We demonstrate how spin-orbit coupling (SOC) controls band topology and induces higher-order topological corner states in a two-dimensional Su-Schrieffer-Heeger (SSH) model. SOC lifts spin degeneracy and, when applied along both lattice directions, generates robust zero-energy corner states. By matching intracell (intercell) hopping strengths to their SOC counterparts, we produce topological flat bands, whose phases are classified by the charge polarization and verified through bulk-edge-corner correspondence. We propose an LC-circuit implementation and show numerically that the corner states remain strongly localized and robust in weak on-site disorder. These results establish SOC as a versatile tool for engineering higher-order topological phases and topological flat bands, offering an experimentally accessible route for realization in circuit systems.

Most topological insulators suffer from intrinsically narrow band gaps, limiting their practical applications. Using tight-binding modeling and first-principles calculations, we show that in the topological state and band gap of the ferroelectric bilayer $\alpha \text{−}\mathrm{I}{\mathrm{n}}_2\mathrm{S}{\mathrm{e}}_3$ ($2\mathrm{L}\text{−}\mathrm{I}{\mathrm{n}}_2\mathrm{S}{\mathrm{e}}_3$) can be effectively tuned by controlling the hybridization strength $\left(V_{\text{hyb}}\right)$ between Se $p_x/p_y$ and $s/p_z$ orbitals through transition metal substitution at In sites. This mechanism is demonstrated in a $\mathrm{ScInS}{\mathrm{e}}_3/\mathrm{I}{\mathrm{n}}_2\mathrm{S}{\mathrm{e}}_3$ van der Waals Heterostructure (vdWH), where enhanced charge transfer from Sc $d$ states to Se atoms strengthens the $\mathit{px}/\mathit{py}\text{−}s/\mathit{pz}$ orbitals' hybridization and enlarges the quantum spin Hall gap to 36 meV. Magnetic Cr substitution further yields a $\mathrm{CrInS}{\mathrm{e}}_3/\mathrm{I}{\mathrm{n}}_2\mathrm{S}{\mathrm{e}}_3$ vdWH with a sizable $\sim 110$ meV quantum anomalous Hall gap and compressive strain-driven topological phase transitions—from a trivial magnetic insulator ($C=0$) to a Chern insulator ($C=1$), and finally, to a magnetic topological metal ($C=2$). These results establish a viable strategy for engineering band gaps and topological phases for low-power spintronic and quantum technologies.

We develop a symmetry-adapted multipolar $\mathbf{k}·\mathbf{p}$ theory close to the bulk $\mathrm{\Gamma }$ point for time-reversal-symmetric, noncentrosymmetric $C_{3v}$ crystals in the strong atomic spin-orbit-coupling ($jj$-coupling) limit. Using a $j\in \{1/2,3/2,5/2\}$ multiplet basis appropriate for heavy-element $p$ and $d$ bands, we systematically construct all symmetry-allowed spin-orbit-coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For $j>1/2$, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-$j$ textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity $W_n$ for every energy band and identify distinct $|W_n|=1,2,5$ phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely, the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.

We study quadrupole orders in heavy-fermion compound ${\mathrm{PrIr}}_2{\mathrm{Zn}}_{20}$ under magnetic fields on the basis of the Landau theory. Assuming $E_g$ electric quadrupolar orders in the cubic symmetry with the ordering wave number at the L points in the face-centered-cubic lattice Brillouin zone as observed experimentally, we construct a Landau free energy and analyze the resulting free energy. We find that the unidentified high-temperature ordered phase under the magnetic field $\mathit{H}\parallel \left[001\right]$ reported earlier arises from the rotation of the quadrupole moments of $f$ electrons on the Pr site. We also discuss the phase diagram for other magnetic-field directions and possible double-$\mathit{q}$ quadrupolar orders in this system.

The electronic structure of materials is fundamentally governed by their crystal symmetry. While most research on two-dimensional materials has focused on hexagonal lattices, such as graphene, hexagonal boron nitride, and transition metal dichalcogenides. This work explores a square-lattice system: the experimentally realized phthalocyanine-based metal–organic framework (ZnPc-MOF). Using group representation theory, we classify the electronic bands of ZnPc-MOF monolayer, AA- and AB-stacked bilayers, and twisted bilayers in terms of the irreducible representations (irreps) of their little groups. We find that bands in the AB-stacked bilayer remain two-fold degenerate along the $Y$ and $Y^{\prime }$ high-symmetry lines, as a consequence of the sole presence of two-dimensional irreps along these directions. We further derive optical transition selection rules to interpret the optical conductivity, revealing pronounced polarization-dependent optical responses. Additionally, we investigate the quasicrystalline electronic states in the ${45}^{\circ }$ twisted bilayer (ZnPc-MOF quasicrystal) using the resonant coupling Hamiltonian. Compared to graphene quasicrystals, ZnPc-MOF quasicrystal exhibits weaker resonant coupling strengths, yet its quasicrystalline states lie closer to the Fermi energy, suggesting a greater contribution to low-energy electronic phenomena.

In this work, we propose a scheme to realize the layer Hall effect in the ferromagnetic topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ via proximity to $d$-wave altermagnets. We show that an altermagnet and an in-plane magnetic field applied near one surface gap the corresponding Dirac cone, yielding an altermagnet-induced half-quantized Hall effect. When altermagnets with antiparallel Néel vectors are placed near the top and bottom surfaces, giving rise to the layer Hall effect with vanishing net Hall conductance, i.e., the altermagnet-induced layer Hall effect. In contrast, altermagnets with parallel Néel vectors lead to a quantized Chern insulating state, i.e., the altermagnet-induced anomalous Hall effect. We further analyze the dependence of the Hall conductance on the orientation of the in-plane magnetic field and demonstrate that the layer Hall effect becomes observable under a perpendicular electric field. Our results establish a route to engineer altermagnet-induced topological phases in ferromagnetic topological insulators.

Nonlinear processes can offer pathways to next-generation sensors and frequency mixing devices to overcome modern imaging, detection, and communication challenges. In this article, we report on strong second- and third-order nonlinear optical responses in visible and terahertz light in single crystals of the noble metal dichalcogenide ${\mathrm{PdTe}}_2$. We find that buried conduction and valence topological surface states of ${\mathrm{PdTe}}_2$ lead to resonant optical second-harmonic generation. On the other hand, although the nonlinear responses obtained with terahertz excitation are not close to this resonance, they can be clearly observed in reflection geometry, even in the presence of broadband excitation, where optical filters are not necessary to observe the enhanced odd-order higher harmonic output. By carefully considering the radiative photocurrent framework of stimulated terahertz emission, we are able to extract fingerprints of both second- and third-order processes in the terahertz regime, and show that ${\mathrm{PdTe}}_2$ is a promising material candidate for radio-frequency rectification, frequency mixing, and beam focusing.

Majorana zero modes (MZMs) are bound midgap topological excitations at the ends of a one-dimensional (1D) topological superconductor, which must come in pairs. If the two MZMs in the pair are sufficiently well-separated by a distance much larger than their individual localization lengths, then the MZMs behave as non-Abelian anyons which can be braided to carry out fault-tolerant topological quantum computation. In this ‘‘topological’’ regime of well-separated MZMs, their overlap is exponentially small, leading to exponentially small Majorana splitting, thus enabling the MZMs to be topologically protected by the superconducting gap. In real experimental samples, however, the existence of disorder and the finite length of the 1D wire considerably complicate the situation, leading to ambiguities in defining ‘‘topology’’ since the Majorana splitting between the two end modes may not necessarily be small in disordered wires of short length. We theoretically study this situation by calculating the splitting in experimentally relevant short disordered wires, and explicitly investigating the applicability of the ‘‘exponential protection’’ constraint as a function of disorder, wire length, and other system parameters in realistic models of nanowires currently being used experimentally. We find that the exponential regime is highly constrained and is suppressed for disorder somewhat less than the topological superconducting gap. We provide detailed results and discuss the implications of our theory for the currently active experimental search for MZMs in superconductor-semiconductor hybrid platforms. A general consequence of our work is that topology in finite disordered wires may not be uniquely defined, necessitating a careful analysis which depends on the context.

Topological photonics provides an unprecedented way to control wave propagation that can eliminate backscattering. However, due to the bulk-boundary correspondence inherent in topological physics, the desired topological state usually manifests at the boundary rather than within the bulk, resulting in a sizeable useless bulk area and hindering applications. Here, we present a unique topological photonic crystal fiber (TPCF) by incorporating a one-dimensional bivortex line within a three-dimensional bilayer photonic crystal, where adjacent layers exhibiting π-phase vortex differences generate artificial spins. The spin-core states along the fiber core and their robust spin-momentum locking against disorders are demonstrated in microwave experiments. Besides the single-mode fiber, we also construct a multimode TPCF. Our research may empower photonic crystal fibers with topological band structures, advancing future high-performance communication devices.

Due to spin-orbit coupling (SOC), a quantized spin Hall conductivity has not yet been reported in realistic materials. Here, we tackle this challenge by predicting robust quantized spin Hall conductivity in monolayer ${\mathrm{Fe}}_2{\mathrm{Te}}_2\mathrm{O}$. The underlying physics originates from the unrecognized mirror-spin coupling (MSC), which couples spin-up and spin-down states into two orthogonal mirror eigenstates. We show that the perfect MSC can naturally emerge in the two-dimensional altermagnets with out-of-plane Néel vector and horizontal mirror but without SOC. Remarkably, the MSC can dramatically weaken the spin hybridization of the systems when SOC is included. When SOC is neglected, ${\mathrm{Fe}}_2{\mathrm{Te}}_2\mathrm{O}$ is an altermagnetic Weyl semimetal with MSC. With SOC, it evolves into the first material candidate for two-dimensional magnetic mirror Chern insulator. Remarkably, under the protection of MSC, the spin hybridization of both bulk and topological edge states in ${\mathrm{Fe}}_2{\mathrm{Te}}_2\mathrm{O}$ with SOC at low energy is negligible. As a consequence, a quantized spin Hall conductivity emerges within the bulk band gap of the system. By unveiling this effect, our findings represent a significant advancement in spin Hall transport, and broaden the material candidates hosting intriguing altermagnetic phenomena.

We explore the nonreciprocal intracell hopping mediated non-Hermitian topological phases of an extended Su-Schrieffer-Heeger model hosting second-nearest-neighbor hopping. We microscopically analyze the phase boundaries using the non-Bloch momentum while the off-critical (critical) phases are directly associated with the gapped (gapless) nature of the non-Bloch bands that we derive from the characteristic equation using the gauge freedom. The non-Bloch momentum accurately reflects the bulk boundary correspondence (BBC) explaining the winding number profile under open boundary conditions. We examine the adiabatic dynamics to promote the concept of adiabatic charge transport in a non-Hermitian scenario justifying the BBC in spatiotemporal Bott index and non-Bloch Chern number. Once the non-Bloch bands experience no (a) gap-closing during the evolution of time, quantized flow of is preserved (broken). Our study systematically unifies the concept of non-Bloch bands for both static and driven situations.

We study a generalization of ‘‘Yukawa models’’ in which Majorana fermions, interacting via all-to-all random couplings as in the Sachdev-Ye-Kitaev (SYK) model, are parametrically coupled to disordered bosonic degrees of freedom described by a quantum $p-\text{spin}$ model. The latter has its own nontrivial dynamics leading to quantum paramagnetic (or liquid) and glassy phases. At low temperatures, this setup results in SYK behavior within each metastable state of a rugged bosonic free-energy landscape, the effective fermionic couplings being different for each metastable state. We show that the boson-fermion coupling enhances the stability of the quantum spin-glass phase and strongly modifies the imaginary-time Green’s functions of both sets of degrees of freedom. In particular, in the quantum spin-glass phase, the imaginary-time dynamics is turned from a fast exponential decay characteristic of a gapped phase into a much slower dynamics. In the quantum paramagnetic phase, on the other hand, the fermions' imaginary-time dynamics get strongly modified, and the critical SYK behavior is washed away.

We reveal a fundamental geometric structure of momentum space arising from the nonadiabatic evolution of Bloch electrons. By extending semiclassical wave-packet theory to incorporate nonadiabatic effects, we introduce a momentum-space metric tensor—the nonadiabatic metric. This metric gives rise to two velocity corrections, dubbed geometric and geodesic velocities, providing a unified and intuitive framework for understanding nonlinear and nonadiabatic transport phenomena beyond Berry phase effects. The geometric velocity is related to the nonadiabatic metric itself, whereas the geodesic velocity is a Christoffel symbol of the nonadiabatic metric. As the nonadiabatic metric is related to the energy-gap renormalized quantum metric, it unifies the broad quantum-metric effects in electronic responses. When the nonadiabatic metric is constant, it reduces to an effective mass, modifying flat-band electron dynamics in confining potentials. In a flat Chern band with harmonic attractive interactions, the two-body wave functions mirror the Landau-level wave functions on a torus. Furthermore, we show that the nonadiabatic metric endows momentum space with a curved geometry, recasting wave-packet dynamics as forced geodesic motion.

The Mpemba effect, where a state prepared farther from equilibrium relaxes faster to equilibrium than one prepared closer, has a quantum counterpart where relaxation is resolved by conserved charge. However, the fate of the quantum Mpemba effect in systems with long-range interactions remains an open question. Here, we study the quantum Mpemba effect in long-ranged, U(1)-symmetric random unitary circuits. Using annealed Rényi-2 entanglement asymmetry computed via replica tensor networks and exact diagonalization, we track the symmetry restoration from three types of tilted product states: ferromagnetic, antiferromagnetic, and ferromagnetic with a central domain wall. The quantum Mpemba effect is present for tilted ferromagnetic states at all interaction ranges, but absent for tilted antiferromagnetic states, and it occurs for the domain-wall state only in effectively short-ranged circuits, where the Mpemba time $t_{\mathrm{M}}$ is found to scale with the subsystem size $N_A$ as $t_{\mathrm{M}}\sim N_A^z$, with the dynamical exponent $z=min(\alpha -1,2)$. These results reveal how the quantum Mpemba effect is governed by the interplay between interaction range and initial-state charge bias in long-ranged chaotic systems.

Terahertz electromagnetic pulses offer a promising route for the ultrafast manipulation of magnetization in ferromagnetic materials. While previous studies have demonstrated the excitation of spin dynamics using linearly polarized terahertz fields, the role of circular polarization and the effects of rapidly oscillating, time-dependent field profiles remain insufficiently understood. We develop a unified theoretical framework for describing the excitation of spin precession via Zeeman interaction in magnetic materials by high-frequency pulses of arbitrary polarization with a temporal Gaussian profile. In the regime of long pulses (at least several oscillations are within the pulse duration), a circularly polarized magnetic field acts as an effective rectified magnetic field along the pulse propagation, while linear polarized pulses excite no free precession. In the regime of short pulses (less than one oscillation is within the pulse duration), pulses of any polarization, including linear, can excite free spin precession. There is an optimal pulse duration which maximizes the amplitude of the spin precession. It depends on magnetic parameters of the sample and the external magnetic field, as well as on the carrier frequency of the pulse and its amplitude. These findings bridge key gaps in the understanding of terahertz-induced spin dynamics and provide insights into the design of light-controlled magnetization schemes using tailored electromagnetic pulses.

We investigate Andreev reflection (AR) in a proximity-induced normal-superconductor (NS) junction within the extended $\alpha –{\mathcal{T}}_3$ lattice, emphasizing the impact of flat bands on AR. Our findings reveal that flat bands significantly enhance AR. Through wave-packet dynamics, we track the real-time evolution of quasiparticle wave packets across the junction, providing deeper insight into electron-hole conversion. Notably, the combination of band flatness and anisotropic dispersion in the $k_x\text{−}{k}_y$ plane induces an electronic analog of Goos-Hänchen (GH) shifts at the NS interface, exhibiting directional asymmetry along the junction. This asymmetry leads to a Hall-like response in a Josephson junction in the superconductor–normal-metal–superconductor geometry, where transport across the junction region is dominated by the quasiflat bands.

We study Anderson localization on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbors with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalization of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalization and analytical renormalized perturbation theory, we establish the resulting localization phase diagram emerging from the interplay of the length scale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localization transition to stronger disorder, and that beyond a critical range the localized phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalized and localized phases, with no evidence for an intervening multifractal phase, for both deterministic and random-hopping models. A scaling analysis based on inverse participation ratios reveals behavior consistent with a Kosterlitz-Thouless–type transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behavior in average and typical correlation functions, reflecting the different scaling properties of generalized inverse participation ratios.

We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time $T$ and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing $T$, from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in $d+1$ space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the “localization” time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a $d=2$ model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.