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Physics 216: Special topics in many-body physics, Spring 2003:
http://socrates.berkeley.edu/˜ jemoore/phys216.html
Lecture V
First let’s finish the flux quantization in vortices from last time. The GL current density is (here n is the total density of electrons)
jx n
ie¯h 2 m
( ψ 1 ∗
∂ψ ∂x − ψ 1
∂ψ∗ ∂x
) −
2 e^2 mc Axψ∗ 1 ψ 1. (1)
and similarly for jy and jz. Consider a single hole containing magnetic flux in a large supercon- ducting body. Far away from the hole, the supercurrent should be zero and the magnitude |ψ| should be constant. For constant |ψ|, the current can be rewritten
j =
e¯hns 2 m
( ∇φ −
2 e ¯hc A
) (2)
. Setting this equal to zero gives
∇φ = 2 e ¯hc A.^ (3)
Now, around any closed loop the phase Φ should change by a multiple of 2π for the wavefunction in the GL equation to be single-valued. Then the line integral of A around the loop must be a multiple of 2 π ¯hc 2 e = Φ^0. Finally, the line integral of^ A^ around the loop is just the integrated magnetic flux through the loop
Φ =
∫ BdS = nΦ 0 = n π¯hc e
hc 2 e
In real units the flux quantum is 2 × 10 −^7 G cm^2.
The above vortex is a simple example of a “topological defect” in a field theory: on a length scale much larger than the vortex, the vortex appears as a point singularity around which the phase wraps by 2πn. Similar topological defects occur in magnets, in liquid crystals (described by a classical field theory) and in high-energy physics (for example, magnetic monopoles in gauge theories).
In the rest of this lecture we introduce the main ideas of the Fermi liquid and motivate many- body perturbation theory. A reference for the Fermi liquid ideas is the first chapter of Landau and Lifshitz volume 9.
The assumption of Fermi liquid theory is that, whatever the interactions may be, we can identify “elementary excitations” that are like particles (and hence are called “quasi-particles”): they have well-defined momentum, spin- 12 , and charge e. Furthermore, they are spatially “local” relative to macroscopic sizes like a sample size, and are long-lived if their excitation energy is small. This last point is quite important: quasiparticles of finite excitation energy E > 0 do interact with each other and can decay, but the lifetime becomes larger as E → 0.
As an intuitive picture of the quasiparticles, think of them as a single electron dressed by a cloud of electron-hole pairs: as a result, the effective mass may be modified by the “screening cloud”. But, since the electron-hole pairs are neutral and bosonic (actually we’ll assume them usually to be spin-zero), the charge and fermionic statistics are unmodified.
The Fermi momentum is determined through
N/V = 2 · 4 πp^3 F /3(2π¯h)^3. (5)
We will assume that this continues to hold because of the adiabatic connection between quasipar- ticles and original fermions. Define the normalized quasiparticle distribution through
∑ α
∫ nαα dτ = tr
∫ n dτ ˆ =
N
V
, dτ = d^3 p (2π¯h)^2
In the following we will use the Einstein summation convention that repeated indices are implicity summed over. The meaning of the indices on the Hermitian density matrix δn is that diagonal elements correspond to the density of spin-up or spin-down electrons.
In case you haven’t seen a density matrix in a while, here is a lightning review. Recall that the off-diagonal term reflects the fact that spin is a quantum variable. For instance, for a single particle in a mixed state with half spin-up and half spin-down, we would have
n =
( (^1) 2 0 (^0 )
) (7)
while for a pure state with spin aligned along some axis in the x-y plane,
n =
( (^1) 2
1 1 2 2
1 2
)
. (8)
The spin operator along the x-axis is given by σx in this two-component space, so
〈sx〉 = tr nσx (9)
where here this is a quantum statistical expectation value, which for a pure state coincides with the ordinary quantum expectation value. Pure states are essentially projection operators, so n^2 = n; any state, pure or mixed, has a trace related to the total number of particles.
The change in energy due to a change δn in the quasiparticle occupancies should be written, for a small change from the equilibrium distribution obtained below, as
δE V
∫ (p)δndτ. (10)
Actually, let us make a slightly more general form to account for situations where the distribution of up-spin particles is different from that of down-spin particles.
δE V
∫ αβ (p)δnβαdτ = tr
∫ ˆ(p)δˆndτ. (11)
For the spin-symmetric case, the tensors ˆ and δnˆ are both diagonal: αβ = δαβ , nαβ = nδαβ.
The assumption of Fermi liquid theory can now be stated: we will allow the effective energy ˆ(p) in equation (11) to depend on the occupancy of other quasiparticle states, in a simple way. Denote by δαβ (p) the change in the effective energy of quasiparticles of momentum p induced by a given nonequilibrium distribution of the other quasiparticles:
δαβ (p) =
∫ fαγ,βδ(p, p′)δn(p′)γδ dτ ′. (12)
and linearizing around the Fermi surface as usual: Ek −μ = vF (k −kF ), we guess that Γ ∼ (E −μ)^2 , which is borne out by a detailed calculation.
Hence in 3D Γ E − μ, so the lifetime of quasiparticles increases rapidly as the quasiparticle moves toward the Fermi level. Another way to look at this, which will be used later, is that the linewidth of the quasiparticle becomes very narrow as E → μ. This gives some support for the contention that the Fermi liquid is a stable picture in 3D.
