Structure by exclusion

It should be known by now to any reader of either academic or popular physics literature that the 1920s marked a decisive decade for the development of quantum mechanics. As the many commentaries associated with the recent centenary celebrations have highlighted1, it was a period in which a remarkable stream of foundational papers helped establish the mathematical framework of quantum theory. Credit: Bettmann / Contributor/ Bettmann/gettyIn those early quantum years, fermions entered the discussion mainly in the form of electrons, which were already a central piece for the understanding of atomic structure and the behaviour of solids. But the emerging quantum formalism offered little guidance on how collections of identical electrons should populate atomic energy levels, and resorting to classical statistical treatments had led to predictions at odds with experiments.For example, models treating conduction electrons in solids as a classical gas suggested a large, temperature-independent electronic contribution to the heat capacity. Nothing of that sort was picked up in experiments on metals — on the contrary, the results indicated that only a small fraction of electrons, becoming negligible at low temperatures, were thermally active.At the atomic scale, spectroscopic observations pointed to well-defined patterns of electron occupancy that did not follow from classical counting rules. Even within semiclassical models, based on the quantization of individual electronic states, there was no consistent explanation for why multiple electrons did not simply pile up in the lowest available states, as bosons do2. Thus, the observed electronic distributions appeared difficult to justify on the basis of repulsive Coulomb interactions alone.The solution came from the works of Enrico Fermi3 (pictured, left) and Paul Dirac4 (pictured, right), who in 1926 independently derived a quantum statistical picture that included both particle indistinguishability and constraints on electronic state occupation.It is instructive that the two physicists reached the same results following distinct conceptual routes. Fermi took a pragmatic approach, reformulating classical statistical mechanics for systems composed of particles obeying the Pauli exclusion principle, which limits single-particle quantum states to host at most one identical fermion. After imposing this constraint, counting the arrangements that fit a fixed energy and particle number was all he needed to derive the correct equilibrium distribution.Dirac proceeded from a more formal starting point, by requiring the many-particle state of identical fermions to be antisymmetric, that is, to switch sign upon the exchange of two particles. He explicitly showed that exclusion follows directly from this principle, as exchanging two fermions in the same state would have no effect, which is incompatible with antisymmetry. From this, he computed the total energy of a system of non-interacting fermions by summing the energies of the single-occupied states. Applying the standard conditions of thermal equilibrium finally led to the same distribution obtained by Fermi.One of the implications of Fermi–Dirac statistics is that electronic states can be filled up to a certain energy — the Fermi energy — and only the electrons in proximity of that boundary can participate in thermal or transport processes. This solved the puzzle of the limited electronic contribution to the heat capacity of solids that emerged from earlier experiments. It also helped explain the distinction between metals, insulators and semiconductors, as the position of the Fermi energy relative to electronic bands or gaps determines whether charge carriers are available for transport.Realizing that fermions and bosons obey different quantum statistics was essential to better understand collective properties of matter in a broader sense. Although solids are built from atoms and electrons, their low-energy behaviour can be more conveniently described in terms of collective excitations with either fermionic or bosonic character. Fermionic excitations range from Bogoliubov quasiparticles in superconductors to spinons in strongly correlated systems, whereas bosonic modes include phonons, magnons and Cooper pairs. Distinguishing these excitations by the statistics they obey represented a key tool to advance the study of condensed matter systems.Beyond condensed matter and microscopic scenarios, the formulation of Fermi–Dirac statistics also had implications in astrophysics. As multiple fermions cannot simultaneously occupy the same low-momentum states, squeezing a dense fermionic system results in high-momentum states being filled quickly. This generates a pressure, called degeneracy pressure, that persists at zero temperature.This effect plays a role in sustaining the structure of compact astronomical objects such as white dwarfs, where it counteracts gravitational collapse, and sets the scale for the characteristic masses and sizes of stellar remnants supported by electrons or, at higher densities, neutrons.The lasting impact of the work of Fermi and Dirac lies in how little it assumes. By just determining how states can be occupied, their statistical framework already explains much of the behaviour of fermionic matter and is a remarkable example of how constraints can be primary explanatory objects in physics.