An investigation is made of the relationship between long-wavelength, low-frequency normal modes and broken symmetry in nonrelativistic many-body systems. In particular, the relationship between broken symmetry as manifested through the so-called Goldstone pole and the normal-mode structure is examined. Through the study of various models, we show that the structure of the normal modes is correlated to the Goldstone pole either completely, partially, or not at all, according to the class of symmetry of the Hamiltonian of the system. For example, in the neutral superconductor, the Hamiltonian has such low symmetry that although the Anderson modes restore the symmetry of the ground state, they have no relationship at all to the Goldstone pole. It is observed that as the symmetry decreases, a dynamical sum rule takes the place of such a correlation and in all systems a sum rule gives the normal-mode frequency $\ensuremath{\omega}$. These sum rules also give the $\ensuremath{\omega}$ distribution of states through the Huang-Klein dispersion relation.
An investigation is made of the relationship between long-wavelength, low-frequency normal modes and broken symmetry in nonrelativistic many-body systems. In particular, the relationship between broken symmetry as manifested through the so-called Goldstone pole and the normal-mode structure is examined. Through the study of various models, we show that the structure of the normal modes is correlated to the Goldstone pole either completely, partially, or not at all, according to the class of symmetry of the Hamiltonian of the system. For example, in the neutral superconductor, the Hamiltonian has such low symmetry that although the Anderson modes restore the symmetry of the ground state, they have no relationship at all to the Goldstone pole. It is observed that as the symmetry decreases, a dynamical sum rule takes the place of such a correlation and in all systems a sum rule gives the normal-mode frequency \ensuremathω. These sum rules also give the \ensuremathω distribution of states through the Huang-Klein dispersion relation.
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.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.
The low-energy dynamics of many-body systems is governed by gapless modes whose properties are dictated by symmetry. Their existence follows from Goldstone's theorem, while their effective description at zero temperature is determined by the pattern of symmetry breaking. At finite temperature, an analogous role is played by hydrodynamics, which describes the universal behavior of many-body systems over long times and large distances. These principles are well understood for uniform symmetries, which act homogeneously in spacetime and lead to a correspondence between massless Goldstone modes and broken generators, as well as gapless hydrodynamic modes and conserved charges. However, this simple picture changes in the presence of nonuniform symmetries, whose generators do not commute with spacetime translations. The low-energy implications of these symmetries remain less understood, as they do not introduce additional gapless modes but instead constrain the dynamics of the existing degrees of freedom. In this thesis, we develop a unified framework for many-body systems with nonuniform symmetries and show that their effects can be understood in terms of additional fields that are not independent at low energies and can be eliminated, leading to kinematic constraints that reshape the infrared dynamics. In systems with spontaneous symmetry breaking, this mechanism modifies the effective theory and often softens the dispersion relations of the remaining modes. At finite temperature, it manifests in hydrodynamics as constraints on macroscopic currents. As a result, nonuniform symmetries give rise to qualitatively new physical phenomena, including modified spectra of collective excitations, exemplified by transverse Tkachenko oscillations in quantum vortex crystals, and unconventional transport phenomena, such as anomalously slow diffusion and softened sound modes in dipole-conserving systems.
Various sum rules, accounting for the coupling between density and particle excitations and emphasizing in an explicit way the role of Bose–Einstein condensation, are discussed. Important consequences on the fluctuations of the particle operator as well as on the structure of elementary excitations are reviewed. These include a recent generalization of the Hohenberg–Mermin–Wagner theorem holding at zero temperature. Introduction The sum rule approach has been employed extensively in the literature in order to explore various dynamic features of quantum many body systems from a microscopic point of view (see [1] and references therein). An important merit of the method is its explicit emphasis on the role of conservation laws and of the symmetries of the problem. Furthermore, the explicit determination of sum rules is relatively easy and often requires only a limited knowledge of the system. Usually the sum rule approach is, however, employed without giving special emphasis to the possible occurrence of (spontaneously) broken symmetries. For example, the most famous f -sum rule [2] holding for a large class of systems is not affected by the existence of an order parameter in the system. The purpose of this paper is to discuss a different class of sum rules which are directly affected by the presence of a broken symmetry. These sum rules can be used to predict significant properties of the system which are the consequence of the existence of an order parameter.