In the past two decades, physicists have made significant progress in the field of cosmology, thanks to precise measurements of the properties of the cosmic microwave background (CMB) and large-scale structure (LSS) of the Universe. The prevailing Λ-cold-dark-matter model, which describes the late-time Universe, has been successful in explaining the interplay of cosmic expansion and structure formation. However, it is not without its limitations. One notable challenge is the fine-tuning required to accommodate the observed value of the cosmological constant, necessitating new developments. Meanwhile, physicists have turned their attention to modified theories of gravity.

By modifying Einstein's theory of general relativity (GR) on cosmological scales, they aim to uncover alternative explanations for cosmic acceleration. While GR has proven its accuracy for small-scale tests, its validity on cosmological scales remains uncertain. Thus, probing gravity at these large scales is crucial for understanding the late-time evolution of the Universe.

In this study, we provide a comprehensive overview of various modified theories of gravity, including scalar-tensor theories in the Horndeski and DHOST family, massive gravity/bigravity, vector-tensor theories, and more of what we have theoretically known. We then highlight cosmological observables of the CMB and LSS and present concrete analytical predictions from well-motivated theories to shed light on their potential deviations from GR.

To further theoretical predictions, we introduce computational tools, such as the CMB Boltzmann code for DHOST theory and emulator codes for LSS observations. Such tools are instrumental in assessing the behavior of gravity models in the nonlinear regime.

Lastly, we highlight the scope of future cosmological observations that hold the key to unraveling the mysteries of gravity. Ground-based CMB experiments, such as the Simons Observatory and the CMB Stage 4 observatory, along with space missions like the LiteBIRD satellite, are anticipated to provide high-quality data for testing gravity at large scales. Additionally, upcoming observations in LSS, including the Subaru Prime Focus Spectroscopy (PFS), the Vera C. Rubin Observatory (LSST), and the Nancy Grace Roman Space Telescope, among others, will offer valuable insights into the evolution of cosmic structures.

By prioritizing well-developed models of gravity like the Horndeski theory, which is the highest-priority model for formulating an observational strategy, physicists can refine their theoretical predictions, address remaining challenges, and ultimately test these models with precise cosmological observations.

Overall, this work emphasizes the need for collaborative efforts encompassing theoretical advancements, forthcoming observations, and computational implementations to advance our understanding of gravity at cosmological scales.

Indeed, by unveiling the mysteries of the cosmos, we inch closer to comprehending the fundamental nature of our vast and awe-inspiring Universe.

In crystals lacking space inversion symmetry, antisymmetric exchange interactions can arise, which are generally expressed in the form of ** D** • (

The experimental observation of the magnetic helicity is challenging. Although polarized neutron diffraction is a powerful technique, it requires a delicate instrumental setup. In addition, it is unsuitable for samples that contain neutron-absorbing elements such as Eu and Gd. In this study, we employed resonant X-ray diffraction using a synchrotron radiation photon source. We used a diamond phase-retarder system inserted in the incident beam path to manipulate the incident polarization to left- and right-handed circularly polarized states, which have direct sensitivity to the magnetic helicity through different scattering cross-sections for the opposite sense of rotation. We applied a phase-retarder scan to observe the cycloidal ordering in the noncentrosymmetric magnet EuIrGe_{3}, with four mirror reflection planes including the four-fold c-axis.

The cycloidal ordering is characterized by a propagation vector (δ, δ, 0.8), where δ=0.012 at 2.0 K. The Fourier component for this structure is either (1, 1, *i*
√2) or (1, 1, - *i* √2), where the sign is related to the cycloidal helicity. Furthermore, there arise four magnetic domains described by (δ, δ, 0.8), (-δ, δ, 0.8), (-δ, -δ, 0.8), and (δ, -δ, 0.8). The helicities of all the four domains were measured.

The helicities perfectly reflected the C_{4v} symmetry of the crystal structure. All four Fourier components were related by the 90° rotations and by the mirror reflections. This result clearly shows that the helicity of the cycloid is uniquely selected by the antisymmetric interactions. We believe that this method can be widely applied to the helicity measurements of various types of spiral magnetic systems.

(Written by T. Matsumura on behalf of all the authors.)

]]>Polyexcitons are composites of multiple electron-hole pairs. In strongly photoexcited intrinsic semiconductors or type II semiconductor heterostructures, an electron in conduction bands and a hole in valence bands naturally form an exciton, and further, a pair of excitons coalesces into a biexciton. These composites have been studied in an analogy to hydrogen atoms or molecules. A common expectation is that excitons cannot form triexcitons, similar to hydrogen atoms that cannot form trimers. However, in multi-valley semiconductors, more than two excitons can be bound, because electrons and holes acquire the additional valley degrees of freedom that, combined with the spins, allow more than two electrons and holes to occupy the same position. This possibility was first identified by Wang and Kittel in 1972 and has been experimentally observed in pure bulk samples of silicon and diamond as the photoluminescence (PL) peaks almost equally spaced at energy intervals.

We consider the two-dimensional simplified model of the type II double-bilayer graphene heterostructures. The electrons and holes stay on different layers separated by a distance and interact with each other via Coulomb potentials. They also acquire four internal degrees of freedom because they have two spin and two valley degrees of freedom. This implies that our system can afford triexcitons and tetraexcitons, and that the quantum diffusion Monte Carlo simulation is free of the negative sign problem and can precisely evaluate the energies.

Notably, the separation energy required to pull out one exciton from the polyexciton grows almost exactly linearly with the exciton number. This behavior resembles the aforementioned PL peaks, indicating that the underlying physics is common between two and three dimensions. This further implies that all pairs of excitons inside the polyexciton are energetically bound by “chemical bonds” of equal strength, irrespective of the bound exciton numbers or the details of the system, for example, the interlayer distances.

(Written by K. Asano on behalf of all the authors)

]]>In type II superconductors under a magnetic field, the magnetic field penetrates as a quantized flux accompanied by the surrounding rotating currents (magnetic vortex). Because the motion of vortices dominates the electromagnetic behavior of superconductors, understanding their kinetics and dynamics is extremely important from scientific and application perspectives. In fact, the kinetics of vortices, flux flow, has been the subject of extensive long-term experimental and theoretical studies. However, our understanding of flux flow, even at the fundamental level, was far from satisfactory until recently. The key component of this problem is the electronic structure of the central part of the vortex (core), where quantized levels of non-superconducting carriers (quasiparticles) are formed. In most superconductors, the quantized nature is almost invisible because of the quantized level spread (dirty limit), which generates ordinary longitudinal flux-flow resistivity. On the other hand, for vortices with clean cores, where the aforementioned quantized nature is prominent, we expect a large Hall effect with a Hall angle of almost π/2. Thus, direct measurement of the Hall angle by the flux flow provides important fundamental information about the kinetics of the vortices. In particular, the study of materials with clean-core vortices is promising, as novel quantum effects are expected.

To experimentally discuss the flux flow, we need to work at high frequencies and low temperatures. However, the Hall effect measurement in microwaves, for instance, in cryogenic circumstances for highly conductive materials, is challenging. We recently developed a cross-shaped bimodal cavity technique to measure the microwave Hall effect in highly conductive materials and observed a large Hall angle of vortex motion in high-Tc cuprate superconductors, in which we expected the quantized nature of the core to be visible. Although our results are surprising and puzzling compared to those of previous flux-flow studies, which used a simpler longitudinal resistivity measurement (effective viscous drag coefficient measurement), it is extremely exciting to see if such a large flux-flow Hall angle is observed in other superconductors that are expected to have a clean vortex core.

In this study, we measured the flux-flow Hall effect in FeSe, which meets the above expectation. However, these results were unexpected. That is, we found that the tangent of the flux-flow Hall angle at low temperatures was approximately 0.5, which is equal to or smaller than that previously evaluated by effective viscous drag coefficient measurements. This contrasts with the results obtained for cuprate superconductors. We focus on the multiband nature of FeSe superconductivity and successfully explain the result in terms of the cancellation of the forces acting on the vortices by the hole and electron bands, which was already considered by Nicolai Kopin in microscopic theories. Thus, our observation of the flux-flow Hall effect in FeSe is a novel characteristic of multiband superconductors with electron and hole bands. This should become an important clue for controlling vortex motion through material design.

(Written by R. Ogawa on behalf of all the authors)

]]>The precise measurement of physical quantities based on the principles of quantum mechanics is called quantum sensing. A remarkable example is magnetic field measurements using nitrogen-vacancy (NV) centers in diamonds as quantum sensors. NV centers are a type of lattice defect in diamond crystals, in which a nitrogen atom and an atomic vacancy replace two adjacent carbon atoms. Among the many lattice defects, NV centers have unique properties that make them suitable for quantum sensing and have been applied to physical property measurements. NV centers are renowned for their high sensitivity. However, to take full advantage of their features, the accuracy (precision) and sensitivity of the measured physical quantities must be improved.

Optically detected magnetic resonance (ODMR) of NV centers measures the red photoluminescence (PL) intensity as a function of microwave frequency when green light is injected. The magnetic field can be determined from the magnetic resonance frequency of the ODMR spectrum. An unexpected phenomenon was recently reported: the ODMR spectra of NV centers in nanodiamonds (NDs) change with the optical power of the excitation light [M. Fujiwara et al., Phys. Rev. Res. 2, 043415 (2020)]. The spectral change is not drastic; hence, the phenomenon has been overlooked thus far. However, it can be problematic because it reduces the accuracy of low magnetic field measurements.

We systematically investigated the optical power dependence of ODMR spectra using NV ensembles in NDs and a bulk diamond single crystal. The experimental ODMR spectrum at zero magnetic field showed a slight splitting of the resonance frequency (Δ) owing to crystal distortion, nuclear spins near the NV centers, and so on. The Zeeman effect increased the frequency splitting in a finite magnetic field, making precise magnetic field measurements possible. We observed that Δ depended on the optical power in both NDs and bulk samples: it decayed exponentially with increasing optical power and saturated at a certain point. The decay amplitude depended on the samples, whereas the optical power at which the decay was saturated was almost independent of the samples.

The mechanism causing this unexpected phenomenon is most likely related to crystal distortion or a change in the charge state near the NV centers induced by photoexcitation. The present findings indicate that accuracy degradation can be suppressed by injecting an excitation light with an optical power above a certain value. Our achievement shows that there are unexplained phenomena even in the NV center, which has been studied for many years, and, at the same time, provides valuable guidelines for accurate magnetic field measurements using diamond quantum sensors.

(Written by Kensuke Kobayashi and Kento Sasaki on behalf of all the authors)

]]>The Toda lattice discovered by Morikazu Toda in 1967 is a one-dimensional lattice model with an exponential-type interaction. Despite the nonlinear interaction, the Toda lattice is completely integrable. The Lax equation introduced by Flaschka in 1974, which is a matrix format of the canonical equations of motion, has led to the proof of the integrability. In 1976, Date and Tanaka obtained generic periodic solutions of the Toda lattice using the eigenvalues of the Lax matrix.

On the other hand, the Thouless pump introduced by Thouless in 1983 is a quantum pump that transports particles in periodically and adiabatically moving potentials. The Thouless pump is often called the topological pump, because the number of transported particles in one period is quantized, given by a topological invariant known as the Chern number.

In our study, we shed light on the topological properties of the periodic Toda lattice. At first glance, the Toda lattice appears to be unrelated to the Thouless pump. However, in 1993, Hatsugai has established the theory of the bulk-edge correspondence (BEC) in topological phenomena in condensed matter physics, based on the techniques developed by Date and Tanaka. Conversely, in the study, we have reconsidered the Lax eigenvalue equation of the Toda lattice from the viewpoint of the BEC. Namely, we have first confirmed that the boundary states that play a crucial role in obtaining the exact solutions of the Toda lattice, have a topological origin. In the spirit of the BEC, we have subsequently shown that the bulk eigenfunctions of the Lax matrix on the periodic Toda lattice yield nontrivial Chern numbers. In particular, the cnoidal wave solution derived by Toda exhibits a Chern number of -1. This implies that the periodic Toda lattice belongs to the same topological class as the Thouless pump.

This result shows that a topological viewpoint can be useful in the study of solvable models of nonlinear waves. As is widely known, solitons are not topologically protected but are due to the balance between nonlinearity and dispersion in the wave propagation. Our result suggests that the topological properties of the Toda lattice may be involved in the stability of solitons. Investigating whether other solvable models of nonlinear waves have the same topological properties will be noteworthy.

(Written by K. Sato and T. Fukui)

]]>Topological materials have attracted great interest because of unique electronic states resulting from their linear energy dispersion. A large number of topological materials have been synthesized and discovered. Among them, some topological materials have unique band structures. The crossing points of the bands with linear dispersion, called Dirac points, extend in k-space and form nodal line loops. One of the most intriguing properties of such topological nodal line materials is the formation of surface states on certain crystal surfaces. Till now, extensive efforts have been made to elucidate the surface states of topological materials.

The nodal line material NaAlSi has a tetragonal unit cell composed of Al-Si tetrahedral layers and Na in between. Conduction bands are formed by the *s* and *p* orbitals of Al and Si. In spite of the low charge carriers (small Fermi surfaces), NaAlSi shows bulk superconductivity with a rather high *T*_{c} of 7 K.

Magnetic torque is defined as the outer product of the magnetic field and magnetization. We have performed magnetic torque experiments on single crystals of NaAlSi using a micro-cantilever, which is a powerful technique for detecting anisotropic magnetic properties. Owing to its layered structure, NaAlSi has anisotropic critical magnetic fields. Thus, it is possible to precisely measure the diamagnetic signal arising from the superconductivity even for a tiny single crystal.

In torque measurements below *T*_{c}, very unique diamagnetic torque signals have been observed in NaAlSi, i.e., a large broad signal arising from the bulk superconductivity and a small sharp signal observed in magnetic fields nearly parallel to the layers. Owing to the high anisotropy of the critical field, the sharp signal is ascribed to the 2D superconductivity in the Na-Si layers (perpendicular to the *c*-axis). The thickness of the 2D superconductivity is only several times the crystal *c*-axis (0.736 nm). As band calculation predicts the presence of a peculiar surface state on the crystal (001) plane, a possible explanation for the torque data is the coexistence of the bulk superconductivity and surface conductivity on the (001) plane. Further studies are necessary to understand the relationship between both superconductivities and the mechanism of 2D superconductivity.

(Written by S. Uji on behalf of all authors)

]]>Superconductivity is the property of certain materials to conduct electricity with no energy losses when cooled below a critical temperature. The discovery of high-temperature copper oxide (or cuprate) superconductors, whose critical temperatures reach about 160 K, has unlocked various potential applications.

However, the mechanisms driving the high-temperature superconductivity are not fully understood, primarily due to challenges in dealing with the electron–electron interactions within the materials. Spectroscopic experiments and nonperturbative theories have been used to explain the high-temperature superconductivity in cuprates.

Recently, in a new study published in the *Journal of the Physical Society of Japan* , researcher Shiro Sakai from the Center for Emergent Matter Science at RIKEN, Japan, reviewed the results of these studies, highlighting that superconductivity in cuprates can be explained by the existence of a self-energy singularity.

The singular self-energy, described as a nonperturbative effect, arises from strong correlations between electrons. Referred to as the ‘missing link,’ the self-energy singularity offers a unifying explanation for the various phases observed in cuprates as well as for a number of experimental observations that have been considered ‘anomalies’ and remain poorly understood.

Cuprates exhibit superconductive properties only within a doping range of 5–25%. At lower doping, hole-doped cuprates behave as Mott insulators, while at higher doping, they show a Fermi-liquid behavior. Additionally, in the underdoped regions lying above the critical temperature, an enigmatic pseudogap state emerges.

It is known that the singular self-energy exists in the Mott insulating state, where it generates a spectral gap. On top of that, nonperturbative calculations have revealed its presence in the finite-doping region as well, where it enhances the superconductivity transition temperature and generates the pseudogap above it.

Therefore, the self-energy singularity is at the origin of the high-temperature superconductivity, the pseudogap, and the Mott insulator phases in cuprates. This review thus sheds light on the complex nature of cuprates, paving the way for the design and discovery of new superconductors with higher critical temperatures.

The quark model, as we know, provides a classification scheme for hadrons, composite subatomic particles made of two or more quarks held together by the strong interaction. While most hadrons are well understood in terms of the quark model, some like the so-called “exotic hadrons” are an exception. These hadrons appear as resonances arising from nonperturbative quantum chromodynamic (QCD) interactions. To understand their properties and internal structure, theoretical studies using first-principles lattice QCD are, therefore, necessary.

In this study, as a preliminary step towards understanding exotic hadrons in lattice QCD, we investigated decuplet baryons (such as Δ and Ω). While they are well described as 3-quark states, we studied them as meson–baryon degrees of freedom in the scattering theory determined by the lattice QCD calculation. Our method has the advantage that it can be applied to the exotic hadrons. In particular, we focused on why Ω appeared as a stable particle below the Ξ*K* threshold while Δ becomes a resonance from the meson–baryon picture. To reduce the numerical cost, we used heavy quark masses, where Δ and Ω are stable particles. In this case, the difference of the decay properties between the two baryons is encoded in the inequality m* _{N}*
+ m

We analyzed Δ and Ω baryons in the HAL QCD method, which helped us extract the meson–baryon interaction potentials directly in lattice QCD. In turn, we obtained the scattering amplitudes from potentials by solving the Schrödinger equations in an infinite volume. This enabled us to study Δ and Ω as bound states of Nπ and Ξ*K* systems.

In our calculations, the binding energies determine the masses of Δ and Ω baryons, while the root-mean-square distances of the bound states qualitatively provides an estimate of the size of these baryons.

Upon extracting the interaction potentials, we found that the Ξ*K* system has a weaker attraction compared to the Nπ system. However, the binding energy was larger for Ω than for the Δ baryon, manifesting in the form of the inequality m* _{N}* + m

The calculations further revealed that the root-mean-squared distances of the bound states were small. This finding indicates tight binding states in Δ and Ω, where the two hadrons could essentially be considered as composite states of three quarks.

Our analyses of binding energies agreed with the results obtained from temporal two-point functions, indicating that our calculation method works well.

In summary, these findings shed light on the internal structure and properties of two decuplet baryons, opening doors to a better understanding of hadron spectroscopy including exotic hadrons, hadron interactions and, ultimately, the mysteries of our universe.

]]>A solid crystal contains an innumerable number of electrons which give rise to various interesting and functional properties.

The geometric nature of electrons in solids, known as Bloch electrons, provides novel nonlinear and nonequilibrium phenomena, making its understanding crucial. Therefore, a group of researchers from Japan has recently reviewed the latest developments in the research of such phenomena in solids, with a focus on their geometrical aspects.

The geometrical phase of Bloch electrons is quantified by a parameter called the Berry connection, which measures the coordinate of a Bloch state located away from the center of the unit cell. Engineering the geometrical phase of electrons in crystals generates shift current, a bulk photovoltaic effect and nonreciprocal current, a nonlinear transport phenomenon.

While the Berry connection describes the nonlinear responses of electrons in bulk crystals, the nonequilibrium phenomena can be accounted for using the Floquet theory. It explains the time-evolution of an electron in an oscillating electromagnetic field, characteristic of periodically-driven systems.

The researchers highlight that by illuminating intense laser light, it is possible to drastically change the geometric properties of electrons in trivial materials to develop materials with interesting properties, such as topological insulators, magnets, and superconductors.

Furthermore, geometric nonlinear phenomena in inversion symmetry-broken materials have potential applications in several areas, such as solar panels made of bulk crystals, highly efficient photodetectors, and novel diodes. Also, light induced topological superconductivity could provide a new platform for quantum computing.

In summary, understanding the geometric aspects of nonlinear optical phenomena in crystals would facilitate advanced materials with novel applications.