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<jats:p>Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.</jats:p>

<jats:p>We conducted measurement and calculation to resolve the long-standing large discrepancy in the metastable state lifetime for the <jats:sup>88</jats:sup>Sr atom between theoretical and experimental results. The present lifetime <jats:inline-formula> <jats:tex-math><?CDATA $\tau ={830}_{-240}^{+600}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>830</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>240</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>600</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpl_39_7_073201_ieqn3.gif" xlink:type="simple" /> </jats:inline-formula> s, measured using the magneto-optical trap as a photon amplifier to detect the weak decay events, is approximately 60% larger than the previous experimental value <jats:inline-formula> <jats:tex-math><?CDATA $\tau ={520}_{-140}^{+310}$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>520</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>140</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>310</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpl_39_7_073201_ieqn4.gif" xlink:type="simple" /> </jats:inline-formula> s. By considering the electron correlation effects in the framework of the multiconfiguration Dirac–Hartree–Fock theory, we obtained a theoretical lifetime of 1079(54) s, which lies in the range of measurements with error bars. Furthermore, we considered the higher-order electron correlation and Breit interaction to control the uncertainty of the theoretical calculation. The significant improvement in the agreement between calculations and measurements is attributed to the updated blackbody radiation-induced decay rate.</jats:p>

<jats:p>We investigate the topological phase transition driven by non-local electronic correlations in a realistic quantum anomalous Hall model consisting of <jats:italic>d<jats:sub>xy</jats:sub> </jats:italic>–<jats:italic>d</jats:italic> <jats:sub> <jats:italic>x</jats:italic> <jats:sup>2</jats:sup> – <jats:italic>y</jats:italic> <jats:sup>2</jats:sup> </jats:sub> orbitals. Three topologically distinct phases defined in the non-interacting limit evolve to different charge density wave phases under correlations. Two conspicuous conclusions were obtained: The topological phase transition does not involve gap-closing and the dynamical fluctuations significantly suppress the charge order favored by the next nearest neighbor interaction. Our study sheds light on the stability of topological phase under electronic correlations, and we demonstrate a positive role played by dynamical fluctuations that is distinct to all previous studies on correlated topological states.</jats:p>

<jats:p>Tuning the bandgap in layered transition metal dichalcogenides (TMDCs) is crucial for their versatile applications in many fields. The ternary formation is a viable method to tune the bandgap as well as other intrinsic properties of TMDCs, because the multi-elemental characteristics provide additional tunability at the atomic level and advantageously alter the physical properties of TMDCs. Herein, ternary Ti<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Zr<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Se<jats:sub>2</jats:sub> single crystals were synthesized using the chemical-vapor-transport method. The changes in electronic structures of ZrSe<jats:sub>2</jats:sub> induced by Ti substitution were revealed using angle-resolved photoemission spectroscopy. Our data show that at a low level of Ti substitution, the bandgap of Ti<jats:sub> <jats:italic>x</jats:italic> </jats:sub>Zr<jats:sub>1 – <jats:italic>x</jats:italic> </jats:sub>Se<jats:sub>2</jats:sub> decreases monotonically, and the electronic system undergoes a transition from a semiconducting to a metallic state without a significant variation of dispersions of valence bands. Meanwhile, the size of spin-orbit splitting dominated by Se 4<jats:italic>p</jats:italic> orbitals decreases with the increase of Ti doping. Our work shows a convenient way to alter the bandgap and spin-orbit coupling in TMDCs at the low level of substitution of transition metals.</jats:p>

<jats:p>As a typical hole-doped cuprate superconductor, Bi<jats:sub>2</jats:sub>Sr<jats:sub>2</jats:sub>CaCu<jats:sub>2</jats:sub>O<jats:sub>8 + <jats:italic>δ</jats:italic> </jats:sub>(Bi2212) carrier doping is mostly determined by its oxygen content. Traditional doping methods can regulate its doping level within the range of hole doping. Here we report the first application of CaH<jats:sub>2</jats:sub> annealing method in regulating the doping level of Bi2212. By continuously controlling the anneal time, a series of differently doped samples can be obtained. The combined experimental results of x-ray diffraction, scanning transmission electron microscopy, resistance and Hall measurements demonstrate that the CaH<jats:sub>2</jats:sub> induced topochemical reaction can effectively change the oxygen content of Bi2212 within a very wide range, even switching from hole doping to electron doping. We also found evidence of a low-<jats:italic>T</jats:italic> <jats:sub>c</jats:sub> superconducting phase in the electron doping side.</jats:p>

<jats:p>We predict ultrafast modulation of the pure molten metal surface stress fields under the irradiation of the single femtosecond laser pulse through the two-temperature model molecular-dynamics simulations. High-resolution and precision calculations are used to resolve the ultrafast laser-induced anisotropic relaxations of the pressure components on the time-scale comparable to the intrinsic liquid density relaxation time. The magnitudes of the dynamic surface tensions are found being modulated sharply within picoseconds after the irradiation, due to the development of the nanometer scale non-hydrostatic regime behind the exterior atomic layer of the liquid surfaces. The reported novel regulation mechanism of the liquid surface stress field and the dynamic surface tension hints at levitating the manipulation of liquid surfaces, such as ultrafast steering the surface directional transport and patterning.</jats:p>

<jats:p>In this paper, the name of the first author was written as Ogundare Rasheed Toyin. To comply with the style of journal publication, the order of given name, Rasheed Toyin, and family name, Ogundare, should be exchanged. The name should be written as Rasheed Toyin Ogundare.</jats:p>

<jats:p>Exploring quantum phenomena beyond predictions of any classical model has fundamental importance to understand the boundary of classical and quantum descriptions of nature. As a typical property that a quantum system behaves distinctively from a classical counterpart, contextuality has been studied extensively and verified experimentally in systems composed of at least three levels (qutrit). Here we extend the scope of experimental test of contextuality to a minimal quantum system of only two states (qubit) by implementing the minimum error state discrimination on a single <jats:sup>171</jats:sup>Yb<jats:sup>+</jats:sup> ion. We observe a substantial violation of a no-go inequality derived by assuming non-contextuality, and firmly conclude that the measured results of state discrimination cannot be reconciled with any non-contextual description. We also quantify the contextual advantage of state discrimination and the tolerance against quantum noises.</jats:p>

<jats:p>We investigate a time-independent many-boson system, whose ground states are quasi-degenerate and become infinitely degenerate in the thermodynamic limit. Out of these quasi-degenerate ground states we construct a quantum state that evolves in time with a period that is logarithmically proportional to the number of particles, that is, <jats:italic>T</jats:italic> ∼ log <jats:italic>N</jats:italic>. This boson system in such a state is a quantum time crystal as it approaches the ground state in the thermodynamic limit. The logarithmic dependence of its period on the total particle number <jats:italic>N</jats:italic> makes it observable experimentally even for systems with very large number of particles. Possible experimental proposals are discussed.</jats:p>

<jats:p>The upper critical dimension of the Ising model is known to be <jats:italic>d</jats:italic> <jats:sub>c</jats:sub> = 4, above which critical behavior is regarded to be trivial. We hereby argue from extensive simulations that, in the random-cluster representation, the Ising model simultaneously exhibits two upper critical dimensions at (<jats:italic>d</jats:italic> <jats:sub>c</jats:sub> = 4, <jats:italic>d</jats:italic> <jats:sub>p</jats:sub> = 6), and critical clusters for <jats:italic>d</jats:italic> ≥ <jats:italic>d</jats:italic> <jats:sub>p</jats:sub>, except the largest one, are governed by exponents from percolation universality. We predict a rich variety of geometric properties and then provide strong evidence in dimensions from 4 to 7 and on complete graphs. Our findings significantly advance the understanding of the Ising model, which is a fundamental system in many branches of physics.</jats:p>