Invariance of on-shell scattering amplitudes under field redefinitions is a well known property in field theory that corresponds to covariance of on-shell amputated connected functions. In recent years there have been great efforts to define a formalism in which the covariance is manifest at all stages of calculation, mainly resorting to geometrical interpretations. In this work covariance is analysed using combinatorial methods relying only on the properties of the tree level effective action, without referring to specific formulations of the Lagrangian. We provide an explicit proof of covariance of on-shell connected functions and of the existence of covariant Feynman rules and we derive an explicitly covariant closed formula for tree level on-shell connected functions with any number of external legs.

Using time-dependent density functional theory, we study how density and size of impurities govern dissipation of persistent currents of fermionic superfluid rings in the BCS regime. The critical winding number for vortex emission increases with impurity density, but this enhancement is impurity size-dependent and capped by the pair-breaking threshold. Below this vortex-emission threshold, the winding number remains constant while flow energy dissipates through impurity-enhanced pair-breaking. Above the threshold, vortex-impurity interactions produce distinct mobility regimes-deflected trajectories, individual pinning, collective pinning, and inter-site hopping, controlled by the impurity size and density, which determine the dominant dissipation channel. These findings provide design principles for ultracold-atom experiments and insights into vortex-pinning dynamics in neutron-star crusts and superconductors.

We investigate the formation of primordial black holes (PBHs) induced by a first-order electroweak phase transition in a realistic renormalizable framework, the complex singlet extension of the Standard Model. We perform a quantitative analysis of the PBH abundance and identify parameter regions consistent with current microlensing constraints. Furthermore, we show that the same parameter space predicts observable stochastic gravitational waves within the sensitivities of future space-based detectors, as well as a sizable deviation in the Higgs triple coupling that can be probed at future lepton colliders. Our results highlight a comprehensive multimessenger framework in which PBH, gravitational wave, and collider observations can jointly test the dynamics of a strongly first-order electroweak phase transition in the early Universe.

Universal machine learning interatomic potentials have emerged as efficient tools for materials simulation, yet their reliability for elastic property prediction remains unclear. Here, we present a systematic benchmark of four uMLIPs -- MatterSim, MACE, SevenNet, and CHGNet -- against first-principles data for nearly 11\,000 elastically stable materials from the Materials Project database. The results show that SevenNet achieves the highest accuracy, MACE and MatterSim balance accuracy with efficiency, while CHGNet performs less effectively overall. To further improve predictive quality, we perform targeted fine-tuning on all four uMLIPs using strained configurations derived from 185 high-error materials. After fine-tuning, CHGNet shows the most substantial improvement in overall accuracy, with MatterSim and SevenNet also benefiting from the fine-tuning, whereas MACE shows limited robustness to this procedure. This work provides quantitative guidance for model selection and data refinement, advancing uMLIPs toward reliable applications in mechanical property prediction.

We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous non-critical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents $β=0.4999$ and $δ=3.007$, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multi-state spin models.

We investigate the accumulation of null matter at the stable photon sphere in the Mannheim-Kazanas metric, the analogue to the Schwarzschild solution in Weyl's conformal theory of gravity. In our toy problem in which we consider an infinitely-thin shell, we find that a jump in radial pressure ${T^r}_r$ is induced across the shell unless the shell has a radius of either the unstable or stable photon sphere radii. We then find that upon loading the stable photon sphere, its area remains invariant. Furthermore, at a critical threshold loading limit for this zero-width null matter shell, we are able to produce a metric containing an extremal horizon with an AdS$_2\times$S$^2$ geometry completely independent of the cosmological curvature. This hitherto unencountered and therefore unexpected result is a phenomenon unseen in standard nonconformal second-order metrics with nonzero cosmological constants.

We investigate the coupling of a multi-mode metal-insulator-metal cavity to a two-dimensional electron gas (2DEG) in a quantum well in the presence of a strong magnetic field. The TM cavity mode is strongly hybridized with an intersubband transition of the 2DEG, forming a polaritonic mode in the ultrastrong coupling regime, while the TE mode remains an almost purely cavity mode. The magnetoplasmon excitation emerging from the presence of the magnetic field couples with both TM and TE modes, exhibiting different coupling strengths and levels of spatial field inhomogeneity. While the strong homogeneity of the bare TE mode gives rise to the standard anticrossing of strong coupling, the inhomogeneous polaritonic TM mode is shown to activate an observable Coulombic effect in the spectral response, often referred to as non-locality. This experiment demonstrates a cavity-induced modification of the 2DEG response and offers a new route to probing the effect of Coulomb interactions in ultrastrongly coupled systems via reshaping of their cavity mode profiles.

The Electron-Ion Collider (EIC) is a next-generation accelerator primarily designed to study the internal structure of nucleons through high-precision electron-hadron collisions. In this work, we explore the feasibility of employing a 1 MW fraction of the EIC proton beam to generate a high-intensity GeV-scale neutrino beam for long-baseline oscillation studies. We have simulated proton-target interactions and optimize the resulting neutrino fluxes for water-based liquid scintillator (WbLS) detectors located at distinct baselines of 900 km and 2900 km. Oscillation analyses performed with GLoBES show that extended baselines allow access to multiple oscillation maxima, significantly enhancing sensitivity to leptonic CP violation. The study also examines the interplay between matter effects and the intrinsic CP violating phase in shaping observable asymmetries. We note that simplified systematics and no backgrounds are used in this analysis to establish the baseline physics potential. These results suggest that the EIC proton beam could provide a novel and complementary source for precision neutrino physics, extending the scientific reach of the EIC program.

In the recent years, argon-based experiments looking for Dark Matter in the Universe have explored the non-standard scenario in which Dark Matter is made by low-mass Weakly Interacting Massive Particles, of mass in the range of 1-10 GeV instead of the canonical hundreds of GeV. Detecting such particles is challenging, as their expected signatures are nuclear recoils with energies below 10 keV, observable solely via ionization. This necessitates a precise understanding of the detector response in this energy regime, which remains incomplete for argon. To address this, the ReD experiment was developed within the framework of the DarkSide-20k Collaboration to produce and characterize few-keV nuclear recoils. A compact dual-phase argon Time Projection Chamber (TPC) was irradiated with neutrons from a Cf252 source, to produce Ar recoils in the energy range of interest via (n,n') elastic scattering. A downstream spectrometer composed of 18 plastic scintillators detected the neutrons scattered off Ar nuclei, enabling recoil energy reconstruction via two-body kinematics. The ionization yield Qy of argon, defined as the number of electrons produced per unit energy deposit, was measured in a model-independent way between 2 and 10 keV. These measurements extend direct experimental coverage well below the previous limit of approximately 7 keV. The results are consistent with existing data above 7 keV, while they indicate a higher Qy at lower energies.

We report the precise X-ray localization of GRB 230906A, a short duration ($T_{90}\sim$0.9 s) burst with no optical or radio counterpart. Deep imaging with the Hubble Space Telescope detects a faint galaxy (G$^\ast$; $F160W\simeq26$ AB mag) coincident with the sub-arcsecond X-ray position. Compared with standard GRB galaxies, its faintness, compact size and color would suggest a high redshift ($z\gtrsim$3) host. However, our observations also reveal the presence of a galaxy group at $z\!\sim$0.453, confirmed spectroscopically with VLT/MUSE, with clear signs of interactions and mergers among group members. The GRB and its putative host project onto an extended ($\approx$180 kpc) tidal tail emerging from the group's central galaxy. The probability of a chance alignment is small ($P_{cc}\!\lesssim\!4$%), we thus argue that the GRB and its galaxy G$^*$ reside within the group. Their peculiar location along the tidal debris suggests that an enhanced burst of star formation, induced by the galaxy merger, might have formed the progenitor compact binary $\lesssim$700 Myr ago. The compact binary later evolved in a neutron star merger which produced GRB 230906A and injected $r$-process material into the surrounding circumgalactic medium.

We report a method to grow B20 MnGe thin films using molecular-beam epitaxy, which employs an ultrathin CrSi template layer on Si(111). This layer is expected to be nonmagnetic, in contrast to MnSi and FeGe buffer layers that have been used previously. This template layer permits an investigation of the intrinsic properties of MnGe in the ultrathin-film limit without the influence of a neighboring magnetic layer. Single-phase MnGe(111) films were grown with thicknesses between 2 and 40 nm, which exhibited low interfacial roughnesses on the order of 0.6 nm. The films crystallized in a B20 structure with a small rhombohedral distortion. Magnetometry measurements in out-of-plane fields are consistent with a cone phase derived from helimagnetic order propagating along the film normal. However, an unexpected remanent moment develops below 35K, concomitant with features in the field dependence of the transport data. This provides indirect evidence for the presence of a low-temperature phase which has been identified by others as either a triple-Q topological spin-hedgehog lattice, or a multidomain single-Q helical state.

In the context of metric measure spaces, we introduce an axiomatic formulation of mixed local and nonlocal $p$-energy forms. Within this framework, we use the Poincaré inequality, the cutoff Sobolev inequality, and mild assumptions on the jump measure to establish the weak and strong elliptic Harnack inequalities for such mixed forms. Our approach is based on the De Giorgi--Nash--Moser method and extends the corresponding results for Dirichlet forms without the killing part, as well as for mixed energy forms on Euclidean spaces. Some consequences of elliptic Harnack inequalities are discussed.

This paper addresses the single-item lot sizing problem with a 1-breakpoint all-units quantity discount in a monotonic setting where the purchase prices are non-increasing over the planning horizon. For this case, we establish several novel properties of the optimal solution and develop a hybrid dynamic programming approach that maintains a compact representation of the solution space by storing only essential information about the states and using linear equations for intermediate values. Our algorithm runs in \(O(n\log n)\) time, where \(n\) denotes the number of periods. Our result is an improvement over the previous state-of-the-art algorithm, which has an \(O(n^2)\) time complexity.

In 2021, Chen, Liu, and Zhandry presented an efficient quantum algorithm for the average-case $\ell_\infty$-Short Integer Solution ($\mathrm{SIS}^\infty$) problem, in a parameter range outside the normal range of cryptographic interest, but still with no known efficient classical algorithm. This was particularly exciting since $\mathrm{SIS}^\infty$ is a simple problem without structure, and their algorithmic techniques were different from those used in prior exponential quantum speedups. We present efficient classical algorithms for all of the $\mathrm{SIS}^\infty$ and (more general) Constrained Integer Solution problems studied in their paper, showing there is no exponential quantum speedup anymore.

We construct relatively bounded toroidal and toric models of relatively bounded fibrations over curves.

The intrinsic width and scattering distributions of fast radio bursts (FRBs) inform on their emission mechanism and local environment, and act as a source of detection bias and, hence, an obfuscating factor when performing FRB population and cosmological studies. Here, we utilise a sample of 29 FRBs with measured high-time-resolution properties and known redshift, which were detected using the Australian Square Kilometre Array Pathfinder (ASKAP) by the Commensal Real-time ASKAP Fast Transients Survey (CRAFT), to model these distributions. Using this sample, we estimate the completeness bias of intrinsic width and scattering measurements, and fit the underlying, de-biased distributions in the host rest-frame. In no case do our model fits prefer a down-turn at high values of the intrinsic distributions of either parameter in the 0.01-40 ms range probed by the data. Rather, when assuming a spectral scattering index of $α= -4$, we find that the intrinsic scattering distribution at 1\,GHz is consistent with a log-uniform distribution above 0.04 ms, and that this functional form is strongly favoured over the lognormal descriptions used by previous works. We also find that the intrinsic width distribution rises as a Gaussian in log-space in the 0.03-0.3 ms range, with a log-uniform distribution above that slightly preferred to a lognormal distribution. This confirms previous works suggesting that FRB observations are currently strongly width- and scattering-limited, and we encourage FRB searches to be extended to higher values of time-width. It also implies a bias in FRB host galaxy studies, although the form of that bias is uncertain. Finally, we find that our updated width and scattering models - when implemented in the zDM code - produce $\sim$10% more FRBs at redshift z=1 than at z=0 when compared to alternative width/scattering models.

Observations have repeatedly confirmed the presence of large-scale peculiar motions in the universe, commonly referred to as ``bulk flows''. These are vast regions of the observable universe, typically spanning scales of several hundred Mpc, that move coherently with speeds of the order of several hundred km/sec. While there is a general consensus on the direction of these motions, discrepancies persist in their reported sizes and velocities, with some of them exceeding the predictions of the standard $Λ$CDM model. The observed large-scale peculiar-velocity fields are believed to have originated as weak peculiar-velocity perturbations soon after equipartition, which have subsequently grown by structure formation and by the increasing inhomogeneity of the post-recombination universe. However, the evolution and the implications of these bulk velocity fields remain poorly understood and they are still a matter of debate. For instance, it remains a challenge for the theoreticians to explain the high velocities measured by several bulk-flow surveys, like those recently reported using the CosmicFlows-4 data. Such extensive and fast velocity fields could have played a non-negligible role during structure formation and they might have also ``contaminated'' our observations. After all, in the history of astronomy, there are examples where relative-motion effects have led us to a serious misinterpretation of reality (shortened abstract due to length limits).

The $Z^0$ bosons produced in electron-positron collisions at the potential Future Circular Collider (FCC-ee) provide unique opportunities for flavour physics. The non-zero polarization of \Lb baryons produced in $Z^0$ decays enables access to a much larger set of observables than at the LHC, where the $Λ_b^0$ baryons are produced unpolarized. This paper presents a toy angular analysis of $Λ_b^0\to Λ(\to pπ^-)μ^+μ^-$ decays using simulation samples of collisions at the FCC-ee reconstructed using the IDEA detector concept and assuming a dataset of $6\times 10^{12}$ $Z^0$ bosons. While the statistical sensitivity achieved for individual angular observables is not expected to significantly exceed that from the LHCb Upgrade II experiment, the addition of the polarized observables leads to a significant improvement of the knowledge on the Wilson coefficients $C_{9^{(\prime)}}$ and $C_{10^{(\prime)}}$.

We investigate the cosmological implications of Tsallis entropy in two widely discussed frameworks: the Cai-Kim thermodynamic derivation of the Friedman equations and the Tsallis holographic dark energy (HDE) scenario, considering both the Hubble scale and the Granda-Oliveros (GO) cutoff as infrared regulators. In both cases, the dynamics introduce a nonextensivity parameter $δ$, with the standard Bekenstein-Hawking entropy-area relation recovered for $δ= 1$. While previous studies have suggested that only small deviations from extensivity are observationally allowed, typically requiring $|1 - δ| \lesssim 10^{-3}$, here we go further and perform a systematic consistency analysis across the entire expansion history. We show that even mild departures from $δ= 1$ lead to pathological behavior in the effective dark energy sector: its density can become negative or complex, its equation of state may diverge, or it can contribute an unacceptably large early-time fraction that spoils radiation domination and violates BBN and CMB constraints. Our results sharpen and unify earlier hints of tension, providing a clear physical interpretation in terms of corrections that grow uncontrollably with the expansion rate toward the past. We conclude that within both the Cai-Kim and HDE formulations, a viable cosmology emerges only in the extensive limit, effectively reducing the models to $Λ$CDM. More broadly, our findings emphasize the importance of dynamical consistency and cosmological viability tests, when assessing nonextensive entropy formalisms as potential frameworks for describing the Universe's dynamics.

The increasing volume of gamma-ray data demands new analysis approaches that can handle large-scale datasets while providing robustness for source detection. We present a Deep Learning (DL) based pipeline for detection, localization, and characterization of gamma-ray sources. We extend our AutoSourceID (ASID) method, initially tested with \textit{Fermi}-LAT simulated data and optical data (MeerLICHT), to Cherenkov Telescope Array Observatory (CTAO) simulated data. This end-to-end pipeline demonstrates a versatile framework for future application to other surveys and potentially serves as a building block for a foundational model for astrophysical source detection.

The learning crisis in the Latin American region (i.e., higher rates of students not reaching basic competencies at secondary level) is worrying, particularly post-pandemic given the stronger role of inequality behind achievement. Within this scenario, the concept of student academic resilience (SAR), students who despite coming from disadvantaged backgrounds reach good performance levels, and an analysis of its determinants, are policy relevant. In this paper, using advancements on explainable machine learning methods (the SHAP method) and relying on PISA 2022 data for 9 countries from the region, we identify leading factors behind SAR using diverse indicators. We find that household inputs (books and digital devices), gender, homework, repetition and work intensity are leading factors for one indicator of academic resilience, whereas for other indicator leading drives fall into the school domain: school size, the ratio of PC connected to the internet, STR and teaching quality proxied by certified teachers and professional development rates and school type. Also, we find negative associations of SAR with the length of school closures and barriers for remote learning during the pandemic. The paper's findings adds to the scare regional literature contributing to future policy designs where key features behind SAR can be used to lift disadvantaged students from lower achievement groups towards being academic resilient.

Utilizing comprehensive experimental data on charmed mesons, we systematically investigate masses of the higher radial and orbital excitations of the $D$ and $D_s$ meson families using the relativistic flux tube model. Our study employs mass splitting induced by spin-dependent interactions within the j-j coupling scheme. Our predicted masses align well with the experimental measurements for the well-established $D$ and $D_s$ states. However, anomalous resonances such as $D_{s0}(2317)$ and $D_{s1}(2460)$ do not align with conventional meson states within our theoretical framework. Based on our reliable mass predictions for low-lying states, we propose spectroscopic assignments for several recently observed high-mass resonances: $D_2(2740)^{0}$, $D^*_3(2750)$, $D_0(2550)^{0}$, $D^*_1(2600)^{0}$, $D_1^*(2760)^0$, $D^*_J(3000)$, $D_J(3000)$, $D^*_2(3000)$, $D^*_{s1}(2860)^{\pm}$ and $D^*_{s3}(2860)^{\pm}$.Additionally, the resonance $D_{sJ}(3040)^+$ is identified as a $2P$ excitation with spin-parity quantum numbers $J^P = 1^+$. Extending our model, we also calculate the mass spectra of doubly charmed $Ξ_{cc}$ and $Ω_{cc}$ baryons within the heavy-diquark-light-quark picture. These theoretical predictions provide crucial guidance for ongoing and future experimental searches for higher radial and orbital excitations in the charmed meson and doubly charmed baryon sectors.

The ability to control and manipulate time-reversal ($T$) symmetry-breaking phases with near-zero net magnetization is a sought-after goal in spintronic devices. The recently discovered hexagonal altermagnet manganese telluride ($α$-MnTe) is a prime example. It has a compensated altermagnetic ground state where the magnetic moments are aligned in each layer and stacked antiparallel along the $c$ axis, yet it exhibits a spontaneous anomalous Hall effect (AHE) that breaks the $T$-symmetry with a vanishingly small $c$-axis ferromagnetic (FM) moment. However, the presence of three 120$^\circ$ separated in-plane magnetic domains presents a challenge in understanding the origin of the AHE and the effective control of the altermagnetic state. Here we use neutron scattering to show that symmetry breaking anisotropic strain, induced by compressive uniaxial pressure along the nearest-neighbor (NN) Mn-Mn bond directions, detwins $α$-MnTe into a single in-plane magnetic domain. This control over in-plane domains allows us to unambiguously establish that the in-plane moments are aligned along the NNN Mn-Mn bond direction, irrespective of the applied strain directions. Mounting the sample on a piezoelectric strain cell along both NN and NNN directions can drive the sample into a single-domain state that significantly sharpens the AHE hysteresis loop and extends the AHE to lower temperatures. Furthermore, tuning the uniaxial strain reverses the sign of the AHE near room temperature. Remarkably, this is achieved without altering the altermagnetic phase-transition temperature or substantially changing the small $c$-axis FM moment. Combined with our phenomenological model, we argue that these effects result from the modification of the electronic Berry curvature by a combination of both spin-orbit coupling and strain. (See the full abstract in the PDF.)

Aluminum oxide is a promising material for visible-light integrated photonics due to its low optical loss and wide transparency window across the RGB spectrum. This work presents the design and experimental demonstration of an integrated RGB beam combiner for applications in AR/VR, holography, 3D displays, and autostereoscopic display systems. The device employs Mach--Zehnder modulators for individual color modulation and gratings for out-of-plane emission. The experimental results demonstrate independent RGB routing, far-field beam combining, and thermo-optic modulation with an extinction ratio of up to 6.3~dB, highlighting the potential of Al$_2$O$_3$ photonic integrated circuits for compact dynamic color control at the pixel level.

The XZZX code is a variant of the surface code tailored to address biased noise in realistic quantum devices. We propose a simulated annealing (SA) decoder for the XZZX code. Our SA decoder is amenable to parallelization because its MCMC updates are simple and local. To initialize SA, we use a recovery configuration produced by our greedy matching decoder. Although $Z$-biased noise is commonly assumed in realistic quantum devices, we instead focus on $Y$-biased noise, where MWPM becomes suboptimal because it neglects correlations induced by $Y$ errors. Our numerical simulations for the code capacity noise model, where only data qubits suffer errors, show that our SA decoder achieves higher accuracy than the MWPM decoder. Furthermore, our SA decoder achieves an accuracy comparable to that of the optimal minimum-energy (MAP-configuration) decoder formulated as an integer programming problem, called the CPLEX decoder. In our greedy matching decoder, we randomize the tie-breaking among equal-weight pairs. This randomness generates a variety of initial configurations for SA, which can lead to faster convergence of our SA decoder. By comparing decoding times of our SA decoder, the CPLEX decoder, and the matrix product state (MPS) decoder, all of which can handle $Y$-biased noise appropriately, we estimate that our SA decoder could be competitive in runtime under an idealized assumption of near-perfect parallel efficiency. These results suggest that combining SA with our greedy matching initializer is a practical approach toward fault-tolerant quantum computation.

The states of a three-mode bosonic system with the restricted Hilbert space are discussed in the context of quantum entanglement and squeezing of quantum fluctuations. The states exhibiting non-zero tripartite entanglement are considered. Mutual relations between the two- and three-mode entanglement quantified by the corresponding negativities and the squeezing described by the corresponding principal squeeze variances are revealed. Entangled three-qubit states exhibiting squeezing are identified.

The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, $N(\varepsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most~$\varepsilon$ in dimension~$s$. It is known that $N(\varepsilon, s)$ depends only linearly on the dimension~$s$. All known proofs of this result are non-constructive. Finding explicit point set constructions that achieve this optimal linear dependence on the dimension remains a major open problem. In this paper, we make progress on this question by analyzing point sets constructed from a multiset union of digitally shifted Korobov polynomial lattice point sets. Specifically, we show the following two results. A union of randomly generated Korobov polynomial lattice point sets shifted by a random digital shift of depth $m$ can achieve a star discrepancy whose inverse depends only linearly on $s$. The second result shows that a union of all Korobov polynomial lattice point sets, each shifted by a different random digital shift, achieves the same star discrepancy bound. While our proof relies on a concentration result (Bennett's inequality) and is therefore non-constructive, it significantly reduces the search space for such point sets from a continuum of possibilities to a finite set of candidates, marking a step towards a fully explicit construction.

Embezzlement of entanglement allows to extract arbitrary entangled states from a suitable embezzling state using only local operations while perturbing the resource state arbitrarily little. A natural family of embezzling states is given by ground states of non-interacting, critical fermions in one spatial dimension. This raises the question of whether the embezzlement operations can be restricted to Gaussian operations whenever one only wishes to extract Gaussian entangled states. We show that this is indeed the case and prove that the embezzling property is in fact a generic property of fermionic Gaussian states. Our results provide a fine-grained understanding of embezzlement of entanglement for fermionic Gaussian states in the finite-size regime and thereby bridge finite-size systems to abstract characterizations based on the classification of von Neumann algebras. To prove our results, we establish novel bounds relating the distance of covariances to the trace-distance of Gaussian states, which may be of independent interest.

Inspired by the recent experimental advances in cold atom quantum simulators, we explore the experimentally implemented bosonic $t$-$t'$-$J$ model on the square lattice using large-scale density matrix renormalization group simulations. By tuning the doping level $δ$ and hopping ratio $t'/t$, we uncover six distinct quantum phases, several of which go far beyond the conventional paradigm of phase-coherent superfluidity (SF) expected for bosonic systems. In particular, in the presence of antiferromagnetic (AFM) order, doped holes are tightly bound into pairs, giving rise to a pair density wave (PDW) phase at low doping and small $|t'/t|$, which is suppressed on the $t'<0$ side, resulting in a disordered PDW state that lacks coherence of either individual bosons or pairs. Upon further doping, bosons can regain phase coherence and form a SF* state, characterized by condensation at emergent incommensurate momenta concurrent with an incommensurate magnetic order. On the $t'>0$ side, the sign-induced kinetic frustration inherently disfavors local AFM correlations, leading to a phase separation in which doped holes cluster into ferromagnetic (FM) domains spatially separated by undoped AFM regions. Upon further doping, this inhomogeneous state evolves into a uniform SF + $xy$-FM phase. Finally, we propose a concrete experimental scheme to realize both signs of $t'/t$ in Rydberg tweezer arrays, with an explicit mapping between model parameters and experimentally accessible regimes. Our results reveal competing and intertwined orders in doped antiferromagnets, which are relevant to central issues in high-$T_c$ superconductivity, reflecting the frustrated interplay between doped holes and spin background.

In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to determine when such conditional measures arise from a metric measure foliation. Additionally, we provide a example demonstrating how this framework can be applied to study perturbations of disintegration-induced foliations.