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Microscopic Physics

Григорий Ефимович Воловик

Published Feb 26, 2009• 0 citations
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Abstract

Abstract This chapter deals with the Fermi systems, where the low-energy effective theory involves both bosonic and fermionic fields. Above the phase transition to the superconducting or superfluid state, the overwhelming majority of systems consisting of fermionic particles (electrons in metals, neutrons in neutron stars, 3He atoms in 3He liquid, etc.) form a so-called Fermi liquid. Below transition new types of fermionic vacua emerge. This chapter discusses the Bardeen–Cooper–Schrieffer (BCS) theory for spin-triplet superfluids, which provide examples of different universality classes of fermionic vacua: fully gapped vacua, vacua with stable and marginal point nodes — Fermi points, and vacua with nodal lines — Fermi lines. It also discusses emergent ‘relativistic’ quasiparticles, fundamental constants and hierarchy of Planck energy scales in fermionic systems, problem of vacuum energy and cosmological term in bi-metric gravity, and mass generation for Standard Model fermions.

DOI
10.1093/acprof:oso/9780199564842.003.0007

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Abstract This chapter deals with the Fermi systems, where the low-energy effective theory involves both bosonic and fermionic fields. Above the phase transition to the superconducting or superfluid state, the overwhelming majority of systems consisting of fermionic particles (electrons in metals, neutrons in neutron stars, 3He atoms in 3He liquid, etc.) form a so-called Fermi liquid. Below transition new types of fermionic vacua emerge. This chapter discusses the Bardeen–Cooper–Schrieffer (BCS) theory for spin-triplet superfluids, which provide examples of different universality classes of fermionic vacua: fully gapped vacua, vacua with stable and marginal point nodes — Fermi points, and vacua with nodal lines — Fermi lines. It also discusses emergent ‘relativistic’ quasiparticles, fundamental constants and hierarchy of Planck energy scales in fermionic systems, problem of vacuum energy and cosmological term in bi-metric gravity, and mass generation for Standard Model fermions.
Abstract This chapter discusses effective theories of quantum fermionic liquids. In the low-energy limit, the type of the effective theory depends on the structure of the quasiparticle spectrum, which in turn is determined by topology in momentum space (p-space topology). The p-space topology distinguishes three main generic classes of the stable fermionic spectrum in the quantum vacuum of a 3+1 fermionic system: vacua with Fermi surfaces, vacua with Fermi points, and vacua with a fully gapped fermionic spectrum. Fermi surface is stable because it represents the topological object — the vortex in momentum space. As a result, Fermi liquids with Fermi surface share the properties of their simplest representative: weakly interacting Fermi gas. The low-energy physics of the interacting particles in a Fermi liquid is equivalent to the physics of a gas of quasiparticles moving in collective Bose fields produced by all other particles. Another topological object in p-space is the hedgehog, which is responsible for stability of Fermi points. Example of vacuum of the Fermi-point universality class is provided by 3He-A. Near the Fermi point the effective relativistic field theory emerges with emerging Weyl fermions, gauge fields, and gravity. The chapter discusses p-space and r-space topology, topological invariant for Fermi surface and Fermi points in terms of Green's function, Landau and non-Landau Fermi liquids, collective modes of the Fermi surface, volume of the Fermi surface as invariant of adiabatic deformations, collective modes of vacuum with Fermi points (electromagnetic and gravitational fields), and manifolds of zeros in p-space of higher dimensions.
Abstract There are two ways to study quantum liquids. One is the fully microscopic treatment, which can be realised either by numerical simulations of the many-body problem, or for some special ranges of the material parameters, analytically. The other is a phenomenological approach in terms of effective theories. The hierarchy of the effective theories corresponds to the low-frequency, long-wavelength dynamics of quantum liquids in different ranges of frequency. This chapter discusses the microscopic (atomic) physics of quantum liquids which is the analog of the Theory of Everything — the Planck scale physics, difference between microscopic particles and emerging quasiparticles, the interplay of microscopic and effective symmetries, the origin of fundamental constants and hierarchy of Planck energy scales in the ‘Theory of Everything’, vacuum pressure and vacuum energy and scenario of vanishing of cosmological constant. These are considered using the model of weakly interacting Bose gas, the model of quantum liquid, and the real liquid 4He.
This short review paper is devoted to 90th anniversary of S.T. Belyaev birthday. Belyaev’s ideas associated with the condensate state in Bose interacting systems have stimulated intensive studies of the possible manifestation of such a condensation in Fermi systems. In many Fermi systems and compounds at zero temperature a phase transition happens that leads to a quite specific state called fermion condensation. As a signal of such a fermion condensation quantum phase transition (FCQPT) serves unlimited increase of the effective mass of quasiparticles that determines the excitation spectrum and creates flat bands. We show that the class of Fermi liquids with the fermion condensate forms a new state of matter. We discuss the phase diagrams and the physical properties of systems located near that phase transition. A common and essential feature of such systems is quasiparticles different from those suggested by L.D. Landau by crucial dependence of their effective mass on temperature, external magnetic field, pressure, etc. It is demonstrated that a huge amount of experimental data collected on different compounds suggest that they, starting from some temperature and down, form the new state of matter, and are governed by the fermion condensation. Our discussion shows that the theory of fermion condensation develops completely good description of the NFL behavior of strongly correlated Fermi systems. Moreover, the fermion condensation can be considered as the universal reason for the NFL behavior observed in various HF metals, liquids, compounds with quantum spin liquids, and quasicrystals. We show that these systems exhibit universal scaling behavior of their thermodynamic properties. Therefore, the quantum critical physics of different strongly correlated compounds is universal, and emerges regardless of the underlying microscopic details of the compounds. This uniform behavior, governed by the universal quantum critical physics, allows us to view it as the main characteristic of the new state of matter.
Superfluidity is one of the most striking manifestations of quantum mechanics at macroscopic scales, characterized by frictionless flow and the emergence of collective quantum phenomena. The theoretical foundation of superfluidity in fermionic systems was revolutionized by the Bardeen-Cooper-Schrieffer (BCS) theory, developed in 1957 to explain superconductivity in metals. This groundbreaking work provided a microscopic description of how attractive interactions between electrons near the Fermi surface can lead to the formation of Cooper pairs-correlated pairs of fermions that condense into a coherent quantum state. The BCS theory not only explained superconductivity but also laid the groundwork for understanding superfluidity in systems like liquid helium-3 and ultracold fermionic gases, becoming a cornerstone of condensed matter physics.
The appearance of the fermion condensation, which can be compared to the Bose-Einstein condensation, in different Fermi liquids is considered; its properties are discussed; and a large amount of experimental evidence in favor of the existence of the fermion condensate (FC) is presented. We show that the appearance of FC is a signature of the fermion condensation quantum phase transition (FCQPT), which separates the regions of normal and strongly correlated liquids. Beyond the FCQPT point, the quasiparticle system is divided into two subsystems, one containing normal quasiparticles and the other, FC, localized at the Fermi level. In the superconducting state, the quasiparticle dispersion in systems with FC can be represented by two straight lines, characterized by effective masses M FC * and M L * and intersecting near the binding energy E 0, which is of the order of the superconducting gap. The same quasiparticle picture and the energy scale E 0 persist in the normal state. We demonstrate that fermion systems with FC have features of a “quantum protectorate” and show that strongly correlated systems with FC, which exhibit large deviations from the Landau Fermi liquid behavior, can be driven into the Landau Fermi liquid by applying a small magnetic field B at low temperatures. Thus, the essence of strongly correlated electron liquids can be controlled by weak magnetic fields. A reentrance into the strongly correlated regime is observed if the magnetic field B decreases to zero, while the effective mass M* diverges as $$M^ * \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt B }}} \right. \kern-\nulldelimiterspace} {\sqrt B }}$$ . The regime is restored at some temperature T∗∝BT^ * \propto \sqrt B T∗∝B​ . The behavior of Fermi systems that approach FCQPT from the disordered phase is considered. This behavior can be viewed as a highly correlated one, because the effective mass is large and strongly depends on the density. We expect that FCQPT takes place in trapped Fermi gases and in low-density neutron matter, leading to stabilization of the matter by lowering its ground-state energy. When the system recedes from FCQPT, the effective mass becomes density independent and the system is suited perfectly to be conventional Landau Fermi liquid.
This chapter first studies the interaction effect of a Fermi sea on a single particle propagator and the related phenomenon of orthogonality catastrophe in X-ray spectrum. It then looks at a weakly interacting Fermi gas using Hartree–Fock approximation and introduces the notion of Fermi liquid. It discusses the concept and the appearance of quasiparticle in Fermi liquid. In addition to the standard quasiparticle description of Fermi liquid, the chapter also introduces a hydrodynamic description of Fermi liquid, which can be viewed as higher dimensional bosonization. Using the spin-density-wave state as an example, it then discusses how interaction can cause instability of Fermi liquid and generate a symmetry breaking phase transition to a spin ordered phase. Several theoretical methods, such as mean-field theory, variatonal approach, and RPA method are introduced within the interacting fermion systems. The low energy O(3) non-linear sigma model for the spin ordered phase is derived, where close attention has been paid to the possible topological terms in the non-linear sigma model. The effect of topological terms on the low energy dynamics of spin waves is discussed.