We analyse the contribution of critical Eu2+ spin fluctuations to the resistivity of the layered pnictide EuFe2As2, using critical parameters established recently from resistivity, magnetic susceptibility and non-resonant microwave absorption of single crystals. Including the reported parameter uncertainties, the fitted fluctuation time scale gives h omega(c)/k(B)T<6 & times;10(-5) at |t|=0.1, so carrier scattering is quasi-static: the Kramers-Kronig relation reduces the transport kernel to the static spin susceptibility, independent of the dynamic exponent z. For a transport kernel smooth around the ordering vector, the singular resistivity follows the magnetic energy (Fisher-Langer); the two-dimensional Ising assignment (alpha=0) gives d rho(fl)/dT proportional to-ln|t|, compatible with the weak bend observed near T-m approximate to 19 K. In a magnetic field, the reported cutoff change xi(sat)/xi(0): 50 -> 20 alone reduces the derivative amplitude only to similar or equal to 0.80; the total suppression of the anomaly involves non-universal, field-dependent amplitudes and requires field-resolved fits of rho(T, H). The increase of z to 2.05 does not directly control the DC resistivity.
Trust is a fundamental cornerstone of socioeconomic systems, underpinning cooperation and stability. Previous researches on the evolution of trust have primarily focused on unidirectional influence from investors to trustees. However, prevalent models frequently fail to mitigate the inherent information asymmetry between these agents. To bridge this gap, we incorporate a levy mechanism and a two-way subsidy mechanism into the networked Nplayer trust game with a reputation evaluation, and investigates its evolutionary dynamics. Through extensive Monte Carlo simulations, we demonstrate that the proposed mechanism can effectively enhance the overall trust level within the population. Furthermore, the two-way subsidy scheme favors the rewarding of investors even under high dilemma strengths, thereby enabling trustworthy trustees to prevail over untrustworthy ones. To refine the closed-loop process of levy-reputation-based subsidy-behavioral reward-feedback to investors, we perform a series of tests to balance the magnitude of levy and subsidy, avoiding excessive intervention in the natural evolutionary dynamics of the system. Finally, our results show that a moderate levy with a relatively low subsidy exhibits the optimal trust level under moderate dilemma strengths, whereas a moderate levy with a higher subsidy is more suitable for the trust model under the higher dilemma strengths.
We investigate the nonlinear dynamical reorganization induced by PT-symmetry breaking in a non-Hermitian spin-12 XXZ chain with a staggered imaginary longitudinal field γ. Using exact diagonalization and focusing on finite-size datasets, primarily N=6, we identify the leading PT transition from the onset of complex spectral weight and track its dynamical consequences through the infinite-temperature out-of-time-ordered correlator (OTOC), from which we extract an effective finite-size Lyapunov exponent λL. We find a sharp two-regime structure: in the PT-unbroken phase the spectrum is real and λL remains zero within numerical resolution, whereas in the broken phase the OTOC develops an extended exponential-growth regime and λL becomes finite. Near the leading transition, the finite-size data are well described by λL∝(γ−γc)ν, with an effective onset exponent ν ≈ 0.60, while deeper in the broken phase λL(γ) develops visible kinks that track the stepped complexification of the many-body spectrum. Extending the analysis into the (γ, Δ) plane reveals a nontrivial finite-size phase boundary γc(Δ) and an approximate deep-broken-phase convergence in which the growth rate becomes only weakly dependent on the exchange anisotropy Δ. We further show that the operator entanglement entropy Sop(t) and Schmidt gap ΔS(t) reorganize in the opposite direction to the OTOC growth: in the broken phase the OTOC grows faster, yet Sop is reduced and ΔS is enhanced. Here Sop is the entropy of the normalized operator-Schmidt spectrum of the vectorized operator, not the entropy of a physical density matrix; the overall nonunitary change of operator norm is separated from this distribution-sensitive quantity. We further expose the local square-root topology of the leading Hamiltonian exceptional point by encircling it in a two-parameter complex staggered field: one circuit exchanges the two coalescing energy sheets, while two circuits restore the original branch assignment. Throughout, λL is used as an effective finite-size growth rate; its positivity alone is not taken as proof of broad unitary-like scrambling. The results therefore identify the PT transition not only as a spectral exceptional-point transition, but also as a finite-size nonlinear dynamical transition with distinct growth and distribution signatures in operator space.
In the weak-coupling regime, we study effects of the competing intrabond-pair hopping (W) and intrabondexchange hopping (F) on the ground state properties of a one-dimensional system modelled by the t-U-W-F Hamiltonian. At half filling and for on-site repulsion, we combine the bosonization and renormalizationgroup analysis to obtain quantum phase diagram. The model shows the occurrence of charge-gap transition, Gaussian transition, and spin-gap transition. We map the phase diagram in the W-F plane, which consists of three insulating phases with a Mott insulator (MI), a Haldane insulator (HI) and a spin dimerized insulator (SDI) as well as two metallic phases with a Luther-Emery (LE) liquid and a Luttinger liquid (LL). The result exhibits that the competition between the W and F hopping closes the charge gap in the region where F > F(c )and W < W-c, with the critical values F-c = (2)/(3) U and W-c = 0.
Phase-imprinting protocols-pulsed potentials that write a target relative phase onto a double well-are often assumed independent of pulse shape and duration. We show this holds only for a gauge-complete Hamiltonian containing both the scalar pulse and a compensating momentum term; the scalar-only imprint used in experiment reproduces the endpoint phase in the impulsive limit or when the phase functional is sufficiently flat across each occupied wavepacket. Direct wavefunction propagation of a continuous quartic double well confirms the gauge complete identity to within 2.17 & times; 10-5 rad across three pulse shapes and durations, and shows the scalar-only discrepancy saturating near 1.77 & times; 10-2, matching a harmonic estimate to within a few percent. Correlated schedule noise cancels analytically, to within discretization error; endpoint miscalibration does not. The results give a flatness criterion for when a scalar imprint suffices and when it cannot reproduce the gauge-complete reference.
