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CRC Press eBooks

Free Fermi sea at finite temperature

Bruno Juliá Díaz, Arnau Rios Huguet, Héctor Briongos Merino, Javier Rozalén Sarmiento, Artur Polls Martí

Published Sep 17, 2025• 0 citations
Publisher

Abstract

At finite temperatures, the sharp distinction between occupied and unoccupied states near the Fermi surface becomes blurred, as the filling of the states is ruled by the Fermi distribution. This leads to significant changes in the physical properties of the system. Analyzing these effects is crucial for understanding phenomena like electrical and thermal conductivity, as well as the response of materials to external perturbations. Moreover, working at nonzero temperatures allows us to study transitions between quantum and classical regimes, shedding light on the statistical mechanics that govern systems from condensed matter to nuclear physics and astrophysics.

DOI
10.1201/9781003191339-7

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At finite temperatures, the sharp distinction between occupied and unoccupied states near the Fermi surface becomes blurred, as the filling of the states is ruled by the Fermi distribution. This leads to significant changes in the physical properties of the system. Analyzing these effects is crucial for understanding phenomena like electrical and thermal conductivity, as well as the response of materials to external perturbations. Moreover, working at nonzero temperatures allows us to study transitions between quantum and classical regimes, shedding light on the statistical mechanics that govern systems from condensed matter to nuclear physics and astrophysics.
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.
Under conditions of high temperature and low density, we shall ˆnd that it does not matter whether classical or quantum mechanical descriptions are used for a system of particles. At high densities and low temperatures this is no longer true, because of the overlap of the particles’ wave functions and quantum mechanics must be used, giving rise to quantum statistics. Under high-density, low-temperature conditions, the properties of a system depend in a crucial way on whether the particles that make up the system are fermions or bosons. Many fascinating phenomena occur in condensed matter as the temperature is lowered. Examples are ferromagnetism, superconductivity, and super uidity. ese important new properties appear fairly abruptly at phase transitions. Progress in the microscopic understanding and description of the behavior of these systems involves quantum mechanics and statistical physics ideas. Applications of quantum statistics are not conˆned to terrestrial systems and include astrophysical phenomena such as the microwave background radiation from the Big Bang and the mass-radius relationships for white dwarf stars and neutron stars. An important concept in thermal physics is that of entropy, which, as we shall see, increases with time as systems become more disordered. e increase of the entropy of the universe with the passage of time provides what is termed time’s arrow. An interesting and unanswered question arises as to why the entropy of the universe was so low at the beginning of time.
Chapter 3 provides a brief review of a few concepts from classical Newtonian physics, and how this fails to explain phenomena at the nanoscopic scale. As is detailed within, the classical mechanical model is able to describe all observed physical phenomena; however, when it comes to the nanoscale this model breaks down, because the properties of the materials change dramatically. At this scale, the surface-to-volume ratio increases, gravitational forces become negligible and random molecular motion becomes more important. This changes the properties of the material and affects the devices that are fabricated at this scale. It is hence necessary to develop a new model that can explain such physical phenomena. This has led to a new branch of physics, i.e. quantum physics (QP) or quantum mechanics, which is able to explain the behavior observed at nanoscale. QP enables us to explain phenomena such as the photoelectric effect, Compton effect. etc. In this chapter, we introduce quantum mechanics and how it evolved in the late 19th and early 20th centuries. Further, light’s behaviour as a particle as well as a wave is described by the photoelectric effect and Young’s double-slit experiment, respectively. Further, the dual nature of the particle is explained using the de Broglie principle. Schrodinger’s equation is then explained, which gives the probability of finding a subatomic-scale particle at a certain position in a quantum well. In addition to the formulation of Schrodinger’s equation, its solution, the concept of a potential well, the characteristics of the wave function and its interpretation and significance, and, finally, the application of Schrodinger’s equation in the structure of an atom form subsequent parts of this chapter. The equation serves as a basis with which to understand band theory. Band theory explains the difference between conductors, insulators and semiconductors. Materials are distinguished according to the differences in their band gaps. Adding impurities or varying temperature changes the band gap in semiconductors, which consequently changes the properties of the semiconductor and its electrical conduction. One more important measure to understand and exploit the properties and behaviour of materials is Fermi–Dirac statistics. It is useful to acquire the probability of electrons at a given energy level at a specific temperature. These concepts can be utilised by physicists and chip makers to make more energy-efficient devices. Hence, this chapter details the fundamentals of quantum physics and how they relate to the electrical properties of nanodevices.
Recent cold atom experiments have observed bad and strange metal behaviors in strongly-interacting Fermi-Hubbard systems. Motivated by these results, we calculate the thermoelectric transport properties of a 2D Fermi-Hubbard system in the weak coupling limit using quantum kinetic theory. We find that many features attributed to strong correlations are also found at weak coupling. In particular, for temperatures T≳tT\gtrsim tT≳t the electrical resistivity is nearly linear in temperature despite the fact that the quasiparticle scattering rate is non-linear and changes by nearly an order of magnitude. We argue that this asymptotic behavior is a general feature of systems with a finite spectral width, which implies that there is no MIR bound on the resistivity in single-band models. Due to nesting, the TTT-linear resistivity persists down to T=0T=0T=0 at half filling. Our work sheds light on the transport regime in ultracold atom experiments, which can differ substantially from that of condensed matter systems. Disentangling these band-structure effects from the physics of strong correlations is a major challenge for future experiments.
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.