Quantum Field Theory I

34 CHAPTER 1. INTRODUCTIONS
Bloch theorem reveals the generic form of eigenfunctions of the Hamiltonian
which is symmetric under the lattice translations. The eigenfunctions are
labeled by two quantum numbers n and ⃗ k, the former (called the zone number)
being discrete and the latter (called the quasi-momentum)being continuousand
discrete for infinite and finite lattices respectively (for large finite lattice, the ⃗ k
is so-called quasi-continuous). The theorem states that the eigenfunctions are
of the form ϕ n, ⃗ k
(⃗r) = e i⃗ k.⃗r u n, ⃗ k
(⃗r), where the function u n, ⃗ k
(⃗r) is periodic (with
the periodicity of the lattice) 38 . In the solid state physics the theorem is supplemented
by the following more or less plausible assumptions:
• The theorem holds not only for the Schrödinger equation, but even for the
Hartree and Hartree-Fock equations with periodic external potentials.
• The energy ε n ( ⃗ k) is for any n a (quasi-)continuous function of ⃗ k.
• The relation ε n ( ⃗ k) = const defines a surface in the ⃗ k-space.
The very important picture of the zone structure of the electron states in
solids results from these assumptions. So does the notion of the Fermi surface,
one of the most crucial concepts of the solid state physic. In the ground
state of the system of non-interacting fermions, all the one-fermion levels with
energies below and above some boundary energy are occupied and empty respectively.
The boundary energy is called the Fermi energy ε F and the surface
corresponding to ε n ( ⃗ k) = ε F is called the Fermi surface.
The reason for the enormous importance of the Fermi surface is that all
the low excited states of the electron system lie in the vicinity of the Fermi
surface. The geometry of the Fermi surface therefore determines a great deal of
the physical properties of the given solid.
Calculationofthe Fermi surfacefrom the first principlesis onlypossible with
some (usually quite radical) simplifying assumptions. Surprisingly enough, yet
the brutally looking simplifications use to give qualitatively and even quantitatively
satisfactory description of many phenomena. However, the most reliable
determinations of the Fermi surface are provided not by the theory, but rather
by the experiment. There are several methods of experimental determination
of the Fermi surface. The mutual consistency of the results obtained by various
methods serves as a welcome cross-check of the whole scheme.
38 The proof is quite simple. The symmetric Hamiltonian commutes with any translation
T ⃗R where R ⃗ is a lattice vector. Moreover, any two translations commutes with each other.
Let us consider common eigenfunctions ϕ(⃗r) of the set of mutually commuting operators
presented by the Hamiltonian and the lattice translations. We denote by λ i the eigenvalues
of the translations along the three primitive vectors of the lattice , i.e.T ⃗ai ϕ(⃗r) = λ i ϕ(⃗r). The
composition rule for the translations (T ⃗A+ ⃗ B = T ⃗A T ⃗B ) implies T ⃗R ϕ(⃗r) = λ n 1
1 λn 2
2 λn 3
3 ϕ(⃗r) for
⃗R = ∑ 3
i=1 n i⃗a i . On the other hand, T ⃗R ϕ(⃗r) = ϕ(⃗r + ⃗R). This requires |λ i | = 1, since
otherwise the function ϕ(⃗r) would be unbounded or disobeying the Born-Karman periodic
boundary conditions for the infinite and finite lattice respectively. Denoting λ i = e ik ia i (no
summation) one has T ⃗R ϕ(⃗r) = e i⃗ k. ⃗R ϕ(⃗r). At this point one is practically finished, as it is now
straightforward to demonstrate that u ⃗k (⃗r) = e −i⃗ k.⃗r ϕ(⃗r) is periodic. Indeed T ⃗R e −i⃗ k.⃗r ϕ(⃗r) =
e −i⃗ k.(⃗r+ ⃗ R) ϕ(⃗r + ⃗ R) = e −i⃗ k.⃗r e −i⃗ k. ⃗R T ⃗R ϕ(⃗r) = e −i⃗ k.⃗r ϕ(⃗r).
For an infinite lattice, ⃗ k can assume any value ( ⃗ k is a continuous). For finite lattice the
allowed values are restricted by the boundary conditions to discrete possible values, nevertheless
for large lattices the ⃗ k-variable turns out to be quasi-continuous (for any allowed value,
the closest allowed value is ”very close”). For a given ⃗ k, the Hamiltonian can have several
eigenfunctions and eigenvalues and so one needs another quantum number n in addition to
the quantum number ⃗ k .