Most of the visible matter around us is made of fermions, e.g. neutrons, protons and electrons. Their properties are thus relevant to understand the behavior of matter in different situations and at very different scales. Fermions, in contrast to bosons, are described by antisymmetric wavefunctions, as explained in Chapter 1.3. This has an important consequence already for non-interacting fermionic systems. Namely, the way a set of non-interacting identical fermions builds its ground state is quite peculiar as compared to bosons and gives rise to the so-called Fermi sea.
Most of the visible matter around us is made of fermions, e.g. neutrons, protons and electrons. Their properties are thus relevant to understand the behavior of matter in different situations and at very different scales. Fermions, in contrast to bosons, are described by antisymmetric wavefunctions, as explained in Chapter 1.3. This has an important consequence already for non-interacting fermionic systems. Namely, the way a set of non-interacting identical fermions builds its ground state is quite peculiar as compared to bosons and gives rise to the so-called Fermi sea.
The concept of the Fermi sea is central to understanding the electronic properties of many-body quantum systems. In one-dimensional systems, the Fermi sea is characterized by a simpler topology, with electrons occupying states along a linear momentum axis. In contrast, in three dimensions, the free Fermi sea takes on a more complex geometry, forming a spherical surface in momentum space. This difference profoundly influences the density of states, interactions between particles, and the emergent phenomena in these systems. Three-dimensional fermionic systems are governed by a rich spectrum of quantum states and exhibit different collective behaviors.
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.
The laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.