Utilizing some conservation laws of the (1+1)-dimensional Camassa–Holm (CH) equation and/or its reciprocal forms, some (n+1)-dimensional CH equations for $n\geq 1$ are constructed by a modified deformation algorithm. The Lax integrability can be proven by applying the same deformation algorithm to the Lax pair of the (1+1)-dimensional CH equation. A novel type of peakon solution is implicitly given and expressed by the LambertW function.

We propose to employ a hierarchical coarse-grained structure in artificial neural networks explicitly to improve the interpretability without degrading performance. The idea has been applied in two situations. One is a neural network called TaylorNet, which aims to approximate the general mapping from input data to output result in terms of Taylor series directly, without resorting to any magic nonlinear activations. The other is a new setup for data distillation, which can perform multi-level abstraction of the input dataset and generate new data that possesses the relevant features of the original dataset and can be used as references for classification. In both the cases, the coarse-grained structure plays an important role in simplifying the network and improving both the interpretability and efficiency. The validity has been demonstrated on MNIST and CIFAR-10 datasets. Further improvement and some open questions related are also discussed.

Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalizations to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus, one is able to see how the phase transition occurs and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition takes place and further exhibits the statistics of Gaussian unitary ensemble with increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.

We systematically investigate the mass spectrum, spatial configuration and magnetic moment of the ground and p-wave states $[cu][\bar{c}\bar{s}]$ with various color-spin configurations in a multiquark color flux-tube model. Numerical results indicate that the state $Z_{cs}(4000)^+$ can be described as the compact state $[cu][\bar{c}\bar{s}]$ with $1^3\!S_1$. Its main color-spin configuration is $[cu]^{1}_{\boldsymbol{6}_c} [\bar{c}\bar{s}]^{1}_{\bar{\boldsymbol{6}}_c}$ and its magnetic moment is 0.73$\mu_{\scriptscriptstyle{N}}$. The state $Z_{cs}(4220)^+$ can be depicted as the compact state $[cu][\bar{c}\bar{s}]$ with $1^1\!P_1$ (or $1^3\!P_1$). Its main color-spin configuration is $[cu]^{0}_{\bar{\boldsymbol{3}}_c}[\bar{c}\bar{s}]^{0}_{\boldsymbol{3}_c}$ (or $[cu]^{0}_{\bar{\boldsymbol{3}}_c}[\bar{c}\bar{s}]^{1}_{\boldsymbol{3}_c}$) and its magnetic moment is 0.12$\mu_{\scriptscriptstyle{N}}$ (or 0.64$\mu_{\scriptscriptstyle{N}}$). The physical state should be the mixture of these two different color-spin configurations and deserves further investigation. In addition, we also predict the properties of the states $[cu][\bar{c}\bar{s}]$ with other quantum numbers in the model.

We investigate the decays of $B^0 \to K^0 X(3872)$ and $B^+ \to K^+ X(3872)$ based on the picture where the $X(3872)$ resonance is strongly coupled to the $D\bar{D}^* + c.c.$ channel. In addition to the decay mechanism where the $X(3872)$ resonance is formed from the $c\bar{c}$ pair hadronization with the short-distance interaction, we have also considered the $D\bar{D}^*$ rescattering diagrams in the long-distance scale, where $D$ and $\bar{D}^*$ are formed from $c$ and $\bar{c}$ separately. Because of the difference of the mass thresholds of charged and neutral $D\bar{D}^*$ channels, and the rather narrow width of the $X(3872)$ resonance, at the $X(3872)$ mass, the loop functions of $D^0\bar{D}^{*0}$ and $D^+\bar{D}^{*-}$ are much different. Taking this difference into account, the ratio of $\mathcal{B}[B^0\to K^0X(3872)]/\mathcal{B}[B^+ \to K^+ X(3872)] \simeq 0.5$ can be naturally obtained. Based on this result, we also evaluate the decay widths of $B_s^0 \to \eta(\eta') X(3872)$. It is expected that future experimental measurements of these decays can be used to elucidate the nature of the $X(3872)$ resonance.

Fiber lasers with different net dispersion cavity values can produce some types of solitons, and rich nonlinear dynamics phenomena can be achieved by selecting different saturable absorbers. A new layered high-entropy van der Waals material (HEX) (Mn,Fe,Co,Ni)PS$_{3}$ was selected as a saturable absorber to achieve a high-power laser output of 34 mW. In addition, the wavelength can be dynamically tuned from 1560 nm to 1531 nm with significant dual-wavelength phenomena at 460 fs pulse duration.

We develop a self-consistent first-principle method based on the density functional theory. Physical quantities such as the density of states, Fermi energy and electron density are obtained using a time-dependent random state method without diagonalization. The numerical error for calculating either global or local variables always scales as $1/\sqrt{SN_{\rm e}}$, where $N_{\rm e}$ is the number of electrons and $S$ is the number of random states, leading to a sublinear computational cost with the system size. In the limit of large systems, one random state could be enough to achieve reasonable accuracy. The accuracy and scaling properties of using the method are derived analytically and verified numerically in different condensed matter systems. Our time-dependent random state approach provides a powerful strategy for large-scale density functional calculations.

The second magnetization peak (SMP) appears in most superconductors and is crucial for the understanding of vortex physics as well as the application. Although it is well known that the SMP is related to the type and quantity of disorder/defects, the mechanism has not been universally understood. We selected three stoichiometric superconducting RbCa$_2$Fe$_4$As$_4$F$_2$ single crystals with identical superconducting critical temperature $T_{\rm c} \sim 31$ K and similar self-field critical current density $J_{\rm c}$, but with different amounts of disorder/defects, to study the SMP effect. It is found that only the sample S2 with moderate disorder/defects shows significant SMP effect. The evolution of the normalized pinning force density $f_{\rm p}$ demonstrates that the dominant pinning mechanism changes from the weak pinning at low temperatures to strong pinning at high temperatures. The microstructure study for sample S2 reveals some expanded Ca$_2$F$_2$ layers and dislocation defects in RbFe$_2$As$_2$ layers. The normalized magnetic relaxation results indicate that the SMP is strongly associated with the elastic to plastic (E-P) vortex transition. As temperature increases, the SMP gradually evolves into a step-like shape and then becomes a sharp peak near the irreversibility field similar to what is usually observed in low-temperature superconductors. Our findings connect the low field SMP of high-temperature superconductors and the high field peak of low-temperature superconductors, revealing the possible universal origin related to the E-P phase transition.

A main task in condensed-matter physics is to recognize, classify, and characterize phases of matter and the corresponding phase transitions, for which machine learning provides a new class of research tools due to the remarkable development in computing power and algorithms. Despite much exploration in this new field, usually different methods and techniques are needed for different scenarios. Here, we present SimCLP: a simple framework for contrastive learning phases of matter, which is inspired by the recent development in contrastive learning of visual representations. We demonstrate the success of this framework on several representative systems, including non-interacting and quantum many-body, conventional and topological. SimCLP is flexible and free of usual burdens such as manual feature engineering and prior knowledge. The only prerequisite is to prepare enough state configurations. Furthermore, it can generate representation vectors and labels and hence help tackle other problems. SimCLP therefore paves an alternative way to the development of a generic tool for identifying unexplored phase transitions.

There have been intensive studies on Kitaev materials for the sake of realization of exotic states such as quantum spin liquid and topological orders. In realistic materials, the Kitaev interaction may coexist with the Dzyaloshinskii–Moriya interaction, and it is of challenge to distinguish their magnitudes separately. Here, we study the topological magnon excitations and related thermal Hall conductivity of kagome magnet exhibiting Heisenberg, Kitaev and Dzyaloshinskii–Moriya interactions exposed to a magnetic field. In a strong magnetic field perpendicular to the plane of the lattice ([111] direction) that brings the system into a fully polarized paramagnetic phase, we find that the magnon bands carry nontrivial Chern numbers in the full region of the phase diagram. Furthermore, there are phase transitions related to two topological phases with opposite Chern numbers, which lead to the sign changes of the thermal Hall conductivity. In the phase with negative thermal conductivity, the Kitaev interaction is relatively large and the width of the phase increases with the strength of Dzyaloshinskii–Moriya interaction. Hence, the present study will contribute to the understanding of related compounds.

Aqueous Na-ion batteries (ANIBs) are considered to be promising secondary battery systems for grid-scale energy storage applications and have attracted widespread attention due to their unique merits of rich resources of Na, as well as the inherent safety and low cost of aqueous electrolytes. However, the narrow electrochemical stability widow and high freezing point of traditional dilute aqueous electrolytes restrict their multi-scenario applications. Considering the charge-storage mechanism of ANIBs, the optimization and design of aqueous Na-based electrolytes dominate their low-temperature performance, which is also hot off the press in this field. In this review, we first systematically comb the research progress of the novel electrolytes and point out their remaining challenges in ANIBs. Then our perspectives on how to further improve the low-temperature performance of ANIBs will also be discussed. Finally, this review briefly sheds light on the potential direction of low-temperature ANIBs, which would guide the future design of high-performance aqueous rechargeable batteries.