Abstract The spontaneous breaking of boost invariance is ubiquitous in nature, yet the associated Goldstone bosons are nowhere to be seen. We discuss why some subtleties are to be expected in the Goldstone phenomenon for spontaneously broken boosts, and derive the corresponding quantum mechanical, non-perturbative Goldstone theorem. Despite similarities with more standard Goldstone theorems, we show by examples that ours can be obeyed by quite unusual spectra of low-energy excitations. In particular, for non- relativistic Fermi liquids, we prove that it is obeyed by the particle-hole continuum. To the best of our knowledge, this is the first example of a Goldstone theorem obeyed by a continuum rather than by (approximately stable) single-particle Goldstone boson states in an interacting theory at zero temperature.
Abstract The spontaneous breaking of boost invariance is ubiquitous in nature, yet the associated Goldstone bosons are nowhere to be seen. We discuss why some subtleties are to be expected in the Goldstone phenomenon for spontaneously broken boosts, and derive the corresponding quantum mechanical, non-perturbative Goldstone theorem. Despite similarities with more standard Goldstone theorems, we show by examples that ours can be obeyed by quite unusual spectra of low-energy excitations. In particular, for non- relativistic Fermi liquids, we prove that it is obeyed by the particle-hole continuum. To the best of our knowledge, this is the first example of a Goldstone theorem obeyed by a continuum rather than by (approximately stable) single-particle Goldstone boson states in an interacting theory at zero temperature.
The spontaneous breaking of boost invariance is ubiquitous in nature, yet the associated Goldstone bosons are nowhere to be seen. We discuss why some subtleties are to be expected in the Goldstone phenomenon for spontaneously broken boosts, and derive the corresponding quantum mechanical, non-perturbative Goldstone theorem. Despite similarities with more standard Goldstone theorems, we show by examples that ours can be obeyed by quite unusual spectra of low-energy excitations. In particular, for non- relativistic Fermi liquids, we prove that it is obeyed by theparticle-hole continuum. To the best of our knowledge, this is the first example of a Goldstone theorem obeyed by a continuum rather than by (approximately stable) single-particle Goldstone boson states in an interacting theory at zero temperature.
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.
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.
Bragg spectroscopy shows the evolution of gapless Goldstone modes and single-particle-like excitations in an atomic Fermi superfluid as it crosses from a Bardeen–Cooper–Schrieffer superfluid to the Bose–Einstein condensate regime. Spontaneous symmetry breaking is a central paradigm of elementary particle physics1, magnetism2, superfluidity3 and superconductivity4. According to Goldstone’s theorem, phase transitions that break continuous symmetries lead to the existence of gapless excitations in the long-wavelength limit5. These Goldstone modes can become the dominant low-energy excitation, showing that symmetry breaking has a profound impact on the physical properties of matter. Here, we present a comprehensive study of the elementary excitations in a homogeneous strongly interacting Fermi gas through the crossover from a Bardeen–Cooper–Schrieffer (BCS) superfluid to a Bose–Einstein condensate (BEC) of molecules using two-photon Bragg spectroscopy. The spectra exhibit a discrete Goldstone mode, associated with the broken-symmetry superfluid phase, as well as pair-breaking single-particle excitations. Our techniques yield a direct determination of the superfluid pairing gap and speed of sound in close agreement with strong-coupling theories.
There are few general physical principles that protect the low-energy excitations of a quantum phase. Of these, Goldstone's theorem and Landau-Fermi liquid theory are the most relevant to solids. We investigate the stability of the resulting gapless excitations--Nambu-Goldstone bosons (NGBs) and Landau quasiparticles--when coupled to one another, which is of direct relevance to metals with a broken continuous symmetry. Typically, the coupling between NGBs and Landau quasiparticles vanishes at low energies, leaving the gapless modes unaffected. If, however, the low-energy coupling is nonvanishing, non-Fermi liquid behavior and overdamped bosons are expected. Here we prove a general criterion that specifies when the coupling is nonvanishing. It is satisfied by the case of a nematic Fermi fluid, consistent with earlier microscopic calculations. In addition, the criterion identifies a new kind of symmetry breaking--of magnetic translations--where nonvanishing couplings should arise, opening a previously unidentified route to realizing non-Fermi liquid phases.