The electronic local density of states of solids, if normalized correctly, represents the probability density that the electron at a specific position has a particular energy. Because this probability density can vary in space in disordered systems, we propose that one can either treat the energy as a random variable and position as an external parameter to construct a real space Fisher information matrix, or treat the position as a random variable and energy as an external parameter to construct an energy space Fisher information, both quantify the variation of local density of states caused by the disorder. The corresponding Cramér-Rao bounds in these two scenarios set a limit on the energy variance and the position variance of electrons, respectively, pointing to new interpretations of STM measurements. Our formalism thus bring the notion of information geometry into STM measurements, as demonstrated explicitly by lattice models of metals and topological insulators.

Space-time singularities represent points where the local structure of a wave field becomes locally undefined, and are central to phenomena ranging from fluid dynamics to cosmology. In optics, spatiotemporal optical vortices (STOVs) provide a versatile setting in which to study such singularities, but their controlled manipulation during nonlinear frequency conversion has remained elusive. Here we show that the evolution of STOV singularities in second-harmonic generation can be continuously engineered through spatial Gouy-phase control. In a 4f system, translating a thin nonlinear crystal about the Fourier plane imposes a tunable Gouy phase that governs the orientation, splitting, and trajectories of singularities in the second-harmonic field. A Hermite-Gaussian modal description identifies Gouy-phase-dependent mechanism underlying these dynamics, in agreement with spatiotemporal interferometry measurements. Beyond integer-order inputs, fractional-order inputs (l=0.5) result in a distinct regime in which Gouy-phase bias drives topological reconfiguration and directional energy redistribution. Our results establish the Gouy phase as a simple yet powerful control knob for space-time singularities, opening new opportunities for structured-light engineering in nonlinear and ultrafast photonics.

A version of quantum Cramér-Rao bound dictates that the covariance of any set of operators is bounded by a product of the derivatives of expectation values and the inverse of quantum metric. We elaborate that because quantum metric itself is the covariance of the generators of translation in the parameter space, quantum metric in any dimension is bounded by a product of itself and Berry curvature. The generator formalism further indicates that the bound is equivalent to a multi-observable uncertainty relation, which in the two-operator case recovers the Robertson-Schrödinger uncertainty relation. The momentum space quantum metric and spin operators of three-dimensional topological insulators under magnetic field are used to demonstrate the validity of the three-operator version of these bounds.

Altermagnets hold great potential for spintronic applications, yet their intrinsic spin dynamics and associated transport properties remain largely unexplored. Here, we investigate spin-resolved quantum transport in a multiterminal setup based on a d-wave altermagnet. It is found that the altermagnetic spin splitting in momentum space induces an interesting spin precession in two-dimensional real space, giving rise to characteristic spin patterns. This altermagnetic spin precession manifests as a spatial modulation of the transverse Hall-like voltage, whose oscillation period provides a direct measure of the spin-splitting strength. When the altermagnetism is electrically tunable, the proposed setup functions as a prototype for a highly efficient spin transistor. The key physical effects are shown to be robust against dephasing and crystalline warping. Our Letter not only identifies a fingerprint signature of altermagnets, offering a direct probe of the altermagnetic spin splitting, but also represents an important step toward bridging their fundamental physics with practical spintronic applications.

Non-Hermitian physics - characterized by complex band spectra - established a new paradigm in condensed matter systems and metamaterials. Combined to nonreciprocity, non-Hermitian gain can be arbitrarily injected in a system to manipulate its valley dynamics, leading to phenomena such as the amplified topological whispering gallery modes for acoustic lasing. Here, we reveal that the coupling loss can also be used to manipulate valley degrees of freedom in a phononic metamaterial. We theoretically and experimentally show three distinct effects induced by valley nonreciprocity, including unidirectional transport of valley bulk states that functions as a valley filter, valley-dependent skin effects, where bulk states from different valleys localize at opposite boundaries, and valley-projected edge states with boundary-dependent lifetimes that leads to an anomalous beam routing. Owing to the simple control over losses demonstrated here, our findings shed light on the interplay between non-Hermitian and valley physics, providing a route to applications of valley-based devices.

Terahertz (THz) structured fields, such as vortex beams carrying orbital angular momentum, Bessel beams, and vector beams, support mode multiplexing, extended depth of focus, and improved scattering tolerance in communication, sensing, and imaging. Metasurfaces, consisting of ultrathin subwavelength arrays, employ metallic or all-dielectric building blocks to enable wavefront manipulation through geometric-phase or resonance mechanisms. Building on these foundations, subsequent research has sought to further enhance functionality and efficiency by developing multilayer configurations that integrate multiple operations and support more complex modulation. In parallel, tunable materials – such as phase-change compounds, liquid crystals, and graphene, whose properties respond to external physical stimuli – have been incorporated to dynamically control optical responses. These approaches enable planar, programmable manipulation of phase, amplitude, and polarization in the THz regime while reducing losses and device footprint, thereby providing reconfigurability beyond that of conventional bulk optics. The chapter summarizes recent advances in THz metasurfaces for structured beams along four themes: high-efficiency beam-shaping mechanisms; control strategies using tunable materials and reconfigurable architectures; multifunctional integration incorporating frequency, polarization, and mode multiplexing; and applications in communication, imaging, and high-field emission. We discuss the future directions for scalable, tunable, and efficient platforms to facilitate chip-to-free-space interfaces, metrology, and manufacturable implementation. Two noteworthy areas of interest are ferroelectric nematic liquid-crystal platforms and spatiotemporally structured THz waves, which are also discussed.

Valley, as a ubiquitous degree of freedom in lattices, has found wide applications in both electronic and classical-wave devices in recent years. However, achieving valley-polarized states, a prerequisite for valley-based operations, still remains challenging. Here, we propose and experimentally demonstrate a universal non-Hermitian mechanism for valley filtering using only uniform background dissipation, which creates a propagation length contrast between valleys through their intrinsic group velocity differences. We implement this concept in an acoustic crystal, observing switchable and robust valley polarization of sound through large-scale field mapping. Remarkably, our approach is solely based on uniform loss, without the need for any special lattice structures, tailored excitations, or external fields. We further provide designs of our non-Hermitian valley filter on photonic and electronic platforms. Our results offer a simple and effective solution to valley-polarized state generation and may advance the development of novel valley-based devices in both classical and quantum regimes.

We show that, under a short optical pulse, the quantum metric of Bloch states in the momentum-time (kx, ky , t) of graphene becomes dynamic and exhibits a wave-like behavior near Dirac points. This quantum metric wave reflects the Floquet-band structure caused by the pulse, as revealed by solving the time-dependent Schrödinger equation assuming that correlations and out-of-equilibrium effects can be ignored. The momentum and temporal components of the metric have very distinct time dependence that persists even after the pulse has passed. In addition, the pulse also generates a Berry curvature wave that is otherwise absent in static graphene. The time-dependent electron densities in conduction and valence bands also give arise to a Fisher information wave that constitutes part of the quantum metric wave, and is readily measurable by pump-probe experiments.

The momentum space of conventional superconductors is recently recognized to possess a quantum metric defined from the overlap of filled quasihole states at neighboring momenta. For multiband superconductors with arbitrary intraband and interband s-wave pairing, we elaborate that their superfluid weight in London equations is given by the momentum integration of the elements of quantum metric times the quasiparticle energy, indicating the quantum geometric origins of Meissner effect and vortex state. The momentum integration of the quantum metric further yields a spread of quasihole Wannier functions that characterizes the stability of the superconducting state. Our formalism allows the diamagnetic response of conventional superconductors to be mapped to individual lattice sites as a superfluid weight marker, which can incorporate the effect of disorder through self-consistently solving the Bogoliubov-de Gennes equations. Using single-band s-wave superconductors in 2D and 3D as examples, our marker reveals a diamagnetic current that becomes turbulent in the presence of nonmagnetic impurities, and the increase of London penetration depth by disorder that is consistent with experiments.

Real-to-complex spectral transitions and the associated spontaneous symmetry breaking of eigenstates are central to non-Hermitian physics, yet a comprehensive and universal theory that precisely describes the underlying physical mechanisms for each individual state remains elusive. Here, we resolve the mystery by employing the complex path integral formalism and developing a generalized Gutzwiller trace formula. These methodologies enable us to establish a universal quantum-classical correspondence that precisely links the real or complex nature of individual energy levels to the symmetry properties of their corresponding semiclassical orbits. Specifically, in systems with a general rl-pseudo-Hermitian symmetry, real energy levels are quantized along periodic orbits that preserve the corresponding classical S-eta symmetry. In contrast, complex conjugate energy levels arise from semiclassical orbits that individually break the S-eta symmetry but together form So-symmetric pairs. This framework provides a unified explanation for the spectral behaviors in various continuous non-Hermitian models and for the PT transition in two-level systems. Moreover, we demonstrate that the exceptional point is inherently a quantum phenomenon, as it cannot be described by a single classical orbit. Our Letter uncovers the physical mechanism of non-Hermitian symmetry breaking and introduces a new perspective with broad implications for the control and application of non-Hermitian phenomena.

Spatiotemporal optical vortices (STOVs) have attracted significant attention for their unique properties. Recently, the second harmonic generation (SHG) of STOV pulses has been experimentally demonstrated, but the phase singularity dynamics during this process remain elusive. Here, we theoretically investigate the separation and tilting of the phase singularities in STOVs during the SHG. Using the nonlinear Maxwell equation, we show that singularity separation is governed by group velocity mismatch, with accurate predictions provided by a Simpson-type integral under weak spatiotemporal walk-off conditions. In addition, paraxial wave equation analysis reveals that propagation induces singularity tilting, driven by spatial phase shifts. Our results not only offer deeper insights into the spatiotemporal coupling induced by complex nonlinear interactions but also reveal the underlying physical mechanisms in frequency up-conversion of space-time light pulses.

We present the first study on the dynamics and manipulation of phase singularity in transverse orbital angular momentum during second harmonic generation driven by spatiotemporal optical vortices with focused beam.

Non-Hermitian physics characterized by complex band spectra has established a new paradigm in condensed matter systems and metamaterials. Recently, non-Hermitian gain and nonreciprocity are deliberately introduced to valley manipulation, leading to various phenomena beyond the Hermitian scenarios, such as the amplified topological whispering gallery modes as an acoustic laser. In contrast, pure loss is inevitable in practice and generally regarded as a detrimental factor. Here, we reveal that the coupling loss can manipulate valley degrees of freedom in a phononic metamaterial. Three distinct valley-related effects, including valley-resolved nonreciprocity that functions as a valley filter, valley-dependent skin effects where bulk states from different valleys localize at opposite boundaries, and valley-projected edge states with boundary-dependent lifetimes that leads to an anomalous beam splitting, are demonstrated through theoretical analysis and airborne sound experiments. Owing to the easy preparation of loss, our findings shed light on both non-Hermitian and valley physics and may facilitate innovative applications of valley-related devices.

A novel collinear magnetic phase, termed “altermagnetism,” has recently been delimited, characterized by zero net magnetization and momentum-dependent collinear spin-splitting. To understand the intriguing physical effects of altermagnets and explore their potential applications, it is crucial to analyze both the geometric and spin configurations of altermagnetic Fermi surfaces. Here, we conduct a comprehensive study of the quasiparticle interference (QPI) effects induced by both nonmagnetic and magnetic impurities in metallic altermagnets, incorporating the influence of Zeeman splitting and spin-orbit coupling. By examining the QPI patterns for various spin polarizations of magnetic impurities and different spin-probe channels, we identify a series of distinctive signatures that can be used to characterize altermagnetic Fermi surfaces. These predicted signatures can be directly compared with experimental results obtained through spin-resolved scanning tunneling spectroscopy.

Reservoir computing (RC), a neural network designed for temporal data, enables efficient computation with low-cost training and direct physical implementation. Recently, quantum RC has opened possibilities for conventional RC and introduced other ideas to tackle open problems in quantum physics and advance quantum technologies. Despite its promise, it faces challenges, including physical realization, output readout, and measurement-induced backaction. Here, we propose to implement quantum RC through quantum transport in mesoscopic electronic systems. Our approach possesses several advantages: compatibility with existing device-fabrication techniques, ease of output measurement, and robustness against measurement backaction. Leveraging universal conductance fluctuations, we numerically demonstrate two benchmark tasks, spoken-digit recognition and time-series forecasting, to validate our proposal. This work establishes another pathway for implementing on-chip quantum RC via quantum transport and expands the mesoscopic physics applications.

By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.

Entanglement of spin degree of freedom can drastically alter the orbital exchange symmetry of electrons, switching their bunching and antibunching behaviors and the resultant current correlations in the Hanbury-Brown-Twiss interferometry. Here, we investigate the exchange correlation of quasiparticles with entanglement encoded in the Nambu spinors, or the electron-hole degree of freedom. In contrast to the conventional correspondence between spin entanglement and current correlation, we find that singlet (triplet) entanglement of Nambu spinors results in suppressed (enhanced) current correlation. This effect arises because the charge degree of freedom itself encodes the entanglement. We propose implementing this phenomenon in the edge states of a quantum Hall system, where the entangled states of the Nambu spinors can be continuously tuned by gate voltages. Our study reveals a novel relationship between entanglement and charge correlations, offering an effective approach for detecting entanglement of Nambu spinors.

The recently delimited altermagnetic phase is characterized by zero net magnetization but momentum-dependent collinear spin-splitting. To explore the intriguing physical effects and potential applications of altermagnets, it is essential to analyze their Fermi surface properties, encompassing both configurations and spin textures. Here, we conduct a Fermiology study on metallic altermagnets and demonstrate that the collinear spin-split features of their Fermi surfaces can be clearly revealed through quantum oscillation measurements. By introducing a transverse Zeeman field to remove the spin-degenerate lines in the momentum space, the Fermi surface undergoes a Lifshitz transition, giving rise to spin-flipped cyclotron motion between orbits with opposite spins. Accordingly, the Lifshitz-Onsager quantization yields two sets of Landau levels, leading to frequency splitting of the Shubnikov-de Haas oscillations in conductivity. In the presence of spin-orbit coupling, the Zeeman field causes two separate cyclotron orbits to merge at the Lifshitz transition point before splitting again. This results in the two original frequencies discontinuously changing into a single frequency equal to their sum. Our work unveils a unique and universal signature of altermagnetic Fermi surfaces that can be probed through quantum oscillation measurements.

The Uhlmann relative amplitude between two density matrices is shown to be a generating function, through which the quantum Fisher tensor that contains both the quantum Fisher information matrix and the mean Uhlmann curvature can be obtained via differentiation over system parameters. In the pure state limit, our generating function recovers that of the quantum geometric tensor proposed by Hetényi and Lévay, and also clarifies the fidelity and phase between two quantum states as the generating functions of the quantum metric and Berry curvature, respectively. A generic expression for the quantum Fisher tensor in terms of the Bloch representation of density matrices is derived, which facilitates the calculation of the tensor, mean Uhlmann curvature, and geometric properties derived from the quantum Fisher information matrix. Canonical ensembles of spins are adopted to demonstrate our formalism, which reveals a constant Ricci scalar, a vacuum Einstein equation, and a cosmological constant on the 3D Euclidean manifold of the magnetic field