We show that Andreev spin qubits can be realized in a Josephson junction based on the helical edge states of a two-dimensional topological insulator (quantum spin Hall system) proximized by superconducting films, in the presence of magnetic doping. We demonstrate that the electric dipole transitions between the Andreev spin states induced by the magnetic doping can be harnessed to manipulate the Andreev spin qubit by microwave radiation pulses, without applying an external Zeeman field or invoking ancillary states. We numerically simulate the realization of not and Hadamard quantum logic gates, and discuss implementations in realistic setups.

We investigate the spectral and topological properties of an atomic chain of red phosphorus. We reveal the separation of flat-band states from the rest of the system and calculate its energy bands analytically. The topological properties of the system are established through the evaluation of the Berry (Zak) phase of the energy bands, revealing nontrivial topology. The Berry phase depends on the relative strength of the hopping parameters and exhibits dual energy-dependent topological phase transitions. Remarkably, the emergence of inert band edges provides a direct spectral signature of these transitions, acting as energy-resolved indicators of the redistribution of topological charge between bands. The existence of the associated edge states is proved numerically for finite lattices. The theoretical predictions, particularly the band structure and the existence of edge states, are further confirmed by numerical simulations of red phosphorus through a phononic lattice in the form of a highly structured aluminum plate.

We present a model for a topological quasicrystalline system that is suitable for realization in cold atom experiments. We define the model in terms of complex momentum-space couplings that break time-reversal symmetry (TRS) and detail how it may be experimentally realized using two-photon Raman couplings. In the weak-potential limit, we study the model analytically by calculating the band structure over a “quasi-Brillouin zone” (QBZ). We find symmetry-protected Dirac cones, which are gapped by a TRS-breaking term, resulting in a Chern number $\mathcal{C}=1$. This provides a direct analogy to the Haldane model, but now in a quasicrystalline setting. We also infer the number of states below the topological gap from the QBZ area. We verify our analysis with numerical calculations of periodic approximants to our system, constructing a phase diagram in parameter space that shows a topological region extending beyond the weak-potential regime. We also find examples of narrow Chern bands with the potential for hosting strongly correlated physics. Our work raises questions about the nature of localization and strongly correlated states in Chern bands in quasiperiodic systems.

We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. While the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices, namely, the Lieb, the double-Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.

Topological wave phenomena in continuous media are fundamentally challenged by unbounded spectra and the absence of a compact Brillouin zone, which obstruct conventional bulk-interface formulations. We develop a unified phase-space framework for screened magnetized plasma based on a pseudo-Hermitian formulation with a positive-definite metric, enabling a generalized Schrödinger description and a Weyl-symbol analysis of the bulk generator. We show that the bulk Weyl symbol hosts isolated band degeneracies acting as Berry-Chern monopoles, including a higher-order spin-1 degeneracy with topological charge $+2$ that generically splits into two spin-$\frac{1}{2}$ Weyl points for finite $k_z$. To characterize topology in this noncompact setting, we introduce a strip-gap Chern number associated with finite real-frequency strips of the spectrum of the phase-space symbol, extending band Chern topology to continuum systems. This invariant governs the spectral flow of interface modes induced by spatial variations of the magnetic field and establishes a bulk-interface correspondence at the level of phase-space symbols. By solving the interface eigenvalue problem, we demonstrate that the net spectral flow across the strip gap is determined by the enclosed monopole charge. We further show that this correspondence persists under weak collisional damping as long as the complex spectrum retains a finite real-part strip gap. Our results provide a systematic phase-space framework for topological wave transport in continuous media beyond compact-band and idealized Hermitian settings.

We present a framework for characterizing higher-order topological phases directly from the one-particle density matrix, without any reference to an underlying Hamiltonian. Our approach extends the mode-shell correspondence, originally formulated for single-particle Hamiltonians, to Gaussian states subject to chiral constraints. In this correspondence, the mode index counts topological boundary modes, while the shell index quantifies the bulk topology in a region surrounding the modes, providing a bulk-boundary diagnostic. In one-dimensional topological insulators, the shell index reduces to the local chiral marker, recovering the winding number in the translation-invariant limit. We apply the mode-shell correspondence to a $C_4$-symmetric higher-order topological insulator with a chiral constraint and show that a fractional shell index implies that the higher-order phase is intrinsic. The one-particle density matrix is formulated in real space, so the mode-shell correspondence also applies to models without translation invariance. By introducing structural disorder into the $C_4$-symmetric higher-order insulator, we show that the mode-shell correspondence remains a meaningful diagnostic in amorphous structures. The mode-shell correspondence generalizes to interacting states with a gapped bulk spectrum in the one-particle density matrix, providing a practical and diverse route to characterize higher-order topology from the quantum state itself.

Strong directional disorder in local magnetic moments coupled to a Chern insulator gives rise to topological phases that cannot be realized in any clean system and are therefore genuinely disorder driven. We demonstrate this in a spinful Qi-Wu-Zhang model of a two-dimensional Chern insulator coupled to disordered classical spins of unit length. The topological phase diagram is computed numerically using two complementary approaches: twisted boundary conditions and the topological Hamiltonian technique. Our results show that strong disorder can act as a fundamental topological mechanism rather than merely a perturbation. For strong exchange coupling, tuning the mass parameter reveals a transition between phases with different Chern numbers $C$. Remarkably, this transition is driven by zeros, rather than poles, of the disorder-averaged Green’s function crossing the chemical potential, and has no analog in any clean system. We further identify a strong-coupling phase with $C=0$ that is nonetheless topologically nontrivial, characterized by a distinct Chern number $C^{\left(\mathrm{S}\right)}\ne 0$ over the manifold of classical spin configurations. This phase is also disorder driven, as $C^{\left(\mathrm{S}\right)}=0$ in the clean limit.

We introduce and experimentally demonstrate an approach for selective mode excitation in non-Hermitian resonant systems using incoherent light. This method eliminates the need for precise phase control that is often required in coherent excitation schemes. Using this technique on a silicon photonic platform with coupled ring resonators, we successfully excite the topological edge state of a non-Hermitian Su-Schrieffer-Heeger model. Our work shows that incoherence-assisted excitation is a robust and passive strategy for topological state preparation, which broadens the scope of non-Hermitian topological photonics, thereby providing a practical and experimentally viable tool for selective mode excitation.

We develop a general framework for classifying fermionic dynamical systems with measurements using symmetry and topology. We introduce two complementary classification schemes based on the Altland-Zirnbauer tenfold way: (1) the many-body evolution operator (mEO) symmetry class, which classifies fermionic dynamics at the many-body level and naturally extends to interacting dynamics, and (2) the single-particle transfer matrix symmetry class, which classifies free-fermion dynamics at the single-particle level and connects to Anderson localization physics. In the free-fermion limit, we show that these two frameworks are equivalent via a dynamical bulk-boundary correspondence: The topology of the dynamical system's spacetime bulk determines the topology of the area-law-entangled steady-state ensemble living on its temporal boundary. Next, we prove that symmetry-invariant, postselection-free Gaussian measurements are realizable in only four of the ten mEO classes (A, AI, BDI, D); the remaining six require either postselection or interacting (non-Gaussian) measurements. Building on these results, we construct general postselection-free topological adaptive circuits that realize topological dynamical phases in any spatial dimension for the four admissible mEO classes. These circuits simultaneously provide a protocol for preparing and stabilizing free-fermion topological states in all ten symmetry classes. As a concrete demonstration, we construct and simulate $2+1\mathrm{d}$ adaptive circuits that realize mEO-class-A topological dynamics, steering toward a steady-state ensemble of Chern insulators in $\mathcal{O}\left(1\right)$ circuit depth. Finally, we numerically characterize topological phase transitions, dynamical domain-wall modes, and robustness to coherent noise, identifying finite error thresholds at which trajectory-resolved and trajectory-averaged quantities undergo distinct phase transitions.

Flat-band states in topological systems provide a unique platform for investigating strongly correlated phenomena and many-body physics. However, in two-dimensional (2D) static tight-binding systems, perfectly flat bands can only exist in the topologically trivial phase, as characterized by a zero Chern number. Here, we show that by introducing periodic driving into a 2D photonic Lieb lattice composed of coupled microring resonators, the resulting Floquet topological insulator can host perfectly flat bands with nontrivial topology. In particular, by tracking the evolution of the flat-band modes over each cycle, we show that the non-Abelian displacements of the flat-band modes are characterized by a nontrivial quantized helicity even though the quasienergy bands have zero Chern number. The helical motion of the flat-band modes can be described by a braiding of the world lines of their trajectories, with a nontrivial winding number directly connected to the helicity. We also propose a scheme to experimentally measure the quantized non-Abelian helicity in a microring lattice subject to a synthetic magnetic field. These results suggest that Floquet topological photonic insulators based on coupled microring resonators can provide a versatile platform for investigating non-Abelian topological physics and strongly correlated phenomena in photonic flat-band systems.

The non-Hermitian skin effect, characterized by the extensive localization of bulk modes at the boundaries, has attracted significant attention as a hallmark feature of non-Hermitian topology. This localization invalidates the conventional Bloch band theory, necessitating an analysis under open boundary conditions even in the thermodynamic limit. The Amoeba formulation addresses this challenge by computing the spectral potential rather than the spectrum itself. Based on the (strong) Szegö limit theorem and its topological generalization, this approach reduces the evaluation of the potential to an optimization problem involving the Ronkin function. However, while the generalized Szegö limit theorem is formally applicable in arbitrary dimensions, its implementation is limited to single-band systems, and its applicability to multiband systems remains unclear even in one-dimensional systems. In this paper, we establish the Wiener-Hopf factorization (WHF) of the non-Bloch Hamiltonian as a powerful framework, providing a unified and rigorous foundation for Amoeba analysis in one-dimensional multiband systems. By combining the WHF with Hermitian doubling, we first elucidate the applicability criteria for the generalized Szegö limit theorem in multiband systems. We then show that the WHF provides the natural mathematical origin for the symmetry-decomposed Ronkin function in symmetry class ${\mathrm{AII}}^{†}$, leading to a rigorous proof of the generalized Szegö limit theorem for these systems and opening a path toward systematic generalizations to other symmetry classes.

Symmetry plays an important role in the topological band theory. Recently, the concept of subsymmetry-protected topological (sub-SPT) phases [Nat. Phys. 19, 992 (2023)] suggests that chiral edge states can exist in a system with broken symmetry. However, sub-SPT theory only focuses on the zero-energy edge states in the half-filling gap, which limits its application. In this work, we extend the scope of sub-SPT phases in both one-dimensional (1D) and two-dimensional (2D) multigap asymmetric systems through redefined sublattices. In three typical asymmetric models without subsymmetry, we find that all the nonzero edge (corner) states can be illustrated by sub-SPT chiral edge (corner) states after redefining sublattices. Moreover, these nonzero chiral boundary states can exhibit some exotic properties, such as the topological bound states in the continuum. As to the bulk-edge correspondence, a universal topological invariant $\overline{Z}$ is defined, which can precisely predict the appearance of chiral boundary states in both 1D and 2D systems. Finally, we propose an acoustic experiment where the nonzero chiral corner states are numerically observed. Our work exhibits a general approach to study the topological properties of asymmetric systems, opening up possibilities for topological phenomena.

Bismuth has been shown to be topological in its different allotropes and compounds, with one of the most notable examples being the Bi-Sb alloy, the first 3D topological insulator ever discovered. In this paper, we explore two-dimensional alloys of Bi and Sb, both crystalline and amorphous, to determine the critical concentrations that render the alloys topological. For the amorphous alloy, we determine the effect of structural disorder on its topological properties, remarkably observing a trivial to topological transition as disorder increases. The alloys are modeled using a Slater-Koster tight-binding model and the topological behavior is assessed through the entanglement spectrum together with artificial neural networks. Additionally, we perform electronic transport calculations with results compatible with those of the entanglement spectrum, which, furthermore, reveal an insulator to metal transition in the highly disordered regime.

Efficient transmissive pure-phase resonances are highly desirable for optical modulation and wave front engineering. Here, we propose a principle to realize a pure-phase resonance in an extremely broad transmission band, as opposed to previous approaches restricted to operating in reflection mode or over a narrow spectral band. We show that a glide-symmetric bilayer metasurface mathematically mimicking a two-dimensional Dirac semimetal induces unidirectional guided-mode excitation and perfect leakage-radiation blazing at the transmission channel. These effects create a peculiar resonant-scattering configuration, similar to the classical reflective Gires-Tournois etalon, but in transmission, providing full 2π phase modulation with constant transmittance near 100%. Most importantly, this effect persists over an extremely wide band, associated with topological effects. Hence, our proposed approach produces a spectrally and parametrically robust pure-phase resonance effect in transmission, which is highly beneficial for practical applications.

The fractional quantum Hall effect provides the first example of multiple quantum phases hosting quasiparticles not being fermions or bosons, but anyons. It is therefore important to investigate whether all these phases can be practical to detect these anyons. Here, we propose revisiting the intriguing physics of hole-conjugate Fractional Quantum Hall (FQH) phases with counter-propagating edge channels in the filling factor range $1/2\le \nu \le 1$ whose properties may prevent anyon interference detection. In the seminal KFP model by Kane, Fisher, and Polchinski, upstream neutral and downstream charge modes, along with random incoherent interchannel tunneling, are considered essential for explaining the observed conductance at $\nu =2/3$. In this work, we introduce a minimal model that qualitatively reproduces the edge physics of the $\nu =2/3$ FQH phase without invoking neutral and charge modes. By inserting $N$ fictive reservoirs—either Landauer reservoirs (LRs) or Energy-Preserving Reservoirs (EPRs)—along the counter-propagating channels to simulate edge channel mixing, we show that the conductance rapidly approaches the KFP fixed point of $2/3\times (e^2/h)$ for both EPRs and LRs. While LRs produce equal charge and thermal relaxation lengths, EPRs exhibit an infinite thermal relaxation length, consistent with recent observations. Our model also predicts current noise in small Hall bars that varies linearly with voltage at large bias, with an apparent Fano factor $\simeq 0.19/\sqrt{N+1}$ for $N$ LRs or an exponentially vanishing Fano factor for EPRs. Moreover, with a Quantum Point Contact at the center of the Hall bar, the model predicts the recently observed $0.5e^2/h$ conductance plateau. Finally, early experiments reporting upstream excitations are found to be conveniently explained by upstream diffusive heat flow and delta-$T$ noise, rather than by specific collective neutral modes. The results obtained in this study can be generalized to more complex hole-conjugate fractional quantum Hall phases.

The coupling between magnetism and topological states at a topological insulator (TI)-magnetic insulator interface can induce exchange gap opening and spin texture change, resulting in many emergent transport phenomena, including anisotropic magnetoresistance (AMR) and planar Hall effect (PHE). While AMR/PHE have been reported in several ferromagnetic insulator-TI heterostructures so far, such effects in antiferromagnetic insulator-TI heterostructure remain elusive. Here, we observe unique features of AMR and PHE in an antiferromagnetic insulator-TI heterostructure $\mathrm{MnSe}\text{/}{\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{T}{\mathrm{e}}_3$. Specifically, at low magnetic fields ($H$), the in-plane AMR exhibits dips under $H$ applied parallel to the current ($H\text{//}I$) but peaks under the perpendicular geometry ($H\text{⊥}I$), opposite to the zero AMR features in ferromagnetic insulator-TI heterostructures. The low-field PHE is most striking at $H\text{//}I$, but gradually suppressed when $H$ deviates from parallel direction. At high magnetic fields, both AMR and PHE show prominent π-period oscillation. Gate-tuning and temperature-dependence measurements demonstrate that the observed AMR/PHE originate from the interplay between the topological surface state of TI with the spin canting and magnetic phase transition of MnSe. These results not only differentiate the AMR/PHE in TI-based magnetic heterostructure from the pristine TI, but also offer a feasible way to distinguish spin Hall effect and AMR in TI-based heterostructures.

Spectra of non-Hermitian systems with skin effects are sensitive to the boundary conditions. In this work, we introduce a system composed of an impurity and a nonreciprocal Su-Schrieffer-Heeger (SSH) chain with odd number of lattice sites. The shift in the energy of the zero mode can be dramatically changed by adding two vanishingly small couplings between the impurity and two ends of the chain. Here, we propose a multiple physical quantities sensor with exponential-power-law amplification mechanism. Sensitivity of the proposed sensor exhibits an exponential scaling with SSH chain size and a power-law scaling with frequency of the impurity. This system is designed for detecting two small physical quantities, i.e., the impurity-chain coupling strengths. If and only if these two physical quantities exist simultaneously, the energy of zero mode will obviously change. Moreover, sensitivity is also robust against on-site disorder or dissipation of the impurity. Sufficiently strong disorder will change the localization of the eigenstate, and this leads to device failure. We also show the parameter range of the proposed sensor, and this range accurately corresponds to a nontrivial spectral winding number. The single physical quantity sensors suffered from repeated construction and low utilization rate. Thus, our findings pave a basic way toward sensors that couple multiple physical quantities to the boundary of the non-Hermitian topological chain.

We present a general recipe to describe topological phase transitions in condensed matter systems with interactions. We show that topological invariants in the presence of interactions can be efficiently calculated by means of a non-Hermitian quasiparticle Hamiltonian introduced on the basis of the Green's function. As an example analytically illustrating the application of the quasiparticle concept, we consider a topological phase transition induced by the short-range electrostatic disorder in a two-dimensional system described by the Bernevig-Hughes-Zhang model. The latter allows us to explicitly demonstrate the change in the ${\mathbb{Z}}_2$ topological invariant and explain the quantized values of the longitudinal conductance in a certain range of the Fermi energy and the disorder strength found previously in numerical calculations.