In the framework of general relativity (GR), gravitational waves (GWs) travel at the speed of light across all frequencies. However, massive gravity and weak equivalence principle (WEP) violation may lead to frequency-dependent variations in the propagation speed of GWs, which can be examined by comparing the theoretical and observed discrepancies in the arrival times of GW signals at various frequencies. This provides us with an opportunity to test these theories. For massive gravity, we consider that gravitons may have a nonzero rest mass. For WEP violations, we hypothesize that different massless particles exposed to the same gravitational source should exhibit varying gravitational time delays. The gravitational time delay induced by massive gravitational sources is proportional to γ + 1, where the parameter γ = 1 in GR. Therefore, we can quantify these two deviations using phenomenological parameters m_g and ∣Δγ∣, respectively. In this study, we use selected GW data from binary black hole coalescences in the LIGO-Virgo catalogs GWTC-2.1 and GWTC-3 to place constraints on the parameters m_g and ∣Δγ∣. We also compute Bayes factors for models that assume the existence of graviton mass and WEP violation compared to the standard GW model, respectively. The absolute value of the natural logarithm of the Bayes factor is generally less than two. Our analysis reveals no significant preference for either model. Additionally, the Bayes factors between these two models do not provide obvious evidence in favor of either one.

This research aims to examine the (1+1)-dimensional Schrödinger–Maxwell–Bloch (SMB) model, which is of great significance for the study of the dynamic properties of optical solitons in the media. We obtain the exact analytical soliton solutions for the given model using the new Kudryashov’s method, which helps to reveal various forms of the nonlinear waves. The proposed approach yields explicit localized wave structures, including bright soliton and dark-bright composite profiles associated with the field components of the SMB system. These solutions describe stable localized pulses and their accompanying field interactions in resonant nonlinear optical media. These solutions are shown using 3D plots, 2D plots, contour plots, as well as polar plots to gain a better understanding of their spatiotemporal properties. These plots also help to show the overlap of the temporal pulses. In addition to the solution development, the dynamical properties of the system are also rigorously investigated. Bifurcation plots are used to explore the stability properties of the system for varying important parameters. Chaotic properties are explored using numerical techniques such as simulations, including properties such as multistability, initial value sensitivity, recurrence plots, and power spectra. These topics illustrate in more detail the complex properties that are inherent in the SMB system. These topics not only discuss properties of optical solitons but could also provide research opportunities in the area of nonlinear optics, optical communication, or laser physics for which controllable soliton propagation is critical. This study provides a uniting methodology that embeds exact solutions and nonlinear dynamical systems analysis to present a well-rounded approach for the study of solitons and related phenomena in the SMB system.

This study investigates the dimensionless quasi-geostrophic potential vorticity (QG-PV) equation with external sources. Employing the Gardner–Morikawa transformation and weakly nonlinear perturbation expansion, we derive the nonlinear Boussinesq equation with external sources. We demonstrate the existence of explicit zero-order and first-order Wronskian solutions for the model equation when α₄ = 0. Furthermore, using a modified Jacobi elliptic function method, we obtain soliton-like solutions for both α₄ = 0 and α₄ ≠ 0. Analysis of these solutions reveals that the generalized β-plane approximation and shear flow are significant factors in inducing nonlinear Rossby waves, and that external sources play a crucial role in influencing Rossby wave behavior.

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.

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.

White dwarfs, one of the compact objects in the Universe, play a crucial role in astrophysical research and provide a platform for exploring nuclear physics. In this work, we extend the relativistic mean field approach by using a Walecka-type quantum hadrodynamics model to capture the intricate structure of white dwarfs. We calculate nuclear properties, Coulomb energy, and photon energy within white dwarfs in a unified framework. By carefully calibrating the model parameters to align with nuclear matter properties, we successfully reproduce the structures of several elements in white dwarfs, such as the isotopes of C and ¹⁶O, except for the unnaturally deeply bound state ⁴He. Furthermore, we predict the characteristics of white dwarfs composed of atom-like units and the gravitational waves stemming from binary white dwarf inspirals incorporating tidal deformability contributions up to the 2.5 post-Newtonian order. These results shed light on the structure of white dwarfs and provide valuable information for future gravitational wave detection. This methodological advancement allows for a cohesive analysis of white dwarfs, neutron stars, and the nuclear pasta within a unified theoretical framework.

In this study, we propose that many different thermodynamic modeling approaches, including the general equation for the non-equilibrium reversible-irreversible coupling (GENERIC), Onsager's variational principle, the energetic variational approach, and classical irreversible thermodynamics, can all be cast into the gradient-conservative structure (GCS). GCS enjoys many nice mathematical properties, has close connection with the large deviations principle and gradient flows in Wasserstein space, and fulfills laws of thermodynamics. Our results demonstrate that the GCS may serve as a unified theoretical framework to model various non-equilibrium thermodynamic processes.

This paper explores the application of Whitham modulation theory to the third-order focusing Kaup–Newell model, offering a complete classification of solutions for step-like initial value problems. Using the finite-gap integration method, we derive periodic solutions and the corresponding Whitham modulation equations. By analyzing the distribution of Riemann invariants, we identify the fundamental wave structures emerging from the step-like initial value problem. Furthermore, we provide a complete classification of solutions for this problem. Our results demonstrate that Whitham modulation theory serves as an effective analytical framework for studying initial value discontinuities in the third-order focusing KN model, offering new insights into its nonlinear dynamical behavior. Moreover, the direct numerical simulations show remarkable agreement with the results from Whitham modulation theory.

This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations. By interpreting conservation laws as closed 1-forms, we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle. This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions, where fiber coordinates are generated by potential functions of the conservation laws. This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-Bäcklund transformations are composite diffeomorphisms. These results are applied to several classical integrable models.

The paper considers applications of Rota–Baxter algebras to renormalization in quantum field theory and quantum integrability to obtain new solutions for the Yang–Baxter equations which can be studied by the method of Renormalization Group.

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.

Among many types of quantum entanglement properties, the entanglement spectrum provides more abundant information than other observables. Exact diagonalization and density matrix renormalization group methods could handle the system in one-dimension properly, while in a higher dimension, it exceeds the capacity of the algorithms. To expand the ability of existing numerical methods, we take a different approach via quantum Monte Carlo algorithm. By exploiting the particle number and spin conservation, we realize an efficient algorithm to solve the entanglement spectrum in the interacting fermionic system. Taking the two-dimensional interacting Su–Schrieffer–Heeger (SSH) model as an example, we verify the existence of topological phase transition under different types of many-body interactions. The calculated particle number distribution and wavefunction of the entanglement Hamiltonian indicate that the two belong to distinct types of topological phase transitions.

Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities, which originates from the parity-symmetric reduction of the Manakov system and has applications in some physical systems with the parity symmetry constraint between two fields/components. Via the Riemann-Hilbert method, the theory of inverse scattering transform with the presence of double poles is extended for this equation under nonzero boundary conditions (NZBCs). Also, the double-pole soliton solutions with NZBCs are derived in the reflectionless case. It is shown that the quasi-periodic beating solitons can be obtained when the double pole lies off the circle $\Gamma$ centered at the origin with radius $\sqrt{2}{q}_{0}$ (where q₀ is the modulus of NZBCs) on the spectrum plane. Moreover, using the improved asymptotic analysis method, the asymptotic solitons are found to be located in some logarithmic curves of the xt plane.

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.

We argue that the hypothesis that positive-parity charm meson resonances exhibit a compact tetraquark structure has some clear tension with recent lattice results for the S-wave πD system for an SU(3) flavor symmetric setting. In particular, we show that such a diquark–anti-diquark tetraquark scenario would call for the presence of a state in the flavor $[\overline{{\bf{15}}}]$ representation, not seen in the lattice analysis. Moreover, we show that analogous lattice data in the axial-vector channel are even more sensitive to the internal structure of these very interesting states.

Using a model-independent Gaussian process (GP) method to reconstruct the dimensionless luminosity distance D and its derivatives, we derive the evolution of the dimensionless Hubble parameter E, the deceleration parameter q, and the state parameter w of dark energy. We utilize the PantheonPlus, SH0ES, and Gamma Ray Burst (GRB) data to derive the dimensionless luminosity distance D. Additionally, we employ observational H(z) data (OHD) and baryon acoustic oscillations (BAO) from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) to obtain the first derivative of the dimensionless luminosity distance ${D}^{{\prime} }$. To obtain the reconstructed D and ${D}^{{\prime} }$, we utilize the fiducial value from each dataset, with particular emphasis on the varying H₀. According to the reconstruction results obtained from PantheonPlus+SH0ES+GRB+OHD and PantheonPlus+SH0ES+GRB+OHD+DESI data, we find that E is consistent with the predictions of the $\Lambda$CDM model at a 2σ confidence level within the redshift range of z < 2. However, the reconstruction results for q exhibit deviations from the $\Lambda$CDM model in the range of z < 0.3. Furthermore, we observe that the mean value of w exhibits evolving behavior, transiting from w < -1 to w > -1 around ${z}_{{\rm{wt}}}=0.46{4}_{-0.120}^{+0.235}$. Combining data from DESI DR2 can slightly enhance the accuracy of our constraints.

We show that in Schwarzschild equivalent mediums, the massless spin particles obey the same dynamical equation, from which we obtain remarkably simple formulae for the frequencies of the quasibound states. We find that the quasibound frequencies of different bosons can be identical at the same quantum number l, and the same is true of different fermions, but a quasibound frequency for bosons can never equal a quasibound frequency for fermions. These results mean that in Schwarzschild equivalent mediums with the quasibound-state boundary conditions, characteristics of electromagnetic waves are the same as those for all the massless bosonic waves, thereby allowing electromagnetic waves to simulate gravitational waves. Our predictions can be tested in future experiments, building upon the successful preparation of Schwarzschild equivalent mediums.

Our analysis is particularly motivated by its relevance to understanding compact object instabilities, gravitational collapse thresholds, and the formation of dense structures under the influence of modified gravity theories. The interplay of anisotropic pressures, perturbative dynamics, and modified gravity contributions offers insight into both the stable configuration of dense fluids and the mechanisms leading to dynamical instability. Such considerations directly contribute to the aims of high energy density profiles, particularly in modeling physical systems where extreme pressure, curvature, and matter interactions co-exist. We consider an axially symmetric, dense structure with anisotropic matter content and employ a specific equation of state (EoS) to examine the interplay between static and dynamic quantities via the adiabatic index. To address the complex dynamics of the collapse process, a perturbative scheme is utilized under Newtonian and post-Newtonian approximations, enabling a detailed examination of the stability and structural evolution of the system under the influence of the considered minimally coupled gravity. Our results demonstrate that hydrostatic equilibrium is maintained when effective pressure, gravitational, and anti-gravitational forces are balanced, while deviations from this balance initiate dynamical instability. Graphical representations of stable and unstable regimes are presented, revealing how the choice of gravity functions significantly affects the outcome. This work provides insight into the behavior of dense, self-gravitating configurations under modified gravity, offering broader implications for the modeling of compact astrophysical objects and contributing to the understanding of gravitational collapse in energy density regimes.

We investigate the impact of including a dynamical charm quark on the properties of light hadrons. Our study compares the calculations performed on 2+1+1 flavor (highly improved staggered quark) ensembles at four lattice spacings to those on 2+1 flavor (clover fermion) ensembles at six lattice spacings, with both sets of ensembles employing the identical Symanzik gauge action. For the light, strange and charm flavor observables, we employ the same tadpole-improved clover fermion action. From numerical results for light and strange quark masses, pion and kaon decay constants, and $\Omega$ and $\Omega_{ccc}$ baryon masses, we find that the values obtained after continuum, chiral, and infinite-volume extrapolations are consistent within uncertainties. Even though the mixed action setup can introduce additional discretization effects, our calculation shows evidence that these effects can cancel with the discretization error in the unitary setup, resulting in better convergence in the continuum extrapolation.

In this study, we present a comprehensive analysis of a modified Frolov black hole (BH) model that incorporates two types of topological defects, a global monopole (GM) and a cloud of strings (CS). This composite BH solution is examined from multiple theoretical perspectives to explore the impact of these modifications on the BH’s geometric, thermodynamic and dynamical properties. We begin by studying the geometrical optics of the spacetime, focusing on the motion of null geodesics. Key features, such as the effective potential, photon sphere, the force acting on photons and the stability of circular photon orbits, are analyzed in detail. Our results show that the presence of GM and CS significantly affects the spacetime geometry and photon dynamics. In addition, the thermodynamic behavior of the modified BH is also investigated. We derive essential quantities such as the Hawking temperature and entropy, demonstrating how the inclusion of GM and CS leads to deviations from the standard thermodynamic relations observed in classical BH solutions. These deviations may offer valuable insights into quantum gravity and the role of topological defects in BH physics. Furthermore, we examine the BH shadow as an observational signature of the underlying geometry. Our analysis shows that the Frolov parameter tends to reduce the apparent size of the shadow, while the presence of topological defects, particularly GM and CS, enlarges it. In addition, we investigate the perturbative dynamics of the BH by studying both scalar (spin-0), fermionic (spin-1/2) and electromagnetic (spin-1) fields through the massless Klein–Gordon and Maxwell equations, respectively. Using the Wentzel–Kramers–Brillouin approximation, we compute the quasinormal modes (QNMs) for scalar and electromagnetic field perturbations. The results confirm the stability of the BH under small perturbations and show that the QNM frequencies and damping rates are strongly influenced by the Frolov parameter, electric charge, GM and CS.

We investigate Hubbard models with bond–charge interactions on general graphs. For a Hamiltonian H of such a model, we provide the condition on its parameters under which the η-pairing method can be employed to construct its exact eigenstates. We arrive at this condition by finding that the requirement for the η-pairing state ${({\eta }^{\dagger })}^{N}| 0\rangle $ to be an eigenstate of H is identical to the requirement for it to be an eigenstate of a Hubbard-type Hamiltonian H_m without bond–charge interactions. When the condition for ${({\eta }^{\dagger })}^{N}| 0\rangle $to be an eigenstate of the Hubbard-type Hamiltonian H_m is satisfied, we demonstrate that there are additional states, distinct from ${({\eta }^{\dagger })}^{N}| 0\rangle $, which are also exact eigenstates of H_m. Our results enhance the understanding of Hubbard models on general graphs, both with and without bond–charge interactions.

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.

The Sasa–Satsuma (S–S) equation, an integrable extension of the nonlinear Schrödinger equation, could provide a more accurate description of ultrashort optical pulse propagation by incorporating third-order dispersion, self-steepening, and delayed nonlinear response. In this work, we establish a quantitative correspondence between the spectral degeneracies of two types of related linear non-Hermitian matrices and the localized wave solutions of the nonlinear Hermitian systems. Compared with the standard nonlinear Schrödinger equation system, the S–S equation displays richer spectral and dynamical structures due to its multi-band Hamiltonian. In particular, we show that exceptional points correspond to rogue waves in the modulational instability regime and to W-shaped solitons in the modulational stability regime. Moreover, higher-order degeneracies, such as third-order exceptional points, reside at the boundary between modulational stability and modulational instability. This configuration enables a coexistence mechanism of W-shaped solitons and rogue waves, where two W-shaped solitons can interact to generate a rogue wave. These results provide a unified non-Hermitian perspective on localized wave phenomena in higher-order integrable systems and reveal the fundamental link between spectral degeneracies and nonlinear wave dynamics.

For loops with UV divergences, assuming that the physical contributions of loops from UV regions are insignificant, a UV-free scheme method described by an equation is introduced to derive loop results without UV divergences in the calculations, i.e., a route of the analytic continuation ${{ \mathcal T }}_{{\rm{F}}}\to {{ \mathcal T }}_{{\rm{P}}}$ besides the traditional route ∞ − ∞ in the mathematical structure. This scheme provides a new perspective to an open question of the hierarchy problem of Higgs mass, i.e., an alternative interpretation without fine-tuning within the standard model.

Investigations into first-order quantum phase transition (QPT) remain unclear in comparison to those of the second-order or continuous QPT, in which the order parameter and associated broken symmetry can be clearly identified and, at the same time, the concepts of universality class and critical scaling can be characterized by critical exponents. Here, we present a comparison study of these two kinds of QPT in the transverse Ising model; the emphasis is on the first-order QPT. In the absence of a longitudinal field, the ground state of the model exhibits a second-order QPT from the paramagnetic phase to the ferromagnetic phase, which is smeared out once the longitudinal field is applied. Surprisingly, the first excited state involves a first-order QPT as the longitudinal field increases, which has not been reported in the literature. Within the framework of the pattern picture, we clearly identify the difference between these two kinds of QPT: for the continuous QPT, only the pattern flavoring the ferromagnetic phase is always dominant over the others. By contrast, there are at least two competitive patterns in the first-order QPT, which is further indicated by the patterns’ occupancies, calculated by pattern projections on the ground- and first excited-state wavefunctions. Our results not only have a fundamental significance in the understanding of the nature of QPTs, but also a practical interest in quantum simulations used to test the present findings.

Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (Lv et al 2025 Nat. Commun. 16 10646). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the forms of Ising and Berezinskii–Kosterlitz–Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling $T \propto h^{2/3}$ and a Grüneisen ratio scaling $\Gamma_h \propto T^{-3/2}$ for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function $\phi_{\Gamma}(x)$ of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression $\phi_{\Gamma}(x) \approx x/(1+x^2)$. Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.

We investigate the impact of the correlation between error gates on the fault-tolerant threshold in quantum circuits under three typical noise models. In the fault-tolerant encoding schemes for two fundamental quantum logical operations, we extend the conventional error injection method by simultaneously applying error gates to both control and target qubits after two-qubit gate operations, with a quantitative characterization of the correlation strength between these error gates. The results show that the fault-tolerant error threshold can be noticeably enhanced by the presence of correlations between error gates, resulting in more robust quantum circuit implementations. We further apply fault-tolerant encoding to the single-qubit teleportation circuit and investigate the effect of correlated noise on its error threshold. The results show that the threshold varies non-monotonically with the correlation strength and that the system's fault-tolerance differs significantly across noise models.

This article investigates the pseudo phase transition behavior of the six-state clock model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection-point analysis, we identified two phase transition points and classified both as continuous transitions within the microcanonical framework. Using Metropolis sampling and canonical ensemble analysis, we verified the accuracy of the transition points obtained from the microcanonical approach and further pinpointed the location of a fourth-order dependent transition with high accuracy. Moreover, the fourth-order dependent transition points extracted from two canonical order parameters——the average cluster perimeter and the ${M}_{{{\mathbb{Z}}}_{6}}$ symmetry——are in excellent agreement and consistently reinforce each other. Through a detailed analysis of the ${M}_{{{\mathbb{Z}}}_{6}}$ symmetry, we further demonstrate that the position of this fourth-order dependent transition may function as a critical transition warning indicator for both primary phase transitions in the system.

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/r^α, 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.

We propose a scheme to achieve nonreciprocal single-photon transmission in a system consisting of a spinning whispering-gallery-mode resonator and a stationary resonator containing a scatterer, both coupled to a one-dimensional waveguide. By tuning the Sagnac−Fizeau shift induced by the spinning resonator, high-contrast nonreciprocal transmission in both forward and backward directions can be realized. Furthermore, we investigate the influences of system parameters including waveguide-resonator coupling strength, inter-mode coupling strengths within two resonators, and inter-cavity coupling strength on nonreciprocal transmissions. The results indicate that the synergistic regulation of these parameters can adjust the position of the nonreciprocal transmission peak and achieve high-contrast nonreciprocal transmission.

We aim to clarify the confusion and inconsistency in our recent works (Luo et al 2023 Commun. Theor. Phys. 75 095702; Liang et al 2024 Phys. Rev. B 110 075125), and to address the incompleteness therein. In order to avoid the ill-defined nature of the free propagator of the gauge field in the ordered states of the t–J model, we adopted a gauge fixing that was not of the Becchi–Rouet–Stora–Tyutin (BRST) exact form in our previous work (Liang et al 2024 Phys. Rev. B 110 075125). This led to the situation where Dirac’s second-class constraints, namely, the slave particle number constraint and the Ioffe–Larkin current constraint, were not rigorously obeyed. Here we show that a consistent gauge fixing condition that enforces the exact constraints is BRST-exact in our theory. An example is the Lorenz gauge. On the other hand, we prove that although the free propagator of the gauge field in the Lorenz gauge is ill-defined, the full propagator is still well-defined. This implies that the strongly correlated t–J model can be exactly mapped to a perturbatively controllable theory within the slave particle representation.

Neural networks, with their high expressiveness in function approximation, have become an important research area for finding exact solutions of partial differential equations (PDEs). This paper employs a neural network-based analytical solutions (NNAS) method, which uses explicit neural networks models as trial functions for PDEs. By substituting the trial functions into the PDEs and solving the resulting algebraic systems symbolically, exact solutions can be obtained. However, this method requires designing specific activation functions for each problem, which often limits the diversity of solutions. To overcome this limitation, this paper extends the Riccati sub-equation neural networks (RSENNs) framework by using different solution forms of the Riccati equation as activation functions in the first hidden layer, and constructing trial functions through feedforward computation. Compared with the NNAS method, the RSENNs method employs the Riccati solutions family as a unified activation function library, reducing the time required for the complex derivation of activation functions. To verify the effectiveness of this approach, the method is applied to the Burgers equation, Fokker–Planck equation, and KdV–mKdV equation, obtaining various solutions forms, including generalized interaction solutions, generalized rational function solutions, and generalized hyperbolic function solutions. These solutions correspond to wave interaction, singular behavior, and localized structures in nonlinear systems, demonstrating the advantages of the method in solutions diversity. Compared with the NNAS method, the RSENNs method also reduces the time required to find suitable activation functions and improves the efficiency of solving PDEs. This work represents an extension and further development of the RSENNs method, providing an effective approach for finding exact solutions to PDEs based on neural networks.

This paper proposes a discrete deep learning method, termed the traveling-wave constrained physics-informed neural networks (TC-PINNs), for addressing both the forward and inverse problems of discrete nonlinear lattice equations (NLEs). The discrete mKdV equation is used as a representative example to validate the effectiveness of the proposed approach. We respectively carry out numerical simulations and parameter inversion for one- and two-soliton solutions of the discrete mKdV equation by employing the traveling-wave transformation and propagation properties through soft constraint and hard constraint schemes, and further extend this approach to rational soliton solutions. Numerical experiments show that the TC-PINNs consistently outperform traditional PINNs and symmetric difference data enhancement physics-informed neural networks (SDE-PINNs) in both the interior and extrapolation predictions of soliton solutions as well as parameter inversion. Compared with soft constraints, TC-PINNs under hard constraints achieve a higher prediction accuracy. This method can also be applied to solve other discrete NLEs and perform parameter inversion.

Both irreversibility and incompatibility are important features of quantum channels. Irreversibility of quantum channels characterizes the fundamental limitation in reconstructing information from their outputs. This concept has traditionally been studied for individual channels in a global sense. In this work we generalize the conventional definition of reversible channels to relatively reversible channels, where a quantum channel is reversible relative to another one; thus it is in a relative sense. Incompatibility refers to the impossibility of simultaneously implementing certain pairs of quantum channels, and lies at the very heart of quantum theory. By leveraging the concept of complementary channels, we obtain a direct connection between relative reversibility and channel compatibility. We further propose a quantifier of channel irreversibility in terms of incompatibility between the complementary channels and the identity channel. To illustrate and compare the quantifier of irreversibility with some other quantifiers in the literature, we evaluate them for some prototypical channels. Our results provide insights into the interplay between irreversibility and incompatibility, which may have potential applications in quantum error correction and the resource theory of incompatibility.

The electronic stopping power of magnesium (Mg) for protons is studied over a wide range of velocities by performing real-time time-dependent density functional theory (TDDFT) simulations. The electronic stopping power of Mg for high-velocity protons traveling along both channeling and off-channeling trajectories is calculated, and the microscopic mechanism responsible for the inner-electron excitation of Mg is revealed. In the low-velocity range (v < 1 au), the stopping power calculated from the channeling geometry aligns closely with the experimental measurements. It captures the departure from the linear relationship between the stopping power and proton velocity, which is attributed to charge exchange effect between protons and host electrons. The threshold velocities for 2p and 2s electron excitation of Mg for protons were found by TDDFT calculations. In the high-velocity range, the dependence of 2s-electron excitation on the impact parameter was investigated. Compared to channeling stopping, the off-channeling stopping exhibits significant enhancement. Furthermore, the impact of ion trajectories on electron excitation was also investigated. The results confirmed that the 3s electron excitation makes a significant contribution, and the decrease of the impact parameter enhances the inner electron excitation.

We search for the stochastic gravitational-wave background (SGWB) predicted by pre-Big-Bang (PBB) cosmology using data from the first three observing runs of Advanced LIGO and Advanced Virgo. PBB cosmology proposes an alternative to cosmic inflation where the Universe evolves from a weak-coupling, low-curvature state to the hot Big Bang through a high-curvature bounce phase, predicting a distinctive SGWB spectrum. We perform a Bayesian analysis of the cross-correlation data to constrain the model parameters characterizing the PBB spectrum. We find no evidence for a PBB-induced SGWB, with a Bayes factor of 0.03 between the PBB and noise-only model, strongly favoring the noise-only hypothesis. Our analysis establishes a lower bound β ≳ −0.19 at 95% confidence level, which is compatible with the theoretical requirement β ≥ 0 for a smooth bounce transition. While we do not detect a signal, our constraints remain consistent with the basic theoretical framework of PBB cosmology, demonstrating the potential of gravitational-wave observations to test early Universe theories.

Loschmidt echo (LE) is a powerful tool for characterizing quantum dynamical properties, yet it lacks a unified definition in non-Hermitian systems due to their intrinsic biorthogonal structure. To address this issue, we start with time-independent non-Hermitian quantum systems and rigorously derive the probability balance equation from the Schrödinger equation. On this foundational basis, we construct a cross inner product that is consistent with quantum probability interpretation and computationally convenient. We then extend this theoretical framework to time-dependent non-Hermitian systems and further derive the corresponding probability balance equation for such systems. By means of the proposed cross inner product, we generalize the Hermitian definition of LE to non-Hermitian systems and obtain its analytical expression in the biorthogonal basis under first-order perturbation theory. We show that the non-Hermitian LE naturally reduces to the standard Hermitian form in the Hermitian limit, which verifies the self-consistency and rationality of our framework. We also compare the proposed cross inner product with the G-inner product for non-Hermitian systems, noting their differences and consistent Hermitian limit. This work establishes a unified definition of LE for general non-Hermitian systems, and provides a consistent theoretical tool for investigating dynamical quantum phase transitions, quantum state stability, and non-equilibrium critical behaviors in both PT-symmetric and non-PT-symmetric non-Hermitian systems.

Shor’s algorithm outperforms its classical counterpart in efficient prime factorization. We explore the coherence and entanglement dynamics of the evolved states within Shor’s algorithm, showing that the coherence in each step relies on the dimension of register or the order, and discuss the relations between geometric coherence and geometric entanglement. We investigate how unitary operators induce variations in coherence and entanglement, and analyze the variations of coherence and entanglement within the entire algorithm, demonstrating that the overall effect of Shor’s algorithm tends to deplete coherence and produce entanglement. Our research not only deepens the understanding of this algorithm but also provides methodological references for studying resource dynamics in other quantum algorithms.

The Wiener path integral framework is proposed to model military combat dynamics by incorporating the neglected stochastic effects to the Lanchester’s square law. This framework is applied to evaluate the empirical 3:1 combat rule, which posits that an attacker requires a threefold force superiority to achieve victory. Specifically, the attacker’s winning probability is computed utilizing a semi-analytical Rayleigh–Ritz method. Numerical results demonstrate that the validity of the rule critically depends on specific parameter regimes, primarily contingent upon the relative combat effectiveness ratio between the opposing forces and the tolerance for attrition. This work establishes a physics-informed theoretical bridge between statistical mechanics and military operations research for analyzing uncertain combat systems.