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BCS-BEC Crossover
Some detailed information for BCS-BEC CrossoverBCS Theory
李泽阳
April 3, 2015
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 1 / 31
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BCS-BEC Crossover
Table of Contents
1 Background Information
2 Whole structureAttraction between electronsThe Mean Field Hamiltonian and Cooper pairApplication
3 Reference
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 2 / 31
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BCS-BEC CrossoverBackground Information
Critics on phenomenological theory
These theories cannot explain the phenomena microscopically.
Hence we need BCS theory, which is basically an electron-phononcorrelation theory.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 3 / 31
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BCS-BEC CrossoverBackground Information
Critics on phenomenological theory
These theories cannot explain the phenomena microscopically.Hence we need BCS theory, which is basically an electron-phononcorrelation theory.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 3 / 31
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BCS-BEC CrossoverBackground Information
What is correlation theory?
Note (Correlation theory)Correlation theory of electron is widely used in Condensed Matter Physics,and mostly have three basic form: e-e correlation (Mott Insulator), e-phcorrelation (BCS) and e-spin correlation (Giant Magnetoresistance), ande-e + e-spin correlation (Colossal Magnetoresistance). With thedevelopment of low dimension systems, other theory like 1D e-e correlation(Luttinger Liquid), 0D e-e correlation (Quantum Dot), 2D e-ph correlation(Charge Density Wave) were also invented.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 4 / 31
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BCS-BEC CrossoverWhole structure
Goals
Before we start, I list the thing that we want to finally have:
The excited spectrum. (Zero temperature)The ground state. (Zero temperature)The critical temperature Tc. (Finite temperature)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 5 / 31
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BCS-BEC CrossoverWhole structure
Goals
Before we start, I list the thing that we want to finally have:The excited spectrum. (Zero temperature)
The ground state. (Zero temperature)The critical temperature Tc. (Finite temperature)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 5 / 31
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BCS-BEC CrossoverWhole structure
Goals
Before we start, I list the thing that we want to finally have:The excited spectrum. (Zero temperature)The ground state. (Zero temperature)
The critical temperature Tc. (Finite temperature)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 5 / 31
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BCS-BEC CrossoverWhole structure
Goals
Before we start, I list the thing that we want to finally have:The excited spectrum. (Zero temperature)The ground state. (Zero temperature)The critical temperature Tc. (Finite temperature)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 5 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Attraction between electrons
We firstly introduce how the well-known attraction between two electronswas invited in this case.
A naïve picture is:p2
p1
p4
p3
phonon scattering
Figure: Due to the so-called ‘non-locality’ of phonon, the two electron gained aeffective attraction by same phonon.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 6 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Attraction between electrons
We firstly introduce how the well-known attraction between two electronswas invited in this case.A naïve picture is:
p2
p1
p4
p3
phonon scattering
Figure: Due to the so-called ‘non-locality’ of phonon, the two electron gained aeffective attraction by same phonon.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 6 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Attraction between electrons
We firstly introduce how the well-known attraction between two electronswas invited in this case.A naïve picture is:
p2
p1
p4
p3
phonon scattering
Figure: Due to the so-called ‘non-locality’ of phonon, the two electron gained aeffective attraction by same phonon.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 6 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
An interacting process between electron and phonon is described well inthe Hamiltonian:
Hint = g∫ψ†α(r)ψα(r)φ(r)d3r
This form is derived based on such a mechanism: the electron ‘perceive’the interaction by addition energy due to polarization by thevibration(phonon):
−e∫
n(r)K(r − r′)divP(r′)d3rd3r′
where K represents the form of interacting, and always replaced byaeδ(r − r′) due to screening effect1, and hence the Hamiltonian is derived.
1Detailed calculation of screening effect is not shown here
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 7 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
An interacting process between electron and phonon is described well inthe Hamiltonian:
Hint = g∫ψ†α(r)ψα(r)φ(r)d3r
This form is derived based on such a mechanism: the electron ‘perceive’the interaction by addition energy due to polarization by thevibration(phonon):
−e∫
n(r)K(r − r′)divP(r′)d3rd3r′
where K represents the form of interacting, and always replaced byaeδ(r − r′) due to screening effect1, and hence the Hamiltonian is derived.
1Detailed calculation of screening effect is not shown here
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 7 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
An interacting process between electron and phonon is described well inthe Hamiltonian:
Hint = g∫ψ†α(r)ψα(r)φ(r)d3r
This form is derived based on such a mechanism: the electron ‘perceive’the interaction by addition energy due to polarization by thevibration(phonon):
−e∫
n(r)K(r − r′)divP(r′)d3rd3r′
where K represents the form of interacting, and always replaced byaeδ(r − r′) due to screening effect1, and hence the Hamiltonian is derived.
1Detailed calculation of screening effect is not shown here
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 7 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Because the system is electron & phonon system, Feynman Diagram inmomentum space has such rule:For solid line (electron fermi line),
G(0)(p) = 1
ω − ϵ(p) + i(0+)sgnε(p)and for wavy line (phonon bose line),
D(0)(k) = ω20(k)
ω2 − ω20(k) + i(0+)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 8 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
After several calculation, we can have the ‘D’ for phonon in the diagram
p2
p1
p4
p3phonon scattering
g2D(ε3 − ε1; p3 − p1) = g2 u2(p3 − p1)2(ε3 − ε1)2 − u2(p3 − p1)2
Mostly, the electron is around Fermi Surface2, upph ∼ ωD, and also|ε3 − ε1| ≪ ωD, (continued by next slide)
2reason can be seen later
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 9 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
After several calculation, we can have the ‘D’ for phonon in the diagram
p2
p1
p4
p3phonon scattering
g2D(ε3 − ε1; p3 − p1) = g2 u2(p3 − p1)2(ε3 − ε1)2 − u2(p3 − p1)2
Mostly, the electron is around Fermi Surface2, upph ∼ ωD, and also|ε3 − ε1| ≪ ωD, (continued by next slide)
2reason can be seen later
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 9 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.
Before we go any further, we should claim something.1 The attraction is an additional term. The Coulomb interaction still
exists.2 Which force dominate causes serious different in our project:
BCS-BEC Crossover.3 BCS theory next consider the overall interaction as a constant local
attraction, but it actually isn’t4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.Before we go any further, we should claim something.
1 The attraction is an additional term. The Coulomb interaction stillexists.
2 Which force dominate causes serious different in our project:BCS-BEC Crossover.
3 BCS theory next consider the overall interaction as a constant localattraction, but it actually isn’t
4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.Before we go any further, we should claim something.
1 The attraction is an additional term. The Coulomb interaction stillexists.
2 Which force dominate causes serious different in our project:BCS-BEC Crossover.
3 BCS theory next consider the overall interaction as a constant localattraction, but it actually isn’t
4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.Before we go any further, we should claim something.
1 The attraction is an additional term. The Coulomb interaction stillexists.
2 Which force dominate causes serious different in our project:BCS-BEC Crossover.
3 BCS theory next consider the overall interaction as a constant localattraction, but it actually isn’t
4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.Before we go any further, we should claim something.
1 The attraction is an additional term. The Coulomb interaction stillexists.
2 Which force dominate causes serious different in our project:BCS-BEC Crossover.
3 BCS theory next consider the overall interaction as a constant localattraction, but it actually isn’t
4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
Attraction between electrons
Continue
In such case, we can have
g2D = −g2
which means the interaction can be constantly considered as an attraction.Before we go any further, we should claim something.
1 The attraction is an additional term. The Coulomb interaction stillexists.
2 Which force dominate causes serious different in our project:BCS-BEC Crossover.
3 BCS theory next consider the overall interaction as a constant localattraction, but it actually isn’t
4 Electrons are almost-free, and have square dispersion relation ω ∼ k2.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 10 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Now we get into BCS theory. The Hamiltonian can be considered as
H = H0 + V =
∫ψ†α(x)
[(−iℏ∇+ eA⃗(x)/c)2
2m − µ
]ψα(x)d3x
− λ
2
∫ψ†α(x)ψ
†β(x)ψβ(x)ψα(x)d3x
where the second term is a constant local attraction.
From another point, we consider the difference between this Hamiltonianand the H0. Obviously, the ground state of H0 can be considered as a‘vacuum’ state, say|vac⟩, or |N⟩ stands for a N-(real) particle and no‘out-of-fermi-sea’ electrons state.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 11 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Now we get into BCS theory. The Hamiltonian can be considered as
H = H0 + V =
∫ψ†α(x)
[(−iℏ∇+ eA⃗(x)/c)2
2m − µ
]ψα(x)d3x
− λ
2
∫ψ†α(x)ψ
†β(x)ψβ(x)ψα(x)d3x
where the second term is a constant local attraction.From another point, we consider the difference between this Hamiltonianand the H0. Obviously, the ground state of H0 can be considered as a‘vacuum’ state, say|vac⟩, or |N⟩ stands for a N-(real) particle and no‘out-of-fermi-sea’ electrons state.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 11 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
The stability of metal is ensured by the energy cost of ‘excitation of twoelectrons’ out of Fermi surface:
∆E =ℏ2k212m +
ℏ2k222m − 2µ ≥ 0
However, if we introduce the attraction, things are different because therecan be some case (near Fermi surface) that
∆E =ℏ2k212m +
ℏ2k222m − 2µ−∆ ≤ 0
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 12 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
The stability of metal is ensured by the energy cost of ‘excitation of twoelectrons’ out of Fermi surface:
∆E =ℏ2k212m +
ℏ2k222m − 2µ ≥ 0
However, if we introduce the attraction, things are different because therecan be some case (near Fermi surface) that
∆E =ℏ2k212m +
ℏ2k222m − 2µ−∆ ≤ 0
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 12 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Due to this, the |N⟩ is not the lowest energy state, and hence the groundstate should be something like
|Ψ0⟩ = |N⟩+♠|N + 2⟩+♡|N + 4⟩+♢|N + 6⟩+♣|N + 8⟩+ · · ·
This causes a ‘big difference’, for the following term may not vanished:⟨Ψ0
∣∣∣ψ†ψ†∣∣∣Ψ0
⟩̸= 0
Now and after, the notation
⟨A⟩def≡ ⟨Ψ0|A|Ψ0⟩
is used.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 13 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Due to this, the |N⟩ is not the lowest energy state, and hence the groundstate should be something like
|Ψ0⟩ = |N⟩+♠|N + 2⟩+♡|N + 4⟩+♢|N + 6⟩+♣|N + 8⟩+ · · ·
This causes a ‘big difference’, for the following term may not vanished:⟨Ψ0
∣∣∣ψ†ψ†∣∣∣Ψ0
⟩̸= 0
Now and after, the notation
⟨A⟩def≡ ⟨Ψ0|A|Ψ0⟩
is used.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 13 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Due to this, the |N⟩ is not the lowest energy state, and hence the groundstate should be something like
|Ψ0⟩ = |N⟩+♠|N + 2⟩+♡|N + 4⟩+♢|N + 6⟩+♣|N + 8⟩+ · · ·
This causes a ‘big difference’, for the following term may not vanished:⟨Ψ0
∣∣∣ψ†ψ†∣∣∣Ψ0
⟩̸= 0
Now and after, the notation
⟨A⟩def≡ ⟨Ψ0|A|Ψ0⟩
is used.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 13 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Consider the electron has addition freedom - spin, and the spin is notinvolved in the calculation, we have to carefully take them into thecontraction of indices.
Definition (□(x) ≡ −λ⟨ψ↑ψ↓⟩ = |□|eiθ)This quantity describe actually the non-trivial part in the situation, and isactually independent of (x). Hereinafter we just use □.
The benefit to introduce such an definition is based on such an statement:occasionally, the ψ†(x)ψ†(x) doesn’t vary too much and can be expressedby
ψ†ψ† = ⟨ψ†ψ†⟩(ψ†ψ† − ⟨ψ†ψ†⟩)
while the second term is relatively small.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 14 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Consider the electron has addition freedom - spin, and the spin is notinvolved in the calculation, we have to carefully take them into thecontraction of indices.
Definition (□(x) ≡ −λ⟨ψ↑ψ↓⟩ = |□|eiθ)This quantity describe actually the non-trivial part in the situation, and isactually independent of (x). Hereinafter we just use □.
The benefit to introduce such an definition is based on such an statement:occasionally, the ψ†(x)ψ†(x) doesn’t vary too much and can be expressedby
ψ†ψ† = ⟨ψ†ψ†⟩(ψ†ψ† − ⟨ψ†ψ†⟩)
while the second term is relatively small.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 14 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Hence, approximately, we can express the Hamiltonian by 3
H =
∫ψ†α(x)
[(−iℏ∇)2
2m − µ
]ψα(x)d3x
− λ
∫ {⟨ψ†
↓(x)ψ†↑(x)⟩ψ↑(x)ψ↓(x) + ψ†
↓(x)ψ†↑(x)⟨ψ↑(x)ψ↓(x)⟩
}d3x
+ λ
∫⟨ψ†
↓(x)ψ†↑(x)⟩⟨ψ↑(x)ψ↓(x)⟩d3x
3Here we omit the potential Am which causes nothing in this case.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 15 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
It’s straight forward to use Fourier Transformation of x, for the ∇ can beeventually written by a number rather than operator. Notice that thetransform is discrete 4
Hence, we have
Heff =∑k,α
ξkc†kαckα +□∗∑
kck↑c−k↓ +□
∑k
c†−k↓c†k↑ −□
∑k⟨c†−k↓c
†k↑⟩
where ξk = ℏ2k2/2m − µ indicates the energy cost to excite an electron,the c† and c for creation and annihilation operator, the eff for effectivemean-field Hamiltonian.
4so that avoid some critical issues happened when there are uncountable infinitedimension
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 16 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
It’s straight forward to use Fourier Transformation of x, for the ∇ can beeventually written by a number rather than operator. Notice that thetransform is discrete 4Hence, we have
Heff =∑k,α
ξkc†kαckα +□∗∑
kck↑c−k↓ +□
∑k
c†−k↓c†k↑ −□
∑k⟨c†−k↓c
†k↑⟩
where ξk = ℏ2k2/2m − µ indicates the energy cost to excite an electron,the c† and c for creation and annihilation operator, the eff for effectivemean-field Hamiltonian.
4so that avoid some critical issues happened when there are uncountable infinitedimension
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 16 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Next we want to diagonalize the Hamiltonian. It’s easy to rewrite it in:
Heff =∑
k
{(c†k↑ c−k↓
)( ξk e−iθ|□|−e−iθ|□| −ξk
)( ck↑c†−k↓
)+ ξk
}−□
∑k⟨c†−k↓c
†k↑⟩
This unitary matrix can be diagonalized simply.We introduce a transformation:(
ck↑c†−k↓
)=
(uk eiθvk
−e−iθvk uk
)(αkβ†−k
)Continued by next slide
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 17 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Next we want to diagonalize the Hamiltonian. It’s easy to rewrite it in:
Heff =∑
k
{(c†k↑ c−k↓
)( ξk e−iθ|□|−e−iθ|□| −ξk
)( ck↑c†−k↓
)+ ξk
}−□
∑k⟨c†−k↓c
†k↑⟩
This unitary matrix can be diagonalized simply.
We introduce a transformation:(ck↑
c†−k↓
)=
(uk eiθvk
−e−iθvk uk
)(αkβ†−k
)Continued by next slide
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 17 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Next we want to diagonalize the Hamiltonian. It’s easy to rewrite it in:
Heff =∑
k
{(c†k↑ c−k↓
)( ξk e−iθ|□|−e−iθ|□| −ξk
)( ck↑c†−k↓
)+ ξk
}−□
∑k⟨c†−k↓c
†k↑⟩
This unitary matrix can be diagonalized simply.We introduce a transformation:(
ck↑c†−k↓
)=
(uk eiθvk
−e−iθvk uk
)(αkβ†−k
)Continued by next slide
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 17 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Continue
with
2ukvk =|□|√
ξ2k + |□|2
u2k − v2k =
ξk√ξ2k + |□|2
And hence can derive a clear diagonalized Hamiltonian:
Heff =∑
k
{√ξ2k + |□|2(a†kαk + β†−kβ−k − 1) + ξk
}−□
∑k⟨c†−k↓c
†k↑⟩
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 18 / 31
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BCS-BEC CrossoverWhole structure
The Mean Field Hamiltonian and Cooper pair
Continue
with
2ukvk =|□|√
ξ2k + |□|2
u2k − v2k =
ξk√ξ2k + |□|2
And hence can derive a clear diagonalized Hamiltonian:
Heff =∑
k
{√ξ2k + |□|2(a†kαk + β†−kβ−k − 1) + ξk
}−□
∑k⟨c†−k↓c
†k↑⟩
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 18 / 31
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BCS-BEC CrossoverWhole structure
Application
Excited Spectrum
Hence, we have a finite-gap spectrum, for exciting a α or β (also calledBogoliubov particle) all cost at least
√ξ2k + |□|2 ≥ |□|, as illustrated
below:
k
E
V
V + |□|
k0
Figure: Excited spectrum; red line for ground BCS state, and solid line for singleexcited state; k0 stands for ξk0 = 0.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 19 / 31
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BCS-BEC CrossoverWhole structure
Application
Next we want to get the relationship between |Ψ0⟩ and |vac⟩.
The |Ψ0⟩ satisfies αk|Ψ0⟩ = 0
βk|Ψ0⟩ = 0
The next derivation is to obtain |Ψ0 in terms of |vac⟩ and c, c†; however,it is not too simple. Firstly, we construct the α, β in terms of c.
αk = ukck↑ − eiθvkc†−k↓
βk = eiθvkc†−k↑ + ukck↓
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 20 / 31
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BCS-BEC CrossoverWhole structure
Application
Next we want to get the relationship between |Ψ0⟩ and |vac⟩.The |Ψ0⟩ satisfies
αk|Ψ0⟩ = 0
βk|Ψ0⟩ = 0
The next derivation is to obtain |Ψ0 in terms of |vac⟩ and c, c†; however,it is not too simple. Firstly, we construct the α, β in terms of c.
αk = ukck↑ − eiθvkc†−k↓
βk = eiθvkc†−k↑ + ukck↓
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 20 / 31
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BCS-BEC CrossoverWhole structure
Application
Next we want to get the relationship between |Ψ0⟩ and |vac⟩.The |Ψ0⟩ satisfies
αk|Ψ0⟩ = 0
βk|Ψ0⟩ = 0
The next derivation is to obtain |Ψ0 in terms of |vac⟩ and c, c†; however,it is not too simple. Firstly, we construct the α, β in terms of c.
αk = ukck↑ − eiθvkc†−k↓
βk = eiθvkc†−k↑ + ukck↓
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 20 / 31
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BCS-BEC CrossoverWhole structure
Application
Here I just give the solution.
|Ψ0⟩ =∏
k(uk + eiθvkc†k↑c
†−k↓)|vac⟩
To verify, we can see that αk|Ψ0⟩ = 0
βk|Ψ0⟩ = 0
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 21 / 31
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BCS-BEC CrossoverWhole structure
Application
Summary
So far, we already have the excited spectrum (though unknown what □is), and obtain the |vac⟩ and |Ψ0⟩. If we want to know the criticaltemperature (i.e., |□| = 0) or know the precise form of the spectrum, wehave to know □ ≡ −λ⟨ψ↑ψ↓⟩. 5
5Going on or not depends on whether the time is enough.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 22 / 31
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BCS-BEC CrossoverWhole structure
Application
After several calculation, we have self-consistent equation
|□| = λ
2
∫ d3k(2π)3
|□|√ξ2k + |□|2
which is called ‘gap equation’.
Note that we have already explained why the attractive interaction in BCSHamiltonian applies only for those two electrons which are lying within anenergy sell of the thickness ℏωD from the Fermi surface.That means, the sum over only |ξk| ≤ ℏωD, and leads to
(uk, vk) = (1, 0) or (0, 1)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 23 / 31
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BCS-BEC CrossoverWhole structure
Application
After several calculation, we have self-consistent equation
|□| = λ
2
∫ d3k(2π)3
|□|√ξ2k + |□|2
which is called ‘gap equation’.Note that we have already explained why the attractive interaction in BCSHamiltonian applies only for those two electrons which are lying within anenergy sell of the thickness ℏωD from the Fermi surface.That means, the sum over only |ξk| ≤ ℏωD, and leads to
(uk, vk) = (1, 0) or (0, 1)
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 23 / 31
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BCS-BEC CrossoverWhole structure
Application
Illustration
ℏωD
ℏωD
Figure: Interaction within a shell, the solid line represents the Fermi Surface, andthe two dashed line represents the boundary of possible interaction-involvedregion.School of Physics, Peking University BCS-BEC Crossover April 3, 2015 24 / 31
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BCS-BEC CrossoverWhole structure
Application
Thus,
|□| = λ
2
∫|ξk|≤ℏωD
d3k(2π)3
|□|√ξ2k + |□|2
Several replacement makes
1 = λ
∫ ℏωD
−ℏωD
dωN(ω)1√
ω2 + |□|2
Occasionally, ωD is relatively small, and hence the fluctuation of N(ω) isnot rapid; a N(0) = mkF/2π2ℏ2 is enough here. So,
1 = gN(0)
∫ ℏωd
0dω 1√
ω2 + |□|2= gN(0) log
[ℏωD +
√ω2 + |□|2
|□|
]
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 25 / 31
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BCS-BEC CrossoverWhole structure
Application
Thus,
|□| = λ
2
∫|ξk|≤ℏωD
d3k(2π)3
|□|√ξ2k + |□|2
Several replacement makes
1 = λ
∫ ℏωD
−ℏωD
dωN(ω)1√
ω2 + |□|2
Occasionally, ωD is relatively small, and hence the fluctuation of N(ω) isnot rapid; a N(0) = mkF/2π2ℏ2 is enough here. So,
1 = gN(0)
∫ ℏωd
0dω 1√
ω2 + |□|2= gN(0) log
[ℏωD +
√ω2 + |□|2
|□|
]
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 25 / 31
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BCS-BEC CrossoverWhole structure
Application
Thus,
|□| = λ
2
∫|ξk|≤ℏωD
d3k(2π)3
|□|√ξ2k + |□|2
Several replacement makes
1 = λ
∫ ℏωD
−ℏωD
dωN(ω)1√
ω2 + |□|2
Occasionally, ωD is relatively small, and hence the fluctuation of N(ω) isnot rapid; a N(0) = mkF/2π2ℏ2 is enough here.
So,
1 = gN(0)
∫ ℏωd
0dω 1√
ω2 + |□|2= gN(0) log
[ℏωD +
√ω2 + |□|2
|□|
]
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 25 / 31
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BCS-BEC CrossoverWhole structure
Application
Thus,
|□| = λ
2
∫|ξk|≤ℏωD
d3k(2π)3
|□|√ξ2k + |□|2
Several replacement makes
1 = λ
∫ ℏωD
−ℏωD
dωN(ω)1√
ω2 + |□|2
Occasionally, ωD is relatively small, and hence the fluctuation of N(ω) isnot rapid; a N(0) = mkF/2π2ℏ2 is enough here. So,
1 = gN(0)
∫ ℏωd
0dω 1√
ω2 + |□|2= gN(0) log
[ℏωD +
√ω2 + |□|2
|□|
]School of Physics, Peking University BCS-BEC Crossover April 3, 2015 25 / 31
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BCS-BEC CrossoverWhole structure
Application
Approximation that □ ≪ ℏωD gives
|□| = 2ℏωDe−1/gN(0) ∼ 0.01ℏωD
The last estimation is based information about some typical metal.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 26 / 31
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BCS-BEC CrossoverWhole structure
Application
Approximation that □ ≪ ℏωD gives
|□| = 2ℏωDe−1/gN(0) ∼ 0.01ℏωD
The last estimation is based information about some typical metal.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 26 / 31
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BCS-BEC CrossoverWhole structure
Application
Generalization to finite temperature
As we already mentioned in the ‘Goals’, the estimation of Tc is based onthe finite temperature field theory. Fortunately, we don’t have to change alot, but the definition of ⟨A⟩ should be taken.
Definition (⟨A⟩ = Tr[e−βHeff A]
Tr[e−βHeff ])
An important application is
⟨ψ†↓(x)ψ
†↑(x)⟩ =
Tr[e−βHeffψ†↓(x)ψ
†↑(x)]
Tr[e−βHeff ]
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 27 / 31
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BCS-BEC CrossoverWhole structure
Application
With some detailed calculation6, we derived similar gap equation:
1 = λ
∫|ξk|≤ℏωD
d3k(2π)3
1
2Ektanh (βℏEk/2)
= λ
∫ ℏωD
−ℏωD
dωN(ω)1
2√ω2 + |□|2
tanh(βℏ√ω2 + |□|2/2
)It’s easy to see that |□| decreases when temperature increases. When|□| = 0, there comes the critical temperature. We then solve it.
6It’s too boring, not easy to understand for a neophyte, and may cost too long for apresentation; we simply gives the result here.
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BCS-BEC CrossoverWhole structure
Application
Critical Temperature
At Tc, naturally, we have □ = 0.
Based on this we have
1 = gN(0)
∫ ℏωD
0
dωω
tanh(ℏω/2kBTc)
= gN(0)
∫ ℏωD2kBTc
0
dxx tanh x
= gN(0)
(ln xc −
∫ xc
0ln x 1
cosh2 xdx)
xc→+∞= gN(0)
(ln xc + ln 4eγ
π+ f(xc)
)∼ gN(0) ln xc
where γ ≈ 0.5772 given by some NB math tool.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 29 / 31
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BCS-BEC CrossoverWhole structure
Application
Critical Temperature
At Tc, naturally, we have □ = 0. Based on this we have
1 = gN(0)
∫ ℏωD
0
dωω
tanh(ℏω/2kBTc)
= gN(0)
∫ ℏωD2kBTc
0
dxx tanh x
= gN(0)
(ln xc −
∫ xc
0ln x 1
cosh2 xdx)
xc→+∞= gN(0)
(ln xc + ln 4eγ
π+ f(xc)
)∼ gN(0) ln xc
where γ ≈ 0.5772 given by some NB math tool.School of Physics, Peking University BCS-BEC Crossover April 3, 2015 29 / 31
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BCS-BEC CrossoverWhole structure
Application
Hence,
ln(
ℏωD2kBTc
× 4eγπ
)=
1
gN(0)
We eventually have
kBTc =2eγπ
ℏωDe−1/gN(0) ∼ 1.13ℏωDe−1/gN(0)
This derivation is confirmed by some simple metals, like Al.BCS theory can also explain critical property of the superconducting andsuperfluity; however, we don’t have enough time. So it’s better to stophere.
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BCS-BEC CrossoverWhole structure
Application
Hence,
ln(
ℏωD2kBTc
× 4eγπ
)=
1
gN(0)
We eventually have
kBTc =2eγπ
ℏωDe−1/gN(0) ∼ 1.13ℏωDe−1/gN(0)
This derivation is confirmed by some simple metals, like Al.
BCS theory can also explain critical property of the superconducting andsuperfluity; however, we don’t have enough time. So it’s better to stophere.
School of Physics, Peking University BCS-BEC Crossover April 3, 2015 30 / 31
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BCS-BEC CrossoverWhole structure
Application
Hence,
ln(
ℏωD2kBTc
× 4eγπ
)=
1
gN(0)
We eventually have
kBTc =2eγπ
ℏωDe−1/gN(0) ∼ 1.13ℏωDe−1/gN(0)
This derivation is confirmed by some simple metals, like Al.BCS theory can also explain critical property of the superconducting andsuperfluity; however, we don’t have enough time. So it’s better to stophere.
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BCS-BEC CrossoverReference
Ryuichi Shindou.Quantum Statistical Physics.PKU Press, v3 edition.
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