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Physics 216: Special topics in many-body physics, Spring 2003
Lecture I
The goal of this course, broadly speaking, is to understand how the seemingly simple laws of quantum mechanics and electromagnetism give rise to a rich variety of highly organized phases of matter. Some of these phases, such as superconductors and superfluids, are so different in their properties from our everyday experience that to their original discoverers they must have seemed nearly magical. This first lecture gives a nontechnical overview of the main topics to be covered and the “big picture” ideas, and has a short discussion/review of second quantization and coherent states.
Useful textbooks are listed on the course information sheet. The two texts closest to the presentation in class are Schrieffer and Auerbach, but there will be some topics discussed, such as the Kondo effect, that are not mentioned in either book. The prerequisites for the course are a solid understanding of quantum mechanics and one semester each of statistical mechanics and solid-state physics.
Some examples of correlated states that we may discuss are superconductors and superfluids, the “Fermi liquid” description of metals, quantum ferromagnetism and antiferromagnetism, the integer and fractional quantum Hall effect, the “Luttinger liquid” theory of one-dimensional systems like carbon nanotubes, and the Kondo effect. A large part of the course will be devoted to understanding both the various instabilities of the Fermi liquid (to attractive interactions, to magnetic order, in one dimension, etc.) and its exceptional stability to repulsive interactions in two and three dimensions. All of these emerge from what is sometimes known as the “theory of almost everything”: nonrelativistic kinetic terms for electrons and ions, plus the instantaneous Coulomb interaction.
The main theoretical techniques used will be second quantization as a way to write new types of many-body states, such as the BCS wavefunction, and many-body perturbation theory (Feynman diagrams) for Green’s functions. Second quantization, which we will begin at the end of this lecture, is a compact way to write states with strong correlations or variable particle number; many-body perturbation theory is a clever way to compute corrections to physical quantities without having to deal with the entire wavefunction of 10^26 particles.
Adiabatic continuity and discontinuity Example I of continuity: In describing most metals and insulators, one starts from a picture of noninteracting electrons in e.g. calculating the band structure and other properties. However, the Coulomb interaction energy is actually very large, and one might wonder why it is appropriate to assume that noninteracting electrons (a free Fermi gas) make a sensible starting point.
The underlying idea, first phrased in these terms by Landau, is that electrons in a real metal form a “Fermi liquid”, which bears the same relation to the “Fermi gas” of free electrons that a normal liquid bears to a normal gas: the interactions are much stronger, but there is no change in symmetry or in the fundamental nature of the state. In particular, the “elementary excitations” of the ground state (those that are found to carry current, heat, and other properties) bear the same quantum numbers as ordinary electrons. We can imagine looking at the full energy spectrum of a many-particle system and trying to identify mobile low-energy excitations: Landau’s theory, which we will justify later in this course, explains how these excitations can wind up as electrons “dressed” by particle-hole pairs, which renormalize the mass (by up to a factor 10^3 in so-called heavy fermion compounds) and some other properties but not the charge e and fermionic statistics.
It turns out that electrons in a typical metal are stable to strong repulsive interactions, but can be unstable to even weak attractive interactions. The resulting superconducting state is an example of how adiabatic continuity can be violated: the lowest-energy charged excitations in a traditional superconductor are “Cooper pairs” of charge 2e.
Example I of discontinuity: The natural energy scale of noninteracting electrons in a solid is the Fermi energy, which can be tens of thousands of kelvins. The natural Coulomb interaction energy scale e^2 n−^1 /^3 is comparable to the Fermi energy. Both these energies are very large in comparison to the superconducting transition temperature Tc, which for an old-fashioned BCS superconductor is of order 10 K. It turns out that this new small energy scale is a signal of adiabatic discontinuity or “nonperturbative” behavior.
The superconducting gap in BCS theory scales as
Tc ∼ De−^1 /λN^ (0), (1)
where D is a bandwidth or Fermi energy, λ is the energy of the attractive electron-electron interac- tion, and N (0) is the electron DOS at the Fermi level. Looking at this formula, suppose we try to expand it as a power series in λ around λ = 0, when the system should be a noninteracting Fermi gas. You will find that all the derivatives at λ = 0 are 0, so the Taylor series looks like
Tc ∼ 0 + λ0 +
λ^2 2!
This is often stated as “Tc is zero to all orders in perturbation theory”. Its practical meaning is that we need to find a new starting point for the description of the superconductor, rather than just starting from the free Fermi gas and trying to incorporate interactions perturbatively. A large part of this course will be devoted to the new starting points or organizational principles that emerge from the simple rules of nonrelativistic QM and the Coulomb interaction.
Example II of continuity: A superfluid is “like” a pure (noninteracting) BEC, even though the strong interactions in the superfluid make its quantitative properties very different. For instance, in a noninteracting bosonic gas, at temperature T = 0 all of the particles are in the lowest eigenstate; for an atomic BEC, about 99 percent or more are in the lowest eigenstate, as the interactions are weak; for superfluid helium-4, only about 10 percent are in the lowest eigenstate. However, helium- 4 still shows amazing properties such as an absence of viscosity for low-velocity flows, because in some sense the interactions do not change the basic nature of the state.
Second quantization and states with variable particle number You are all familiar, I’m sure, with Slater determinants for states of multiple identical fermions. A convenient way to write the overall wavefunction of three electrons in states ψ 1 , ψ 2 , ψ 3 is
Ψ(r 1 , r 2 , r 3 ) =
∣∣ ∣∣ ∣∣
ψ 1 (r 1 ) ψ 1 (r 2 ) ψ 1 (r 3 ) ψ 2 (r 1 ) ψ 2 (r 2 ) ψ 2 (r 3 ) ψ 3 (r 1 ) ψ 3 (r 2 ) ψ 3 (r 3 )
∣∣ ∣∣ ∣∣.^ (3)
This satisfies the requirement of asymmetry; note that the wavefunction vanishes if any two of the ri or ψi are equal.
An example of when we might want to use Slater determinants is when we study a many-body system in the Hartree approximation. Suppose we want to approximate the many-body state of N electrons moving in some constant background potential Vb (for example, from the ions of a
Another requirement is that trying to put two fermions into the same single-particle state should also give 0, as should trying to get rid of two fermions:
c† kc† k = ckck = 0, (10)
where this notation means that the operators give 0 applied to any many-body state.
The most important condition on the ck is the anticommutation relation:
{c† k, ck} = c† kck + ckc† k = 1. (11)
This essentially says that the probability of single-particle state k being empty in many-body state |Ψ〉, which is 〈Ψ|c† kck|Ψ〉, plus the probability of its being occupied, which is 〈Ψ|c† kck|Ψ〉, should sum to 1. Recall that this anticommutation relation holds only for fermions; clearly something different will be required for bosons.
Now we need to add the spin of the fermions and ask how the operators of different states k alter each other. For instance, is the state c† k 1 c† k 2 | 0 〉 the same as c† k 1 c† k 2 | 0 〉? There is an element
of choice here, but the simplest way to think about it is to consider the operator c† k 1 as “adding on” a new row to the Slater determinant (3). Then, since interchanging two rows of a determinant changes the sign of a determinant, we should have
c† k 1 c† k 2 | 0 〉 = −c† k 1 c† k 2 | 0 〉, (12)
which we summarize as {c† σ 1 k 1 , c† σ 2 k 2 } = {cσ 1 k 1 , cσ 2 k 2 } = 0. (13)
so that the two sequences create the same many-body state but with a sign difference. As you might expect, this sign difference is only there for fermions, and will not be present for bosons. Finally, restoring the spin variable σ, we write the full anticommutation relation as
{c† σ 1 k 1 , cσ 2 k 2 } = δk 1 ,k 2 δσ 1 ,σ 2. (14)
Hence these operators just anticommute unless both the momentum and the spin are the same, in which case there is a number 1 on the right side, expressing the idea that the fermionic state should have probability 1 of being occupied or empty.
Now let’s see how the above need to be modified for bosons. The main examples of bosonic operators that will appear in this course are the modes of the electromagnetic field, or of phonon excitations in a solid. For bosonic operators, we have a commutation relation instead of an anti- commutation relation: [bk 1 , b† k 2 ] = δk 1 ,k 2. (15)
From this you can show that the number of quanta in mode k, i.e., the expectation value of the number operator nk ≡ b† kbk, is increased by 1 by the creation operator b† k, and decreased by 1 by the annihilation operator
We are now in a position to understand the simplest example of coherent states, which are useful in taking the classical limit of a harmonic oscillator or in setting up Feynman path integrals over classical configurations.
Let’s return to fermions. We can write the filled Fermi sea as ∏
|k|≤kF
c† k↑c† k↓| 0 〉. (16)
Here all states of momentum below the Fermi momentum are filled by the creation operators, while all those above the Fermi momentum are empty. We could equally well have written this state as a Slater determinant, of course.
As motivation for next time, consider the following famous example of a strongly correlated wavefunction: ΨBCS =
∏
k
(uk + vkc† k↓c†−k↑)| 0 〉, |uk|^2 + |vk|^2 = 1. (17)
Here uk and vk are some k-dependent complex numbers satisfying the normalization constraint above. Now this will just give the filled Fermi sea above for a particular choice of the uk, vk: the filled Fermi sea results from
uk =
{ 0 if k ≤ kf 1 if k > kf ,^ vk^ =
{ 1 if k ≤ kf 0 if k > kf.^ (18)
However, suppose we smear out the sharp boundary at the Fermi level by an energy ∆, so that now over some interval, uk and vk are both between 0 and 1. This might seem to be similar to the Fermi gas at finite temperature, because then the occupancy is also smeared out over a distance kT near the Fermi level.
However, the smeared BCS state is very different on a fundamental level because it has perfect pair correlations. For every momentum k, even in the smeared state, if the spin-up orbital is occupied then the spin-down orbital at momentum −k is automatically also occupied. This does not occur in the Fermi gas at finite temperature, where the spin-up and spin-down orbitals are independent. The next lecture will begin the discussion of how this amazing state winds up as the ground state of a semi-realistic Hamiltonian.
