A central mystery in high-temperature cuprate superconductors is the coexistence of multiple exotic orders, which is presumably associated with strong electronic correlation. The ongoing interest in this enigmatic phenomenon is further energized when similar electronic orders and states emerge and coexist in less correlated kagome superconductors. Here, by utilizing angle-resolved photoemission spectroscopy (ARPES), nuclear magnetic resonance (NMR) spectroscopy, scanning tunneling microscopy (STM) and first-principles calculations, we reveal the sudden emergence of a distinct short-range charge stripe order in Sn-doped ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ kagome superconductors when the long-range $2\times 2$ charge density wave order in pristine ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ is suppressed. This short-range stripe order features a modulation vector of approximately $1/3$ along one of the three lattice directions, induces remarkable quasiparticle scattering between the original quasi-1D kagome $d$ bands and their replica of folding, and clearly suppresses the electron density of states at the Fermi level. Our first-principles calculations reveal that $3\times 1$ supermodulation represents a hidden secondary instability in pristine ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. This instability is further enhanced by in-plane chemical pressure induced by Sn substitution, and coupled to the electronic correlation, leading to a unique charge stripe order in the system. As such, our results reveal a new route toward emergent electronic orders via cooperative interactions between the lattice and electronic degrees of freedom.

Magnetic skyrmions are spatially localized whirls of spin moments in two dimensions, featuring a nontrivial topological charge and a well-defined topological charge density. We demonstrate that the quantum dynamics of magnetic skyrmions is governed by a dipole conservation law associated with the topological charge, akin to that in fracton theories of excitations with constrained mobility. The dipole conservation law enables a natural definition of the collective coordinate to specify the skyrmion’s position, which ultimately leads to a greatly simplified equation of motion in the form of the Thiele equation. In this formulation, the skyrmion mass, whose existence is often debated, actually vanishes. As a result, an isolated skyrmion is intrinsically pinned to be immobile and cannot move at a constant velocity. In a spin-wave theory, we show that such dynamics corresponds to a precise cancellation between a highly nontrivial motion of the quasiclassical skyrmion spin texture and a cloud of quantum fluctuations in the form of spin waves. Given this quenched kinetic energy of quantum skyrmions, we identify close analogies to the bosonic quantum Hall problem. In particular, the topological charge density is shown to obey the Girvin-MacDonald-Platzman algebra that describes neutral modes of the lowest Landau level in the fractional quantum Hall problem. Consequently, the conservation of the topological dipole suggests that magnetic skyrmion materials offer a promising platform for exploring fractonic phenomena with close analogies to fractional quantum Hall states.

The quantum spin liquid is a state manifesting extraordinary many-body entanglement, and the material ${\mathrm{NaYbSe}}_2$ is thought to be one of the most promising candidates for its realization. Through low-temperature heat capacity and thermal conductivity measurements, we identify an apparent contradiction familiar to many quantum spin liquid candidates: While entropy is stored by apparently gapless excitations, the itinerant carriers of entropy are gapped. By studying the compositional series ${\mathrm{NaYb}}_x{\mathrm{Lu}}_{1-x}{\mathrm{Se}}_2$ across a percolation transition of the magnetic lattice, we suggest that this contradiction can be resolved by the presence of entanglement scales of random sizes. Moreover, as we truncate the scale of entanglement by magnetic dilution, we show that the itinerant magnetic entropy carrier in ${\mathrm{NaYbSe}}_2$ does not arise from a uniform globally entangled spin ground state but rather materializes through the stochastic propagation of boundaries between locally entangled spin objects.

While extensively studied in normal metals, semimetals, and semiconductors, the superconducting (SC) proximity effect remains elusive in the emerging field of flat-band systems. In this study, we probe proximity-induced superconductivity in Josephson junctions (JJs) formed between superconducting NbTiN electrodes and twisted bilayer graphene (TBG) weak links. Here, the TBG acts as a highly tunable topological flat-band system, which, due to its twist-angle-dependent bandwidth, allows us to probe the SC proximity effect at the crossover from the dispersive to the flat-band limit. Contrary to our original expectations, we find that the induced superconductivity remains strong even in the flat-band limit and gives rise to broad, dome-shaped SC regions, in the filling-dependent phase diagram. In addition, we find that, unlike in conventional JJs, the critical current $I_c$ strongly deviates from a scaling with the normal state conductance $G_N$. We attribute these findings to the onset of strong electron interactions, which can give rise to an excess critical current. By also studying the dependence of $I_c$ on the filling and twist angle across multiple samples, we further uncover the importance of quantum geometric terms as well as multiband pairing mechanisms in describing the induced superconductivity in the TBG flat bands as their bandwidth decreases. To the best of our knowledge, our results present the first detailed study of the SC proximity effect in the flat-band limit and shed new light on the mechanisms that drive the formation of SC domes in flat-band systems.

Gate operations composed in quantum circuits form the basis for digital quantum simulation and quantum processing. While two-qubit gates generally operate on nearest neighbors, many circuits require nonlocal connectivity and necessitate some form of quantum information transport. Yet, connecting distant nodes of a quantum processor still remains challenging, particularly for neutral atoms in optical lattices. Here, we create singlet pairs of two magnetic states of fermionic potassium-40 atoms in an optical lattice and use a bidirectional topological Thouless pump to transport, coherently split, and separate the pairs, as well as to demonstrate interaction between them via tuneable $(\mathrm{SWAP}{)}^{\alpha }$-gate operations. We achieve pumping with a single-shift fidelity of 99.78(3)% over 50 lattice sites and split the pairs within a decoherence-free subspace. Gates are implemented by superexchange interaction, allowing us to produce interwoven atomic singlets. For readout, we apply a magnetic field gradient, resulting in single- and multifrequency singlet-triplet oscillations. Our work shows avenues to create complex patterns of entanglement and new approaches to quantum processing, sensing, and atom interferometry.

Layered oxide materials, with their two-dimensional crystalline architectures and tunable interlayer interaction, serve as a fertile field for harnessing emergent quantum phenomena. Among these materials, metallic delafossites (e.g., ${\mathrm{PdCoO}}_2$) have emerged as a prominent system with extraordinary two-dimensional electronic properties, though their intrinsic lack of ferromagnetism has remained a fundamental constraint. Here, we report the creation of robust, bulk high-temperature ferromagnetism ($T_c>420\mathrm{K}$) in inherently nonmagnetic ${\mathrm{PdCoO}}_2$ through controlled hydrogenation while preserving the delafossite structure. This process induces layer-selective electron doping into ${\mathrm{CoO}}_2$ layers, stabilizing Ising-type ferromagnetism with pronounced perpendicular magnetic anisotropy while preserving the material’s exceptional metallicity. Remarkably, the system self-assembles into a superlattice of alternating metallic Pd and insulating ferromagnetic hydrogenated ${\mathrm{CoO}}_2$ layers, enabling an unconventional anomalous Hall effect mediated by interlayer spin-charge coupling. These findings demonstrate that bulk ferromagnetism can be achieved in delafossite oxides while preserving their structural integrity, positioning hydrogenated delafossites as a versatile platform for exploring correlated quantum effects and designing multifunctional devices.

${\mathrm{TiSe}}_2$ has long been considered one of the best candidate materials to host the elusive excitonic insulator (EI) phase. However, a finite coupling to the lattice can generically be expected, while a lack of “smoking-gun” signatures for the importance of the electron-hole interaction in driving the phase transition has rendered it challenging to distinguish the EI from the conventional charge-density wave (CDW) phase. Here, we demonstrate a new approach, exploiting the susceptibility of excitons to dielectric screening. We combine mechanical exfoliation with molecular-beam epitaxy to fabricate ultraclean van der Waals heterostructures of monolayer (ML) ${\mathrm{TiSe}}_2$/graphite and ML ${\mathrm{TiSe}}_2/h$-BN. We observe how the modified substrate screening environment drives a renormalization of the quasiparticle band gap of the ${\mathrm{TiSe}}_2$ layer, signifying its susceptibility to Coulomb engineering. The temperature-dependent evolution of its electronic structure, however, remains unaffected, indicating that excitons are not required to drive the CDW transition in ${\mathrm{TiSe}}_2$.

We consider the universal aspects of two problems: (i) the singular value structure of a product $M_t=m_t{m}_{t-1}\dots m_1$ of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices $m_i$, representing time steps in the evolution, but the $m_i$ are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.

In the past couple of decades, there have been significant advances in measuring quantum properties of light, such as quadratures of squeezed light and single-photon counting. Here, we explore whether such tools can be leveraged to probe electronic correlations in the many-body quantum regime. Specifically, we show that it is possible to probe certain spin, charge, and topological orders in an electronic system by measuring the correlation functions of scattered photons. We construct a mapping from the correlators of the scattered photons to those of a correlated insulator, particularly for Mott insulators described by a single-band Fermi-Hubbard model at half filling. We show that frequency filtering before photodetection plays a crucial role in determining this mapping. We find that if the ground state of the insulator is a gapped spin liquid, a photon-pair correlation function, i.e., $G^{(2)}$, can detect the presence of anyonic excitations with fractional mutual statistics. Moreover, we show that correlations between electromagnetic quadratures can be used to detect expectation values of static spin chirality operators on both the kagome and triangular lattices, thus being useful in detecting chiral spin liquids. More generally, we show that a series of hitherto unmeasured spin-spin and spin-charge correlation functions of the material can be extracted from photonic correlations. This work opens up access to probe correlated materials, beyond the linear-response paradigm, by detecting quantum properties of scattered light.

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter and can be revealed by nonanalytic points in statistical order parameters. A prime example are quantum speed limits, which mark the onset of controllability as the protocol duration is increased. Here, we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: Utilizing path-integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the quantum speed limit to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable and compare the landscapes for bang-bang and piecewise continuous protocols. Our work provides the first steps toward a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.

Collective modes in multilayer graphene, such as plasmons and phonons, exhibit sensitivity to displacement fields and interlayer coupling, distinguishing them from their counterparts in single-layer graphene. Here, we engineer collective modes in charge-transfer heterostructures composed of multilayer graphene and $\alpha \text{−}{\mathrm{RuCl}}_3$. In heterostructures with a single $\alpha \text{−}{\mathrm{RuCl}}_3$ interface, the charge transfer generates displacement fields up to 7 V/nm at the interface between $\alpha \text{−}{\mathrm{RuCl}}_3$ and the adjacent graphene layer—the highest value achieved through charge-transfer methods. As a result of the broken inversion symmetry, we discover enhanced nonlinear optical response and modified phonon selection rules. Conversely, we find that multilayer graphene sandwiched between two $\alpha \text{−}{\mathrm{RuCl}}_3$ flakes causes displacement fields to cancel. There, we achieve carrier densities as high as $8\times {10}^{13}{\mathrm{cm}}^{-2}$ in multilayer graphene and restore the phonon selection rules to their unperturbed state. Meanwhile, we demonstrate that plasmonic properties derive from the depletion of multiple valence bands. As a result of the quasilinear band dispersion, these “Dirac multiband plasmons” are relatively unaffected by displacement fields. On the other hand, the inverted heterostructure sequence—two multilayer graphene sheets encapsulating $\alpha \text{−}{\mathrm{RuCl}}_3$—activates significant alteration of the plasmons via interlayer plasmon-plasmon coupling. Hence, multilayer graphene and $\alpha \text{−}{\mathrm{RuCl}}_3$ heterostructures offer a gate-free platform for engineering collective modes derived from inversion symmetry and interlayer coupling.

Pauli blocking in Fermi liquids imposes strong phase-space constraints on quasiparticle lifetimes, leading to a well-known quadratic-in-temperature decay rate of quasiparticle modes at low temperatures. In two-dimensional systems, however, even longer-lived modes are predicted (dubbed “odd-parity” modes) that involve a collective deformation of the Fermi distribution. Here, we present an efficient method to evaluate the full spectrum of relaxational eigenmodes of a Fermi liquid within kinetic theory. We employ this method to study the experimentally relevant case of a Fermi liquid with screened Coulomb interactions and map out the decay rates of quasiparticle modes beyond the asymptotic low-temperature limit up to the Fermi temperature, thus covering the entire temperature range of typical experiments. We confirm the existence of anomalously long-lived odd-parity modes and provide a comprehensive classification and detailed analysis of the relaxation spectrum. In particular, we find that (i) the odd-parity effect in the decay rates extends to temperatures as large as $T=0.15T_F$, (ii) there is only a small number of long-lived odd-parity modes, with an infinite number of remaining modes that show standard Fermi-liquid scaling, and (iii) the ratio between the odd- and even-parity lifetimes is tunable with the Coulomb interaction strength, in addition to temperature, which reflects a difference in the microscopic relaxation mechanism of the modes. Our findings provide a comprehensive description of the nonequilibrium relaxation behavior of two-dimensional electron gases and bridge a significant gap in our understanding of these systems.

Tensor networks provide a natural language for noninvertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a noninvertible operator implementing the Wegner duality in $3+1\mathrm{D}$ lattice ${\mathbb{Z}}_2$ gauge theory. The noninvertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the ${\mathbb{Z}}_2$ gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the noninvertible duality operators (including noninvertible parity and reflection symmetries) of generalized Ising models based on graphs, encompassing the $1+1\mathrm{D}$ Ising model, the three-spin Ising model, the Ashkin-Teller model, and the $2+1\mathrm{D}$ plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.

Beginning with high-$T_c$ cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density ${\rho }_{s0}$ is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature ${\sigma }_{\mathrm{dc}}{T}_c$. For conventional, $s$-wave superconductors, such scaling has been shown to be a natural consequence of elastic-scattering disorder, not only in the extreme dirty limit, but across a broad range of scattering parameters. Here we show that when an analogous calculation is carried out for elastic scattering in $d$-wave superconductors, a stark contrast emerges, with ${\rho }_{s0}\propto {({\sigma }_{\mathrm{dc}}{T}_c)}^2$ in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal-Eliashberg treatment of inelastic scattering, we show how the observed Homes scaling is recovered. The normal-state behavior of near-optimally-doped cuprates is dominated by inelastic scattering, but significant deviations from Homes scaling occur for disorder-dominated cuprate systems, such as underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.333}$ and overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, and in very clean materials with little inelastic scattering, such as ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight ${\omega }_{p,D}^2$ and show that a new universal scaling emerges, in which the superfluid fractions of dirty $s$-wave and dirty $d$-wave superconductors coalesce to a single point at which normal-state scattering is occurring at the Planckian bound. The combined result is a new tool for classifying superconductors in terms of order parameter symmetry, as well as scattering strength and character. Although our model starts from a Fermi-liquid assumption, it describes underdoped cuprates surprisingly well.

Electrons trapped on the surface of cryogenic substrates (liquid helium, solid neon, or hydrogen) are an emerging platform for quantum information processing made attractive by the inherent purity of the electron environment, the scalability of trapping devices, and the predicted long lifetime of electron spin states. Here we demonstrate the spatial control and detection of single electrons above the surface of liquid helium at temperatures above 1 K. A superconducting coplanar waveguide resonator is used to read out the charge state of an electron trap defined by gate electrodes beneath the helium surface. Dispersive frequency shifts are observed as the trap is loaded with electrons, from several tens down to single electrons. These frequency shifts are in good agreement with our theoretical model that treats each electron as a classical oscillator coupled to the cavity field. This sensitive charge readout scheme can aid efforts to develop large-scale quantum processors that require the high cooling powers available in cryostats operating above 1 K.

The optical properties of atomically thin semiconductors are dominated by excitons, tightly bound electron-hole pairs, which give rise to particularly rich and remarkable physics. Despite their importance, the microscopic formation mechanisms of excitons remain very poorly understood due to the complex interplay of concurrent phenomena occurring on an ultrafast timescale. Here, we investigate the exciton formation processes in 2D materials based on transition-metal dichalcogenide (TMD) monolayers using a technique based on the control of excitation light polarization. It allows us to distinguish between the two competing models of exciton formation: geminate and bimolecular formation. The geminate process is the direct formation of the exciton from the initially photogenerated electron-hole pair before the loss of correlation between them, whereas the bimolecular process corresponds to the random binding of free-electron-hole pairs from the initially photogenerated plasma. These processes control the exciton formation time. Our findings reveal that the luminescence intensity is higher by up to 40% for circularly polarized excitation compared to linearly polarized excitation for laser energy above the free-carrier gap. We show that this spin-dependent exciton emission is a fingerprint of the bimolecular formation process. Importantly, we observe that exciton linear polarization (valley coherence) persists even for laser excitation energies exceeding the gap. We demonstrate that it is the result of a fraction of excitons formed by a geminate process. This shows that two formation processes coexist for excitation energies above the gap, where both mechanisms operate concurrently. Similar results obtained on the two most emblematic materials of the TMD semiconductor family, ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ monolayers, confirm this dual formation mechanism. These findings overturn the prevailing simplistic view of purely geminate or bimolecular exciton formation and provide crucial insights into exciton physics and means to preserve spin or valley coherence. Our methodology can be applied to a broader range of semiconductor for investigating the intricate interplay between nonresonant excitation conditions, Coulomb interactions, electron-phonon couplings, excitonic dynamics, and quantum coherence.

Two-dimensional (2D) crystals proved revolutionary soon after graphene was discovered in 2004. However, 2D amorphous materials down to a single layer of atoms only became accessible in 2020, and they remain largely unexplored. In particular, the thermophysical properties of amorphous materials are of great interest upon transition from 3D to 2D. Here, we probe and simulate thermal transport in monolayer amorphous carbon (MAC). An ultralow cross-plane thermal conductivity ($\kappa$) is measured for van der Waals stacked multilayers, which is comparable to that of randomly stacked graphene despite the extra disorder in MAC. This result reveals the predominant role of the weak interlayer interactions in 2D materials. Meanwhile, an unexpectedly high in-plane $\kappa$ is obtained for freestanding monolayers, which is a few times higher than what is predicted by conventional wisdom for 3D amorphous carbon with a similar ${\mathrm{sp}}^2$ fraction. This observation is primarily attributed to the dimensionality-induced reduction of anharmonicity and the unique low-frequency out-of-plane vibrational modes in MAC. Amorphous materials at the 2D limit open up new avenues for understanding and manipulating heat at the atomic scale.

Antiferromagnets that break both space-time reversal and translation-spin-rotation symmetries were recently predicted [L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020)] to possess splitting between the otherwise spin-degenerate energy bands even without the relativistic spin-orbit coupling (SOC). Here, we point out that such nonrelativistic spin splitting (NRSS)—in particular, “spin splitting type 4” (SST-4) symmetry-broken antiferromagnets—can be divided into subgroups having distinct patterns of spin splitting and spin textures, depending on additional auxiliary symmetries of spin interconversion and polarity. These SST-4 subgroups include the $\alpha$-type (no spin-interconverting symmetry) having spin splitting at the Brillouin zone center, as well as the $\beta$ subgroup in which a rotation symmetry is applied and determines the alternating spin texture and the $\gamma$ subgroup having exclusively reflection spin-interconverting symmetry. Unlike ferrimagnets, the $\alpha$-type compounds are shown to have tiny net magnetization at finite temperature and thus avoid the adverse effect of the stray field. The $\alpha$ and $\beta$ subgroups can be either polar or nonpolar, whereas the $\gamma$ subgroup is polar only, providing a basis for possible switching by external fields. The combination of NRSS-enabling and auxiliary symmetries is used here as a filter for identifying previously synthesized compounds as specific prototypes. Their characteristic splitting and spin polarization are calculated by density functional theory to the benefit of potential future experimental testing. Interesting results are as follows: (i) SOC-independent NRSS can exceed the magnitude of the SOC-induced Rashba and Dresselhaus spin splitting in semiconductors. (ii) Examples of predicted $\alpha$-type insulating compounds include ${\mathrm{BiCrO}}_3$ (nonpolar) and ${\mathrm{Mn}}_2{\mathrm{ScSbO}}_6$ (polar), the latter having spin splitting of 158 meV and 160 meV in the valence and conduction bands, respectively. (iii) The $\beta$-type (${\text{Cu}}_2{\text{Y}}_2{\text{O}}_5$ and ${\text{FeF}}_2$) and $\gamma$-type compounds (${\text{Mn}}_4{\text{Nb}}_2{\text{O}}_9$ and ${\text{FeScO}}_3$) are distinguished both by their auxiliary symmetries and polarity. The spin textures of $\gamma$-type compounds are mirror reflected with spin degeneracy of the wave vectors on that mirror. These observations will likely broaden the experimental playing field of NRSS physics significantly.

We study novel itinerant phases that can be accessed by doping a fractional quantum anomalous Hall (FQAH) insulator, with a focus on the experimentally observed Jain states at lattice filling $\nu =p/(2p+1)$. Unlike in the lowest Landau level, where charge motion is confined into cyclotron orbits, the charged excitations in the FQAH occupy Bloch states with well-defined crystal momenta. At a nonzero doping density, this feature enables the formation of itinerant states of the doped anyons just beyond the FQAH plateau region. Focusing on the vicinity of $\nu =2/3$, we describe a few possible itinerant states, including a topological superconductor with chiral neutral fermion edge modes as well as a more exotic pair density wave (PDW) superconductor with non-Abelian topological order. A Fermi liquid metal with a doping-induced period-3 charge density wave also occurs naturally in our analysis. This Fermi liquid (as well as the PDW) arises from pairing instabilities of a composite Fermi liquid metal that can emerge near filling $2/3$. Though inspired by the theory of anyon superconductivity, we explain how our construction is qualitatively different. At a general Jain filling $\nu =p/(2p+1)$, the same analytical framework leads to a wider variety of phases, including higher-charge superconductors and generalized composite Fermi liquids. We predict unusual physical signatures associated with each phase and analyze the crossover between different temperature regimes. These results provide a proof-of-principle that exotic itinerant phases can be stabilized by correlations intrinsic to the FQAH setup.

The Středa formula establishes a fundamental physical connection between the topological invariants characterizing the bulk of topological matter and the presence of gapless edge modes. In this work, we extend the Středa formula to periodically driven systems, providing a rigorous framework to elucidate the unconventional bulk-boundary correspondence of Floquet systems while offering a physically grounded link between Floquet winding numbers and tractable response functions. Using the Sambe representation of periodically driven systems, we analyze the response of the unbounded Floquet density of states to a magnetic perturbation. This Floquet-Středa response is regularized through Cesàro summation, yielding a well-defined, quantized result within spectral gaps. The response features two physically distinct contributions: a quantized charge flow between edge and bulk and an anomalous energy flow between the system and the drive, offering new insight into the nature of anomalous edge states. This fundamental result rigorously connects Floquet winding numbers to the orbital magnetization density of Floquet states and holds broadly, from clean to disordered and inhomogeneous systems. This is further supported by providing a real-space formulation of the Floquet-Středa response, which introduces a local topological marker suited for periodically driven settings. In translationally invariant systems, the framework yields a remarkably simple expression for Floquet winding numbers involving geometric properties of Floquet-Bloch bands. A concrete experimental protocol is proposed to extract the Floquet-Středa response via particle-density measurements in systems coupled to engineered baths. Finally, by expressing the topological invariants through the magnetic response of the Floquet density of states, this approach opens a promising route toward the topological characterization of interacting driven phases.