We propose a realization of tunable long-range mediated interactions generated by Nambu-Goldstone (NG) modes near spontaneous symmetry breaking. An effective exchange interaction emerges between two distinguishable impurities proportional to the retarded susceptibility mediated by the bath particles. The interaction is mediated by emergent NG modes associated with symmetry breaking in systems, namely the magnon mode in a repulsive two-component Fermi or Bose gas, or the goldstino mode in a Bose-Fermi mixture. When the NG mode is gapped by explicit symmetry breaking, the interaction has a Yukawa-type short-ranged interaction, whereas the gapless limit gives a Coulomb law. Above the NG mode threshold, the Dzyaloshinskii-Moriya-like cross product coupling emerges in a long-range form. Our proposal paves a way towards the realization of a quantum simulator of spin models and polarons with controllable non-local interactions.
We propose a realization of tunable long-range mediated interactions generated by Nambu-Goldstone (NG) modes near spontaneous symmetry breaking. An effective exchange interaction emerges between two distinguishable impurities proportional to the retarded susceptibility mediated by the bath particles. The interaction is mediated by emergent NG modes associated with symmetry breaking in systems, namely the magnon mode in a repulsive two-component Fermi or Bose gas, or the goldstino mode in a Bose-Fermi mixture. When the NG mode is gapped by explicit symmetry breaking, the interaction has a Yukawa-type short-ranged interaction, whereas the gapless limit gives a Coulomb law. Above the NG mode threshold, the Dzyaloshinskii-Moriya-like cross product coupling emerges in a long-range form. Our proposal paves a way towards the realization of a quantum simulator of spin models and polarons with controllable non-local interactions.
In modern physics, spontaneous symmetry breaking is one of the most important and ubiquitous ideas. It is well known that the breakdown of global and continuous symmetry gives rise to gapless excitations called Nambu-Goldstone (NG) bosons. This is called NG theorem. Since the generators of broken symmetries are bosonic, the excitations are also bosons. On the other hand, excitations are thought of as fermionic when a generator of broken symmetry is fermionic. The most famous example of fermionic symmetry is supersymmetry (SUSY). SUSY is a symmetry that exchanges bosons and fermions. In high-energy physics, it is known that spontaneous SUSY breaking gives rise to massless modes called NG fermions or Nambu- Goldstinos. In non-relativistic systems, fewer examples of models with SUSY are studied. Therefore, spontaneous SUSY breaking in non-relativistic systems is less understood than that in nonrelativistic systems or spontaneous symmetry breaking. In this thesis, we introduce interacting Majorana models with supersymmetry (SUSY) and discuss the properties in terms of spontaneous SUSY breaking and NG modes.
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.
Abstract We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu–Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward–Takahashi identity and the effective action formalism, we establish the Nambu–Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu–Goldstone modes. Using these coefficients, we classify the Nambu–Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.
Non-Abelian vortices arise when a non-Abelian global symmetry is exact in the ground state but spontaneously broken in the vicinity of their cores. In this case, there appear (non-Abelian) Nambu-Goldstone (NG) modes confined and propagating along the vortex. In relativistic theories, the Coleman-Mermin-Wagner theorem forbids the existence of a spontaneous symmetry breaking, or a long-range order, in 1+1 dimensions: quantum corrections restore the symmetry along the vortex and the NG modes acquire a mass gap. We show that in non-relativistic theories NG modes with quadratic dispersion relation confined on a vortex can remain gapless at quantum level. We provide a concrete and experimentally realizable example of a three-component Bose-Einstein condensate with U(1) × U(2) symmetry. We first show, at the classical level, the existence of S 3 ≃ S 1 ⋉ S 2 (S 1 fibered over S 2) NG modes associated to the breaking U(2) → U(1) on vortices, where S 1 and S 2 correspond to type I and II NG modes, respectively. We then show, by using a Bethe ansatz technique, that the U(1) symmetry is restored, while the SU(2) symmery remains broken non-pertubatively at quantum level. Accordingly, the U(1) NG mode turns into a c = 1 conformal field theory, the Tomonaga-Luttinger liquid, while the S 2 NG mode remains gapless, describing a ferromagnetic liquid. This allows the vortex to be genuinely non-Abelian at quantum level.
The physics of long-range-interacting quantum systems is currently living through a renaissance driven by the fast progress in quantum simulators. In these systems many paradigms of statistical physics do not apply and also the universal long-wavelength physics gets substantially modified by the presence of long-ranged forces. Here we explore the low-energy excitations of several long-range-interacting quantum systems, including spin models and interacting Bose gases, in the ordered phase associated with the spontaneous breaking of U(1) and SU(2) symmetries. Instead of the expected Goldstone modes, we find three qualitatively different regimes, depending on the range of the interaction. In one of these regimes the Goldstone modes are gapped, via a generalization of the Higgs mechanism. Moreover, we show how this effect is realized in current experiments with ultracold atomic gases in optical cavities.