Magnetic skyrmions exhibit unique dynamical properties. However, it is unclear whether the Brownian motion of skyrmions exhibits chiral properties in thermal equilibrium derived from the Magnus force. For example, when a skyrmion is driven by an electric current, the driving force is parallel to the current. However, its trajectory is bent by the Magnus force, which is proportional to the skyrmion number. Consequently, the skyrmion Hall effect is observed. Such chiral motion of skyrmions has been observed in the presence of a driving force. However, whether the Brownian motion of skyrmions exhibits chiral properties in thermal equilibrium remains unclear. (Note that the chiral property in this study is not that of skyrmions but of their motion.) In this study, we show that chiral properties play an essential role in the diffusion of skyrmions by performing the following two experiments:

Diffusion of skyrmions in one-dimensional and two-dimensional channels

The diffusion coefficient of skyrmions in one-dimensional narrow channels is a factor of 1.3–1.9 larger than that in two-dimensional films. This is because the confinement of skyrmions in one-dimensional channels suppresses their gyration.

Observation of angular momentum of diffusion in two-dimensional systems

We evaluated chiral properties by analyzing the diffusion of skyrmions using the spontaneous velocity–position correlation functions, i.e., <

**v**(*t*)・**x**(*t*)> and <(**v**(*t*) ×**x**(*t*))> [1], and obtained the finite value of <(_{z}**v**(*t*)×**x**(*t*))as a thermal average. This result indicates the existence of the off-diagonal term of the diffusion coefficient of the skyrmions' motion. In addition, the sign of the observed <(_{z}>**v**(*t*)×**x**(*t*))> is opposite to the sign of the simple theoretical prediction, i.e., the direction of rotation is opposite to that predicted by theory. This can be explained by considering the shallow potential fluctuation in the films. The orbit of skyrmions captured by the harmonic potential was a hypotrochoid, a combination of two circular motions, with thermal fluctuation. The small, high-frequency circular motion was the intrinsic gyration of skyrmions. In contrast, a low-frequency orbit that rotated along the edge of the potential exhibited reverse rotation. Because the camera's frame rate was set at 30 ms_{z}^{−}^{1}in the experiments, only the global rotation in the direction opposite to that of the local gyration was observed. The observation of the intrinsic gyration and mass of skyrmions under Brownian motion should be examined in the future [2].

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

[1] Y. Suzuki, S. Miki, Y. Imai, and E. Tamura, Phys. Lett. A, **413**, 127603 (2021), and references therein.

[2] I. Makhfudz, B. Krüger, and O. Tchernyshyov, Phys. Rev. Lett. **109**, 217201 (2012).

The Hamiltonian formulation of classical mechanics is an elegant formalism. It is characterized by the Hamiltonian function, which represents the total energy of the system, and a Poisson bracket acting on it to give the Hamilton’s equation of motion. From a geometrical perspective, these equations can be interpreted as the flow of a vector field in phase space, or the space of all possible states of the physical system, such that it always conserves the phase space volume, representing the conservation of energy.

Typically, the phase space is “two-dimensional,” in the sense that it is characterized by the positions and momenta only. But, can the formalism be generalized for a higher dimensional phase space? The physicist Yoichiro Nambu proposed such a formalism for a three-dimensional phase space with “Nambu brackets” replacing the Poisson bracket. However, constructing an algebraic framework analogous to the Poisson bracket proved to be difficult because the Jacobi identity, representing the closure property of Poisson bracket, could not be generalized for Nambu brackets.

In a recent research article, Prof. Naoki Sato from the University of Tokyo managed to evade this problem. Starting from a differential geometric approach, he constructed a framework for a three-dimensional phase space with generalized Poisson brackets characterized by anti-symmetric, third-order, contravariant tensors. His approach not only led to a generalized Jacobi identity that conserved the phase space volume, but also showed that it was a weaker condition than that represented by the substitute for Jacobi identity for Nambu brackets.

Such a generalization could have profound implications for theoretical physics, providing new ways of understanding the laws of physics and possibly leading to real-world applications.

]]>Fig. 1: Pair of Dirac cones realized in a quasi-two-dimensional organic compound, α-(BEDT-TTF)_{2}I_{3}. They are tilted and located in the two distinct positions in the momentum space (Brillouin zone). When an electric current is applied, there appears a nonequivalence between the paired Dirac cones. The blue color shows how the electrons occupy the states under a current. This figure is adapted from Fig. 1 of J. Phys. Soc. Jpn. **90**, 053704 (2021).

Extensive research has shown that these Dirac electrons in solids can carry a topological nature called the Berry curvature. This induces effects such as the anomalous Hall effect (that is, the Hall effect without a magnetic field). However, because each pair of Dirac cones (Fig. 1) gives the Berry curvature of the opposite sign, the anomalous effect is, unfortunately, exactly canceled. However, there exists a method to resolve this cancelation problem. If an electric current is applied, the equivalence between the paired Dirac cones is broken. Thus, the complete cancelation of the Berry curvature is disrupted. This type of current-induced phenomenon was proposed by Sodemann and Fu and has been observed in several materials. Recently, Osada and Kiswandhi extended this idea to thermoelectric phenomena and proposed a novel nonlinear anomalous Ettingshausen effect. This enabled a new approach to observe the topological nature using the thermoelectric properties. As an explicit model, Osada and Kiswandhi studied the organic conductor α-(BEDT-TTF)_{2}I_{3} and obtained several interesting results. In particular, they claim that the tilting, which is typical in the Dirac electron systems of organic conductors, plays an essential role in the nonlinear anomalous Ettingshausen effect. This study by Osada and Kiswandhi demonstrated an interesting method to confirm the topological nature of Dirac electrons in solids.

Superconductors comprise a condensate of electron pairs, known as Cooper pairs, capable of electricity conduction without energy loss. Conventional superconductivity involves the formation of electron pairs via the attractive interactions mediated by lattice vibrations. Excitons are the bound states generated via Coulomb attraction between electrons and holes, similar to that for the Cooper pairs in superconductivity. The ineffective screening of Coulomb interactions in narrow-gap semiconductors and semimetals is expected to induce the spontaneous condensation of excitons with a decrease in temperature. Consequently, the system becomes an excitonic insulator.

Superconductivity was initially observed in mercury in 1911; subsequently, various superconducting materials, including high-temperature superconductors, were discovered. Conversely, the theoretical proposal of excitonic insulators over 50 years ago was not subsequently validated by conclusive experimental evidence of the excitonic state.

Ta_{2}NiSe_{5} is a layered chalcogenide located near the semiconductor–semimetal boundary. Recently, this compound has attracted significant attention as a promising candidate for excitonic insulators. Furthermore, there is renewed interest in the formation of a gapped state driven by electron–lattice coupling. The applied pressure is an ideal controlling parameter to tune the electronic states. To the best of our knowledge, the present research was the first to report a high-pressure phase diagram for Ta_{2}NiSe_{5} encompassing the entire range i.e., from the semiconducting to semimetallic region. The investigations revealed pressure-induced superconductivity in the semimetallic phase.

A transition to another semimetal with a partial gap was observed in the pressure-induced semimetallic phase. This was accompanied by lattice distortion analogous to that occurring during excitonic transition in the low-pressure semiconducting phase. An increase in the carrier density facilitated effective screening of the Coulomb interactions between electrons and holes. Therefore, the gap characteristics changed from excitonic dominant to hybridization-gap dominant with an increase in the band overlap under an applied pressure. The results revealed the critical role of electron–lattice coupling in the occurrence of superconductivity and excitonic transition in the low-pressure phase.(Written by Kazuyuki Matsubayashi on behalf of all authors.)

]]>Spintronics is a field that addresses science and engineering at the intersection of magnetism and transport phenomena in small structures and devices. By focusing on the electron’s spin rather than its charge, spintronics enables faster and more energy-efficient information and communication technologies.

Ferrimagnetic materials stand out among the spintronics community. Discovered in 1948 by Louis Néel, ferrimagnetism occurs when a material is composed of atoms with opposing magnetic moments of unequal magnitude, resulting in net spontaneous magnetization. Ferrimagnetic materials are attractive because they combine the controllability of ferromagnets and the fast dynamics of antiferromagnets.

The latest Special Topics edition of the *Journal of the Physical Society of Japan* presents articles covering a broad spectrum of spintronics research on ferrimagnetic materials.

On the theoretical side, Barker and Atxitia review computer modelling and simulation techniques for the magnetic excitations of complex magnets at finite temperatures. Nakata and Kim review a formalism for spin transport in ferrimagnets involving the topological Hall effect.

On the experimental side, Nambu and Shamoto study single crystals of yttrium iron garnet using polarized and unpolarized inelastic neutron scattering, giving unprecedented insights into the collective precessional motion called spin waves or magnons in this complex material.

Chudo and coworkers focus on the angular momentum compensation temperature measured via nuclear magnetic resonance and the Barnett effect, which is the magnetization induced by mechanical rotation.

Sheng and coworkers report the generation of fast and chiral spin waves in yttrium iron-garnet nanostructures by microwaves and electric currents.

Zhou and coworkers focus on the current-induced fast magnetization dynamics near the magnetic compensation point of both insulating and conducting ferromagnets that enable fast domain wall motion.

Avci emphasizes the electrical control of magnetic excitations in ferrimagnetic insulators by heavy metal contacts.

Stupakiewicz and Satoh discuss the ultrafast magneto–optical response of ferrimagnetic insulators, while Iihama and coworkers discuss the optical magnetization switching in magnetic metals close to compensation.

Suemasu and coworkers report progress in the fabrication of rare-earth free ferrimagnetic manganese nitride films.

Tanabe and Ohe report enhanced electric voltages induced by magnetization dynamics in nearly compensated ferrimagnetic gadolinium–iron–cobalt alloys.

The impressive progress reported by the global research community in this special issue raises the hope that ferrimagnet-based spintronics will contribute to a sustainable information society for the benefit of all mankind.

]]>Magnetic skyrmions with swirling topological spin textures are of significant interest owing to their peculiar magnetic, transport, and optical properties. In magnetic materials, skyrmions usually appear in a periodically ordered state, which is referred to as the skyrmion crystal (SkX). The SkX can be considered as a multiple-*q*
order described by the superposition of multiple spiral waves. The SkX was originally discovered in 2009 in *d*-electron compounds like MnSi, where the Dzyaloshinskii–Moriya interaction plays an important role. Recently, SkXs have been observed in centrosymmetric magnets via other mechanisms, such as frustrated exchange interactions and/or multiple-spin interactions in itinerant magnets. The resulting difference due to the individual origins appears in the magnetic modulation period of the SkXs, with the original mechanism usually leading to much longer magnetic periods than those of the latter ones. Specifically, a small skyrmion induces a large emergent magnetic field, and hence engineering small skyrmions is relevant for future spintronics applications, as they may potentially aid in realizing energy-efficient devices based on high density topological objects.

The itinerant chiral antiferromagnet EuPtSi is the first *f*-electron compound with a noncentrosymmetric lattice structure found to host a SkX with a small magnetic period. The origin of the SkX in EuPtSi is not explained by the conventional mechanism based on the Dzyaloshinskii–Moriya interaction, as the magnetic periods and the magnetic-field-direction dependencies of the SkXs are different from those found in other noncentrosymmetric magnets. Therefore, we need to consider other mechanisms that have not yet been theoretically clarified. Understanding the origin of short-period SkXs in noncentrosymmetric itinerant magnets found in EuPtSi is crucial in enabling small skyrmion engineering for practical applications.

In this study, we theoretically investigated the origin of SkXs in *f*-electron compounds by constructing a new model based on EuPtSi. We discovered the following two important elements in inducing a field-direction sensitive SkX with a small magnetic period: (1) the multiple-*q* superpositions of the spirals with low-symmetric ordering vectors and (2) the synergy between the long-range Dzyaloshinskii–Moriya interaction arising from the spin-orbit coupling and the biquadratic interaction arising from the spin-charge coupling in itinerant magnets. We demonstrated that the theoretical model which satisfies the two conditions accurately describes the SkX physics in EuPtSi through unbiased annealing simulations. Our study provides a reference for both engineering short-period SkXs and for exploring further skyrmion-hosting materials in noncentrosymmetric itinerant magnets.

(Written by Satoru Hayami on behalf of all authors.)

]]>At atmospheric pressure, the magnetic properties can be easily measured using commercial instruments such as the magnetic property measurement system (MPMS®) from Quantum Design (USA). To measure the magnetization under a high-pressure custom-designed pressure cell, a clear fitting to the available sample space of an MPMS is required. The typical available sample space in an MPMS has a small diameter of approximately 9.0 mm.

Currently, most high-pressure cells are self-clamp-type piston cylinder apparatus with a relatively large sample space with a maximum pressure range of approximately 3.0 GPa. Furthermore, for magnetization measurements over 3.0 GPa, a ceramic anvil-type pressure cell is used [1]. However, the available sample space that can be used in these opposed-anvil-type devices is severely reduced. Therefore, the measurable materials were only ferromagnets and superconductors, showing a large magnetization signal.

To overcome these bottlenecks, Hiraoka et al. newly developed an opposed-anvil-type pressure cell that combines a large sample volume and is capable of generating higher pressures up to 6.3 GPa. Figure 1 demonstrates a conceptual schematic of a high-pressure cell with a clamp cell body. In addition, using this pressure cell, they successfully demonstrated the detection of weak volume susceptibilities of approximately 10^{-4}, which is sufficient to resolve the magnetism of a paramagnet. Measurements of such high-quality magnetic signals were possible using a technique called truncated singular value decomposition linear algebra.

Fig. 1: Schematic of a high-pressure cell that can be used in a commercial SQUID magnetometer (MPMS, Quantum Design). (a) Cross-section of the cell assembly. Gray and brown indicate the two species of materials, WC (binder less) alloy and Cu–Be alloy, respectively. Zoomed in view of the cupped WC (binder less) anvils (two culet sizes: Φ1.6 mm and Φ1.2 mm) and conical shaped Cu–Be alloy gasket.

Successful implementation of this new technique, with measurements of paramagnetic susceptibilities under pressure revealed a drastic improvement in the magnetic signal. With the advent of such high-precision magnetization measurements under high pressure, it is expected to clarify several unsolved problems, such as magnetic quantum critical phenomena.

[1] N. Tateiwa, Y. Haga, T. D. Matsuda, Z. Fisk, S. Ikeda, and H. Kobayashi, Rev. Sci. Instrum. **84**, 046105 (2013).

Elementary quantum physics usually tackles quantum systems that are energy-conserving and described well by a Hermitian Hamiltonian. In reality, however, many quantum systems are dissipative in nature and can only be described effectively using a non-Hermitian Hamiltonian. Consequently, non-Hermitian quantum physics has garnered considerable attention from researchers across diverse subfields of physics.

In condensed matter physics, for instance, many-body systems are a widely researched topic. The electrons in these systems interact strongly with one another, giving rise to quantum states that cannot be described by knowing the equation of motion of a single electron. Recent theoretical studies have now revealed that many-body systems can be dissipative with parity-time (or PT)-symmetry and show unique exotic phases with no counterpart in conservative systems.

Against this backdrop, a team of physicists from Japan have recently experimentally realized a PT-symmetric, non-Hermitian, many-body system from ultracold Ytterbium atoms trapped in an optical lattice formed by interfering counter-propagating laser beams. In their study, the team experimentally investigated the ideal conditions for one-and two-body dissipation and developed methods to measure and control relative phases between on- and off-resonant lattices for PT symmetry. Additionally, they constructed a new theoretical framework to predict the appearance of interesting loss dynamics.

The experimental system developed by the team could serve as a future platform for investigating novel and uniquely non-Hermitian quantum phenomena as well as extend the non-Hermitian perspective to other quantum systems, potentially revolutionizing our understanding of many-body systems as a whole.

]]>Following the discovery of topological matter—or materials characterized by boundary conduction properties that are “topologically protected” (immune to continuous deformations or defects)—there has been a renewed interest in strongly correlated systems—many-body systems showing a complicated interplay between spin, charge, lattice, and orbital degrees of freedom—and their properties, from a topological perspective.

In addition to boundary conduction, topological semi-metals are known to exhibit “topologically protected band touching” in which the valence and conduction bands touch each other at discrete points or along closed loops.

Interestingly, such band touching has recently been extended to non-Hermitian systems, where it is called “exceptional band touching”. Exceptional band touching occurs for *non-real* energy values, implying that the underlying Hamiltonian is *non-Hermitian*. From a physical viewpoint, the non-Hermiticity in equilibrium systems is a manifestation of ephemeral “quasiparticles” representing the quanta of collective excitations.

Thus, in this paper, we take stock of the recent advances in non-Hermitian correlated systems in equilibrium, focusing on exceptional band touching in heavy-fermion systems (materials where the effective mass of the quasiparticles is much heavier than free electrons) and correlated systems with chiral symmetry, i.e., systems with the combined symmetry of time-reversal and particle-hole symmetries (electrons and holes feel the same interaction).

We start by demonstrating numerically, using a technique called “dynamical mean-field theory”, that exceptional band touching points appear in a correlated system represented by a non-Hermitian Hamiltonian due to finite lifetimes of quasiparticles. We then move on to correlated systems with chiral symmetry and demonstrate novel structures of exceptional band touching, namely *“symmetry protected exceptional rings”* (SPERs) and *“symmetry protected exceptional surfaces”* (SPESs) in two and three dimensions, respectively. Finally, we address a ten-fold way classification of exceptional band touching for a single-particle spectrum, taking into account chiral symmetry, PT symmetry (product of parity and time-reversal symmetry), and CP symmetry (product of particle-hole and parity symmetry).

Our study shows that non-Hermitian properties, typically expected in non-equilibrium systems, can arise even under equilibrium for strongly correlated systems. Moreover, the richer topological physics provided by non-Hermiticity could help solve some of the standing puzzles in condensed matter physics, such as the pseudo-gap of cuprate superconductors and the unusual quantum oscillations in samarium hexaboride (SmB_{6}).

However, we usually insist that these operators give back real values, since, in the real world, we associate measurable quantities with real numbers. Consequently, physical operators are normally restricted to being what are called “Hermitian operators”. However, in my article I have explored two scenarios where this dogma can be questioned.

One of the most important operators in quantum mechanics is the Hamiltonian operator, which corresponds to physical energy. It may seem intuitively obvious that the Hamiltonian operator must be Hermitian, but this is, in fact, only true so long as energy is conserved. Now while the energy of the *entire* universe is certainly conserved, it may not be necessarily so for a *part* of it, such as a radioactive nuclide. This system loses energy to the outside world whenever it emits an alpha or a beta particle and therefore, does not conserve energy. Such an *open* system can be described by an effective non-Hermitian Hamiltonian that yields complex energy values corresponding to short-lived “resonant states.” While the eigenfunctions of these states are not normalizable, it simply means that the states extend over the entire universe.

Another instance is where the theory of parity-time (PT) symmetry (symmetry under the combined operations of spatial inversion and time reversal) considers the Hamiltonian of the entire universe as non-Hermitian but operating in a parameter space where its eigenvalues are exclusively real. For example, a system of two closed-shell atoms can be described using a non-Hermitian Hamiltonian that is PT symmetric and yields energy eigenvalues that can be rendered real by parameter tuning. This shows that Hermiticity is only a sufficient condition for real values, not a necessary one.

The non-Hermitian formulation has since been picked up and applied by researchers in various fields, most notably in topological matter and many-body systems. Just as generalizing to complex numbers can often help describe physical systems more elegantly, a non-Hermitian generalization of quantum mechanics can help broaden our perspective of quantum systems.

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