The BCS Ground State

This state has the correct symmetry properties and it is clear that, if we choose the functionϕproperly, so that the two particles in the pair get the maximum possible binding energy, this will automatically happen for all the particles in the conden-sate. This was the original line of thinking of Cooper, before the BCS condensation in the space of the states was introduced.

It will be the purpose of the next section to describe the BCS GS and its zero tem-perature properties.

whereΘ(x) = 1forx >0and0elsewhere, andφis a generic phase.

Except from the special case (4.18), one peculiar aspect of the BCS state (4.15) is to be asuperposition of many-body stateswithdifferent(but even) number of particles.

Therefore the total number of electrons in a state described by (4.15) isnotfixed.

Rather, its mean valueNcan be centered around a preferred number by tuning the parametersu_kandv_k. It can be shown that the width of the distributions of the par-ticle numbers scales as√

N, implying that the deviations get smaller and smaller in the thermodynamic limit of largeN. At the end of the section we will comment on what is the meaning of not having a fixed number of particles in a closed physical system (the 3D superconducting sample).

The normalization constraint onu_k andv_k and their symmetry relations allow us to define asinglecomplex parameter, called ”the gap” (we will see why)∆_k deter-mining the GS, via the relations

u_k = E_k+ε_k p|∆_k|²+ (E_k+ε_k)²

v_k = ∆_k

p|∆_k|²+ (E_k+ε_k)² (4.19) having definedE_k =q

ε²_k+|∆_k|²(also this notation will acquire a physical mean-ing in the next section), with∆_k= ∆_−k.

The previous relations imply

|u_k|² = 1 2

1 + ε_k

E_k

|v_k|² = 1 2

1− ε_k

E_k

. (4.20)

Unless macroscopic phase coherence is considered, as in the Josephson’s effect, the functionsu_k andv_kcan be chosen to be real. We will however keep trace of their complex nature, although no physical effect presented here will depend on it.

The final step in the BCS analysis is to determine the function∆_k as to minimize the expectation value of the Hamiltonian on the proposed GS|ΨBCSi.

Before doing that we briefly comment about the nature of the creation and annihi-lation operators of the pairs.

Indeed we can introduce the second quantization operators creating and destroy-ing a pair of fermions with opposite momenta and spin as

b^†_k = c^†_k,↑c^†_−k,↓

b_k = c_−k,↓c_k,↑. (4.21)

In the BCS state these operators repeatedly act on the vacuum generating states with even number of particles. If we study their commutation relations directly we realize that







hb_k, b^†_k0

i=δ_kk⁰(1−n_k,↑−n_−k,↓) b_k, b_k0

=h b^†_k, b^†_k0

i= 0 (4.22)

wheren_k,s ≡c^†_k,sc_k,sis the particle number operator in the single particle electron state with momentumkand spins.

We can therefore see that the pair operators fulfill usual bosonic commutation rela-tions when acting on states with different momenta but not when their momentum coincide. It can be directly verified thatb^†_k²=b²_k= 0, so that the pair operators have a bosonic behaviour for different momenta and fulfill the Pauli principle when ap-plied on the same state twice.

If a full Bose-Einstein statistics were satisfied by theb_k’s the GS would be a bosonic condensate of pairs in the state with vanishing momentum. This isnotthe case for the superconductor, and this is the reason why the Fermi surface due to the exclu-sion principle still plays a crucial role in the properties of the SC GS.

Among the many curious properties of the BCS state, one which has deep conse-quences is the fact that the operatorb_k(destroying two electrons in the GS) has a non vanishing expectation value on|ΨBCSi.

This clearly has to do with the fact that the SC GS does not have a fixed number of particles and it represents a dramatic difference with respect to the usual ”normal”

Fermi liquid-like states. The existence of such ”anomalous” averages will force us to consider an extension to the usual Green’s functions techniques to account for the superconducting condensation effects, as we will see in the second part of this chapter. The anomalous average effect leads to the introduction and development of the concept of the spontaneous symmetry breaking mechanism which has now a broad range of applications in the theory of the phase transitions.

A simple example can directly show the existence of anomalous averages in the BCS state. Let us imagine to consider just a single momentum statek₀, in the BCS GS. Thus

|Ψ⁽¹⁾_BCSi=

u_k0+v_k0b^†_k

0

|vaci=u_k0|0i+v_k0|1i (4.23) where|0iand|1iindicate the states where thepair(k₀ ↑,−k₀↓)is empty or occu-pied, respectively. It is then easy to evaluate

hΨ⁽¹⁾_BCS|b_k

0|Ψ⁽¹⁾_BCSi=hΨ⁽¹⁾_BCS|c_−k

0,↓c_k

0,↑|Ψ⁽¹⁾_BCSi=u^∗_k

0v_k

0 . (4.24)

If the productu^∗_k

0v_k

0does not vanish the anomalous average survives: in a normal Fermi liquid state the same expectation value would be trivially zero.

Normal averages on the same|Ψ⁽¹⁾_BCSistate give hΨ⁽¹⁾_BCS|c^†_k

0,sc_k

0,s|Ψ⁽¹⁾_BCSi=hΨ⁽¹⁾_BCS|c^†_−k

0,sc_−k

0,s|Ψ⁽¹⁾_BCSi=|v_k

0|². (4.25) Let us now turn to the direct determination of the GS parameters u_k and v_k (or alternatively to the single order parameter ∆_k) by minimizing the energy of the system on the BCS state.

To be more precise, working in the grandcanonical ensemble, we plan to mini-mize the expectation value ofH−µNon the BCS state, whereHis the Hamiltonian of our system,µthe chemical potential andN the operator of the total number of particles.

Since we work atT = 0and the single particle energyε_k is measured with respect

to the Fermi level, we can write H−µN =X

p,s

εpc^†_p,sc_p,s+1 2

X

p,p⁰,q,s,s⁰

V(p,p⁰,q)c^†_p+q,sc^†_p0−q,s⁰c_p⁰_,s⁰c_p,s. (4.26) We will take expectation values on|Ψ_BCSiconsidering the ”normal” scattering terms (corresponding to Hartree and Fock energy corrections) as already incorporated in the single particle energy ε_p. In the interaction part of the Hamiltonian we will then isolate only scattering processes where the pairing structure of both incoming and outgoing pairs of particles is preserved. This leads us to the relationss⁰ =−s, p=−p⁰. Renamingp≡kandp+q≡k⁰and definingV(k,−k,k⁰−k) =V_k⁰_kwe get the so-calledreducedHamiltonian

H_red−µN=X

k,s

ε_kc^†_k,sc_k,s+1 2

X

k,k⁰,s

V_k0kc^†_k0,sc^†_−k0,−sc_−k,−sc_k,s. (4.27) The scattering processes in the interaction parts of (4.26) and (4.27) are diagram-matically depicted in Fig (4.1.2).

q

p, s p⁰, s⁰ p+q, s p⁰−q, s⁰

Fig. 4.2:The four fermion interaction diagram. Momentum conservation has been implemented. Notice the absence of spin-flip processes in the vertices. The interaction term in the reduced Hamiltonian (4.27) is obtained withp=k, p⁰=−kandq=k⁰−k.

The expectation value of (4.27) on (4.15) is then easily evaluated using what we learned in the simple examples (4.24,4.25). Indeed, since the occupations of differ-ent pairs are uncorrelated, the expectation value of the operator

c^†_k0,sc^†_−k0,−sc_−k,−sc_k,sis factorized into

hc^†_k0,sc^†_−k0,−sc_−k,−sc_k,si=hc^†_k0,sc^†_−k0,−sihc_−k,−sc_k,si= (u_k⁰v^∗_k⁰)(u^∗_kv_k)≡F_k^∗0F_k. (4.28) withF_−k =F_k≡u^∗_kv_k. Thus we deduce

hH_red−µNi=X

k

2ε_k|v_k|²+X

k,k⁰

V_kk⁰F_k^∗F_k⁰. (4.29) We then have to minimize (4.29) with respect to the function∆_kenteringF_k. In order to do that it is convenient to notice that|∆_k|²/E_k= 2∆_kF_k^∗and dE_k/d∆_k =

2F_k^∗(the derivative being done so that∆_kand∆^∗_kremain complex conjugates, and notat constant∆^∗_k), whenceE_k= 2R d∆_kF_k^∗. Using (4.20) we get

hH_red−µNi = X

k

ε_k− ε²_k

E_k

+X

k,k⁰

V_kk⁰F_k^∗F_k0

= X

k

ε_k+X

k

(−E_k+ 2∆_kF_k^∗) +X

k,k⁰

V_kk⁰F_k^∗F_k0 , (4.30) so that, integratingE_kby parts,

hH_red−µNi=X

k

ε_k+ 2X

k

Z

dF_k^∗∆_k+X

k,k⁰

V_kk⁰F_k^∗F_k0 . (4.31) Differentiating with respect toF_k^∗and imposing the vanishing of the derivative, we finally get the BCS equation atT = 0

∆_k=−X

k⁰

V_kk⁰F_k0 =−X

k⁰

V_kk⁰ ∆_k0

2E_k0

. (4.32)

The solution of this equation can be difficult, depending on the form of the matrix elementV_kk⁰. However, following the single-pair Cooper problem, we can use the factorized interaction (4.6), i.e.

V_kk⁰ =

λ <0, for|ε_k|and|ε_k⁰|< ω_D

0, elsewhere . (4.33)

With this choice we get

∆_k

∆0, |ε_k|< ω_D

0, elsewhere (4.34)

where

∆0= ω_D sinh

1 N0|λ|

. (4.35)

We can then substitute the obtained expression for∆_k into (4.19) and finally into (4.29) to get the condensation energy. In fact, by taking the difference between the GS energyEfor the normal phase (4.18) and the resulting (4.29) in the SC state, we obtain

EN− ESC=1

2N0∆²₀. (4.36)

We therefore see that the total energy of the system has been decreased by the phe-nomenon of condensation. The reorganization of the normal GS into the BCS state leads to a net gain in energy atT = 0.

We can understand how the BCS state differs from a normal phase by directly look-ing at the occupation distribution of thekeigenstates. The probability amplitude for the pair(k↑,−k↓)to be occupied is given byv_k. In Fig (4.3) we plot the typical form of|v_k|as a function of|k|.

Fig. 4.3:The typical occupation distribution of the momentum eigenstates in the BCS Ground State.

We see that the condensation implies a smoothening of the occupation distribution around the Fermi energy, in comparison with the sharp Fermi-liquid like drop.

In the following section we will investigate the spectrum of the excitations out of the BCS GS and see that it is indeed gapped. This accounts for the rigidity of the superconducting phase to the external perturbations of low enough energy, and finally to the vanishing resistance. Moreover we will present the temperature de-pendence of the gap and show that there is a critical temperature above which the SC state becomes unstable and a normal Fermi-liquid like phase is preferred.

Im Dokument On the role of spin, pairing and statistics for composite fermions in the fractional quantum hall effect (Seite 98-103)