The Navier–Stokes (NS) equation with Coriolis force and density-dependent viscosity is an important physical model, which has been widely used to understand and analyze a wide array of phenomena, including behaviors of the Gulf stream, dynamics of hurricanes, operation of chemical reactors and functionality of rotating machines. In this paper, based on the matrix and curve integration techniques, we build a sufficient condition for the existence of Cartesian vector solutions u = b(t) + A(t)x for the N-dimensional NS equation, in which A satisfies appropriate matrix equations. Then, we discuss two special cases of A and thereby explicit analytical solutions are obtained. To shed light on these solutions, we give some illustrative examples. Among them, some examples form the generalization previously obtained by other authors and some examples are quite new. Finally, we analyze the properties of Cartesian vector solutions in a special case.
Nonlinear option pricing models are of significant research value as they better reflect the realities of financial markets, yet their numerical solution remains highly challenging. On the one hand, such models typically involve strong nonlinearity, multi-scale features of small parameters, and high sensitivity to initial data, which often make it difficult for traditional numerical methods to maintain stability and accuracy. On the other hand, many deep learning methods rely on boundary data, while in real financial markets boundary conditions are often unavailable, thereby limiting their applicability. Thus, deep learning methods that rely solely on initial data still face significant challenges in efficiently solving nonlinear option pricing models and achieving effective numerical predictions. To address these difficulties, this work employs the respecting causality physics-informed neural network (RCPINN), which depends solely on initial data and respects the inherent spatiotemporal causal structure of system evolution, enabling it to effectively handle the complex characteristics of nonlinear financial models. In the Ivancevic nonlinear option pricing model, the RCPINN successfully predicts the formation and evolution of both low-order and high-order financial rogue waves, revealing the underlying dynamical mechanisms of extreme localized waves. In the nonlinear Black–Scholes transaction cost model, the RCPINN effectively captures the dynamical evolution of European call and put option prices. This work not only validates the effectiveness and applicability of RCPINN in predicting financial rogue waves and European option prices within the framework of nonlinear option pricing models, but also demonstrates its advantages in handling initial data sensitivity, multi-scale features, and strong nonlinearity in financial problems. More importantly, this approach provides a novel technical pathway for both early warning of extreme financial risks and the simulation of European option price evolution, while the findings offer valuable insights for applications in derivative pricing, exchange rate forecasting and financial market risk monitoring and early warning.
The theory of multipartite entanglement provides a robust framework for understanding complex phenomena in quantum many-body systems, while multipartite entanglement also serves as a key resource in numerous quantum information processing tasks. Using density matrix renormalization group calculations, we study the multipartite entanglement in one-dimensional quantum long-range ferromagnetic and antiferromagnetic Ising chains with algebraically decaying interactions 1/rα, respectively. To quantitatively probe the system's multipartite entanglement structure, we employ the quantum Fisher information (QFI). Our results reveal two key behaviors: as the system size increases, the QFI rises, while the multipartite entanglement density decreases and converges toward a constant value. The latter trend points to a fundamental constraint on entanglement distribution in the thermodynamic limit. We further discuss the effects of the magnetic field and the exponent α on the QFI. We find that the peak values of the QFI increase with increasing α. Additionally, the discontinuity in the QFI can be regarded as a powerful tool for characterizing quantum phase transitions in the systems.
The growing demand to understand non-equilibrium dynamics in open quantum many-body systems—motivated by progress in quantum simulation and error correction—requires theoretical frameworks that avoid the exponential scaling of the full density matrix. In this work, we develop a first-principles approach to the dissipative hydrodynamical equations of a finite-temperature superfluid. We show that dominant external dissipation not only simplifies the description of intrinsic damping (Beliaev and Landau) but also stabilizes a hydrodynamic formulation. Starting from a density-matrix method aligned with the Zaremba–Nikuni–Griffin formalism, a closed set of dissipative quantum Liouville equations (DQLE) can be obtained. We then introduce a systematic hydrodynamic expansion beyond the conventional Wigner transform, yielding a closed set of dissipative hydrodynamic equations (DHE). Through detailed comparison with the full DQLE, we demonstrate that retaining only the zeroth-order term in the expansion of the anomalous density matrix offers the most accurate and stable DHE implementation, in close agreement with the DQLE under strong dissipation and weak interaction. Our results establish a stabilized hydrodynamic theory for dissipative quantum fluids and significantly reduce the computational complexity of modeling open many-body dynamics.
Magic states play an important role in fault-tolerant quantum computation, and so the quantification of magic for quantum states is of great significance. In this work, we propose two new magic quantifiers by introducing two versions of quantum (α, β) Jensen–Shannon divergence, based on the quantum (α, β) entropy and the quantum (α, β)-relative entropy, respectively. We derive many desirable properties for our magic quantifiers, and find that they are efficiently computable in low-dimensional Hilbert spaces. We also show that the initial nonstabilizerness of the input state can boost the magic-generating power for our magic quantifiers with appropriate parameter ranges for a certain class of quantum gates. Our magic quantifiers may provide new tools for addressing some specific problems in magic resource theory.
The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system's robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.
For pure states, the quantum Berry curvature has been well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.
In this study, we propose an experimental scheme to generate macroscopic quantum cat states of a mechanical oscillator within a hybrid atom-optomechanical system. The system consists of a mechanical oscillator harmonically coupled to an optical cavity, where the atom-cavity interaction operates in the ultrastrong-coupling regime described by the Rabi–Stark model. The cavity photon and atom state measurements project the mechanical mirror into cat states, with the process enhanced by Fock-state initialization, nonlinear atom-field coupling and modulated mechanical tuning. Wigner-function analysis confirms nonclassicality. The key findings are: (1) higher initial photon number n increases coherent-state amplitudes and separations of the cat states; (2) smaller detuning improves distinguishability; (3) both the nonlinear atom-photon coupling term and stronger optomechanical coupling accelerate the generation of cat states. This approach provides a parameter-tunable framework for quantum coherence control.
For even–even nuclei 180−184Yb, 182−186Hf and 184−188W located on an island of hexadecapole-deformation archipelago, the structure properties, especially under rotation, are reinvestigated by using the Hartree–Fock–Bogliubov–Cranking (HFBC) calculation with a fixed shape (e.g., the ground-state equilibrium shape). The equilibrium deformations, extracted from the potential energy surface, are calculated based on the phenomenological Woods–Saxon (WS) mean-field Hamiltonian within the framework of macroscopic-microscopic (MM) model. The impact of different deformation degrees of freedom on, e.g., single-particle levels, total energy, and moment of inertia (MoI), is revealed, especially concentrating on the hexadecapole-deformation effects and the quadrupole-hexadecapole coupling. Considering the axially hexadecapole deformation, the present calculations can reproduce available experimental data well, including the quadrupole deformations and moments of inertia. Interestingly, it is found that the impact of different deformation degrees of freedom on MoI exhibits a similar trend in the HFBC and rigid-body calculations though the latter ignores the pairing effects. Before starting or constructing a complex theory-model, to some extent, such a similarity can provide an alternative way of understanding the effect of, e.g., exotic deformations, on the MoI by the calculation of a simple rigid-body approximation. The present findings could offer insights into the static and dynamic effects of hexadecapole deformations, contributing valuable information for the corresponding research in nuclear structure and reaction.
In this paper, using the ensemble-averaged theory, we define the thermodynamic free energy of Einstein–Gauss–Bonnet (EGB) black holes in anti-de Sitter (AdS) spacetime. This approach derives the gravitational partition function by incorporating non-saddle geometries besides the classical solutions. Unlike the sharp transition points seen in free energy calculated via saddle-point approximation, the ensemble-averaged free energy plotted against temperature shows a smoother behavior, suggesting that black hole phase transitions may be viewed as a small-GN (Newton's gravitational constant) limit of the ensemble theory. This is similar to the behavior of black hole solutions in Einstein's gravity theory in AdS spacetime. We have obtained an expression for the quantum-corrected free energy for EGB-AdS black holes, and in the six-dimensional case, we observe a well-defined local minimum after the transition temperature which was absent in the earlier analysis of the classical free energy landscape. Furthermore, we expand the ensemble-averaged free energy in powers of GN to identify non-classical contributions. Our findings indicate that the similarities in the thermodynamic behavior between five-dimensional EGB-AdS and Reissner–Nordström-AdS black holes, as well as between six-dimensional EGB-AdS and Schwarzschild-AdS black holes, extend beyond the classical regime.
Recent studies have demonstrated that AdS black holes possess the basic characteristics of a standard thermodynamic system. Concurrently, the thermodynamic properties of spacetimes featuring multiple horizons with distinct radiation temperatures have also attracted research interest. In this work, considering the high-order quantum electrodynamics correction, we initially establish an equivalent thermodynamic system for the coexistence region of black hole and cosmological horizons. On this basis, we conduct a detailed investigation into the thermodynamic properties of this dual-horizon coexistence region. Our results demonstrate that this equivalent thermodynamic system exhibits van der Waals-like thermodynamic behavior. Furthermore, we introduce a nonlinear parameter γ to analyze its impact on phase transitions within the equivalent thermodynamic system. Under specific conditions, the system undergoes first- or second-order phase transitions for γ = 0, and zeroth- or second-order phase transitions for γ ≠ 0. Finally, by utilizing the generalized off-shell Helmholtz free energy within the thermodynamic topological framework for black holes, we extend this methodology to investigate the topological properties of de Sitter (dS) spacetime. We compute the topological number characterizing the coexistence region of dual horizons in Euler–Heisenberg dS spacetime using equivalent thermodynamic state parameters. Additionally, we investigate the influence of the nonlinear parameter γ on the thermodynamic characteristics of the equivalent system.
We model the inspiral and merger dynamics of two co-planar rings in Newtonian mechanics with GR motivated corrections and illustrate their similarity with those of black hole (BH) binary systems on the orbital plane. Our simulation reveals a banana-shape deformation of the ‘BHs' involved, and a typhoon-like spiral structure in the merger product. Using an eXact One-Body approach, we compute the full gravitational waveform of this process and qualitatively reproduce results consistent with those of numerical relativity (NR). Our simulation offers a transparent link between the feature of gravitational waveforms and the internal structure of BHs, thus a complementary interpretation of physics behind NR.
In this paper, we investigate the Sharma-Mittal Holographic Dark Energy (SMHDE) model within the framework of the recently proposed f(Q) gravity theory. Using the reconstruction technique, we derive the corresponding f(Q) functions for the SMHDE model under three different cosmological scale factors: power-law, intermediate, and unified scenarios. Furthermore, we conduct a comprehensive graphical analysis of these reconstructed models based on key cosmological parameters, including the equation-of-state (EoS) parameter, deceleration parameter, the $\omega -\omega ^{\prime} $ phase plane and statefinder parameters. The results indicate that all reconstructed models are dynamically stable and consistent with the current accelerated expansion of the Universe. Hence, the SMHDE model in f(Q) gravity provides a viable and consistent framework for describing the present dark energy phenomenon.
We investigate a class of spatially covariant vector field theories on a flat background, where the Lagrangians are constructed as polynomials of first-order derivatives of the vector field. Because Lorentz and U(1) invariances are broken, such theories generally propagate three degrees of freedom (DOFs): two transverse modes and one longitudinal mode. We examine the conditions under which the additional longitudinal mode is eliminated so that only two DOFs remain. To this end, we perform a Hamiltonian constraint analysis and identify two necessary and sufficient degeneracy conditions that reduce the number of DOFs from three to two. We find three classes of solutions satisfying these degeneracy conditions, corresponding to distinct types of theories. Type-I theories possess one first-class and two second-class constraints, type-II theories have four second-class constraints, and type-III theories contain two first-class constraints. The Maxwell theory is recovered as a special case of the type-III theories, where Lorentz symmetry is restored.
This paper investigates the perturbation dynamics of massless scalar and electromagnetic fields on magnetically charged de Sitter black holes within the framework of string-inspired Euler–Heisenberg (EH) gravity. We calculate the quasinormal frequencies (QNFs) and discuss the influences of black hole magnetic charge Qm, the cosmological constant Λ, coupling parameter ε and multipole number l on QNFs, emphasizing the relationships between these parameters and quasinormal modes behavior. We find that the results obtained through the asymptotic iteration method are in good agreement with those obtained by the WKB method. Importantly, the Bernstein spectral method is employed as a rigorous cross-check for QNFs in the l = 0 scalar perturbation sector, where the WKB approximation is often unreliable. The greybody factors (GFs) are calculated using WKB method. The effects of the parameters Qm and ε on the GF are also studied.
We consider quantum droplets in Bose–Einstein condensates with three-body recombination and external feeding in a spherically symmetric harmonic trap. A quantum droplet is a self-bound state which arises due to the delicate balance between the mean-field effect and quantum fluctuation. We find effective potentials for flat- and sharp-top droplets and show that the potentials in both cases are minimized for certain values of the width. The width of a droplet oscillates periodically with time and the oscillation amplitude depends sensitively on the initial width of a droplet around the minimum of the effective potential. The maximum width of a droplet decreases gradually with time due to the loss of atoms arising from the three-body recombination. It shows that this loss of atoms can be managed by appropriately feeding the condensate from an external source to generate stable droplets. We examine the stability of a droplet by fixed point analysis and find that the region of stability is affected with the change of external feeding and three-body loss.
We discover three novel classes of pulse-trains in an optical Kerr nonlinear medium possessing all orders of dispersion up to the fourth order. We show that both single- and double-humped pulse-trains can be formed in the nonlinear medium. A distinguishing property is that these structures have different amplitudes, widths and wave numbers but equal velocity which depends on the three dispersion parameters. More importantly, we find that the relation between the amplitude and duration of all the newly obtained pulse-trains is determined by the sign of a joint parameter solely. The results show that those optical waves are general, in the sense that no specified conditions on the material parameters are taken on. Considering the long-wave limit, the derived pulse-trains degenerate to soliton pulses of the quartic and dipole kinds. Results in this study may be useful for experimental realization of pulse-trains in highly dispersive optical fibers and further understanding of their optical transmission properties.
Voting is an important social activity for expressing public opinions. By conceptually considering a group of voting agents to be intelligent matter, the impact of real-time information on voting results is quantitatively studied by an intelligent Ising model, which is formed by adding nonlinear instantaneous feedback of the overall magnetization to the conventional Ising model. In the new model, the interaction strength becomes a variable depending on the total magnetization rather than a constant, which mimics the scenario that the decision of an individual during vote influenced by the dynamically changing polling result during the election process. Our analytical derivations along with Monte Carlo simulations reveal that, with a positive feedback, the intelligent Ising model exhibits phase transitions at any finite temperatures, a feature lacked in the conventional one-dimensional Ising model. In all dimensions, by varying the feedback strength, the system changes from going through a second-order phase transition to going through a first-order phase transition with increasing temperature, and the two types of phase transitions are connected by a tricritical point. This study on the one hand demonstrates that the intelligent matter with a nonlinear adaptive interaction can exhibit qualitatively different phase behaviors from conventional matter, and on the other hand shows that, during voting, even unbiased feedback may possibly induce spontaneous symmetry breaking, leading to a biased outcome where one side of the vote becomes favored.
The convergence of cycle expansions in non-hyperbolic systems has long been an outstanding challenge, primarily due to the presence of singularities in natural measures and the attributed weak shadowing properties of unstable periodic orbits. Here, we introduce a novel layered approximation scheme that effectively decomposes the natural measure into distinct singular and hyperbolic layers. For the singular components, we employ analytical approximations, whereas for the hyperbolic components, we utilize cycle expansions. A key innovation of our approach is the incorporation of dynamical information beyond periodic orbits, which is essential for achieving a more accurate approximation of the natural measure and for precisely calculating observable averages. By integrating additional dynamical details, the new scheme more effectively captures the subtleties of the system's behavior that are not fully represented by a limited set of periodic orbits alone. This invariance allows us to refine the approximation of the natural measure in both the singular and the hyperbolic layers, thereby yielding a more reliable estimation of observable averages.
We investigate magnetic correlation in the triangular lattice Hubbard model using determinant quantum Monte Carlo simulations. Focusing on the role of next-nearest-neighbor hopping ${t}^{{\prime} }$ and electron filling ⟨n⟩, we demonstrate that regions of high density of states, particularly near van Hove singularity (VHS) points, significantly enhance short-range ferromagnetic correlations as measured by the uniform spin susceptibility χ(Γ). Specifically, χ(Γ), quantifying ferromagnetic fluctuations, is amplified at fillings corresponding to the VHS for a given ${t}^{{\prime} }$. Increasing the on-site Coulomb repulsion U further strengthens these ferromagnetic correlations, with a more pronounced effect at lower temperatures, and the observed ferromagnetic correlations are found to be short-ranged. Conversely, near half-filling ⟨n⟩ = 1.0, larger ${t}^{{\prime} }$ values promote antiferromagnetic (AFM) fluctuations, evidenced by an increase in χ(K). Our results reveal that ${t}^{{\prime} }$and filling act as effective tuning parameters for manipulating the competition between ferromagnetic and AFM fluctuations in the triangular lattice.
