Quantum matter is everywhere, from the interiors of neutron stars to the electrons in everyday metals. Like ordinary, classical matter, it is made up of many interacting particles. In classical matter, however, it is possible to think of each particle as an individual entity, whereas in quantum matter Heisenberg's uncertainty principle prevents us from telling individual particles apart: their behaviour can only be described collectively. In spite of this, many types of quantum matter are well understood from a theoretical point of view. For example, the “electron liquid” that is responsible for the flow of electricity through ordinary metals, the magnetic properties of many insulating materials and the normal and superfluid phases of helium at very low temperatures have all succumbed to the probing of theorists.
Quantum matter is everywhere, from the interiors of neutron stars to the electrons in everyday metals. Like ordinary, classical matter, it is made up of many interacting particles. In classical matter, however, it is possible to think of each particle as an individual entity, whereas in quantum matter Heisenberg's uncertainty principle prevents us from telling individual particles apart: their behaviour can only be described collectively. In spite of this, many types of quantum matter are well understood from a theoretical point of view. For example, the “electron liquid” that is responsible for the flow of electricity through ordinary metals, the magnetic properties of many insulating materials and the normal and superfluid phases of helium at very low temperatures have all succumbed to the probing of theorists.
Abstract In the 1930s and 1940s, the quantum theory of solids dealt primarily with singleelectron phenomena. Although surprisingly many problems were solvable within this framework, some were not-for example, the explanation from first principles of superconductivity and the calculation of the cohesive energy of metals. A related problem, whose connection to superconductivity was recognized almost immediately, was the superfluidity of4He, discovered experimentally in 1937. More generally, phase transitions-ordinary liquid-gas and melting, as well as those to the superconducting, superfluid, and ferromagnetic states-lacked adequate theory. These various phenomena, unexplainable within a single-particle framework, depend strongly on the interactions between particles in the systems; today they are recognized as “collective” or “cooperative” phenomena.
It has become ever more clear over the past 70 years that reality is best described by a theory that no one fully understands and which cannot be applied to most everyday phenomena. Quantum theory has been subjected to many stringent tests and all of its predictions, however strange, have always been borne out. Originally developed to account for why atoms do not collapse and why they radiate and absorb energy in discrete amounts (quanta), it has now gone far beyond these rather simple beginnings. It is essentially a mathematical theory, so trying to describe it in words is analogous to transforming a picture into a poem. Inaccuracies, distortions and over-simplifications inevitably creep in. Nevertheless, one can convey the essence of the picture, and that is what is attempted here. Some grasp of what the material world is really like, and of the unfamiliar ways in which it can behave, is probably necessary to reaching an understanding of awareness and is certainly needed to comprehend the experimental evidence set out in Chapter 5.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
Quantum mechanics represents one of the great conceptual revolu- tions of the 20th century. It has raised a huge number of fundamental questions of both physical and philosophical kind. • What does matter mean at all? • What are the main properties or characteristics of matter? • Can matter be reduced to information? • Is our universe probabilistic at the most fundamental level? • Are there non-local correlations in nature? • Are non-causal interconnections between physical systems possible?
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
We have seen in Chapter 3 , that as temperature is increased quantum statistics, both Bose-Einstein or Fermi-Dirac, tend to classical. High temperatures blur out quantum effects in a system, which can be safely described in classical terms. In this chapter, we will consider some systems which manifest quantum properties at macroscopic level, but at very low temperatures. We recall that helium is the only substance which remains liquid at low temperature, down to absolute zero (see Chapter 6 ), a phenomenon related to the quantum behavior of its constituent atoms. Liquid helium was instrumental to the discovery that some metals are able to sustain an electrical current for long periods of time without any driving force. This phenomenon is named superconductivity, and is also a macroscopic manifestation of the quantum behavior of the electrons in metals. Later on it was discovered that liquid helium itself shows a similar property, named superfluidity, consisting in its ability to flow through a capillary without viscosity. Hence, a persistent flow of liquid can be maintained without any driving force. While the quantum agents of superconductivity are electrons, which are spin- 1/2 fermions, those of superfluidity are 4 He atoms, which are spin- 0 bosons. There are other systems able to display similar behavior as liquid 3 He, atomic nuclei and nuclear matter. In fact, superfluidity and superconductivity are new phases of matter related to the bosonic character of 4 He atoms or of pairs of fermions (either electrons, 3 He atoms or nucleons), the same as Bose-Einstein condensation in ultracold atomic gases. The temperatures where these new phases manifest range from 165 K for superconductivity down to a few nano-Kelvin for Bose-Einstein condensation.