Nambu-Goldstone Research Articles

Interpolating relativistic and nonrelativistic Nambu-Goldstone and Higgs modes

When a continuous symmetry is spontaneously broken in non-relativistic theories, there appear Nambu-Goldstone (NG) modes, whose dispersion relations are either linear (type-I) or quadratic (type-II). We give a general framework to interpolate between relativistic and non-relativistic NG modes, revealing a nature of type-I and II NG modes in non-relativistic theories. The interpolating Lagrangians have the nonlinear Lorentz invariance which reduces to the Galilei or Schrodinger invariance in the non-relativistic limit. We find that type-I and type-II NG modes in the interpolating region are accompanied with a Higgs mode and a chiral NG partner, respectively, both of which are gapful. In the ultra-relativistic limit, a set of a type-I NG mode and its Higgs partner remains, while a set of type-II NG mode and gapful NG partner turns to a set of two type-I NG modes. In the non-relativistic limit, the both types of accompanied gapful modes become infinitely massive, disappearing from the spectrum. The examples contain a phonon in Bose-Einstein condensates, a magnon in ferromagnets, and a Kelvon and dilaton-magnon localized around a skyrmion line in ferromagnets.

Effective field theory for spacetime symmetry breaking

We discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view. By gauging spacetime symmetries, the identification of Nambu-Goldstone (NG) fields and the construction of the effective action are performed based on the breaking pattern of diffeomorphism, local Lorentz, and (an)isotropic Weyl symmetries as well as the internal symmetries including possible central extensions in nonrelativistic systems. Such a local picture distinguishes, e.g., whether the symmetry breaking condensations have spins and provides a correct identification of the physical NG fields, while the standard coset construction based on global symmetry breaking does not. We illustrate that the local picture becomes important in particular when we take into account massive modes associated with symmetry breaking, whose masses are not necessarily high. We also revisit the coset construction for spacetime symmetry breaking. Based on the relation between the Maurer-Cartan one form and connections for spacetime symmetries, we classify the physical meanings of the inverse Higgs constraints by the coordinate dimension of broken symmetries. Inverse Higgs constraints for spacetime symmetries with a higher dimension remove the redundant NG fields, whereas those for dimensionless symmetries can be further classified by the local symmetry breaking pattern.

Dispersion relations of Nambu-Goldstone modes at finite temperature and density

We discuss the dispersion relations of Nambu-Goldstone (NG) modes associated with spontaneous breaking of internal symmetries at finite temperature and/or density. We show that the dispersion relations of type-A (I) and type-B (II) NG modes are linear and quadratic in momentum, whose imaginary parts are quadratic and quartic, respectively. In both cases, the real parts of the dispersion relations are larger than the imaginary parts when the momentum is small, so that the NG modes can propagate far away. We derive the gap formula for NG modes in the presence of a small explicit breaking term. We also discuss the gapped partners of type-B NG modes, when the expectation values of a charge density and a local operator that break the same symmetry coexist.

Topological interactions of Nambu-Goldstone bosons in quantum many-body systems

We classify effective actions for Nambu-Goldstone (NG) bosons assuming the absence of anomalies. Special attention is paid to Lagrangians that are invariant only up to a surface term, which are shown to be in a one-to-one correspondence with Chern-Simons (CS) theories for unbroken symmetry. Without making specific assumptions on the spacetime symmetry, we give explicit expressions for these Lagrangians, generalizing the Berry and Hopf terms in ferromagnets. Globally well-defined matrix expressions are derived for symmetric coset spaces of broken symmetry. The CS Lagrangians exhibit special properties, on both the perturbative and the global topological levels. The order-one CS term is responsible for the noninvariance of the canonical momentum density under internal symmetry, known as the linear momentum problem. The order-three CS term gives rise to a novel type of interaction among NG bosons. All the CS terms are robust against local variations of microscopic physics.

Nambu–Goldstone and Higgs Modes in the Kondo Insulator

It is shown that collective modes can emerge in the Kondo insulator which is described as an ordered state of conduction–f electron–hole (\(\bar{c}\)–f) pair fields in the symmetric case of the Kondo lattice model (KLM). The one-loop approximation is adopted in a functional integral formalism similar to that developed by Popov for the s-state superfluid Fermi system (SFS). The massless Nambu–Goldstone (NG) mode due to the phase fluctuation of \(\bar{c}\)–f pair fields possesses a quadric dispersion, differently from a linear one for the SFS, and the massive Higgs mode due to the amplitude fluctuation has a gap 2Δ, twice the fermionic gap, similarly to the SFS. Because of the quadric dispersion of the NG mode, similar to the ferromagnetic spin wave mode, the specific heat of the KLM is expected not to be an activation-type but to exhibit a T3/2-dependence, which may be confirmed experimentally.

Quantum exact non-abelian vortices in non-relativistic theories

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) x U(2) symmetry. We first show, at the classical level, the existence of S^3 = S^1 |x 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.

Nonrelativistic Nambu-Goldstone modes propagating along a Skyrmion line

We study Nambu-Goldstone (NG) modes or gapless modes propagating along a Skyrmion (lump) line in a relativistic and nonrelativistic $O(3)$ sigma model, the latter of which describes isotropic Heisenberg ferromagnets. We show for the nonrelativistic case that there appear two coupled gapless modes with a quadratic dispersion. In addition to the well-known Kelvin mode consisting of two translational zero modes transverse to the Skyrmion line, we show that the other consists of a magnon and dilaton, that is, a NG mode for a spontaneously broken internal $U(1)$ symmetry and a quasi-NG mode for a spontaneously broken scale symmetry of the equation of motion. We find that the commutation relations of Noether charges admit a central extension between the dilatation and phase rotation, in addition to the one between two translations found recently. The counting rule is consistent with the Nielsen-Chadha and Watanabe-Brauner relations only when we take into account quasi-NG modes.

Charge quantization in the [formula omitted] nonlinear σ-model

We investigate the consistency conditions for matter fields coupled to the four-dimensional (N=1 supersymmetric) CP(1) nonlinear sigma model (the coset space SU(2)G/U(1)H). We find that consistency requires that the U(1)H charge of the matter be quantized, in units of half of the U(1)H charge of the Nambu–Goldstone (NG) boson, if the matter has a nonsingular kinetic term and the dynamics respect the full group SU(2)G. We can then take the linearly realized group U(1)H to comprise the weak hypercharge group U(1)Y of the Standard Model. Thus we have charge quantization without a Grand Unified Theory (GUT), completely avoiding problems like proton decay, doublet–triplet splitting, and magnetic monopoles. We briefly investigate the phenomenological implications of this model-building framework. The NG boson is fractionally charged and completely stable. It can be naturally light, avoiding constraints while being a component of dark matter or having applications in nuclear physics. We also comment on the extension to other NLSMs on coset spaces, which will be explored more fully in a followup paper.

Spontaneous Symmetry Breaking and Coherence in Two-Dimensional Electron–Hole and Exciton Systems

The spontaneous breaking of the continuous symmetries of the two-dimensional(2D) electron-hole systems in a strong perpendicular magnetic field leads to the formation of new ground states and determines the energy spectra of the collective elementary excitations appearing over these ground states. In this review the main attention is given to the electron-hole systems forming coplanar magnetoexcitons in the Bose-Einstein condensation(BEC) ground state with the wave vector k=0, taking into account the excited Landau levels, when the exciton-type elementary excitations coexist with the plasmon-type oscillations. At the same time properties of the two-dimensional electron gas(2DEG) spatially separated as in the case of double quantum wells(DQWs) from the 2D hole gas under conditions of the fractional quantum Hall effect(FQHE) are of great interest because they can influence the quantum states of the coplanar magnetoexcitons when the distance between the DQW layers diminishes. We also consider in this review the bilayer electron systems under conditions of the FQHE with the one half filling factor for each layer and with the total filling factor for two layers equal to unity because the coherence between the electron states in two layers is equivalent to the formation of the quantum Hall excitons(QHExs) in a coherent macroscopic state. The breaking of the global gauge symmetry as well as of the continuous rotational symmetries leads to the formation of the gapless Nambu-Goldstone(NG) modes while the breaking of the local gauge symmetry gives rise to the Higgs phenomenon characterized by the gapped branches of the energy spectrum. The conditions in which the spontaneous coherence could appear in a system of indirect excitons in a double quantum well structures are discussed. The experimental attempts to achieve these conditions, the main results and the accumulated knowledge are reviewed.

True, quasi and unstable Nambu–Goldstone modes of the two-dimensional Bose–Einstein condensed magnetoexcitons

The collective elementary excitations of two-dimensional magnetoexcitons in a Bose–Einstein condensate (BEC) with wave vector k→=0 were investigated in the framework of the Bogoliubov theory of quasi-averages. The Hamiltonian of the electrons and holes lying in the lowest Landau levels (LLLs) contains supplementary interactions due to virtual quantum transitions of the particles to the excited Landau levels (ELLs) and back. As a result, the interaction between the magnetoexcitons with k→=0 does not vanish and their BEC becomes stable. The energy spectrum contains only one gapless, true Nambu–Goldstone (NG) mode of the second kind with dependence ω(k)≈k2 at small values k describing the optical-plasmon-type oscillations. There are two exciton-type branches corresponding to normal and abnormal Green's functions. Both modes are gapped with roton-type segments at intermediary values of the wave vectors and can be named as quasi-NG modes. The fourth branch is the acoustical plasmon-type mode with absolute instability in the region of small and intermediary values of the wave vectors. All branches have a saturation-type dependencies at great values of the wave vectors. The number and the kind of the true NG modes are in accordance with the number of the broken symmetry operators.

Nambu-Goldstone modes of the two-dimensional Bose-Einstein condensed magnetoexcitons

The collective elementary excitations of two-dimensional magnetoexcitons in a Bose-Einstein condensate (BEC) with wave vector k = 0 were investigated in the framework of the Bogoliubov theory of quasiaverages. The Hamiltonian of the electrons and holes lying in the lowest Landau levels (LLLs) contains supplementary interactions due to virtual quantum transitions of the particles to the excited Landau levels (ELLs) and back. As a result, the interaction between the magnetoexcitons with k = 0 does not vanish and their BEC becomes stable. The equations of motion for the exciton operators d(P) and d†(P) are interconnected with equations of motion for the density operators ρ(P) and D(P). Instead of a set of two equations of motion, as in the case of usual Bose gas, corresponding to normal and abnormal Green’s functions, we have a set of four equations of motion. This means we have to deal simultaneously with four branches of the energy spectrum, the two supplementary branches being the optical plasmon branch represented by the operator ρ(P) and the acoustical plasmon branch represented by the operator D(P). The perturbation theory on the small parameter v2(1 − v2), where v2 is the filling factor and (1 − v2) is the phase space filling factor was developed. The energy spectrum contains only one gapless, true Nambu-Goldstone (NG) mode of the second kind with dependence ω(k) ≈ k2 at small values k describing the optical-plasmon-type oscillations. There are two exciton-type branches corresponding to normal and abnormal Green’s functions. Both modes are gapped with roton-type segments at intermediary values of the wave vectors and can be named as quasi-NG modes. The fourth branch is the acoustical plasmon-type mode with absolute instability in the region of small and intermediary values of the wave vectors. All branches have a saturation-type dependencies at great values of the wave vectors. The number and the kind of the true NG modes is in accordance with the number of the broken symmetry operators. The comparison of the results concerning two Bose-Einstein condensates namely of the coplanar magnetoexcitons and of the quantum Hall excitons in the bilayer electron systems reveals their similarity.

Number of Nambu-Goldstone bosons and its relation to charge densities

The low-energy physics of systems with spontaneous symmetry breaking is governed by the associated Nambu-Goldstone (NG) bosons. While NG bosons in Lorentz-invariant systems are well understood, the precise characterization of their number and dispersion relations in a general quantum many-body system is still an open problem. An inequality relating the number of NG bosons and their dispersion relations to the number of broken symmetry generators was found by Nielsen and Chadha. In this paper, we give a presumably first example of a system in which the Nielsen-Chadha inequality is actually not saturated. We suggest that the number of NG bosons is exactly equal to the number of broken generators minus the number of pairs of broken generators whose commutator has a nonzero vacuum expectation value. This naturally leads us to a proposal for a different classification of NG bosons.

Higgs mechanism with type-II Nambu-Goldstone bosons at finite chemical potential

When the spontaneous symmetry breaking occurs for systems without Lorentz covariance, there arises possible mismatch, ${N}_{\mathrm{NG}}<{N}_{\mathrm{BG}}$, between numbers of Nambu-Goldstone (NG) bosons (${N}_{\mathrm{NG}}$) and the numbers of broken generators (${N}_{\mathrm{BG}}$). In such a situation, so-called type-II NG bosons emerge. We study how the gauge bosons acquire masses through the Higgs mechanism under this mismatch by employing gauge theories with complex scalar field at finite chemical potential and by enforcing ``charge'' neutrality. To separate the physical spectra from unphysical ones, the ${R}_{\ensuremath{\xi}}$ gauge is adopted. Not only massless NG bosons but also massive scalar bosons generated by the chemical potential are absorbed into spatial components of the gauge bosons. Although the chemical potential induces a nontrivial mixings among the scalar bosons and temporal components of the gauge bosons, it does not affect the structure of the physical spectra, so that the total number of physical modes is not modified even for ${N}_{\mathrm{NG}}<{N}_{\mathrm{BG}}$.

Collective modes in the color flavor-locked phase

We study the low-energy effective action for some collective modes of the color flavor-locked (CFL) phase of QCD. This phase of matter has long been known to be a superfluid because by picking a phase its order parameter breaks the quark-number U(1)B symmetry spontaneously. We consider the modes describing fluctuations in the magnitude of the condensate, namely the Higgs mode, and in the phase of the condensate, namely the Nambu–Goldstone (NG) (or Anderson–Bogoliubov) mode associated with the breaking of U(1)B. By employing as microscopic theory the Nambu–Jona-Lasinio model, we reproduce known results for the Lagrangian of the NG field to the leading order in the chemical potential and extend such results evaluating corrections due to the gap parameter. Moreover, we determine the interaction terms between the Higgs and the NG field. This study paves the way for a more reliable study of various dissipative processes in rotating compact stars with a quark matter core in the CFL phase.

Bogoliubov theory and Lee-Huang-Yang corrections in spin-1 and spin-2 Bose-Einstein condensates in the presence of the quadratic Zeeman effect

We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is found that in the spin-2 nematic phase there are special Bogoliubov modes that have gapless linear dispersion relations but do not belong to the NG modes.

Investigation of phase transition in QED3

In this Letter we investigate dynamical mass generation and phase transition in QED3 for number of fermion flavors N=1,2 at finite chemical potential μ under the rainbow approximation of the Dyson–Schwinger equations. Employing three Ansätze of the fermion–photon vertex, we numerically solve the coupled fermion and photon gap equation at zero chemical potential in the Landau gauge. It is found that the fermion and photon propagators in the Nambu–Goldstone (NG) phase and Wigner (WN) phase thus obtained have distinct behaviors. It is expected that the similar result holds for the quark and gluon propagators in QCD. We then derive a model-independent expression for P(μ) (the pressure at finite μ) in QED3. In this expression, P(μ) is the sum of two terms: the first term is the pressure at μ=0 which is a constant; the second term is totally expressed in terms of the fermion propagator at finite chemical potential and contains all the nontrivial μ-dependence. The order parameter for the phase transition between NG phase and WN phase is the difference between the pressure of NG phase and that of WN phase Δ(μ)=PNG(μ)−PWN(μ). By solving the fermion gap equation at finite μ, the numerical results for the order parameter are obtained. The obtained Δ(μ) is a monotonously decreasing function of μ. When μ reaches a critical value μc which is about 0.015 at N=1 and 0.00163 at N=2, Δ(μ) begins to turn from positive to negative, which means the system goes into the deconfined WN phase and chiral symmetry gets restored.

Gravity from breaking of local Lorentz symmetry

We present a model of gravity based on spontaneous Lorentz symmetry breaking. We start from a model with spontaneously broken symmetries for a massless 2-tensor with a linear kinetic term and a nonderivative potential, which is shown to be equivalent to linearized general relativity, with the Nambu-Goldstone (NG) bosons playing the role of the gravitons. We apply a bootstrap procedure to the model based on the principle of consistent coupling to the total energy energy-momentum tensor. Demanding consistent application of the bootstrap to the potential term severely restricts the form of the latter. Nevertheless, suitable potentials exists that permit stable vacua. It is shown that the resulting model is equivalent, at low energy, to General Relativity in a fixed gauge.

NAMBU–GOLDSTONE MODES IN GRAVITATIONAL THEORIES WITH SPONTANEOUS LORENTZ BREAKING

Spontaneous breaking of Lorentz symmetry has been suggested as a possible mechanism that might occur in the context of a fundamental Planck-scale theory, such as string theory or a quantum theory of gravity. However, if Lorentz symmetry is spontaneously broken, two sets of questions immediately arise: What is the fate of the Nambu–Goldstone (NG) modes, and can a Higgs mechanism occur? A brief summary of some recent work looking at these questions is presented here.

Mass spectrum of diquarks and mesons in the color–flavor locked phase of dense quark matter

The spectrum of meson and diquark excitations of dense quark matter is considered in the framework of the Nambu–Jona-Lasinio model with three types of massless quarks in the presence of a quark number chemical potential μ. We investigate the effective action of meson and diquark fields both at sufficiently large values of μ>μc≈ 330 MeV, where the color–flavor locked (CFL) phase is realized, and in the chirally broken phase of quark matter (μ<μc). In the latter case all nine pseudoscalar mesons are Nambu–Goldstone (NG) bosons, whereas the mass of the scalar meson nonet is twice the dynamical quark mass. In the chirally broken phase the pseudoscalar diquarks are not allowed to exist as stable particles, but the scalar diquarks might be stable only at a rather strong interaction in the diquark channel. In the case of the CFL phase, all NG bosons of the model are realized as scalar and pseudoscalar diquarks. Moreover, it turns out that massive diquark excitations are unstable for this phase. In particular, for the scalar and pseudoscalar octets of diquark resonances a mass value around 230 MeV was found numerically. In contrast, mesons are stable particles in the CFL phase. Their masses lie in the interval 400–500 MeV for not too large values of μ>μc.

Quark description of the Nambu-Goldstone bosons in the color-flavor locked phase

We investigate the color-singlet order parameters and the quark description of the Nambu-Goldstone (NG) bosons in the color-flavor locked (CFL) phase. We put emphasis on the NG boson (phason) called $H$ associated with the ${\mathrm{U}}_{\mathrm{B}}(1)$ symmetry breaking. We qualitatively argue the nature of $H$ as the second sound in the hydrodynamic regime. We articulate, based on a diquark picture, how the structural change of the condensates and the associated NG bosons occurs continuously from hadronic to CFL quark matter if the quark-hadron continuity is realized. We sharpen the qualitative difference between the flavor octet pions and the singlet phason. We propose a conjecture that superfluid $H$ matter undergoes a crossover to a superconductor with tightly bound diquarks, and then a crossover to superconducting matter with diquarks dissociated.