Anomalous tunneling refers to the phenomenon in which the transmission coefficient through a potential barrier approaches unity as the energy of an incident particle or quasiparticle tends to zero. This counterintuitive effect has been reported in systems exhibiting spontaneous symmetry breaking (SSB), such as superfluids, yet the general conditions for its occurrence remain unclear. In this Letter, we establish that anomalous tunneling of Nambu-Goldstone (NG) modes is a universal low-energy theorem dictated solely by symmetry and scaling, using a low-energy effective field theory (EFT) framework. We formulate the scattering of NG modes by external potentials in terms of spatially dependent EFT coefficients and demonstrate that symmetry-preserving localized potentials are irrelevant in the long-wavelength limit, leading to perfect transmission. In contrast, symmetry-breaking perturbations are relevant and suppress transmission, resulting in the absence of anomalous tunneling. We illustrate this universal behavior with explicit examples of superfluid phonons and magnons.
Anomalous tunneling refers to the phenomenon in which the transmission coefficient through a potential barrier approaches unity as the energy of an incident particle or quasiparticle tends to zero. This counterintuitive effect has been reported in systems exhibiting spontaneous symmetry breaking (SSB), such as superfluids, yet the general conditions for its occurrence remain unclear. In this Letter, we establish that anomalous tunneling of Nambu-Goldstone (NG) modes is a universal low-energy theorem dictated solely by symmetry and scaling, using a low-energy effective field theory (EFT) framework. We formulate the scattering of NG modes by external potentials in terms of spatially dependent EFT coefficients and demonstrate that symmetry-preserving localized potentials are irrelevant in the long-wavelength limit, leading to perfect transmission. In contrast, symmetry-breaking perturbations are relevant and suppress transmission, resulting in the absence of anomalous tunneling. We illustrate this universal behavior with explicit examples of superfluid phonons and magnons.
The Nambu-Goldstone (NG) mode in a Bose-Einstein condensate (BEC) transmits a potential barrier with probability 1 in the zero-energy limit, which is known as the anomalous tunneling. In this paper, we investigate the tunneling properties of quasiparticles in a dynamically unstable BEC. We prepare a multi-component BEC (binary and spin-1 BEC) in a dynamically unstable state and solve the tunneling problem of the spin-wave excitation from the condensate. We find that the perfect transmission occurs even when the BEC is dynamically unstable if the spin-wave is the NG mode. Here, the mode that exhibits the perfect transmission is the dynamically unstable spin-wave mode, which is a pure-imaginary-eigenvalue solution of the Bogoliubov-de Gennes equation. Hence, we should take the zero-energy limit along not the real axis but the imaginary axis. We also demonstrate the existence of the perfect reflection of a dynamically unstable mode at the point where the imaginary part of the eigenvalue takes its maximum. In this case, the incident and reflected waves destructively interfere, and the amplitude of the quasiparticle wave function is strongly suppressed. We numerically confirm that the perfect reflection is a generic nature of dynamically unstable modes and not related to the NG mode.
Abstract We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu–Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward–Takahashi identity and the effective action formalism, we establish the Nambu–Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu–Goldstone modes. Using these coefficients, we classify the Nambu–Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.
In modern physics, spontaneous symmetry breaking is one of the most important and ubiquitous ideas. It is well known that the breakdown of global and continuous symmetry gives rise to gapless excitations called Nambu-Goldstone (NG) bosons. This is called NG theorem. Since the generators of broken symmetries are bosonic, the excitations are also bosons. On the other hand, excitations are thought of as fermionic when a generator of broken symmetry is fermionic. The most famous example of fermionic symmetry is supersymmetry (SUSY). SUSY is a symmetry that exchanges bosons and fermions. In high-energy physics, it is known that spontaneous SUSY breaking gives rise to massless modes called NG fermions or Nambu- Goldstinos. In non-relativistic systems, fewer examples of models with SUSY are studied. Therefore, spontaneous SUSY breaking in non-relativistic systems is less understood than that in nonrelativistic systems or spontaneous symmetry breaking. In this thesis, we introduce interacting Majorana models with supersymmetry (SUSY) and discuss the properties in terms of spontaneous SUSY breaking and NG modes.
We study collective modes of superfluid Bose gases in optical lattices combined with potential barriers. We assume that the system is in the vicinity of the quantum phase transition to a Mott insulator at a commensurate filling, where emergent particle-hole symmetry gives rise to two types of collective mode, namely a gapless Nambu-Goldstone (NG) phase mode and a gapful Higgs amplitude mode. We consider two kinds of potential barrier: One does not break the particle-hole symmetry while the other does. In the presence of the former barrier, we find Higgs bound states that have binding energies lower than the bulk Higgs gap and are localized around the barrier. We analyze tunneling properties of the NG mode incident to both barriers to show that the latter barrier couples the Higgs bound states with the NG mode, leading to Fano resonance mediated by the bound states. Thanks to the universality of the underlying field theory, it is expected that Higgs bound states may be present also in other condensed-matter systems with a particle-hole symmetry and spontaneous breaking of a continuous symmetry, such as quantum dimer antiferromagnets, superconductors, and charge-density-wave materials.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.
The low-energy dynamics of many-body systems is governed by gapless modes whose properties are dictated by symmetry. Their existence follows from Goldstone's theorem, while their effective description at zero temperature is determined by the pattern of symmetry breaking. At finite temperature, an analogous role is played by hydrodynamics, which describes the universal behavior of many-body systems over long times and large distances. These principles are well understood for uniform symmetries, which act homogeneously in spacetime and lead to a correspondence between massless Goldstone modes and broken generators, as well as gapless hydrodynamic modes and conserved charges. However, this simple picture changes in the presence of nonuniform symmetries, whose generators do not commute with spacetime translations. The low-energy implications of these symmetries remain less understood, as they do not introduce additional gapless modes but instead constrain the dynamics of the existing degrees of freedom. In this thesis, we develop a unified framework for many-body systems with nonuniform symmetries and show that their effects can be understood in terms of additional fields that are not independent at low energies and can be eliminated, leading to kinematic constraints that reshape the infrared dynamics. In systems with spontaneous symmetry breaking, this mechanism modifies the effective theory and often softens the dispersion relations of the remaining modes. At finite temperature, it manifests in hydrodynamics as constraints on macroscopic currents. As a result, nonuniform symmetries give rise to qualitatively new physical phenomena, including modified spectra of collective excitations, exemplified by transverse Tkachenko oscillations in quantum vortex crystals, and unconventional transport phenomena, such as anomalously slow diffusion and softened sound modes in dipole-conserving systems.