Hilbert Series and Operator Counting on the Higgs Effective Field Theory

We present a systematic procedure for determining the Hilbert series that counts the number of independent operators in the Higgs effective field theory. After removing the redundancies from equation-of-motion and integration-by-part, we provide an algorithm of treating the redundancy from the operators involving in spurion fields parametrizing the custodial symmetry breaking. Furthermore, we utilize the outer automorphism of the Lorentz and internal symmetries to separate operators with different CP properties. With these new implements, the Hilbert series up to chiral dimension 10 are obtained, and CP-even and CP-odd operators can be further divided. Extensions to all orders are straightforward.Comment: 28 pages, 3 figures, 6 table

Quench Dynamics of Topological Maximally-Entangled States

We investigate the quench dynamics of the one-particle entanglement spectra (OPES) for systems with topologically nontrivial phases. By using dimerized chains as an example, it is demonstrated that the evolution of OPES for the quenched bi-partite systems is governed by an effective Hamiltonian which is characterized by a pseudo spin in a time-dependent pseudo magnetic field $\vec{S}(k,t)$. The existence and evolution of the topological maximally-entangled edge states are determined by the winding number of $\vec{S}(k,t)$ in the $k$-space. In particular, the maximally-entangled edge states survive only if nontrivial Berry phases are induced by the winding of $\vec{S}(k,t)$. In the infinite time limit the equilibrium OPES can be determined by an effective time-independent pseudo magnetic field \vec{S}_{\mb{eff}}(k). Furthermore, when maximally-entangled edge states are unstable, they are destroyed by quasiparticles within a characteristic timescale in proportional to the system size.Comment: 5 pages, 3 figure

Bacteria photosensitized by intracellular gold nanoclusters for solar fuel production.

The demand for renewable and sustainable fuel has prompted the rapid development of advanced nanotechnologies to effectively harness solar power. The construction of photosynthetic biohybrid systems (PBSs) aims to link preassembled biosynthetic pathways with inorganic light absorbers. This strategy inherits both the high light-harvesting efficiency of solid-state semiconductors and the superior catalytic performance of whole-cell microorganisms. Here, we introduce an intracellular, biocompatible light absorber, in the form of gold nanoclusters (AuNCs), to circumvent the sluggish kinetics of electron transfer for existing PBSs. Translocation of these AuNCs into non-photosynthetic bacteria enables photosynthesis of acetic acid from CO2. The AuNCs also serve as inhibitors of reactive oxygen species (ROS) to maintain high bacterium viability. With the dual advantages of light absorption and biocompatibility, this new generation of PBS can efficiently harvest sunlight and transfer photogenerated electrons to cellular metabolism, realizing CO2 fixation continuously over several days

Dimensions of fractals related to languages defined by tagged strings in complete genomes

A representation of frequency of strings of length K in complete genomes of many organisms in a square has led to seemingly self-similar patterns when K increases. These patterns are caused by under-represented strings with a certain "tag"-string and they define some fractals when K tends to infinite. The Box and Hausdorff dimensions of the limit set are discussed. Although the method proposed by Mauldin and Williams to calculate Box and Hausdorff dimension is valid in our case, a different and simpler method is proposed in this paper.Comment: 9 pages with two figure

Mean-squared displacement and variance for confined Brownian motion

For one-dimension Brownian motion in the confined system with the size $L$, the mean-squared displacement(MSD) defined by $\left \langle (x-x_0)^2 \right\rangle$ should be proportional to $t^{\alpha(t)}$. The power $\alpha(t)$ should range from $1$ to $0$ over time, and the MSD turns from $2Dt$ to $c L^2$, here the coefficient $c$ independent of $t$, $D$ being the diffusion coefficient. The paper aims to quantitatively solve the MSD in the intermediate confinement regime. The key to this problem is how to deal with the propagator and the normalization factor of the Fokker-Planck equation(FPE) with the Dirichlet Boundaries. Applying the Euler-Maclaurin approximation(EMA) and integration by parts for the small $t$, we obtain the MSD being $2Dt(1-\frac{2\sqrt{\xi} }{3\pi\sqrt{\pi}})$, with $t_{ch}=\frac{L^2}{4\pi^2D},\xi\equiv \frac{t}{t_{ch}}$, and the power $\alpha(t)$ being $\frac{1-0.18\sqrt{\xi}}{1-0.12\sqrt{\xi}}$. Further, we analysis the MSD and the power for the $d$-dimension system with $\gamma$-dimension confinement. In the case of $\gamma< d$, there exists the sub-diffusive behavior in the intermediate time. The universal description is consistent with the recent experiments and simulations in the micro-nano systems. Finally, we calculate the position variance(PV) meaning $\left\langle (x-\left\langle x \right\rangle)^2 \right\rangle$. Under the initial condition referring to the different probability density function(PDF) being $p_{0}(x)$, MSD and PV should exhibit different dependencies on time, which reflect corresponding diffusion behaviors.As examples, the paper discusses the representative initial PDFs reading $p_{0}(x)=\delta(x-x_0)$, with the midpoint $x_0=\frac{L}{2}$ and the endpoint $x_0=\epsilon$(or $0^+$).The MSD(equal to PV) reads $2Dt(1-\frac{5\pi^3 Dt}{L^2})$,and $\frac{4}{\pi}(2Dt)[1+\frac{2\sqrt{\pi Dt}}{L}]$for the small $t$,respectively.Comment: 18 pages, 4 figure