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

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




Be able to deal with self-boundsystems;

V(r

)

(s

ome

units

)

r (some units)

ε

σ


ρ (some units)

E/N

(som

e un

its)

bosons

fermions






Have no answers if “mothernature” does not have them;

V(r

)

(s

ome

units

)

r (some units)

ε

σ


ρ (some units)

E/N

(som

e un

its)

bosons

fermions






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


ρ (some units)

E/N

(som

e un

its)

bosons

fermions


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;




“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


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



Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)


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,




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)


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




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)



−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;




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)



−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);




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)



−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;




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)



−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;




Ψ0(1, . . . ,N) = exp12

[∑

i

u1(ri) +∑

i<j

u2(ri , rj) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)



−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.


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





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






VJL(r) = 4ǫ[(σ

r

)12−

(σ

r

)6]


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






VJL(r) = 4ǫ[(σ

r

)12−

(σ

r

)6]


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

)



spinodal points


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




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


−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)





“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


−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


E/N

ρ (σ−3)






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


−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


E/N

ρ (σ−3)






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


−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


E/N

ρ (σ−3)


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 !




ELY = 4π~

2ρa2m

[

1 +128

15π1/2

(

ρa3)1/2

+ . . .

]

.


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

+ . . .




ELY = 4π~

2ρa2m

[

1 +128

15π1/2

(

ρa3)1/2

+ . . .

]

.



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




ELY = 4π~

2ρa2m

[

1 +128

15π1/2

(

ρa3)1/2

+ . . .

]

.



EN

=~

2k2F

2m

35+

23

akF

π+

4(11 − 2 ln 2)35

︸︷︷︸

=1.098

(akF

π

)2

+ . . .


2nd order correlated basis functions perturbation theory repairs




ELY = 4π~

2ρa2m

[

1 +128

15π1/2

(

ρa3)1/2

+ . . .

]

.



EN

=~

2k2F

2m

35+

23

akF

π+

4(11 − 2 ln 2)35

︸︷︷︸

=1.098

(akF

π

)2

+ . . .


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).


Low-density limitThe moment of truth....


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


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





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





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


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



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





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






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

)



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


Microscopic ground state calculationsWhat we expect – what we get

For a0 > 0, no bound state→ repulsive Fermi gas; X




−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

)



spinodal points

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1 10


mc2

(ρ)/m

c2 F

ρ




For a0 < 0 (“BCS” regime):BCS pairing; X



−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

)



spinodal points

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1 10


mc2

(ρ)/m

c2 F

ρ





For a0 > 0 (“BEC” regime):No homogeneous solution ?dimerization ? X


−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

)



spinodal points

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1 10


mc2

(ρ)/m

c2 F

ρ





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

)



spinodal points

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1 10


mc2

(ρ)/m

c2 F

ρ


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

)



spinodal point




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

)



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





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

)



spinodal points






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

)



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






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

)



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


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




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





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






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






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


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



⟩=

∏

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.




⟩=

∏

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



Getting it right:

∣∣CBCS

⟩=

∑

N,m

FN

∣∣∣m(N)

⟩

⟨

m(N)∣∣∣F 2

N

∣∣∣m(N)

⟩1/2

⟨

m(N)∣∣∣BCS

⟩


H.-H. Fan. Ph. D. Thesis: Carry out full FHNC summation andoptimization.



Getting it right:

∣∣CBCS

⟩=

∑

N,m

FN

∣∣∣m(N)

⟩

⟨

m(N)∣∣∣F 2

N

∣∣∣m(N)

⟩1/2

⟨

m(N)∣∣∣BCS

⟩


H.-H. Fan. Ph. D. Thesis: Carry out full FHNC summation andoptimization.

The problem solved:

∆k = −12

∑

k′

Pkk′

∆k′

√

(ek′ − µ)2 +∆2k′

.


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.


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.


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.


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





∆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)





∆k =1

2V

∑

k′

U(k, k′)∆k′

ǫk′

ǫ2k = ∆2k + (ǫ0k − µ)2


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

]





∆k =1

2V

∑

k′

U(k, k′)∆k′

ǫk′

ǫ2k = ∆2k + (ǫ0k − µ)2


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 ′)

]


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..):



At low density, let

aF =m

4πρ~2WF

∆F ≈8e2 eF exp

(π

2aF kF

)

.


[

1 + αa0kFπ

]

then

∆F ≈8e2 eF exp

(

−α

2

)

exp(

π

2a0kF

)

.

Questions:


How accurate is the solution ?



At low density, let

aF =m

4πρ~2WF

∆F ≈8e2 eF exp

(π

2aF kF

)

.


[

1 + αa0kFπ

]

then

∆F ≈8e2 eF exp

(

−α

2

)

exp(

π

2a0kF

)

.

Questions:


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



Corrections:


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+)




Corrections:


W(0+) =4πρ~2

ma =

4πρ~2

ma0

[

1 + α′a0kF

π

]


WF ≡1

2k2F

∫ 2kF

0dkkW(k) 6= W(0+)

The value of WF is influenced by the regime 0 ≤ k ≤ 2kF




Corrections:


W(0+) =4πρ~2

ma =

4πρ~2

ma0

[

1 + α′a0kF

π

]


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 !


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


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


∆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



∆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



∆F =8e2 eF exp

(π

2a0kF

)

Can be far off

∆F =8e2 eF exp

(π

2aFkF

)

Not too bad


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.



∆F =8e2 eF exp

(π

2a0kF

)

Can be far off

∆F =8e2 eF exp

(π

2aFkF

)

Not too bad


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.


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 ?



⟩=

∑

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




⟩=

∑

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


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

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

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