Search

Higgs and Goldstone modes manifest as fluctuations in the order parameter of system, offering insights into its phase transitions and symmetry properties. Exploring the dynamics of these collective excitations in a Rydberg atoms system advances various branches of condensed matter, particle physics, and cosmology. Here, we report an experimental signature of Higgs and Goldstone modes in a U(1) symmetry-broken Rydberg atomic gases. By constructing two probe fields to excite atoms, we observe the distinct phase and amplitude fluctuations of Rydberg atoms collective excitations under the particle-hole symmetry. Due to the van der Waals interactions between the Rydberg atoms, we detect a symmetric variance spectrum divided by the divergent regime and phase boundary, capturing the full dynamics of the additional Higgs and Goldstone modes. Studying the Higgs and Goldstone modes in Rydberg atoms allows us to explore fundamental aspects of quantum phase transitions and symmetry breaking phenomena, while leveraging the unique properties of these highly interacting systems to uncover new physics and potential applications in quantum simulation.

We consider systems in which a continuous symmetry $G$, which may be anomalous, is spontaneously broken to an anomaly-free subgroup $H$ such that the effective action for the Goldstone modes contains topologically non-trivial terms. If the original system has trivial $G$ anomaly, it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual $H$ symmetry. Here we address the more general setting in which the $G$ symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the "compatibility relation" between the Goldstone invariants and the $G$ anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e. without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking $\text{U}(1)$ symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number., Comment: 40 + 5 pages

We investigate a pseudo-Nambu-Goldstone (pNG) dark matter (DM) model based on a gauged $SU(2)_x$ and a global $SU(2)_g$ symmetries. These symmetries are spontaneously broken to a global $U(1)_D$ symmetry by a vacuum expectation value of an $SU(2)_x \times SU(2)_g$ bi-fundamental scalar field. The global $SU(2)_g$ symmetry is also softly broken to a global $U(1)_D$ symmetry. Under the setup, a complex pNG boson arises. It is stabilized by $U(1)_D$ and is a DM candidate. Its scattering cross section off a nucleon is highly suppressed by small momentum transfer and thus evades the stringent constraints from DM direct detection experiments. Assuming all the couplings in the dark sector are real, a discrete symmetry arises. Consequently, in addition to the pNG DM, the lighter one of an $SU(2)_x$ gauge boson $V^0$ and a CP-odd scalar boson $a_0$ from the bi-fundamental scalar field can also serve as a DM candidate. Therefore, the model provides two-component DM scenarios. We find that the relic abundance of the DM candidates explains the measured value of the DM energy density. We also find that the pNG DM is the dominant DM component in large regions of the parameter space. In contrast to the pNG DM, both $V^0$ and $a_0$ scatter off a nucleon, and their scattering cross sections are not suppressed. However, their scattering event rates are suppressed by their number densities. Thus, the scattering cross section is effectively reduced. We show that the effective WIMP-nucleon scattering cross sections in the two-component scenarios are smaller than the current upper bounds and above the neutrino fog., Comment: 36 pages, 13 figures

We conduct a mode analysis of a general U(1)-charged first-order relativistic hydrodynamics within the framework of the effective field theory of dissipative fluids in a Minkowski background. The most general quadratic action for collective excitations around hydrodynamic solutions -- specifically, the Nambu-Goldstone (NG) modes associated with the symmetry-breaking pattern induced by external fields -- is derived. It is found that the hydrodynamical frame invariants write the first-order dispersion relations in the low energy limit. In thermodynamical viscous fluids realized by integrating out fast modes, unitarity and local KMS symmetry for its underlying UV theory are guaranteed. Then, we find that first-order hydrodynamics are stable if the null energy condition is satisfied. As the NG modes nonlinearly realize the diffeomorphism symmetries, our mode analysis is valid in any coordinate systems, including Lorentz-transformed references. We also comment on the causal structure for the diffusive modes based on the retarded Green functions for the stochastic NG mode., Comment: 44 pages

We present a general framework to implement massive Nambu-Goldstone quasi-particles in driven many-body systems. The underlying mechanism leverages an explicit Lie group structure imprinted into an effective Hamiltonian that governs the dynamics of slow degrees of freedom; the resulting emergent continuous symmetry is weakly explicitly broken, giving rise to a massive Nambu-Goldstone mode, with a spectral mass gap scaling linearly with the drive period. We discuss explicit and experimentally implementable realizations, such as Heisenberg-like spin models that support gapped spin-wave excitations. We provide a protocol to certify the existence of the massive Nambu-Goldstone mode from the dynamics of specific observables, and analyse the dispersion spectrum and their lifetime in the presence of weak explicit symmetry breaking.

The one-dimensional spin-$s$ ${\rm SU}(2)$ ferromagnetic Heisenberg model, as a paradigmatic example for spontaneous symmetry breaking (SSB) with type-B Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying the ground state subspace. This intrinsic abstract fractal is here revealed from a systematic investigation into the entanglement entropy for a linear combination of factorized (unentangled) ground states on a fractal decomposable into a set of the Cantor sets. The entanglement entropy scales logarithmically with the block size, with the prefactor being half the fractal dimension of a fractal, as long as the norm for the linear combination scales as the square root of the number of the self-similar building blocks kept at each step $k$ for a fractal, under an assumption that the maximum absolute value of the coefficients in the linear combination is chosen to be around one, and the coefficients in the linear combination are almost constants within the building blocks. Actually, the set of the fractal dimensions for all the Cantor sets forms a {\it dense} subset in the interval $[0,1]$. As a consequence, the ground state subspace is separated into a disjoint union of countably infinitely many regions, each of which is labeled by a decomposable fractal. Hence, the interpretation of the prefactor as half the fractal dimension is valid for any support beyond a fractal, which in turn leads to the identification of the fractal dimension with the number of type-B GMs for the orthonormal basis states. Our argument may be extended to any quantum many-body systems undergoing SSB with type-B GMs., Comment: 23 pages, 8 figures

Stringent constraints from the dark matter (DM) direct detection experiments can be naturally evaded for a pseudo-Nambu-Goldstone boson (pNGB) DM. We propose a realization of pNGB DM in the context of a left-right symmetric model, wherein the neutrinos are pseudo-Dirac in nature. The Dirac mass term for neutrinos arises from two-loop quantum corrections, whereas the Majorana mass terms are generated from Planck-induced corrections. This class of model also provides a parity solution to the strong CP problem without the need for an axion. We show an interesting correlation between the lifetime of the DM and the mass-squared differences between active and sterile neutrinos while maintaining a solution to the strong CP problem., Comment: 5 pages, 4 figures and references

A coordinate-free formulation of first order effective field theory, in which Nambu-Goldstone fields are described as sections on associated bundle, is presented. This construction, which is based only on symmetry considerations, allows for a direct derivation of number and types of Nambu-Goldstone fields in a classical field theory without any reference to effective Lagrangian. A central role in classification is shown to be played by Lorentz-symmetry breaking order parameter which induces symplectic structure in the field space of the theory., Comment: 20 pages

It is well known that massless Goldstone bosons have not yet been observed. In the Goldstone boson model, after the spontaneous symmetry breaking under $U(1)$, two coupled nonlinear equations are obtained for which we present the exact solitonic solutions. These solutions completely localize the energy density of the model and show the existence of two boson fields, one massive and the other massless. Also, it is seen that the solitonic waves are modulated and the massless wave rides on the massive wave. Finally, we calculate the charge density of the model, which again confirms the neutrality of these boson fields. Since massive bosons can be observed in the laboratory, these solitonic waves may be useful in tracking and detecting Goldstone bosons., Comment: 9 pages, 3 figures. Accepted in International Journal of Modern Physics A

The Goldstone-Apple Valley Radio Telescope (GAVRT) project conducts a regular monitoring program of the Sun. The GAVRT Solar Patrol project uses a 34 m diameter antenna to produce raster-scan maps of the Sun simultaneously at 4 frequencies ranging from approximately 3 GHz to 14 GHz. On 2024 May 14, as part of regular GAVRT Solar Patrol observations, raster maps were produced when an X8.7 solar flare occurred in active region AR13664. Here we present the GAVRT maps of the May 14 flare along with microwave flux density spectra showing the non-thermal microwave burst emission from mildly relativistic electrons produced in this largest flare of Solar Cycle 25 to date. AR13664 reappeared as AR13697 and continued to be very active, producing X flares while GAVRT monitored its activity. GAVRT microwave data provide a powerful complement to the energetic electrons tracked by X-ray, millimeter-wave and {\gamma}-ray emissions., Comment: 4 pages; 1 figure

The existence of pseudo-Nambu-Goldstone boson (pNGB) fields is a common feature in many models beyond the Standard Model, characterized by their exclusive derivative couplings. This paper investigates a scenario where a pNGB is coupled to the inflaton field during the reheating phase of the early universe. We calculate the perturbative decay rate of a coherently oscillating inflaton into pNGBs on a general basis, considering both constant and field-dependent couplings with monomial potentials at the minimum. As a concrete application, we explore the production of axions when the radial mode of the Peccei-Quinn (PQ) scalar serves as the inflaton, particularly in the presence of a large gravitational non-minimal coupling. Our findings suggest that the presence of pNGBs during reheating can lead to significant non-thermal relics, offering new constraints on inflationary reheating models and providing potential observational signatures in the form of dark radiation., Comment: 19 pages, 3 figures

From the quasisymmetry-group perspective [Phys. Rev. Lett. 126, 120604 (2021)], we show the universal existence of collective, coherent modes of excitations in many-body scar models in the degenerate limit, where the energy spacing in the scar tower vanishes. The number of these modes, as well as the quantum numbers carried by them, are given, not by the symmetry of the Hamiltonian, but by the quasisymmetry of the scar tower: hence the name quasi-Nambu-Goldstone modes. Based on this, we draw a concrete analogy between the paradigm of spontaneous symmetry breaking and the many-body scar physics in the degenerate limit., Comment: 27 pages, 14 figures

The recently discovered ferroelectric nematic (NF) liquid crystals (LC) have been reported to show an extraordinarily large value of the real part of the dielectric constant at low frequencies. However, it was argued by Clark et al in Physical Review Research 6, 013195 (2024) that what was measured was the capacitance of the insulating layer at LC/electrode surface and not that of the liquid crystal. Here we describe the results of dielectric spectroscopy measurements of an NF material in cells with variable thickness of the insulating layers. Our measurements quantitatively verify the model by Clark et al. Additionally, our measurements in cells with bare conducting indium tin oxide surface provide a crude estimate of epsilon perpendicular~100 in the NF phase., Comment: 17 pages, 6 figures, Presented in Workshop on Ferroelectric Nematic Liquid Crystals in Ljubljana

The effective field theory of single-field inflation characterizes the inflationary epoch in terms of a pattern of symmetry breaking. An operator acquires a time-dependent vacuum expectation value, defining a preferred spatial slicing. In the absence of dynamical gravity, the fluctuations around the time-dependent background are described by the Goldstone boson associated with this symmetry breaking process. With gravity, the Goldstone is eaten by the metric, becoming the scalar metric fluctuation. In this paper, we will show that in general single-field inflation, the statistics of scalar metric fluctuations are given by the statistics of this Goldstone boson decoupled from gravity up to corrections that are controlled as an expansion in slow-roll parameters. This even holds in the presence of additional parameters, like the speed of sound, that naively enhance the impact of the gravitational terms. In the process, we derive expressions for leading and sub-leading gravitational corrections to all-orders in the Goldstone boson., Comment: 38 pages

Recently, in studies of holographic models and hydrodynamics with spontaneous breaking of approximate symmetries, it has been proposed that the damping of pseudo-Goldstone modes at finite temperatures is universally constrained in the way that $\Omega_{\varphi}/m_{\varphi}^2\simeq D_{\varphi}$ in the broken phase, where $\Omega_{\varphi}$ and $m_{\varphi} $ are the relaxation rate at zero wavenumber and the mass of pseudo-Goldstones, $D_{\varphi}$ is the Goldstone diffusivity in the limit of purely spontaneous breaking. In this paper, we investigate the pseudo-Goldstone damping in a purely relaxational O($N$) model by performing the functional renormalization group calculations at the full quantum and stochastic level within the Schwinger-Keldysh formalism. We find that, away from the critical temperature, the proposed relation is always valid. When the temperature is very close to the critical value such that the mass of the Higgs mode is comparable to the mass of the pseudo-Goldstone modes, the pseudo-Goldstone damping displays a novel scaling behavior that follows $\Omega_\varphi/m_\varphi^2\propto m_{\varphi}^{\Delta_\eta}$ with a correction $\Delta_\eta$ controlled by the critical universalities. Moreover, we study how the correction depends on the value of $N$ and show that $\Delta_\eta \rightarrow 0$ when fluctuations are infinitely suppressed in the large $N$ limit. In this case, the proposed relation works even in the critical region. Finally, we match our results to the dissipative sector of the pion dynamics near the chiral phase transition., Comment: V2:minor revision, references added, discussion on the pure SSB case has been moved to the supplement

We study simultaneous symmetry-breaking in a spontaneous Floquet state, focusing on the specific case of an atomic condensate. We first describe the quantization of the Nambu-Goldstone (NG) modes for a stationary state simultaneously breaking several symmetries of the Hamiltonian by invoking the generalized Gibbs ensemble, which enables a thermodynamical description of the problem. The quantization procedure involves a Berry-Gibbs connection, which depends on the macroscopic conserved charges associated to each broken symmetry and whose curvature is not invariant under generalized gauge transformations. We extend the formalism to Floquet states simultaneously breaking several symmetries, where Goldstone theorem translates into the emergence of Floquet-Nambu-Goldstone (FNG) modes with zero quasi-energy. In the case of a spontaneous Floquet state, there is a genuine temporal FNG mode arising from the continuous time-translation symmetry breaking, whose quantum amplitude provides a rare realization of a time operator in Quantum Mechanics. Furthermore, since they conserve energy, spontaneous Floquet states can be shown to possess a conserved Floquet charge. Conventional Floquet systems also admit a thermodynamic description in terms of the Floquet enthalpy, the Legendre transform of the energy with respect to the Floquet charge. We apply our formalism to a particular realization of spontaneous Floquet state, the CES state, which breaks $U(1)$ and time-translation symmetries, representing a time supersolid. Using the Truncated Wigner method, we numerically compute its quantum fluctuations, which are theoretically predicted to be dominated by the temporal FNG mode at long times, observing a remarkable agreement between simulation and theory. Based on these results, we propose a feasible experimental scheme to observe the temporal FNG mode of the CES state., Comment: 29 pages, 5 figures. Minor modifications added to further improve the presentation of the text and correct minor typos

We investigate the scaling behavior of Nambu-Goldstone (NG) modes in the ordered phase of the Vicsek model, introducing a phenomenological equation of motion (EOM) incorporating a previously overlooked non-linear term. This term arises from the interaction between velocity fields and density fluctuations, leading to new scaling behaviors. We derive exact scaling exponents in two dimensions, which reproduce the isotropic scaling behavior reported in a prior numerical simulation., Comment: 5 pages, 1 figure. This paper is a revised version of arXiv:2401.01603 with modified derivation of the scaling exponents

The reformulation of gravity as the Nambu-Goldstone mode associated with broken gauge symmetry is given and applied to cosmology. We find that; the new gravity consists of a scalar mode and a tensor mode, of which the tensor mode corresponds to the Einstein gravity; the universe pulsates without spacial curvature; the gravitational constant increases from zero as the universe expands and decreases to zero as the universe contracts; the dark energy is identified with the sum of the gravitational energies of scalar and tensor modes, while the dark matter is the mirage of the gravitational energy caused by the projection into the $\Lambda$CDM spectrum; the gravitational equation dictates the cosmological constant to be zero; the universe begins with inflation, and the present age of the universe is approximately 550Gy., Comment: 16 pages, 5 figures

Pseudo-Nambu-Goldstone (pNG) dark matter (DM) is a promising DM candidate and able to explain the measured DM abundance by the thermal freeze-out mechanism evading the stringent bound from DM direct detection experiments. We propose a new model providing a pNG DM by introducing two Standard-Model-singlet complex scalars with the same charges of a dark $U(1)$ gauge symmetry. They are also charged under a $U(1)$ global symmetry corresponding to their relative phase rotations, which is explicitly broken by a soft-breaking term in the scalar potential. The both $U(1)$ symmetries are spontaneously broken by their vacuum expectation values, giving rise to one real pNG boson. We also introduce a discrete $\mathbb{Z}_2$ symmetry exchanging the two scalars to stabilize the pNG boson as DM. It is shown that this model reproduces the DM abundance consistently with the current bound from the direct detection experiments. The model has a gauge kinetic mixing between the dark and $U(1)_Y$ gauge fields, which allows the dark gauge boson to decay even with a relatively light mass and prevents it from being an additional DM component. The Landau pole is avoided thanks to the small gauge coupling constant. In addition, a DM pair dominantly annihilates into a pair of the dark gauge bosons if the gauge boson mass is lighter than the DM mass, and thus its cross section has significantly different parameter dependence from other pNG DM models. We also calculate the DM-nucleon scattering cross section at the loop level. It turns out that it is necessary to probe region covered by the neutrino fog in order to test this model., Comment: 31 pages, 10 figures; v2 published version

In this work, we revisit pseudo Nambu Goldstone boson (pNGB) DM model where global $U(1)$ dark symmetry is spontaneously broken as well as explicitly with broken by with linear symmetry breaking, focusing on the dark matter mass range in $500 {\rm GeV} \lesssim m_{\rm DM} \lesssim O(10)$ TeV. This model is interesting not only in its own in the context of pNGB DM, but also in the context of cosmological collider signatures from heavy particle mass regime without Boltzmann suppressions, $H \lesssim m_{\rm heavy} \lesssim 60 H$, through the chemical potential type interaction. After imposing perturbative unitarity, perturbativity, correct thermal relic density and constraints from colliders and (in)direct detection experiments, we find that pNGB DM mass is allowed up to $\sim 1 (10)$ TeV for the dark Higgs mass $m_{H_2} = 1 (100)$ TeV for the Higgs-dark Higgs mixing $\sin \theta = 0.1$. We also consider the case where global $U(1)$ dark symmetry is not spontaneously broken, where DM is no longer pNGB. In this case, DM and dark Higgs masses are nearly degenerate in the range of a few TeV $\lesssim m_{{\rm DM}, H_2} \lesssim \sim 70$ TeV. Low mass region for pNGB DM and $H_2$ could be directly probed in colliders or indirect DM detections, whereas the heavy mass regime could be probed through the non-Gaussianity at the level of $f_{\rm NL} \sim O(0.01-10)$ if $H$ is as low as $O(1-10)$ TeV., Comment: 21 pages, 3 figures

We consider a $U(1)$ gauge theory on $M^4\times T^4$ with background magnetic fluxes. We show that a theory including arbitrary fluxes can always be studied in a theory involving only diagonal fluxes by appropriate coordinate transformations. It is found that the number of independent magnetic fluxes is equal to the rank of the classical value of the field strength matrix, ${\rm rank}\langle F\rangle$. The number of massless zero modes induced from extra components of higher-dimensional gauge field (Wilson-line scalar field), is also determined by ${\rm rank}\langle F\rangle$. We explicitly confirm that the quantum corrections due to the matter fermion to the squared mass of Wilson-line scalar field cancel out at one-loop level. For this purpose, we derive the fermion mass spectrum on $M^4\times T^4$ with arbitrary fluxes. By taking the flux diagonal basis, creation and annihilation operators for Kaluza-Klein quantum numbers are defined appropriately. Our results are easily generalized to the case of $M^4\times T^{2n}~(n\geq3)$., Comment: 17 pages, 2 figures

An emergent gapless Goldstone mode originates from continuous spontaneous symmetry breaking, which has become a doctrine since the pioneering work by Goldstone [J. Goldstone, Nuovo Cimento \textbf{19}, 154 (1961)]. However, we argue that it is possible for a continuous symmetry group $\rm{U}(1)$ to make an exceptional case, simply due to the well-known mathematical result that a continuous symmetry group $\rm{U}(1)$ may be regarded as a limit of a discrete symmetry group $Z_q$ when $q$ tends to infinity. As a consequence, spontaneous symmetry breaking for such a continuous symmetry group $\rm{U}(1)$ does not necessarily lead to any gapless Goldstone mode. This is explicitly explained for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. In a sense, this model provides an illustrative example regarding the dichotomy between continuity and discreteness., Comment: 13 pages, 9 figures, 9 tables

Collective dynamics of many particle systems is tightly linked to their underlying symmetry and phase transitions. Higgs and Nambu-Goldstone modes are, respectively, collective amplitude and phase modes of the order parameter that are widely observed in various physical systems at different energy scales, ranging from magnets, superfluids, superconductors to our universe. The Higgs mode is a massive excitation, which is a condensed-matter analog of Higgs particle in high-energy physics, while the Nambu-Goldstone mode is a massless excitation that appears when a continuous symmetry is spontaneously broken. They provide important information on the fundamental aspects of many particle systems, such as symmetry, phases, dynamics, response to external fields, and so on. In this article, we review the physics of Higgs and Nambu-Goldstone modes in condensed matter physics. Especially, we focus on the development on the study of collective modes in superconductors and cold-atom systems., Comment: 12 pages, 7 figures

We study a lattice Nambu-Jona-Lasinio model with SU(2) and SU(3) flavor symmetries of staggered fermions in the Kogut-Susskind Hamiltonian formalism. This type of four-fermion interactions has been widely used for describing low-energy behaviors of strongly interacting quarks as an effective model. In particular, we focus on the Nambu-Goldstone modes associated with the spontaneous breakdown of the flavor symmetries. In the strong coupling regime for the interactions, we prove the following: (i) For the spatial dimension $\nu \ge 5$, the SU(3) model shows a long-range order at sufficiently low temperatures. (ii) In the case of the SU(2) symmetry, there appears a long-range order in the spatial dimension $\nu \ge 3$ at sufficiently low temperatures. (iii) These results hold in the ground states as well. (iv) In general, if a long-range order emerges in this type of models, then there appear gapless excitations above the sector of the infinite-volume ground states. These are nothing but the Nambu-Goldstone modes associated with the spontaneous breakdown of the global rotational symmetry of flavors. (v) In particular, we establish that the number of the lineraly independent Nambu-Goldstone modes is equal to the number of the broken symmetry generators on the Hilbert space constructed from a certain symmetry-breaking infinite-volume ground state., Comment: 63 pages, final version to appear in Journal of Mathematical Physics

We discuss the Goldstone mode of skyrmion crystal in a model of two-dimenssional ferromagnet with Dzyaloshinskii-Moriya interaction in magnetic field. We use stereographic projection approach to construct skyrmion crystal and consider skyrmion's displacement field. The small overlap of the individual skyrmion images restricts the potential energy to the interaction of nearest neighboring displacements. The closed form of the Goldstone mode dispersion is found and its dependence on the magnetic field is studied. We use semiclassical quantization to define the Green's function and show that the propagation of displacements through the crystal changes its tensorial form from isotropic to anisotropic one at large distances., Comment: 6 pages, 4 figures

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 $\eta$ meson, consisting of a stack of sine-Gordon solitons carrying a baryon number, due to the anomalous coupling of the $\eta$ 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 $\eta$ soliton decays into a pair of non-Abelian sine-Gordon solitons carrying $S^2$ moduli originated from Nambu-Goldstone (NG) modes localized around it, corresponding to the spontaneously broken vector symmetry SU$(2)_{\rm V}$. There, the $S^2$ 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 $S^2$ modes of neighboring solitons. In this paper, we calculate excitations including gapless NG modes and excited modes around non-Abelian and Abelian ($\eta$) CSLs, and find three gapless NG modes with linear dispersion relations (type-A NG modes): two isospinons ($S^2$ modes) and a phonon corresponding to the spontaneously broken vector SU$(2)_{\rm V}$ and translational symmetries around the non-Abelian CSL, respectively, and only a phonon for the Abelian CSL because of the recovering SU$(2)_{\rm V}$. We also find in the deconfined phase that the dispersion relation of the isospinons becomes of the Dirac type, {\it i.~e.~} linear even at large momentum., Comment: 32 pages, 11 figures

In a dimensionless multi-scalar extension of the Standard Model, Gildener and Weinberg assumed that along a flat direction, there is only one classically massless scalar, known as the {\it scalon}, which acquires mass through radiative corrections \`a la Coleman-Weinberg, while all other scalars remain heavy. In this paper, by introducing a toy model with four scalar degrees of freedom we demonstrate the existence of {\it two} scalons along a specific flat direction that we construct. We present the effective potential for the model and provide the masses of the heavy scalars and two radiatively light pseudo-Goldstone bosons., Comment: 4 pages, V1: minor modifications, results unchanged V2: discrete symmetry argument in the model modified, results remain unchanged, matches the published version

We analytically study friction and dissipation of a driven bead in a 1D harmonic chain, and analyze the role of internal damping mechanism as well as chain length. Specifically, we investigate Dissipative Particle Dynamics and Langevin Dynamics, as paradigmatic examples that do and do not display translational symmetry, with distinct results: For identical parameters, the friction forces can differ by many orders of magnitude. For slow driving, a Goldstone mode traverses the entire system, resulting in friction of the driven bead that grows arbitrarily large (Langevin) or gets arbitrarily small (Dissipative Particle Dynamics) with system size. For a long chain, the friction for DPD is shown to be bound, while it shows a singularity (i.e. can be arbitrarily large) for Langevin damping. For long underdamped chains, a radiation mode is recovered in either case, with friction independent of damping mechanism. For medium length chains, the chain shows the expected resonant behavior. At the resonance, friction is non-analytic in damping parameter $\gamma$, depending on it as $\gamma^{-1}$. Generally, no zero frequency bulk friction coefficient can be determined, as the limits of small frequency and infinite chain length do not commute, and we discuss the regimes where "simple" macroscopic friction occurs., Comment: 10 pages, 3 figures

We propose a grand unified theory (GUT) pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on $SU(7)$ gauge symmetry. In the GUT model, the Standard Model (SM) gauge symmetry $G_{\rm SM} := SU(3)_C\times SU(2)_L\times U(1)_Y$ and the ``dark'' gauge symmetry $SU(2)_D$ are unified, where the $SU(2)_D$ symmetry plays an important role in the stability of DM. The unification of SM fermions and dark sector fermions is partially realized. The gauge symmetry $SU(7)$ is spontaneously broken to $SU(5)\times SU(2)\times U(1)$ gauge symmetry at the GUT scale by the nonvanishing vacuum expectation values of an $SU(7)$ adjoint scalar field. The symmetry is further broken to $G_{\rm SM}\times SU(2)_D$ at an intermediate scale. Furthermore, the $SU(2)_D$ symmetry is broken by the $SU(2)_D$ doublet and triplet scalar fields at the TeV scale. In the pNGB DM model based on $G_{\rm SM}\times SU(2)_D$, the residual global $U(1)_V$ dark custodial symmetry guarantees DM stability. On the other hand, in the $SU(7)$ pNGB DM model, this global symmetry is explicitly broken by the Yukawa interaction and the effective Majorana mass terms. To maintain $U(1)_V$ symmetry and thus the DM stability, we need to tune Yukawa coupling constants and cubic scalar couplings at high accuracy. We find that the allowed DM mass region is quite restricted as the gauge coupling constant of $SU(2)_D$ is determined by the condition of the gauge coupling unification. To satisfy gauge coupling unification and the current experimental constraint on proton lifetime, we find that three generations of $SU(3)_C$ adjoint fermions and another three generations of $SU(2)_L$ adjoint fermions with the intermediate mass scale are required. We also find that there is no other solution to satisfy simultaneously the gauge coupling unification and the proton decay constraint if one assumes the other symmetry breaking schemes., Comment: 25 pages, 6 tables, 4 figures; some paragraphs added; typos corrected; accepted for publication in Physical Review D

Second-order phase transitions are governed by spontaneous symmetry breaking, which yield collective excitations with a gapless spectrum called Nambu-Goldstone (NG) modes. While NG modes in conservative systems are propagating excitations, non-equilibrium phase transitions have been predicted to feature a diffusive NG mode. We present the first experimental evidence of a diffusive NG mode in a non-equilibrium Bose-Einstein condensate of microcavity polaritons. The NG mode is observed as a spectral narrowing in the spectroscopic response of the condensate. Additionally, explicitly breaking the symmetry causes the opening of a gap in the spectrum and the disappearance of the NG mode. Our observations confirm the diffusive dynamics of the NG mode of non-equilibrium phase transitions and establish a promising framework to investigate fundamental questions in statistical mechanics., Comment: 8 pages + refs + appendix, 6 figures

We investigate in detail the $\nu=+1$ displacement-field-tuned interacting phase diagram of $L=3,4,5,6,7$ layer rhombohedral graphene aligned to hBN (R$L$G/hBN). Our calculations account for the 3D nature of the Coulomb interaction, the inequivalent stacking orientations $\xi=0,1$, the effects of the filled valence bands, and the choice of `interaction scheme' for specifying the many-body Hamiltonian. We show that the latter has a dramatic impact on the Hartree-Fock phase boundaries and the properties of the phases, including for pentalayers (R5G/hBN) with large displacement field $D$ where recent experiments observed a Chern insulator at $\nu=+1$ and fractional Chern insulators for $\nu<1$. In this large $D$ regime, the low-energy conduction bands are polarized away from the aligned hBN layer, and are hence well-described by the folded bands of moir\'eless rhombohedral graphene at the non-interacting level. Despite this, the filled valence bands develop moir\'e-periodic charge density variations which can generate an effective moir\'e potential, thereby explicitly breaking the approximate continuous translation symmetry in the conduction bands, and leading to contrasting electronic topology in the ground state for the two stacking arrangements. Within time-dependent Hartree-Fock theory, we further characterize the strength of the moir\'e pinning potential in the Chern insulator phase by computing the low-energy $\mathbf{q}=0$ collective mode spectrum, where we identify competing gapped pseudophonon and valley magnon excitations. Our results emphasize the importance of careful examination of both the single-particle and interaction model for a proper understanding of the correlated phases in R$L$G/hBN.

It is found that the staggered $\rm SU(4)$ ferromagnetic spin-orbital model accommodates highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. The spontaneous symmetry breaking pattern is ${\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}$, with three type-B Goldstone modes. An abstract fractal underlies the ground state subspace with the fractal dimension identified with the number of type-B Goldstone modes. This connection is established by evaluating the entanglement entropy, which exhibits a universal finite system-size scaling behaviour. The latter in turn implies that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. In addition, the ground state degeneracies, depending on the boundary conditions adopted, constitute the two Fibonacci-Lucas sequences. In the limit of large system size their asymptotic forms become a self-similar logarithmic spiral. As a result, the model has a non-zero residual entropy $S_{\!r} = -2 \ln R $, where $R=(\! \sqrt{6}-\!\sqrt{2})/2$., Comment: 16 pages, 5 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2302.13126

A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-$s$ Heisenberg ferromagnetic model, the $\rm{SU}(3)$ ferromagnetic model and the $\rm{SO}(4)$ spin-orbital model., Comment: 14 pages, 24 figures, 11 tables

We consider electronic systems with a spontaneously broken continuous symmetry. The scattering vertex between electrons and Goldstone modes is calculated over the entire Brillouin zone using the random phase approximation. This calculation reveals two things: (1) electrons always couple to both $\phi$ and $\partial_t \phi$, where $\phi$ is the Goldstone field, and (2) quasi-particles in a state with continuous symmetry breaking have to be defined in a rotating frame, which locally follows the fluctuations of the order parameter. The implications of these findings for electron spectral functions in both symmetry-broken and thermally disordered systems are discussed, and the examples of anti-ferromagnetism in the Hubbard model and spin spiral order in the three-band model are worked out in detail.

We study the vacuum dynamics of pseudo-Nambu-Goldstone bosons (pNGBs) for $SO(N+1) \rightarrow SO(N)$ spontaneous and explicit symmetry breaking. We determine the magnitude of explicit symmetry breaking consistent with an EFT description of the effective potential at zero and finite temperatures. We expose and clarify novel additional vacuum transitions that can arise for generic pNGBs below the initial scale of $SO(N+1) \rightarrow SO(N)$ spontaneous symmetry breaking, which may have phenomenological relevance. In this respect, two phenomenological scenarios are analyzed: thermal and supercooled dark sector pNGBs. In the thermal scenario the vacuum transition is first-order but very weak. For a supercooled dark sector we find that, depending on the sign of the explicit symmetry breaking, one can have a symmetry-restoring vacuum transition $SO(N-1) \rightarrow SO(N)$ which can be strongly first-order, with a detectable stochastic gravitational wave background signal., Comment: 28 pages, 10 figures

We perform a systematic classification of scalar field theories whose amplitudes admit a double copy formulation and identify two building blocks at 4-point and 13 at 5-point. Using the 4-point blocks as bootstrap seeds, this naturally leads to a single copy theory that is a gauged NLSM. Moreover, its double copy includes a novel theory that can be written in terms of Lovelock invariants of an induced metric, and includes Dirac-Born-Infeld and the special Galileon in specific limits. The amplitudes of these Goldstone modes have two distinct soft behaviour regimes, corresponding to a hybrid of non-linear symmetries., Comment: 8 pages, 3 tables; v2: version to appear in PRL

We introduce a model to study magnon scattering in skyrmion crystals, sandwiched between ferromagnets which act as the source of magnons. Skyrmions are topological objects while skyrmion crystals break internal and translational symmetries, thus our setup allows us to study the interplay of topology and symmetry breaking. Starting from a basis of holomorphic theta functions, we construct an analytical ansatz for such a junction with finite spatially modulating topological charge density in the central region and vanishing in the leads. We then construct a suitably defined energy functional for the junction and derive the resulting equations of motion, which resemble a Bogoliubov-de Gennes-like equation. Using analytical techniques, field theory, heuristic models and microscopic recursive transfer-matrix numerics, we calculate the spectra and magnon transmission properties of the skyrmion crystal. We find that magnon transmission can be understood via a combination of low-energy Goldstone modes and effective emergent Landau levels at higher energies. The former manifests in discrete low-energy peaks in the transmission spectrum which reflect the nature of the Goldstone modes arising from symmetry breaking. The latter, which reflect the topology, lead to band-like transmission features, from the structure of which further details of the excitation spectrum of the skyrmion crystal can be inferred. Such characteristic transmission features are absent in competing phases of the quantum Hall phase diagram, and hence provide direct signatures of skyrmion crystal phases and their spectra. Our results directly apply to quantum Hall heterojunction experiments in monolayer graphene with the central region doped slightly away from unit filling, a $\nu = 1:1 \pm \delta \nu: 1$ junction and are also relevant to junctions formed by metallic magnets or in junctions with artificial gauge fields., Comment: 20+6 pages, 9+2 figures, comments welcome

We consider pion condensation in QCD at finite isospin density $\mu_I$ and zero temperature using two-flavor chiral perturbation theory ($\chi$PT). The pressure is calculated to next-to-leading order (NLO) in the low-energy expansion. In the nonrelativistic limit, we recover the classic result by Lee, Huang, and Yang for the energy density of a dilute Bose gas with an $s$-wave scattering length that includes loop corrections from $\chi$PT. In the chiral limit, higher-order calculations are tractable. We calculate the pressure to next-to-next-to-leading order (NNLO) in the low-energy expansion, which is an expansion in powers of $\mu_I^2/(4\pi)^2f^2$, where $f$ is the (bare) pion decay constant. The spontaneous breakdown of the global internal symmetry $U(1)_{I_3}$ gives rise to a massless Goldstone boson or phonon. We discuss the properties of the low-energy effective theory describing this mode. Finally, we compare our results for the pressure and the speed of sound with recent lattice simulations with 2+1 flavors. The agreement is very good for isospin chemical potentials up to 180-200 MeV, depending on the physical quantity., Comment: 11 pages, one figure. V2: few typos fixed and added references. V3: Added two figures of pressure and speed of sound. Comparison with lattice. New abstract. Matches printed version in PRD

We consider a scale-invariant inverse seesaw model with dynamical breaking of gauge symmetry and lepton number. In some regions of the parameter space, the Majoron - the pseudo-Goldstone of lepton number breaking - is a viable dark matter candidate. The bound on the Majoron decay rate implies a very large dilaton vacuum expectation value, which also results in a suppression of other dark matter couplings. Because of that, the observed dark matter relic abundance can only be matched via the freeze-in mechanism. The scalar field which gives mass to heavy neutrinos can play the role of the inflaton, resulting in a tensor-to-scalar ratio $r \lesssim 0.01$ for metric inflation and $r \lesssim 0.21$ for Palatini gravity., Comment: 31 pages, 3 figures, version published in JHEP, typos corrected

We explore the gravitational wave spectrum generated by string-wall structures in an $SO(10)$ ($Spin(10)$) based scenario of pseudo-Goldstone boson dark matter (pGDM) particle. This dark matter candidate is a linear combination of the Standard Model (SM) singlets present in the 126 and 16 dimensional Higgs fields. The Higgs $126$-plet vacuum expectation value (VEV) $\left<126_H\right>$ leaves unbroken the $\mathbb{Z}_2$ subgroup of $\mathbb{Z}_4$, the center of $SO(10)$. Among other things, this yields topologically stable cosmic strings with a string tension $\mu \sim \left<126_H\right>^2$. The subsequent (spontaneous) breaking of $\mathbb{Z}_2$ at a significantly lower scale by the $16$-plet VEV $\left<16_H\right>$ leads to the appearance of domain walls bounded by the strings produced earlier. We display the gravitational wave spectrum for $G \mu$ values varying between $10^{-15}$ and $10^{-9}$ ($\left<126_H\right>\sim 10^{11}$ - $10^{14}$ GeV), and $\left<16_H\right>\sim 0.1$ - $10^2$ TeV range ($G$ denotes Newton's constant.) These predictions can be tested, as we show, by a variety of (proposed) experiments including LISA, ET, CE and others., Comment: 8 pages, 3 figures, version published in PLB

We study a simple extension of the Standard Model featuring a dark sector with an ultralight pseudo Nambu-Goldstone boson as dark matter candidate. We focus on the mass range $\mathcal{O}(10^{-20} - 10^{-10})$ eV, relevant for strong gravity applications, and explore its production and evolution in the early Universe. The model is formulated in such a way that dark matter does not couple directly to photons or other Standard Model particles avoiding some of the most stringent cosmological bounds related to axion-like particles. In this work, two different scenarios are considered depending on whether dark matter is produced in a pre-inflationary or post-inflationary regime. We also discuss the effect from emergent topological defects such as cosmic strings and domain walls, and estimate the spectrum of stochastic gravitational waves produced by their decay, enabling to test the model at current and future gravitational-wave experiments., Comment: 22 pages, 5 figures

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-$N$ 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-$N$ 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-$N$ limit., Comment: 33 pages, 2 figures. V2: minor editings in response to Sci-Post Core peer-review process

Ordered phases realized through broken continuous symmetries embrace long-range order-parameter fluctuations as manifest in the power-law decays of both the transverse and longitudinal correlations, which are similar to those at the second-order transition point. We calculate the transverse one-loop correction to the dispersion relation of nonrelativistic Nambu-Goldstone (NG) bosons at finite temperatures, assuming that they have well-defined dispersion relations with some integer exponents. It is found that the transverse correlations make the lifetime of the NG bosons vanish right on the assumed dispersion curve below three dimensions at finite temperatures. Combined with the vanishing of the longitudinal "mass", the result indicates that the correlations generally bring both an intrinsic lifetime to the excitations and a qualitative change in the dispersion relation at long wavelengths below three dimensions at finite temperatures, unless it is protected by some conservation law. A more detailed calculation performed specifically on a single-component Bose-Einstein condensate reveals that the gapless Bogoliubov mode predicted by the perturbative treatment is modified substantially after incorporating the correlations to have (i) an intrinsic lifetime and (ii) a peak shift at long wavelengths from the linear dispersion relation. Thus, the NG bosons may be fluctuating intrinsically into and out of the ordered "vacuum" to maintain the order temporally., Comment: 7 pages, three figures

We study stochastic gravitational waves from cosmic strings generated in an ultraviolet-complete model for pseudo-Nambu-Goldstone dark matter with a hidden $\mathrm{U(1)}$ gauge symmetry. The dark matter candidate in this model can naturally evade direct detection bounds and easily satisfy other phenomenological constraints. The bound on the dark matter lifetime implies an ultraviolet scale higher than $10^9~\mathrm{GeV}$. The spontaneous $\mathrm{U(1)}$ symmetry breaking at such a high scale would induce cosmic strings with high tension, resulting in a stochastic gravitational wave background with a high energy density. We investigate the constraints from current gravitational wave experiments as well as the future sensitivity. We find that most viable parameter points can be well studied in future gravitational wave experiments., Comment: 24 pages, 6 figures; revisions to match the published version

We study the spectrum of scalar charged operators in Conformal Field Theories (CFTs) with a $U(1)$ global symmetry. The charged operators are dual, by the state-operator correspondence, to homogenous charged states on the sphere. Such states can break the $U(1)$ symmetry, and we define what we call the large $f$ regime in the CFT as one where the symmetry breaking scale is much higher than the scale of the CFT sphere. In such a regime, there is (an approximate) Goldstone boson associated to the breaking. We show that consistency of the Goldstone boson physics implies that the spectrum of states, and therefore of operators, must be convex in charge. More precisely, we show that any family of operators of different charges, which are lowest dimension of their charge, and which additionally share the same realisation of the Goldstone boson in terms of the degrees of freedom of the CFT, must be convex., Comment: 44 pages

Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from ${\rm SU}(3)$ to ${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes. The ground state degeneracies are characterized as the Fibonacci-Lucas sequences -- an ancient mathematical gem, under open and periodic boundary conditions, with the residual entropy being non-zero. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit, with the prefactor being half the number of type-B Goldstone modes. The latter in turn is identified to be the fractal dimension., Comment: 20 pages, 3 figures,1 table, additional references to earlier related work added, a minor change

We show that a conformal extension of the standard model with local B-L symmetry and two complex scalars breaking B-L can provide a unified description of neutrino mass, origin of matter and dark matter. There are two hierarchical B-L breaking vacuum expectation value (VEV) scales in the model, the higher denoted by $v_B$ and the lower by $v_A$. The higher breaking scale is dynamically implemented via the Coleman-Weinberg mechanism and plays a key role in the model since it induces electroweak symmetry breaking as well as the lower B-L breaking scale. It is also responsible for neutrino masses via the seesaw mechanism and origin of matter. The imaginary part of the complex scalar with lower B-L breaking VEV plays the role of a pseudo-Goldstone dark matter (DM). The DM particle is unstable with its lifetime naturally longer than $10^{28}$ seconds. We show that its relic density arises from the freeze-in mechanism for a wide parameter domain. Due to the pseudo-Goldstone boson nature of the DM particle, the direct detection cross section is highly suppressed. The model also predicts the dark matter to be heavier than 100 TeV and it decays to two high energy neutrinos which can be observable at the IceCube, providing a test of this model., Comment: 16 pages and seven figures; references updated