Nambu-Goldstone Modes Research Articles

Modified dilepton production rate from charged pion-pair annihilation in the inhomogeneous chiral condensed phase

We investigate the dilepton production rates from annihilation processes of charged pion pairs with modified pion dispersion relations in the inhomogeneous chiral condensed phase. We assume a dual chiral density wave as an inhomogeneous chiral condensate, and obtain the dispersion relations of the Nambu-Goldstone modes in the inhomogeneous chiral condensed phase. We use a low energy effective Lagrangian based on the O(4) symmetry which is expanded by the order parameter up to the sixth order. The obtained dispersion relations are anisotropic and quadratic for the momentum. We evaluate the electron-positron production rates by charged pion-pair annihilations as functions of an invariant mass using the obtained dispersion relations. Basically, the production rate in the inhomogeneous chiral condensed phase has a steeper overall slope with respect to an invariant mass than that in the homogeneous chiral condensed phase. Therefore, the production rate may be enhanced when the invariant mass is around twice the pion mass. Published by the American Physical Society 2024

Nambu–Goldstone modes in a lattice Nambu–Jona-Lasinio model with multi flavor symmetries

Nambu–Goldstone modes in a lattice Nambu–Jona-Lasinio model with multi flavor symmetries

Bestiary of 6D (1, 0) SCFTs: Nilpotent orbits and anomalies

Many six-dimensional (1, 0) superconformal field theories (SCFTs) are known to fall into families labeled by nilpotent orbits of certain simple Lie algebras. For each of the three infinite series of such families, we show that the anomalies for the continuous zero-form global symmetries of a theory labelled by a nilpotent orbit O of g can be determined from the anomalies of the theory associated to the trivial nilpotent orbit (the parent theory), together with the data of O. In particular, knowledge of the tensor branch field theory is bypassed completely. We show that the known anomalies, previously determined from the geometric/atomic construction, are reproduced by analyzing the Nambu-Goldstone modes inside of the moment map associated to the g flavor symmetry of the parent SCFT. This provides further evidence for the physics underlying the labeling of the SCFTs by nilpotent orbits. We remark on some consequences, such as the reinterpretation of the 6D a-theorem for such SCFTs in terms of group theory. Published by the American Physical Society 2024

The Schwinger-Keldysh coset construction

The coset construction is a tool for systematically building low energy effective actions for Nambu-Goldstone modes. This technique is typically used to compute time-ordered correlators appropriate for S-matrix computations for systems in their ground state. In this paper, we extend this technique to the Schwinger-Keldysh formalism, which enables one to calculate a wider variety of correlators and applies also to systems in a mixed state. We focus our attention on internal symmetries and demonstrate that, after identifying the appropriate symmetry breaking pattern, Schwinger-Keldysh effective actions for Nambu-Goldstone modes can be constructed using the standard rules of the coset construction. Particular emphasis is placed on the thermal state and ensuring that correlators satisfy the KMS relation. We also discuss explicitly the power counting scheme underlying our effective actions. We comment on the similarities and differences between our approach and others that have previously appeared in the literature. In particular, our prescription does not require the introduction of additional “diffusive” symmetries and retains the full non-linear structure generated by the coset construction. We conclude with a series of explicit examples, including a computation of the finite-temperature two-point functions of conserved spin currents in non-relativistic paramagnets, antiferromagnets, and ferromagnets. Along the way, we also clarify the discrete symmetries that set antiferromagnets apart from ferromagnets, and point out that the dynamical KMS symmetry must be implemented in different ways in these two systems.

Non-Abelian chiral soliton lattice in rotating QCD matter: Nambu-Goldstone and excited modes

The ground state of QCD with two flavors at a finite baryon chemical potential under rapid rotation is a chiral soliton lattice (CSL) of the η meson, consisting of a stack of sine-Gordon solitons carrying a baryon number, due to the anomalous coupling of the η meson to the rotation. In a large parameter region, the ground state becomes a non-Abelian CSL, in which due to the neutral pion condensation each η soliton decays into a pair of non-Abelian sine-Gordon solitons carrying S2 moduli originated from Nambu-Goldstone (NG) modes localized around it, corresponding to the spontaneously broken vector symmetry SU(2)V. There, the S2 modes of neighboring solitons are anti-aligned, and these modes should propagate in the transverse direction of the lattice due to the interaction between the S2 modes of neighboring solitons. In this paper, we calculate excitations including gapless NG modes and excited modes around non-Abelian and Abelian (η) CSLs, and find three gapless NG modes with linear dispersion relations (type-A NG modes): two isospinons (S2 modes) and a phonon corresponding to the spontaneously broken vector SU(2)V and translational symmetries around the non-Abelian CSL, respectively, and only a phonon for the Abelian CSL because of the recovering SU(2)V. We also find in the deconfined phase that the dispersion relation of the isospinons becomes of the Dirac type, i.e. linear even at large momentum.

A symmetry principle for gauge theories with fractons

Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.

Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum

We demonstrate the general existence of a local dipole conservation law in bosonic field theory. The scalar charge density arises from the symplectic form of the system, whereas the tensor current descends from its stress tensor. The algebra of spatial translations becomes centrally extended in presence of field configurations with a finite nonzero charge. Furthermore, when the symplectic form is closed but not exact, the system may, surprisingly, lack a well-defined momentum density. This leads to a theorem for the presence of additional light modes in the system whenever the short-distance physics is governed by a translationally invariant local field theory. We also illustrate this mechanism for axion electrodynamics as an example of a system with Nambu-Goldstone modes of higher-form symmetries.

Dipole symmetry breaking and fractonic Nambu-Goldstone mode

We introduce a family of quantum field theories for fields carrying monopole and dipole charges. In contrast to previous realizations, fields have quadratic two-derivative kinetic terms. The dipole symmetry algebra is realized in a discretized internal space and connected to the physical space through a background gauge field. We study spontaneous symmetry breaking of dipole symmetry in 1+1 dimensions in a large-NN limit. The trivial classical vacuum is lifted by quantum corrections into a vacuum which breaks dipole symmetry while preserving monopole charge. By means of a Hubbard-Stratonovich transformation, heat-kernel and large-NN techniques, we compute the effective action for the low-energy modes. We encounter a fractonic immobile Nambu-Goldstone mode whose dispersion characteristics avoid Coleman-Hohenberg-Mermin-Wagner theorem independently of the large-NN limit.

Nambu-goldstone Modes in Two Segregated Bose-einstein Condensates Limited by Two Hard Walls

The Nambu-Goldstone (NG) modes in the system of two segregated Bose-Einstein condensates (BECs) limited by two hard walls are studied by means of the Gross-Pitaevskii (GP) theory. Based on the double-parabola approximation (DPA) combining with the Bogoliubov-de Gennes (BdG) equations we found four NG modes that proves the failure of the Watanabe-Brauner counting rule and, furthermore, their dispersion relations depend explicitly on the geometrical structure.
 
 

Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.

A possibility of existence of a pseudovector-type quark–antiquark condensate in the quark matter and Nambu–Goldstone modes on that condensate in the Nambu–Jona–Lasinio model

A possibility of a pseudovector-type quark–antiquark condensed phase, which leads to a quark spin polarized phase, in the quark matter is investigated taking account of the vacuum effects leading to the chiral symmetry breaking by using the Nambu–Jona–Lasinio model. Also, possible Nambu–Goldstone modes on the pseudovector-type quark–antiquark condensate and the tensor-type quark–antiquark condensate, which also leads to the quark spin polarized phase, are investigated.

Theory of Energy Dispersion of Chiral Phonons

We have developed a microscopic theory on phonon energy dispersion in chiral crystals within a harmonic approximation. One of the main issues is about the splitting of sound velocity of acoustic phonons with opposite ``crystal'' angular momentum. We have shown that the splitting must be zero even in chiral crystals and the difference starts from the order of at least $k^2$ or higher in their energy dispersion. Splitting is evident for chiral optical phonons, and we have derived a formula for their $k$-linear splitting. Another important finding is about possible interactions of atomic displacements in microscopic models. We have found that antisymmetric interactions of $\mathbf{D} _{ij} \cdot (\mathbf{d} _i \times \mathbf{d} _j) $ type are not allowed in microscopic Hamiltonians for chiral phonons in compatible with the stability against the Nambu-Goldstone mode. We have identified that the splitting in both acoustic and optical modes arises from the harmonic potentials with the electric toroidal quadrupole of $G_{u}$-type symmetry. These constraint are important for modeling real materials. Most of our microscopic calculations have been performed for (quasi-)one-dimensional systems with a trigonal crystal symmetry including Te, but these results generally hold also for other chiral phonon systems.

Hydrodynamic signatures and spectral properties of the quantum vortex

We characterize the low-lying excitations of a quantum vortex in a quasi-two-dimensional Bose-Einstein condensate (BEC) using the standard definition of the density of states (DOS) and a modified version that is sensitive to complementary aspects of the excitation's spectrum. The latter proves to be particularly relevant to studying the polaronic state realized when an impurity is embedded in a quantum vortex. We establish that the impurity becomes sensitive to the transversal fluctuations of the vortex, via its remnant kelvon mode, and to the phase fluctuations of the BEC Nambu-Goldstone mode. The presence of the vortex yields an anomalous excitation spectrum with a finite energy gap and non-linear DOS at low energies. We find that the high sensitivity of the kelvon mode to external potentials provides a channel of quantum-level control over impurities trapped in a vortex. This extra control channel may be of practical use for the proposal of using vortex-trapped impurities as qubit units for quantum information processing.

Unstable Nambu-Goldstone modes

Nambu-Goldstone (NG) modes for 0-form and higher-form symmetries can become unstable in the presence of background fields. Examples include the instability of a photon with a time-dependent axion background or with a chirality imbalance, known as the chiral plasma instability, and the instability of a dynamical axion with a background electric field. We show that all these phenomena can be universally described by a symmetry algebra for 0-form and higher-form symmetries. We prove a counting rule for the number of unstable NG modes in terms of correlation functions of broken symmetry generators. Based on our unified description, we further give a simple new example where one of the NG modes associated with the spontaneous 0-form symmetry breaking $U(1) \times U(1) \to \{1\}$ becomes unstable.

Heavy + heavy and heavy + light pseudoscalar to vector semileptonic transitions

Using a symmetry-preserving regularisation of a vector times vector contact interaction (SCI), we complete a systematic treatment of twelve semileptonic transitions with vector meson final states: Drightarrow rho , D_{(s)}rightarrow K^*, D_srightarrow phi , Brightarrow rho , B_srightarrow K^*, B_{(s)}rightarrow D_{(s)}^*, B_c rightarrow B_{(s)}^*, J/psi , D^*; and thereby finalise a unified analysis of semileptonic decays of heavy + heavy and heavy + light pseudoscalar mesons to both pseudoscalar and vector meson final states. The analysis is marked by algebraic simplicity, few parameters, and the ability to consistently describe systems from Nambu-Goldstone modes to heavy + heavy mesons. Regarding the behaviour of the transition form factors, the SCI results compare well wherever sound experimental or independent theory analyses are available; hence, the SCI branching fraction predictions should be a reasonable guide. Considering the ratios R(D_{(s)}^{(*)}), R(J/psi ), R(eta _c), whose values are key tests of lepton universality in weak interactions, the SCI values agree with Standard Model predictions. The B_{(s)}rightarrow D_{(s)}^* transitions are used to predict the precursor functions that evolve into the universal Isgur–Wise function in the heavy-quark limit, with results that conform with those from other sources where such are available. The study also exposes effects on the transition form factors that flow from interference between emergent hadron mass from the strong interaction and Higgs boson couplings via current-quark masses, including flavour symmetry violation.

Supersolidity of α cluster structure in Ca40

$\alpha$ cluster structure in nuclei has been long understood based on the geometrical configuration picture. By using the spatially localized Brink $\alpha$ cluster model in the generator coordinate method, it is shown that the $\alpha$ cluster structure has the apparently opposing duality of crystallinity and condensation, a property of supersolids. To study the condensation aspects of the $\alpha$ cluster structure a field theoretical superfluid cluster model (SCM) is introduced, in which the order parameter of condensation is incorporated by treating rigorously the Nambu-Goldstone mode due to spontaneous symmetry breaking of the global phase. The $\alpha$ cluster structure of $^{40}$Ca, which has been understood in the crystallinity picture, is studied by the SCM with ten $\alpha$ clusters. It is found that the $\alpha$ cluster structure of $^{40}$Ca is reproduced by the SCM in addition to $^{12}$C reported in a previous paper, which gives support to the duality of the $\alpha$ cluster structure. The emergence of the mysterious $0^+$ state at the lowest excitation energy near the $\alpha$ threshold energy is understood to be a manifestation of the Nambu-Goldstone zero mode, a soft mode, due to the condensation aspect of the duality similar to the Hoyle state in $^{12}$C. The duality of $\alpha$ cluster structure with incompatible crystallinity and coherent wave nature due to condensation is the consequence of the Pauli principle, which causes clustering.

Tensor network renormalization study on the crossover in classical Heisenberg and RP^{2} models in two dimensions.

We study the classical two-dimensional RP^{2} and Heisenberg models, using the tensor-network renormalization (TNR) method. The determination of the phase diagram of these models has been challenging and controversial due to the very large correlation lengths at low temperatures. The finite-size spectrum of the transfer matrix obtained by TNR is useful in identifying the conformal field theory describing a possible critical point. Our results indicate that the ultraviolet fixed point for the Heisenberg model and the ferromagnetic RP^{2} model in the zero-temperature limit corresponds to a conformal field theory with central charge c=2, in agreement with two independent would-be Nambu-Goldstone modes. On the other hand, the ultraviolet fixed point in the zero-temperature limit for the antiferromagnetic Lebwohl-Lasher model, which is a variant of the RP^{2} model, seems to have a larger central charge. This is consistent with c=4 expected from the effective SO(5) symmetry. At T>0, the convergence of the spectrum is not good in both the Heisenberg and ferromagnetic RP^{2} models. Moreover, there seems to be no appropriate candidate of conformal field theory matching the spectrum, which shows the effective central charge c∼1.9. These suggest that both models have a single disordered phase at finite temperatures, although the ferromagnetic RP^{2} model exhibits a strong crossover at the temperature where the dissociation of Z_{2} vortices has been reported.

Axion cosmology in the presence of nontrivial Nambu-Goldstone modes

Axion cosmology is reexamined taking into account effect of kinetic pseudo Nambu-Goldstone modes, with its importance recently pointed out. When Peccei-Quinn (PQ) symmetry is broken by a chiral U(1) singlet, it is found that the effect of kinetic Nambu-Goldstone mode makes the axion dark matter untenable. When PQ symmetry is extended and is broken by two singlets, we find axion cosmology to work, but there are several differences from the axion cosmology studied in the literature. The differences are (1) ordinary type of dark matter scaling as $1/{\rm cosmic \; scale\; factor}^3$ arising from a modulus field and not from the usual angular field, (2) mass of the dark matter quantum in the ultralight range, $(10^{-32} \sim 10^{-14})\,$eV, (3) emergence of dark energy with the present density of order (a few meV)$^4$ consistent with observations, (4) presence of a long range spin-dependent force, (5) slow-roll inflation after PQ symmetry breaking when conformal coupling to gravity is introduced.

Effective field theory of fluctuating wall in open systems: from a kink in Josephson junction to general domain wall

We investigate macroscopic behaviors of fluctuating domain walls in nonequilibrium open systems with the help of the effective field theory based on symmetry. Since the domain wall in open systems breaks the translational symmetry, there appears a gapless excitation identified as the Nambu-Goldstone (NG) mode, which shows the non-propagating diffusive behavior in contrast to those in closed systems. After demonstrating the presence of the diffusive NG mode in the (2+1)-dimensional dissipative Josephson junction, we provide a symmetry-based general analysis for open systems breaking the one-dimensional translational symmetry. A general effective Lagrangian is constructed based on the Schwinger-Keldysh formalism, which supports the presence of the gapless diffusion mode in the fluctuation spectrum in the thin wall regime. Besides, we also identify a term peculiar to the open system, which possibly leads to the instability in the thick-wall regime or the nonlinear Kardar-Parisi-Zhang coupling in the thin-wall regime although it is absent in the Josephson junction.

Bifurcated symmetry breaking in scalar-tensor gravity

We present models that simultaneously predict the presence of dark energy and cold dark matter along with slow-roll inflation. The dark energy density is found to be of order $(\text{a few meV}{)}^{4}$, and the mass of dark matter constituent is $\ensuremath{\approx}1\text{ }\text{ }\mathrm{meV}$. These numbers are given in terms of the present value of Hubble constant ${H}_{0}$ and the Plank energy ${M}_{\mathrm{P}}\text{ }=\text{ }1/\sqrt{16\ensuremath{\pi}{G}_{N}}$: they are $({H}_{0}{M}_{\mathrm{P}}{)}^{2}$ for the dark energy density and $({H}_{0}{M}_{\mathrm{P}}{)}^{1/2}$ for the dark matter constituent mass. The basic theoretical framework we work in is a multiscalar tensor gravity with nontrivial conformal coupling to the Ricci scalar curvature in the Lagrangian density. The key for a right amount of dark energy is to incorporate in a novel way the spatially homogeneous kinetic contribution of Nambu-Goldstone modes in a spontaneously broken multiscalar field sector. Proposed theories are made consistent with general relativity tests at small cosmological distances, yet are different from general relativity at cosmological scales. Dark matter is generated as a spatially inhomogeneous component of the scalar system, with a roughly comparable amount to the dark energy. In some presented models a cosmological bifurcation of symmetry breaking of the scalar sector is triggered by the spontaneous breaking of electroweak $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge symmetry, hence the separation occurring simultaneously at the electroweak phase transition. The best experimental method to test presented models is to search for the fifth-force type of scalar exchange interaction with a force range, $O({10}^{\ensuremath{-}2})\text{ }\text{ }\mathrm{cm}$, whose coupling to matter is basically of gravitational strength.