic many-body calculationsprovide energetics, structure, and stability.
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
Methods Verification: Bulk quantum fluids
Microscopic ground state calculationsAn overkill at low densities ?
Assume an attractive square-well potential
Vsc(r) = ǫθ(σ − r)
or a Lennard-Jones potential
VJL(r) = 4ǫ[(σ
r
)12−
(σ
r
)6]
Adjust potential depth ǫ to produce thedesired scattering length,
Microscopic many-body calculationsprovide energetics, structure, and stability.
Interested in “low-density regime” asfunction of the scattering length a.
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
Methods Verification: Bulk quantum fluids
Verification I: The Lennard-Jones liquidHow well it works (Bragbook)
Equation of state for Bosons
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.10 0.20 0.30 0.40
4He
E/N
ρ (σ−3)
e =10e = 7e = 6e = 5e = 2e = 1
PIGS−MCHNC−EL/0HNC−EL/5+T
Methods Verification: Bulk quantum fluids
Verification I: The Lennard-Jones liquidHow well it works (Bragbook)
Equation of state for Bosons
Equation of state for Fermions
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.10 0.20 0.30 0.40
4He
E/N
ρ (σ−3)
e =10e = 7e = 6e = 5e = 2e = 1
PIGS−MCHNC−EL/0HNC−EL/5+T
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.10 0.20 0.30 0.40
3Henuclear matter
FHNC−EL//0FHNC−EL/5+T
E/N
ρ (σ−3)
Methods Verification: Bulk quantum fluids
Verification I: The Lennard-Jones liquidHow well it works (Bragbook)
Equation of state for Bosons
Equation of state for Fermions
“Quick and dirty” version haspercent accuracy below0.25*(saturation density). Nonew physics is learned fromdoing a better calculation.
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.10 0.20 0.30 0.40
4He
E/N
ρ (σ−3)
e =10e = 7e = 6e = 5e = 2e = 1
PIGS−MCHNC−EL/0HNC−EL/5+T
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.10 0.20 0.30 0.40
3Henuclear matter
FHNC−EL//0FHNC−EL/5+T
E/N
ρ (σ−3)
Methods Verification: Bulk quantum fluids
Verification I: The Lennard-Jones liquidHow well it works (Bragbook)
Equation of state for Bosons
Equation of state for Fermions
“Quick and dirty” version haspercent accuracy below0.25*(saturation density). Nonew physics is learned fromdoing a better calculation.
Works the same in 2D
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.10 0.20 0.30 0.40
4He
E/N
ρ (σ−3)
e =10e = 7e = 6e = 5e = 2e = 1
PIGS−MCHNC−EL/0HNC−EL/5+T
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.10 0.20 0.30 0.40
3Henuclear matter
FHNC−EL//0FHNC−EL/5+T
E/N
ρ (σ−3)
Methods Verification: Bulk quantum fluids
Verification I: The Lennard-Jones liquidHow well it works (Bragbook)
Equation of state for Bosons
Equation of state for Fermions
“Quick and dirty” version haspercent accuracy below0.25*(saturation density). Nonew physics is learned fromdoing a better calculation.
Works the same in 2D
FHNC-EL (or parquet) has nosolutions if “mother nature”cannot make the system: Theequation of state ends at thespinodal points.
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
0.00 0.10 0.20 0.30 0.40
4He
E/N
ρ (σ−3)
e =10e = 7e = 6e = 5e = 2e = 1
PIGS−MCHNC−EL/0HNC−EL/5+T
−4.0
−2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.00 0.10 0.20 0.30 0.40
3Henuclear matter
FHNC−EL//0FHNC−EL/5+T
E/N
ρ (σ−3)
Methods Verification: Bulk quantum fluids
Low-density limitsSince everybody talks about cold gases....
Equation of state for Bosons (Lee-Yang)
ELY = 4π~
2ρa2m
[
1 +128
15π1/2
(
ρa3)1/2
+ . . .
]
.
. . . if the system exists !
Methods Verification: Bulk quantum fluids
Low-density limitsSince everybody talks about cold gases....
Equation of state for Bosons (Lee-Yang)
ELY = 4π~
2ρa2m
[
1 +128
15π1/2
(
ρa3)1/2
+ . . .
]
.
. . . if the system exists !
Equation of state for Fermions (Huang-Yang)
EN
=~
2k2F
2m
35+
23
akF
π+
4(11 − 2 ln 2)35
︸ ︷︷ ︸
=1.098
(akF
π
)2
+ . . .
Methods Verification: Bulk quantum fluids
Low-density limitsSince everybody talks about cold gases....
Equation of state for Bosons (Lee-Yang)
ELY = 4π~
2ρa2m
[
1 +128
15π1/2
(
ρa3)1/2
+ . . .
]
.
. . . if the system exists !
Equation of state for Fermions (Huang-Yang)
EN
=~
2k2F
2m
35+
23
akF
π+
4(11 − 2 ln 2)35
︸ ︷︷ ︸
=1.098
(akF
π
)2
+ . . .
Local correlation operator (“fixed node”) gives 1.53517
Methods Verification: Bulk quantum fluids
Low-density limitsSince everybody talks about cold gases....
Equation of state for Bosons (Lee-Yang)
ELY = 4π~
2ρa2m
[
1 +128
15π1/2
(
ρa3)1/2
+ . . .
]
.
. . . if the system exists !
Equation of state for Fermions (Huang-Yang)
EN
=~
2k2F
2m
35+
23
akF
π+
4(11 − 2 ln 2)35
︸ ︷︷ ︸
=1.098
(akF
π
)2
+ . . .
Local correlation operator (“fixed node”) gives 1.53517
2nd order correlated basis functions perturbation theory repairs
Methods Verification: Bulk quantum fluids
Low-density limitsSince everybody talks about cold gases....
Equation of state for Bosons (Lee-Yang)
ELY = 4π~
2ρa2m
[
1 +128
15π1/2
(
ρa3)1/2
+ . . .
]
.
. . . if the system exists !
Equation of state for Fermions (Huang-Yang)
EN
=~
2k2F
2m
35+
23
akF
π+
4(11 − 2 ln 2)35
︸ ︷︷ ︸
=1.098
(akF
π
)2
+ . . .
Local correlation operator (“fixed node”) gives 1.53517
2nd order correlated basis functions perturbation theory repairs
The same problem, same solution with electrons:(0.05690 ln rs instead of the exact value 0.06218 ln rs).
Methods Verification: Bulk quantum fluids
Low-density limitThe moment of truth....
Equation of state for Fermions (Huang-Yang)
EN
=~
2k2F
2m
35+
23
akF
π+
4(11 − 2 ln 2)35
︸ ︷︷ ︸
=1.098
(akF
π
)2
+ . . .
+ BCS corrections for a < 0.
0.980
0.985
0.990
0.995
1.000
0.010 0.100
square−well
E/E
HY
a kF
a = −1a = −2a = −3a = −4
0.980
0.985
0.990
0.995
1.000
0.010 0.100
Lennard−Jones
E/E
HY
a kF
a = −1a = −2a = −3a = −4
Methods Verification: Bulk quantum fluids
Microscopic ground state calculations– the many-body effects
Three (not two) range regimes
Methods Verification: Bulk quantum fluids
Microscopic ground state calculations– the many-body effects
Three (not two) range regimes
Short-ranged correlations0 ≤ r ≤ λσ determined byinteraction(λσ a typical interaction range)
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Lennard−Jones
Γ dd(
r)
r/σ
kF=0.001kF=0.010kF=0.040
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Square−well
Γ dd(
r)r/σ
kF=0.001kF=0.010kF=0.040
Methods Verification: Bulk quantum fluids
Microscopic ground state calculations– the many-body effects
Three (not two) range regimes
Short-ranged correlations0 ≤ r ≤ λσ determined byinteraction(λσ a typical interaction range)
Medium-ranged correlationsλσ ≤ r ≤ 1/kF determined byvaccum properties, ψ(r) ∝ a0/r
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Lennard−Jones
Γ dd(
r)
r/σ
kF=0.001kF=0.010kF=0.040
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Square−well
Γ dd(
r)r/σ
kF=0.001kF=0.010kF=0.040
Methods Verification: Bulk quantum fluids
Microscopic ground state calculations– the many-body effects
Three (not two) range regimes
Short-ranged correlations0 ≤ r ≤ λσ determined byinteraction(λσ a typical interaction range)
Medium-ranged correlationsλσ ≤ r ≤ 1/kF determined byvaccum properties, ψ(r) ∝ a0/r
Long-ranged correlations1/kF ≤ r ≤ ∞ determined bymany-body properties:ψ(r) ∝ F s
0 /(r2k2
F)
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Lennard−Jones
Γ dd(
r)
r/σ
kF=0.001kF=0.010kF=0.040
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Square−well
Γ dd(
r)r/σ
kF=0.001kF=0.010kF=0.040
Methods Verification: Bulk quantum fluids
Microscopic ground state calculationsWhat we expect –
For a0 > 0, no bound state→ repulsive Fermi gas;
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect –
For a0 > 0, no bound state→ repulsive Fermi gas;
For a0 < 0 (“BCS” regime):BCS pairing;
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect –
For a0 > 0, no bound state→ repulsive Fermi gas;
For a0 < 0 (“BCS” regime):BCS pairing;
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ?
−4
−2
0
2
4
8
10
0 5 10 15 20
a 0
BEC
BCS
BEC
BCS
e
LJSW
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect –
For a0 > 0, no bound state→ repulsive Fermi gas;
For a0 < 0 (“BCS” regime):BCS pairing;
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ?
For a0 < 0 (“BCS” regime):spinodal instabilites.
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.2
0.4
0.6
0.8
1.0
0.0001 0.001 0.01 0.1 1 10
What one might expect
mc2
(ρ)/m
c2 F
ρ
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect – what we get
For a0 > 0, no bound state→ repulsive Fermi gas; X
For a0 < 0 (“BCS” regime):BCS pairing;
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ?
For a0 < 0 (“BCS” regime):spinodal instabilites.
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.2
0.4
0.6
0.8
1.0
0.0001 0.001 0.01 0.1 1 10
What one might expect
mc2
(ρ)/m
c2 F
ρ
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect – what we get
For a0 > 0, no bound state→ repulsive Fermi gas; X
For a0 < 0 (“BCS” regime):BCS pairing; X
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ?
For a0 < 0 (“BCS” regime):spinodal instabilites.
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.2
0.4
0.6
0.8
1.0
0.0001 0.001 0.01 0.1 1 10
What one might expect
mc2
(ρ)/m
c2 F
ρ
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect – what we get
For a0 > 0, no bound state→ repulsive Fermi gas; X
For a0 < 0 (“BCS” regime):BCS pairing; X
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ? X
For a0 < 0 (“BCS” regime):spinodal instabilites.
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.2
0.4
0.6
0.8
1.0
0.0001 0.001 0.01 0.1 1 10
What one might expect
mc2
(ρ)/m
c2 F
ρ
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we expect – what we get
For a0 > 0, no bound state→ repulsive Fermi gas; X
For a0 < 0 (“BCS” regime):BCS pairing; X
For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ? X
For a0 < 0 (“BCS” regime):spinodal instabilites. ✗
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.2
0.4
0.6
0.8
1.0
0.0001 0.001 0.01 0.1 1 10
What one might expect
mc2
(ρ)/m
c2 F
ρ
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we got – and what it means
For SC potentials, there is nohigh-density homogeneousphase and no upper spinodalpoint
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal point
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we got – and what it means
For SC potentials, there is nohigh-density homogeneousphase and no upper spinodalpoint
It is impossible to get close tothe lower instability;
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal point
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1e−08 1e−07 1e−06 1e−05 0.0001 0.001 0.01 0.1 1
mc2
(ρ)/m
c2 F
ρ (σ−3)
strong − weak
square−well potential
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we got – and what it means
For SC potentials, there is nohigh-density homogeneousphase and no upper spinodalpoint
It is impossible to get close tothe lower instability;
For LJ potentials, we have arepulsive high-density regimeand an upper spinodal point;
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we got – and what it means
For SC potentials, there is nohigh-density homogeneousphase and no upper spinodalpoint
It is impossible to get close tothe lower instability;
For LJ potentials, we have arepulsive high-density regimeand an upper spinodal point;
It is easy to get close to theupper instability,
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1e−08 1e−07 1e−06 1e−05 0.0001 0.001 0.01 0.1 1
mc2
(ρ)/m
c2 F
ρ (σ−3)
strong − weak
Lennard−Jones potential
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat we got – and what it means
For SC potentials, there is nohigh-density homogeneousphase and no upper spinodalpoint
It is impossible to get close tothe lower instability;
For LJ potentials, we have arepulsive high-density regimeand an upper spinodal point;
It is easy to get close to theupper instability,
It is impossible to get close tothe lower instability;
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.0 0.5 1.0 1.5 2.00.0
0.2
0.4
0.6
0.8
1.0
1.2
E/N
(kF)
(a
rbitr
ary
units
)
c/c F
(kF)
(a
rbitr
ary
units
)
kF (arbitrary units)
Schematic equation of stateof a self−bound Fermi Fluid
spinodal points
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1e−08 1e−07 1e−06 1e−05 0.0001 0.001 0.01 0.1 1
mc2
(ρ)/m
c2 F
ρ (σ−3)
strong − weak
Lennard−Jones potential
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat it means
The pair distribution functiong↑↓(r) develops a huge peakat the origin:
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
g ↑↓
(r)
r/σ
SWLJkF = 0.01/σ
a0 = −3.5a0 = −3.0a0 = −2.5a0 = −2.0a0 = −1.5a0 = −1.0a0 = −0.5
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat it means
The pair distribution functiong↑↓(r) develops a huge peakat the origin:
Hard to see for hard-coreinteractions;
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
g ↑↓
(r)
r/σ
SWLJkF = 0.01/σ
a0 = −3.5a0 = −3.0a0 = −2.5a0 = −2.0a0 = −1.5a0 = −1.0a0 = −0.5
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat it means
The pair distribution functiong↑↓(r) develops a huge peakat the origin:
Hard to see for hard-coreinteractions;
Seen in the divergence of thein-medium scattering length⇒Formation of dimers ?
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
g ↑↓
(r)
r/σ
SWLJkF = 0.01/σ
a0 = −3.5a0 = −3.0a0 = −2.5a0 = −2.0a0 = −1.5a0 = −1.0a0 = −0.5
1.0
1.2
1.4
1.6
1.8
2.0
0.0 0.1 0.2 0.3 0.4
a/a 0
−kF a0
LJSW
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat it means
The pair distribution functiong↑↓(r) develops a huge peakat the origin:
Hard to see for hard-coreinteractions;
Seen in the divergence of thein-medium scattering length⇒Formation of dimers ?
This is a many-body effect(“phonon-exchange drivendimerization”) !
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
g ↑↓
(r)
r/σ
SWLJkF = 0.01/σ
a0 = −3.5a0 = −3.0a0 = −2.5a0 = −2.0a0 = −1.5a0 = −1.0a0 = −0.5
1.0
1.2
1.4
1.6
1.8
2.0
0.0 0.1 0.2 0.3 0.4
a/a 0
−kF a0
LJSW
Structure Calculations The normal Fermi Liquid
Microscopic ground state calculationsWhat it means
The pair distribution functiong↑↓(r) develops a huge peakat the origin:
Hard to see for hard-coreinteractions;
Seen in the divergence of thein-medium scattering length⇒Formation of dimers ?
This is a many-body effect(“phonon-exchange drivendimerization”) !
A similar effect seen in 2D3He-4He mixtures
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
g ↑↓
(r)
r/σ
SWLJkF = 0.01/σ
a0 = −3.5a0 = −3.0a0 = −2.5a0 = −2.0a0 = −1.5a0 = −1.0a0 = −0.5
1.0
1.2
1.4
1.6
1.8
2.0
0.0 0.1 0.2 0.3 0.4
a/a 0
−kF a0
LJSW
Structure Calculations The normal Fermi Liquid
On the BCS side: Theory with strong correlationsHow (not to) derive a BCS wave function with correlations
Early attempt:∣∣BCS
⟩=
∏
k
[
uk + vka†k↑a
†−k↓
] ∣∣0⟩,
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩⟨
m(N)∣∣∣BCS
⟩
EK and J. W. Clark, Nucl. Phys. A333, 77 (1980):Expand in the deviation of the Bogoljubov amplitudes uk, vk fromtheir normal state values;
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow (not to) derive a BCS wave function with correlations
Early attempt:∣∣BCS
⟩=
∏
k
[
uk + vka†k↑a
†−k↓
] ∣∣0⟩,
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩⟨
m(N)∣∣∣BCS
⟩
EK and J. W. Clark, Nucl. Phys. A333, 77 (1980):Expand in the deviation of the Bogoljubov amplitudes uk, vk fromtheir normal state values;S. Fantoni, Nucl. Phys. A363, 381 (1981):Attempt full FHNC summation.
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow (not to) derive a BCS wave function with correlations
Early attempt:∣∣BCS
⟩=
∏
k
[
uk + vka†k↑a
†−k↓
] ∣∣0⟩,
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩⟨
m(N)∣∣∣BCS
⟩
EK and J. W. Clark, Nucl. Phys. A333, 77 (1980):Expand in the deviation of the Bogoljubov amplitudes uk, vk fromtheir normal state values;S. Fantoni, Nucl. Phys. A363, 381 (1981):Attempt full FHNC summation.
The problem:
∆k = −12
∑
k′
Pkk′
∆k′
√
(ek′ − µ)2 +∆2k′/z2(k′)
.
where z(k) → ∞ for long ranged correlations.Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow to derive a BCS wave function with correlations
Getting it right:
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩
⟨
m(N)∣∣∣F 2
N
∣∣∣m(N)
⟩1/2
⟨
m(N)∣∣∣BCS
⟩
EK, R. A. Smith and A. D. Jackson: Phys Rev. B24, 6404 (1981):Correlated Basis Functions corrections
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow to derive a BCS wave function with correlations
Getting it right:
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩
⟨
m(N)∣∣∣F 2
N
∣∣∣m(N)
⟩1/2
⟨
m(N)∣∣∣BCS
⟩
EK, R. A. Smith and A. D. Jackson: Phys Rev. B24, 6404 (1981):Correlated Basis Functions corrections
H.-H. Fan. Ph. D. Thesis: Carry out full FHNC summation andoptimization.
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow to derive a BCS wave function with correlations
Getting it right:
∣∣CBCS
⟩=
∑
N,m
FN
∣∣∣m(N)
⟩
⟨
m(N)∣∣∣F 2
N
∣∣∣m(N)
⟩1/2
⟨
m(N)∣∣∣BCS
⟩
EK, R. A. Smith and A. D. Jackson: Phys Rev. B24, 6404 (1981):Correlated Basis Functions corrections
H.-H. Fan. Ph. D. Thesis: Carry out full FHNC summation andoptimization.
The problem solved:
∆k = −12
∑
k′
Pkk′
∆k′
√
(ek′ − µ)2 +∆2k′
.
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow about a fixed particle number state ?
Just look at an uncorrelated system, let uk = cos ηk, vk = sin ηk,
δ2
δηkδηk′
⟨
BCS∣∣∣H − µN
∣∣∣BCS
⟩∣∣∣∣0
= (1 − 2n0(k))(1 − 2n0(k′))
[2 |ek − µ| δkk′ +
⟨k ↑,−k ↓
∣∣V
∣∣k′ ↑,−k′ ↓
⟩]
where n0(k) = θ(kF − k) is the normal Fermi distribution.
Pairing with strong correlations
On the BCS side: Theory with strong correlationsHow about a fixed particle number state ?
Just look at an uncorrelated system, let uk = cos ηk, vk = sin ηk,
δ2
δηkδηk′
⟨
BCS∣∣∣H − µN
∣∣∣BCS
⟩∣∣∣∣0
= (1 − 2n0(k))(1 − 2n0(k′))
[2 |ek − µ| δkk′ +
⟨k ↑,−k ↓
∣∣V
∣∣k′ ↑,−k′ ↓
⟩]
where n0(k) = θ(kF − k) is the normal Fermi distribution.
Same for number-projected state:∣∣∣BCS(N)
⟩
we get
δ2
δηkδηk′
⟨
BCS(N)∣∣∣H − µN
∣∣∣BCS(N)
⟩
⟨
BCS(N) | BCS(N)⟩
∣∣∣∣∣∣0
= −⟨k ↑,−k ↓
∣∣V
∣∣k′ ↑,−k′ ↓
⟩
for k > kF and k ′ < kF or vice versa, zero otherwise.
Pairing with strong correlations
BCS Theory with strong correlationsAnalysis of the pairing interaction:
At low density we can ignore non-localities:
Pkk′ =⟨k ↑,−k ↓
∣∣W(1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
+(|ek − µ|+ |ek ′ − µ|)⟨k ↑,−k ↓
∣∣N (1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
≡1N
[W(k − k′) + (|ek − µ|+ |ek ′ − µ|)N (k − k′)
].
The gap is (mostly) determined by the matrix element⟨k ↑,−k ↓
∣∣W(1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsAnalysis of the pairing interaction:
At low density we can ignore non-localities:
Pkk′ =⟨k ↑,−k ↓
∣∣W(1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
+(|ek − µ|+ |ek ′ − µ|)⟨k ↑,−k ↓
∣∣N (1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
≡1N
[W(k − k′) + (|ek − µ|+ |ek ′ − µ|)N (k − k′)
].
The gap is (mostly) determined by the matrix element⟨k ↑,−k ↓
∣∣W(1, 2)
∣∣k′ ↑,−k′ ↓
⟩
a
The “energy numerator” term regularizes the integral forzero-range interactions.
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsCompare with what is known:
cf. Pethick and Smith, Bose-Einstein Condensation in Dilute Gases,Cambridge University Press 2008
Zero temperature gap equation:
∆k =1
2V
∑
k′
U(k, k′)∆k′
ǫk′
ǫ2k = ∆2k + (ǫ0k − µ)2
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsCompare with what is known:
cf. Pethick and Smith, Bose-Einstein Condensation in Dilute Gases,Cambridge University Press 2008
Zero temperature gap equation:
∆k =1
2V
∑
k′
U(k, k′)∆k′
ǫk′
ǫ2k = ∆2k + (ǫ0k − µ)2
Eliminate bare interaction by sero-energy T -matrix:
T0(k, k′, 0) = U(k, k′)−1
2V
∑
k′′
U(k, k′′)1
2ǫ0k′′ − iδT0(k′′, k, 0)
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsCompare with what is known:
cf. Pethick and Smith, Bose-Einstein Condensation in Dilute Gases,Cambridge University Press 2008
Zero temperature gap equation:
∆k =1
2V
∑
k′
U(k, k′)∆k′
ǫk′
ǫ2k = ∆2k + (ǫ0k − µ)2
Eliminate bare interaction by sero-energy T -matrix:
T0(k, k′, 0) = U(k, k′)−1
2V
∑
k′′
U(k, k′′)1
2ǫ0k′′ − iδT0(k′′, k, 0)
Take zero-range limit
∆k =U0
2V
∑
k′
[
1ǫk′
−1
ǫ0k − µ
]
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsCompare with what is known:
cf. Pethick and Smith, Bose-Einstein Condensation in Dilute Gases,Cambridge University Press 2008
Zero temperature gap equation:
∆k =1
2V
∑
k′
U(k, k′)∆k′
ǫk′
ǫ2k = ∆2k + (ǫ0k − µ)2
Eliminate bare interaction by sero-energy T -matrix:
T0(k, k′, 0) = U(k, k′)−1
2V
∑
k′′
U(k, k′′)1
2ǫ0k′′ − iδT0(k′′, k, 0)
Take zero-range limit
∆k =U0
2V
∑
k′
[
1ǫk′
−1
ǫ0k − µ
]
Second term regularizes k → ∞ limit.Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsAnalysis of the gap equation:
Recall
Pkk′ =1N
[W(k − k′) + (|ek − µ|+ |ek ′ − µ|)N (k − k′)
].
If the gap is small, let
WF ≡1
2k2F
∫ 2kF
0dkkW(k) = NWkF,kF .
Then
1 = −WF
∫d3k ′
(2π)3ρ
[
1√
(ek ′ − µ)2 +∆2kF
−|ek ′ − µ|
√
(ek ′ − µ)2 +∆2kF
SF(k ′)
t(k ′)
→ −WF
∫d3k ′
(2π)3ρ
[
∼µ
t2(k ′)
]
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsApproximate solution of the gap equation
At low density, let
aF =m
4πρ~2WF
∆F ≈8e2 eF exp
(π
2aF kF
)
.
Corrections: aF → a0 for ρ→ 0+If aF = a0
[
1 + αa0kFπ
]
then
∆F ≈8e2 eF exp
(
−α
2
)
exp(
π
2a0kF
)
.
Questions:
What influences the pre-factor (Gorkov-corrrections etc..):
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsApproximate solution of the gap equation
At low density, let
aF =m
4πρ~2WF
∆F ≈8e2 eF exp
(π
2aF kF
)
.
Corrections: aF → a0 for ρ→ 0+If aF = a0
[
1 + αa0kFπ
]
then
∆F ≈8e2 eF exp
(
−α
2
)
exp(
π
2a0kF
)
.
Questions:
What influences the pre-factor (Gorkov-corrrections etc..):
How accurate is the solution ?
Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsApproximate solution of the gap equation
At low density, let
aF =m
4πρ~2WF
∆F ≈8e2 eF exp
(π
2aF kF
)
.
Corrections: aF → a0 for ρ→ 0+If aF = a0
[
1 + αa0kFπ
]
then
∆F ≈8e2 eF exp
(
−α
2
)
exp(
π
2a0kF
)
.
Questions:
What influences the pre-factor (Gorkov-corrrections etc..):
How accurate is the solution ?
Are there non-universal effects ?Pairing with strong correlations Pairing interaction
BCS Theory with strong correlationsWhat’s new ?
The gap is determined by aF
Corrections:
Interaction corrections (“phonon exchange”)
W(0+) =4πρ~2
ma =
4πρ~2
ma0
[
1 + α′a0kF
π
]
Pairing with strong correlations Many-Body effects
BCS Theory with strong correlationsWhat’s new ?
The gap is determined by aF
Corrections:
Interaction corrections (“phonon exchange”)
W(0+) =4πρ~2
ma =
4πρ~2
ma0
[
1 + α′a0kF
π
]
Finite-range corrections: Note that
WF ≡1
2k2F
∫ 2kF
0dkkW(k) 6= W(0+)
Pairing with strong correlations Many-Body effects
BCS Theory with strong correlationsWhat’s new ?
The gap is determined by aF
Corrections:
Interaction corrections (“phonon exchange”)
W(0+) =4πρ~2
ma =
4πρ~2
ma0
[
1 + α′a0kF
π
]
Finite-range corrections: Note that
WF ≡1
2k2F
∫ 2kF
0dkkW(k) 6= W(0+)
The value of WF is influenced by the regime 0 ≤ k ≤ 2kF
Pairing with strong correlations Many-Body effects
BCS Theory with strong correlationsWhat’s new ?
The gap is determined by aF
Corrections:
Interaction corrections (“phonon exchange”)
W(0+) =4πρ~2
ma =
4πρ~2
ma0
[
1 + α′a0kF
π
]
Finite-range corrections: Note that
WF ≡1
2k2F
∫ 2kF
0dkkW(k) 6= W(0+)
The value of WF is influenced by the regime 0 ≤ k ≤ 2kF
The value of WF is influenced real space correlations in theinteraction regime r > 1/kF !
Pairing with strong correlations Many-Body effects
BCS Theory with strong correlationsFinite-range effects
Pair correlations
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000 10000
Γ dd(
r)
r/σ
kF=0.001kF=0.010kF=0.040
0.0001
0.001
0.01
0.1
1
10
Γ dd(
r)
kF=0.001kF=0.010kF=0.040
Pairing interaction
1.00
1.10
1.20
1.30
0.0 0.5 1.0 1.5 2.0 2.5 3.0
W(k
)/W
(0)
k/kF
LJ SWkFσ = 0.04
LJ SWkFσ = 0.01
1.00
1.02
1.04
1.06
W(k
)/W
(0)
Interaction dominated regime
Pairing with strong correlations Many-Body effects
Solution of the gap equation... and what approximations do
∆F =8e2 eF exp
(π
2a0kF
)
10−12
10−10
10−8
10−6
10−4
10−2
100
0.01 0.1 1
∆ F/E
F
kF σ
SW Potential
full solutionaFa0
Pairing with strong correlations Results
Solution of the gap equation... and what approximations do
∆F =8e2 eF exp
(π
2a0kF
)
Can be far off
∆F =8e2 eF exp
(π
2aFkF
)
10−12
10−10
10−8
10−6
10−4
10−2
100
0.01 0.1 1
∆ F/E
F
kF σ
SW Potential
full solutionaFa0
Pairing with strong correlations Results
Solution of the gap equation... and what approximations do
∆F =8e2 eF exp
(π
2a0kF
)
Can be far off
∆F =8e2 eF exp
(π
2aFkF
)
Not too bad
Full solution 10−12
10−10
10−8
10−6
10−4
10−2
100
0.01 0.1 1
∆ F/E
F
kF σ
SW Potential
full solutionaFa0
Pairing with strong correlations Results
Solution of the gap equation... and what approximations do
∆F =8e2 eF exp
(π
2a0kF
)
Can be far off
∆F =8e2 eF exp
(π
2aFkF
)
Not too bad
Full solution 10−12
10−10
10−8
10−6
10−4
10−2
100
0.01 0.1 1
∆ F/E
F
kF σ
SW Potential
full solutionaFa0
Exponential behavior becomes universal, prefactor not.
Pairing with strong correlations Results
Solution of the gap equation... and what approximations do
∆F =8e2 eF exp
(π
2a0kF
)
Can be far off
∆F =8e2 eF exp
(π
2aFkF
)
Not too bad
Full solution 10−12
10−10
10−8
10−6
10−4
10−2
100
0.01 0.1 1
∆ F/E
F
kF σ
SW Potential
full solutionaFa0
Exponential behavior becomes universal, prefactor not.
Low density expansion valid only for physically uninterestingcases.
Pairing with strong correlations Results
What’s next ?Strong coupling ?
Go back to∣∣CBCS
⟩=
∑
N,m
⟨m(N)
∣∣F 2
N
∣∣m(N)
⟩−1/2FN∣∣m(N)
⟩⟨m(N)
∣∣BCS
⟩
Pairing with strong correlations Quo vadis ?
What’s next ?Strong coupling ?
Go back to∣∣CBCS
⟩=
∑
N,m
⟨m(N)
∣∣F 2
N
∣∣m(N)
⟩−1/2FN∣∣m(N)
⟩⟨m(N)
∣∣BCS
⟩
Develop diagrammatic expansions at the “parquet” level1
2−
1
2+
1
2+ − − − − +
− +1
2+
1
2−
1
2−
1
2+ + + −
+ + − −2
+2
2−
2×2
2+
1
2−
2×2
2+
2×2
2− − +2 +
− +2 −2 +4 + −2 + −2 +
−2 −
1
2+
2×2
2−
2×2
2−
2
2+
2×2
2
Pairing with strong correlations Quo vadis ?
What’s next ?Strong coupling ?
Go back to∣∣CBCS
⟩=
∑
N,m
⟨m(N)
∣∣F 2
N
∣∣m(N)
⟩−1/2FN∣∣m(N)
⟩⟨m(N)
∣∣BCS
⟩
Develop diagrammatic expansions at the “parquet” level1
2−
1
2+
1
2+ − − − − +
− +1
2+
1
2−
1
2−
1
2+ + + −
+ + − −2
+2
2−
2×2
2+
1
2−
2×2
2+
2×2
2− − +2 +
− +2 −2 +4 + −2 + −2 +
−2 −
1
2+
2×2
2−
2×2
2−
2
2+
2×2
2
Solve simultaneously Euler and BCS equation
Pairing with strong correlations Quo vadis ?
A visitor... to my office on a cold spring afternoon
A visitor
A visitor... came back and brought something
A visitor
Summary... is good many body theory an overkill ?
There are non-universal corrections to the low-densityHuang-Yang equation of state
Long–ranged properties are determined by many-body effects,not by vacuum
FHNC-EL diverges for phase transitions:in the particle-hole channel for spinodal decomposition,in the particle-particle channel for BCS pairing
Many-Body effects for long-ranged correlations r > 1/kF implyMany-Body effects for long wavelengths k < kF
Non-universal behavior of the pairing matrix element.
Summary
Thanks to collaborators in this project
Hsuan Hao Fan University at BuffaloRobert Zillich JKU Linz
Thanks for your attentionand thanks to our funding agency:
Acknowledgements