correlations in the low-density fermi gas: fermi-liquid ...bqmc.upc.edu/pdfs/doc898.pdfcorrelations...
TRANSCRIPT
![Page 1: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/1.jpg)
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
![Page 2: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/2.jpg)
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
![Page 3: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/3.jpg)
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
![Page 4: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/4.jpg)
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
![Page 5: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/5.jpg)
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
![Page 6: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/6.jpg)
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
![Page 7: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/7.jpg)
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
![Page 8: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/8.jpg)
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
![Page 9: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/9.jpg)
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
![Page 10: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/10.jpg)
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
![Page 11: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/11.jpg)
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
![Page 12: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/12.jpg)
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
![Page 13: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/13.jpg)
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
![Page 14: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/14.jpg)
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
![Page 15: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/15.jpg)
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
![Page 16: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/16.jpg)
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
![Page 17: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/17.jpg)
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
![Page 18: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/18.jpg)
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
![Page 19: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/19.jpg)
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
![Page 20: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/20.jpg)
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
![Page 21: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/21.jpg)
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
![Page 22: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/22.jpg)
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
![Page 23: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/23.jpg)
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
![Page 24: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/24.jpg)
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
![Page 25: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/25.jpg)
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
![Page 26: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/26.jpg)
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
![Page 27: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/27.jpg)
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
![Page 28: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/28.jpg)
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
![Page 29: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/29.jpg)
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
![Page 30: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/30.jpg)
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
![Page 31: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/31.jpg)
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
![Page 32: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/32.jpg)
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
![Page 33: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/33.jpg)
Microscopic ground state calculations– the many-body effects
Three (not two) range regimes
Methods Verification: Bulk quantum fluids
![Page 34: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/34.jpg)
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
![Page 35: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/35.jpg)
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
![Page 36: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/36.jpg)
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
![Page 37: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/37.jpg)
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
![Page 38: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/38.jpg)
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
![Page 39: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/39.jpg)
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
![Page 40: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/40.jpg)
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
![Page 41: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/41.jpg)
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
![Page 42: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/42.jpg)
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
![Page 43: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/43.jpg)
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
![Page 44: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/44.jpg)
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
![Page 45: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/45.jpg)
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
![Page 46: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/46.jpg)
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
![Page 47: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/47.jpg)
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
![Page 48: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/48.jpg)
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
![Page 49: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/49.jpg)
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
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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
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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
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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
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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
![Page 54: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/54.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 73: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/73.jpg)
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
![Page 74: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/74.jpg)
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
![Page 75: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/75.jpg)
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
![Page 76: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/76.jpg)
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
![Page 77: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/77.jpg)
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
![Page 78: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/78.jpg)
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
![Page 79: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/79.jpg)
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
![Page 80: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/80.jpg)
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
![Page 81: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/81.jpg)
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
![Page 82: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/82.jpg)
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
![Page 83: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/83.jpg)
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 ?
![Page 84: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/84.jpg)
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 ?
![Page 85: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/85.jpg)
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 ?
![Page 86: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/86.jpg)
A visitor... to my office on a cold spring afternoon
A visitor
![Page 87: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/87.jpg)
A visitor... came back and brought something
A visitor
![Page 88: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/88.jpg)
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
![Page 89: Correlations in the Low-Density Fermi Gas: Fermi-Liquid ...bqmc.upc.edu/pdfs/doc898.pdfCorrelations in the Low-Density Fermi Gas: Fermi-Liquid state, BCS Pairing, and Dimerization](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff9554aad53786de173a493/html5/thumbnails/89.jpg)
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