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Complement A_XV Particles and holes

1  1 Ground state of a non-interacting fermion gas

2  2 New definition for the creation and annihilation operators

3  3 Vacuum excitations

Creation and annihilation operators are frequently used in solid state physics where the notion of particle and hole plays an important role. A good example is the study of metals or semiconductors, where we talk about an electron-hole pair created by photon absorption. A hole means an absence of a particle, but it has properties similar to a particle, like a mass, a momentum, an energy; the holes obey the same fermion statistics as the electrons they replace. Using creation or annihilation operators allows a better understanding of the hole concept. We will remain in the simple framework of a free particle gas, but the concepts can be generalized to the case of particles placed in an external potential or a Hartree-Fock mean potential (Complement E_XV).

1. Ground state of a non-interacting fermion gas

Consider a system of non-interacting fermions in their ground state. We assume for simplicity that they are all in the same spin state, and thus introduce no spin index (generalization to several spin states is fairly simple). As we showed in Complement C_XIV, this system in its ground state is described by a state where all the occupation numbers of the individual states having an energy lower than the Fermi energy EF are equal to 1, and all the other individual states are empty. In momentum space, the only occupied states are all the individual states whose wave vector k is included in a sphere (called the “Fermi sphere”) of radius kF (the “Fermi radius”) given by¹:

(1)

where we have used the notation of formula (7) in Complement C_XIV: EF is the Fermi energy (proportional to the particle density to the power 2/3), and L the edge length of the cube containing the N particles. When the system is in its ground state, all the individual states inside the Fermi sphere are occupied, whereas all the other individual states are empty. Choosing for the individual states basis {|ui〉} the plane wave basis, noted {|uk〉} to explicit the wave vector k_i, the occupation numbers are:

(2)

In a macroscopic system, the number of occupied states is very large, of the order of the Avogadro number (≃10²³). The ground state energy is given by:

(3)

with:

(4)

The sum over k_i in (3) must be interpreted as a sum over all the k_i values that obey the boundary conditions in the box of volume L³, as well as the restriction on the length of the vector k_i which must be smaller or equal to kF.

2. New definition for the creation and annihilation operators

We now consider this ground state as a new “vacuum” and introduce creation operators that, acting on this vacuum, create excited states for this system. We define:

(5)

Outside the Fermi sphere, the new operators and c_ki are therefore simple operators of creation (or annihilation) of a particle in a momentum state that is not occupied in the ground state. Inside the Fermi sphere, the results are just the opposite: operator creates a missing particle, that we shall call a “hole”; the adjoint operator b_ki repopulates that level, hence destroying the hole. It is easy to show that the anticommutation relations for the new operators are:

(6)

as well as:

(7)

which are the same as for ordinary fermions. Finally, the cross anticommutation relations are:

(8)

3. Vacuum excitations

Imagine, for example, that with this new point of view we apply an annihilation operator b_ki, with |k_i| ≤ kF, to the “new vacuum” . The result must be zero since it is impossible to annihilate a non-existent hole. From the old point of view and according to (5), this amounts to applying the creation operator to a system where the individual state |k_i〉 is already occupied, and the result is indeed zero, as expected. On the other hand, if we apply the creation operator , with |k_i| ≤ kF, to the new vacuum, the result is not zero: from the old point of view, it removes a particle from an occupied state, and in the new point of view it creates a hole that did not exist before. The two points of view are consistent.

Instead of talking about particles and holes, one can also use a general term, excitations (or “quasi-particles”). The creation operator of an excitation of |k_i| ≤ kF is the creation operator of a hole ; the creation operator of an excitation of |k_i| > kF is the creation operator of a particle. The vacuum state defined initially is a common eigenvector of all the particle annihilation operators, with eigenvalues zero; in a similar way, the new vacuum state is a common eigenvector of all the excitation annihilation operators. We therefore call it the “quasi-particle vacuum”.

As we have neglected all particle interactions, the system Hamiltonian is written as:

(9)

Taking into account the anticommutation relations between the operators b_ki and . we can rewrite this expression as:

(10)

where E₀ has been defined in (3) and simply shifts the origin of all the system energies. Relation (10) shows that holes (excitations with |k_i| ≤ kF ) have a negative energy, as expected since they correspond to missing particles. Starting from its ground state, to increase the system energy keeping the particle number constant, we must apply the operator that creates both a particle and a hole: the system energy is then increased by the quantity ej — ei ; inversely, to decrease the system energy, the adjoint operator c_kj b_ki must be applied.

Comments:

(i) We have discussed the notion of hole in the context of free particles, but nothing in the previous discussion requires the one-particle energy spectrum to be simply quadratic as in (4). In semi-conductor physics for example, particles often move in a periodic potential, and occupy states in the “valence band” when their energy is lower than the Fermi level EF whereas the others occupy the “conduction band”, separated from the previous band by an “energy gap”. Sending a photon with an energy larger than this gap allows the creation of an electron-hole pair, easily studied in the formalism we just introduced.

A somewhat similar case occurs when studying the relativistic Dirac wave equation, where two energy continuums appear: one with energies greater than the electron rest energy mc² (where m is the electron mass, and c the speed of light), and one for negative energies less than —mc² associated with the positron (the antiparticle of the electron, having the opposite charge). The energy spectrum is relativistic, and thus different from formula (4), even inside each of those two continuums. However, the general formalism remains valid, the operators and b_ki describing now, respectively, the creation and annihilation of a positron. The Dirac equation however leads to difficulties by introducing for example an infinity of negative energy states, assumed to be all occupied to avoid problems. A proper treatment of this type of relativistic problems must be done in the framework of quantum field theory.

(ii) An arbitrary N-particle Fock state |Φ〉 does not have to be the ground state to be formally considered as a “quasi-particle vacuum”. We just have to consider any annihilation operator on an already occupied individual state as a creation operator of a hole (i.e. of an excitation); we then define the corresponding hole (or excitation) annihilation operators, which all have in common the eigenvector |Φ〉 with eigenvalue zero. This comment will be useful when studying the Wick theorem (Complement C_XVI). In § E of Chapter XVII, we shall see another example of a quasi-particle vacuum, but where, this time, the new annihilation operators are no longer acting on individual states but on states of pairs of particles.

1 ¹ In Complement CXIV we had assumed that both spin states of the electron gas were occupied, whereas this is not the case here. This explains why the bracket in formula (1) contains the coefficient 6π2N instead of 3π2N.