AI for Science

In this paper, we study the Bloch and $\mathcal Q_p$--Carleson measure problems on the unit disc $\mathbb D$. In the Bloch case, for a positive Borel measure $μ$ on $\mathbb D$, we give a complete characterization of the boundedness and compactness of the embedding $$ \operatorname{id}:\mathcal B \longrightarrow L^2(μ) $$ in terms of the Bloch capacity $\mathfrak B_{\mathcal R}(μ)$ associated with an admissible dyadic resolution $\mathcal R$ of $\mathbb D$. The proof is based on the Bergman projection representation of Bloch functions, conditional expectations on admissible dyadic resolutions, and a finite-dimensional semidefinite programming argument. We also adapt this dyadic framework to the more general $\mathcal Q_p$--Carleson measure problem and obtain a corresponding complete boundedness and compactness characterization for $$ \operatorname{id}:\mathcal Q_p \longrightarrow L^2(μ), \qquad 0<p\le1. $$ This work further develops the dyadic approach introduced in our recent work on composition operators on $\mathcal Q_p$ spaces, but in a different setting where the embedding involves recovering function values from derivative information.

We develop a control-theoretic framework for understanding Quantum Advantage (QA), providing a systematic route to characterize when and how QA can arise. The bilinear controlled Schrödinger equation is the common thread: the target quantum computation is recast as an operator controllability problem on the special unitary group $SU(N)$, and QA is identified with a polynomial-in-$n$ upper bound on the associated minimal-time function. We illustrate the framework on two paradigmatic problems: a) the Quantum Fourier Transform (QFT) on superconducting digital quantum processors (such as IBM's ibm_brisbane), for which we prove operator controllability by a Lie-algebraic argument and derive an $O(n^2)$ upper bound on the minimal time via a gate-concatenation lemma combined with the standard QFT circuit decomposition; b) the Maximum Independent Set (MIS) problem on neutral-atom analog quantum processors (such as Pasqal's hardware), for which we analyze the Rydberg-blockade Hamiltonian as a bilinear control system and reformulate the Quantum Approximate Optimization Algorithm (QAOA) as a continuous-time optimal control problem. By a controllability result, we show how the problem can be solved on Pasqal Quantum Computers and we introduce a control-based definition of Quantum Advantage for MIS. We conclude by outlining several open problems that chart directions for future research at the intersection of Control Theory and Quantum Computing.

We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-Hölderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular.

A proper-time method for constructing models of dynamic gravitational-wave fields is presented. Using the proper-time method, analytical (not numerical) models of secondary gravitational waves are constructed as perturbative solutions of linearized field equations against the background of the exact wave solution of Einstein's equations for the vacuum in the Bianchi VI universe in a privileged wave coordinate system. Relations for the proper time of test particles against the background of a strong gravitational wave are used. The analytical form of the metric components for secondary gravitational waves is found from compatibility conditions for the field equations. From the field equations, an explicit form of ordinary differential equations and their solutions is obtained for functions included in small corrections to the metric for secondary gravitational waves. It is shown that there exists a continuum of gravitational wave parameters for which the perturbative solutions are stable.

The rapid decay of target signal strength with distance from the sensor presents a key challenge in nanoscale magnetic sensing with nitrogen-vacancy (NV) centers in diamond, limiting both sensitivity and spatial resolution. Here we introduce a strategy to overcome this limitation by using radical anions formed from rhodamine-derived dyes as reporter spins localized to the diamond surface. These radicals, generated through photoreduction, are optically identifiable and stable on timescales exceeding an hour. We experimentally demonstrate their coherent manipulation and detection using single, shallow NV centers for readout. We observe heterogeneity in the local magnetic environments of the photoactivated spins from site to site, likely due to variations in inter-radical couplings across our measurements. Looking forward, our approach enables correlative nanoscale magnetic and optical imaging, and opens new pathways toward single-molecule magnetic resonance studies.

Metastable atomic qubits are a highly promising platform for the realization of quantum computers, owing to their scalability and the possibility of converting leakage errors to erasure errors mid-circuit. Here, we demonstrate and characterize a universal gate set for the metastable fine-structure qubit encoded between the $^3\text{P}_0$ and $^3\text{P}_2$ states in bosonic strontium-88. We find single-qubit gate fidelities of 0.993(1), and two-qubit gate fidelities of 0.9945(6) after correcting for losses during the gate operation. Furthermore, we present a novel state-resolved detection scheme for the two fine-structure states that enables high-fidelity detection of qubit loss. Finally, we leverage the existence of a stable ground state outside the qubit subspace to perform mid-circuit erasure conversion using fast destructive imaging. Our results establish the strontium fine-structure qubit as a promising candidate for near-term error-corrected quantum computers, offering unique scaling perspectives.

The Schur index of a $4$ dimensional $\mathcal{N}=2$ superconformal field theory counts (with sign) bosonic and fermionic states that preserve $4$ supercharges. We consider the Schur indices of $4$d $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=2$ circular quiver gauge theories with gauge groups $U(N)$ or $SU(N)$. We calculate the exponentially dominant part of their asymptotic expansions as the index parameter $q$ approaches any root of unity. We find that some of the indices exhibit ``small" ($\mathcal{O}(N^0)$ as $N \rightarrow \infty$) exponential growth, which is much smaller than an $\mathcal{O}(N^2)$ exponential growth of states that is indicative of a black hole. This implies that the indices do not capture a growth of states that would correspond to a supersymmetric black hole that preserves 4 supercharges in the holographic dual AdS theory. Interestingly, the exponentially dominant part in the Schur asymptotics we consider, depends on the parity of the rank $N$.

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and only uses the properties of the Kolmogorov operator. The new second FDT expresses the memory kernel of the generalized Langevin equation as the correlation function of the fluctuation force plus an additional term. In particular, we show that for nonequilibrium states such as heat transport between two thermostats with different temperatures, the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The obtained theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that the new model of heat transfer yields better predictions than the Green-Kubo formula for systems far from the equilibrium.

Laminar-to-turbulent transition delay is a key challenge in hypersonic boundary-layer flows. Unstable disturbances-most prominently the first and second modes-trigger the onset of turbulence and pose a fundamental technological barrier to hypersonic transport. While existing control strategies target the second mode, simultaneous mitigation of the first mode has long appeared physically impossible. A new flow-control concept is introduced in which phase relations between wall pressure and velocity fluctuations are tailored using subsurface phonon engineering to control both modes concurrently. The outcome is substantial drag reduction and alleviation of the extreme thermal loads associated with turbulence.

Urban venues serve as arenas for social mixing. While residential and activity-space segregation have been extensively studied, less is known about how the spatial structure of cities, particularly public transit infrastructure, shapes the geography of social mixing at specific locations. This study examines how transit accessibility associates with visitor diversity -- the compositional heterogeneity of visitors sharing a venue, used here as an indicator of social mixing potential -- at points of interest (POIs) in nine cities in Sweden and three cities in the United States (New York, Washington DC, Atlanta). Using mobile phone GPS data in 2024, we compute visitor diversity indices based on the birth background composition of visitors' home neighborhoods. Transit catchment diversity positively predicts visitor diversity, but this association is robust only in the largest metropolitan areas; in smaller Swedish cities, the coefficient attenuates to insignificance once geographic catchment composition, centrality, and venue density are controlled. Transit-diversity hotspots concentrate not in already diverse venues, but in lower-diversity POIs with lower commercial density, greater distance from transit in US cities, and greater centrality in Sweden. These patterns are consistent with transit infrastructure playing a bridging role, linking diverse populations to venues where alternative pathways are limited.

This article presents a fast direct solver, termed Algebraic Inverse Fast Multipole Method (from now on abbreviated as AIFMM), for linear systems arising out of $N$-body problems. AIFMM relies on the following three main ideas: (i) Certain sub-blocks in the matrix corresponding to $N$-body problems can be efficiently represented as low-rank matrices; (ii) The low-rank sub-blocks in the above matrix are leveraged to construct an extended sparse linear system; (iii) While solving the extended sparse linear system, certain fill-ins that arise in the elimination phase are represented as low-rank matrices and are "redirected" though other variables maintaining zero fill-in sparsity. The main highlights of this article are the following: (i) Our method is completely algebraic (as opposed to the existing Inverse Fast Multipole Method~\cite{ arXiv:1407.1572,doi:10.1137/15M1034477,TAKAHASHI2017406}, from now on abbreviated as IFMM). We rely on our new Nested Cross Approximation~\cite{arXiv:2203.14832} (from now on abbreviated as NNCA) to represent the matrix arising out of $N$-body problems. (ii) A significant contribution is that the algorithm presented in this article is more efficient than the existing IFMMs. In the existing IFMMs, the fill-ins are compressed and redirected as and when they are created. Whereas in this article, we update the fill-ins first without affecting the computational complexity. We then compress and redirect them only once. (iii) Another noteworthy contribution of this article is that we provide a comparison of AIFMM with Hierarchical Off-Diagonal Low-Rank (from now on abbreviated as HODLR) based fast direct solver and NNCA powered GMRES based fast iterative solver. (iv) Additionally, AIFMM is also demonstrated as a preconditioner.

We develop indexed Bellman information complexity, a representation-level theory of interactive decision making centered on information indices and reference histories. The representation strips away problem-specific syntax and retains only the ingredients needed for dynamic programming and information accounting, thereby unifying the earlier framework of indexed algorithmic information ratios (AIR). On the upper-bound side, regret is controlled by Bellman supersolutions or potential identities whose gradient bracket is paid for by indexed information. Upper-confidence-bound (UCB), estimation-to-decision/decision-estimation-coefficient (E2D/DEC), and adaptive-minimax-sampling or exploration-by-optimization (AMS/EBO) methods appear as three relaxations of this same identity. On the lower-bound side, the posterior-reference trajectory supplies both the information telescope and the ghost quantile of small-regret trajectories. The resulting critical radius in the lower bound is an effective-dimension-scale quantity, as in Fano and local-prior-mass lower bounds, rather than the constant radius of a two-point Le Cam argument. The examples show that DEC is best viewed as a one-step relaxation of indexed Bellman information complexity, not as a universally tight conversion mechanism. We illustrate the framework through several applications, with particular emphasis on kernel bandits. In this setting, the active action marginal provides a concrete basis for comparing UCB, E2D, and AMS/EBO.

We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic of McCarthy (the non-commutative version of Kleene three-valued logics). Our analysis will be conducted in a very general algebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) operation $'$ satisfying $x''\approx x$; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provides an algebraic common ground for several non-classical logics. Our main contributions consist in providing for McCarthy algebras: reduced and equivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.

We prove that the minimal size $M(π_n)$ of a maximal matching in the permutahedron $π_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(π_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(π_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(π_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.

Equitable access to essential urban services is a pillar of modern planning, yet most accessibility models rely strictly on static residential locations, ignoring how demand shifts throughout the daily loop. This study introduces a population-based, temporally differentiated framework to examine the resulting "moving target" of urban equity, focusing on medical facilities and green spaces in Hefei, China. Utilising large-scale mobile phone GPS data, we construct dynamic residential and workplace population exposure surfaces to capture shifting hourly demand. We then evaluate accessibility via network-based travel times paired with a novel per-capita provision metric that accounts for real-time demand competition. We define \textit{double disadvantage} as the co-occurrence of poor spatial accessibility and insufficient per-capita service availability. Counterintuitively, the results reveal that double-disadvantaged areas cluster primarily along the inner suburban belt rather than the remote periphery, where per-capita service provision remains relatively sufficient. Furthermore, temporal shifts drastically alter equity landscapes: daytime workplace concentrations intensely exacerbate demand competition in urban job centres. These findings demonstrate that urban inequality depends heavily on spatiotemporal population flows rather than just the fixed location of services. Ultimately, achieving true urban equity requires dynamic planning interventions that address time-varying demand rather than focusing solely on static, home-based metrics.

Increasing optical communication capacity without proportionally increasing receiver complexity remains a key challenge for direct-detection links. Conventional systems typically assign wavelength, polarization and intensity to fixed, separately recovered functions, so that alphabet expansion is accompanied by additional demultiplexing, polarization handling, receiver branches and electronic processing. Here we introduce a unified resource-pool architecture for high-dimensional direct-detection optical communication, in which wavelength, polarization and intensity are jointly organized as a composite optical symbol space and recovered through optical-domain joint projection rather than dimension-by-dimension separation. The receiver is implemented with an integrated disordered photonic processor that transforms each composite optical state into a reproducible multi-output electrical fingerprint for single-shot direct recovery. In a dual-wavelength transmission experiment, the system resolves 4096 composite symbols, corresponding to 12 bits per symbol slot, with a bit error rate of 4.25e-4 after 10 km standard-fiber transmission. Additional experiments demonstrate dense polarization alphabets, wavelength-indexed state-space expansion and high-launch-power operation over hollow-core fiber. These results establish disorder-enabled joint projection in an integrated photonic processor as a route to hardware-efficient high-dimensional direct-detection communication beyond conventional dimension-partitioned receiver architecture.

In the last years, an anomalously high spreading of West Nile virus (WNV) has been observed in Italy, with particularly high peaks of infections in southern Lazio, Campania and Veneto regions. The main disease vector for WNV is represented by Culex pipiens mosquitoes, which spread human infections through their bites. Here, we investigate WNV fever epidemic diffusion during summer season 2025 in Italy through a computational approach based on a quantum version of the Game of Life (GOL) cellular automaton model. Specifically, human dynamics evolves according to the GOL rules, while stochastic dynamics of disease vectors, i.e., mosquitoes, as well as their interaction with humans, simultaneously occur. We show that this model fits the curves of cumulative infected individuals with high accuracy, either at local and average-regional level, with only optimization of mosquito birth and removal rates parameters. Furthermore, leveraging model flexibility, we show that changes in model parameters values elucidate system response to environmental variations. For instance, we quantify, e.g., the impact of mosquito spreading containment measures or sudden mosquito increasing abundance due to climatic and ecological changes. Overall, we provide a general, quantitative description of WNV infection spreading in Italy which could represent a supportive tool to test different environmental scenarios and could be useful to devise strategies for decision makers to monitor disease vector dynamics and to control consequent virus diffusion.

Superconductors are many-body quantum states in which current flows without dissipation. Theory predicts that supercurrents follow a relatively simple spatial pattern in both conventional and unconventional superconductors. Recent studies into the AV3Sb5 (A = Cs, K, Rb) family of Kagome superconductors indicate that CsV3Sb5 has unconventional transport properties that cannot be accounted for with these simple theories, including reports of intrinsic Josephson junctions, higher order Cooper pairing and the zero field diode effect. Attempts to interpret these findings have focused on the interplay of superconductivity with the unconventional charge density wave (CDW) order in these materials, with which superconductivity competes. A current roadblock to understanding how these kagome superconductors give rise to their intriguing properties is the lack of spatially resolved information about transport. Here we show, using a recently developed superconducting quantum interference device (SQUID) microscope, that flakes of CsV3Sb5-xSnx host a network of narrow supercurrent channels. These supercurrent channels emerge at the critical temperature and remain stable for all temperatures and currents. Their non-linear behaviour is consistent with a network of Josephson junctions linked by narrow supercurrent filaments, which naturally leads to the observed transport anomalies. Intriguingly, these observations are much weaker in undoped samples, which suggests links to the physics of charge density waves, disorder, and electronic correlations, all of which are greatly influenced by the doping strength. These results establish new frontiers for the local investigation of charge transport and competing orders in strongly correlated electron systems, and shine a new light on the anomalous transport properties of the AV3Sb5 kagome superconductors.

We investigate terahertz (THz) harmonic generation in the InSb/CdTe heterostructure, demonstrating, for the first time, efficient in-plane magnetic field-induced second-harmonic generation (SHG). We also achieve significant third-harmonic generation (THG), rivalling Dirac materials such as graphene and Cd3As2. Our theoretical analysis identifies the primary SHG mechanism as the orbital-Zeeman correction to Drude conductivity, while the dominant THG contribution also shows Drude-like behavior. The results provide a general route to efficient THz harmonic generation in high mobility materials.

We propose a scheme for realizing thermal logic gates at the nanoscale using a phononic ring system. Two atomic sites, placed in close proximity to the ring, serve as the inputs for two-input logic operations, while a single proximity site is employed for single-input logic functionality. The logic output is encoded in the phonon transmission probability, which is calculated within the framework of non-equilibrium Green's function formalism. By appropriately tuning the ring-electrode junction configuration, all seven standard logic gates, comprising three fundamental and four combinatorial operations, are successfully realized in different phonon frequency regimes. Our results suggest that the proposed logic operations remain valid over a broad range of phonon frequencies, highlighting the generality and reliability of the proposed approach.

A magnetic skyrmion tube is a three-dimensional topological soliton formed by stacking two-dimensional skyrmions along the out-of-plane direction. Recent real-space observations of skyrmion tubes have stimulated growing interest in their dynamics and emergent properties. Here, we go beyond simple skyrmion stacking and investigate how a ``twist" introduced along the tube direction affects the dynamics and emergent responses of skyrmion tubes. We find that such a twist can be created as a localized texture, termed a twist soliton, through thermal quench dynamics. By complementarily combining large-scale numerical simulations with analytical calculations based on collective coordinates, we clarify its current-driven nonlinear motions that depend on its twist chirality. Remarkably, its velocity can be substantially enhanced by a magnetic-field component perpendicular to the tube. Furthermore, the associated emergent electric field enables identification of the twist soliton, including the sign of its chirality, through Hall measurements. Our results reveal the twist degree of freedom as an essential ingredient of skyrmion-tube physics and pave the way for the development of spintronic devices exploiting the three-dimensional nature of spin textures.

Given a closed, oriented Lagrangian submanifold $L$ in a Liouville domain $\overline{M}$, one can define a Maurer-Cartan element with respect to a certain $L_\infty$-structure on the string homology $\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R})$, completed with respect to the action filtration. When the first Gutt-Hutchings capacity of $\overline{M}$ is finite, and $L$ is a $K(π,1)$ space, we show that $L$ bounds a pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin's conjecture and generalizes the works of Fukaya and Irie in the case of $\mathbb{C}^n$ to a wide class of Liouville manifolds, which includes low degree smooth affine hypersurfaces in $\mathbb{C}^{n+1}$. In particular, when $\dim_\mathbb{R}(\overline{M})=6$, every closed, orientable, prime Lagrangian 3-manifold $L\subset\overline{M}$ is diffeomorphic either to a spherical space form, or $S^1\timesΣ_g$, where $Σ_g$ is a closed oriented surface.

Merge trees are a type of graph-based topological summary that tracks the evolution of connected components in the sublevel sets of scalar functions. They enjoy widespread applications in data analysis and scientific visualization. In this paper, we consider the problem of comparing two merge trees via the notion of interleaving distance in the metric space setting. We investigate various theoretical properties of such a metric. In particular, we show that the interleaving distance is intrinsic on the space of labeled merge trees and provide an algorithm to construct metric 1-centers for collections of labeled merge trees. We further prove that the intrinsic property of the interleaving distance also holds for the space of unlabeled merge trees. Our results are a first step toward performing statistics on graph-based topological summaries.

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze the topological persistence over spatial scales, persistent homology has rarely been devised for time evolution. We propose the use of a new filtration function for persistent homology which takes as input the adjacent oscillator trajectories of a dynamical system. We also regulate the dynamical system by a weighted graph Laplacian matrix derived from the network of interest, which embeds the topological connectivity of the network into the dynamical system. The resulting topological signatures, which we call evolutionary homology (EH) barcodes, reveal the topology-function relationship of the network and thus give rise to the quantitative analysis of nodal properties. The proposed EH is applied to protein residue networks for protein thermal fluctuation analysis, rendering the most accurate B-factor prediction of a set of 364 proteins. This work extends the utility of dynamical systems to the quantitative modeling and analysis of realistic physical systems.

A comprehensive methodology is developed to extract electron and hole effective masses in degenerate semiconductors through a simultaneous global fit of carrier concentration dependence of bandgap and plasma energy, explicitly incorporating band nonparabolicity. Broadband spectroscopic ellipsometry combined with Hall effect analyses enables accurate determination of the bandgap, plasma energy and carrier concentrations. The dielectric function of sputtered Al-doped ZnO thin films are modeled in the fundamental absorption region using an Elliott based model with overlapping excitonic transitions and Urbach tails, while free carrier absorption is described by a modified sernelius formula. Wide carrier concentrations are achieved via controlled deposition and post-annealing, revealing changes in electron effective masses and deviations from parabolic dispersion. Two nonparabolic models are compared, Pisarkiewicz, assuming spherically symmetric band with a step-function approximation of the Fermi-Dirac distribution and Nilsson, incorporating thermal and impurity effects. The latter is shown to capture accurately band nonparabolicity, yielding effective masses and nonparabolicity parameter consistent with bandgap evolution. This approach quantitatively separates Burstein-Moss shift and bandgap renormalization, reproducing carrier dependent bandgap shifts across a wide concentration range. Neglecting valence band contributions introduces systematic bias. Bandgap renormalization is further evaluated using plasmon pole and random phase approximations, underscoring the importance of many-body screening. This framework also enables determination of the Mott critical concentration and the fundamental absorption edge onset. Collectively, these results establish a reliable methodology for extracting band-structure parameters and bandgap shifts, extendable to other transparent conducting oxides.

Superatomic graphene platforms host a rich portfolio of flat-band-driven exotic quantum properties, yet their experimental realization remains challenging. Here, we report the bottom-up on-surface synthesis of superatomic graphene using phosphorus-doped triangulene as building blocks. Scanning tunneling microscopy and spectroscopy measurements resolve the well-defined honeycomb lattice of as-fabricated superatomic graphene and demonstrate the characteristic Dirac band and flat band electronic structures. Density functional theory calculations reveal that the flat bands originate from the in-plane p$_x,_y$-like frontier orbitals of the phosphorus-doped triangulene units, leading to intrinsic half-metallic behavior. Furthermore, oxygen functionalization of the molecular precursor enables deterministic modulation of the electronic structure and magnetic ordering. This work establishes a general platform for designing correlated quantum materials with tunable flat band properties.

Solution-processed thin films of colloidal lead halide perovskite (LHP) nanocrystals (NCs) show great potential for the implementation into optoelectronic devices such as light-emitting diodes (LEDs), lasers, and solar cells. However, these hybrid LHP NC solids exhibit non-negligible size and shape polydispersity, which introduces both structural and energetic disorder. Here, we resolve the exciton dynamics in space, time, and energy to elucidate the impact of different forms of disorder (structural and energetic) on exciton transport. We show that the disorder depends sensitively on the length of the alkylamine ligand used in the synthesis. While shorter alkyl chain lengths lead to high polydispersity, longer alkyl chains lead to more monodispersed and smaller particles where quantum confinement becomes more pronounced and, consequently, lead to increased energetic disorder. Strikingly, we find that exciton transport is less efficient in NC solids with long alkyl chain ligands, despite having a significantly more monodisperse ensemble. This demonstrates that energetic disorder, rather than structural disorder, is the dominant factor for predicting exciton transport within these materials. These findings reveal the critical role of ligand engineering in designing high-performance optoelectronic devices based on hybrid LHP NCs, providing new insights into energy transport dynamics in disordered systems and highlighting the versatility of these materials for advanced photonic and optoelectronic applications.