By virtue of their unique physicochemical properties, gold nanoparticles (AuNPs) have gained significant interest in a broad range of biomedical applications such as sensors, diagnosis, and therapy. AuNPs are generally synthesized via different conventional physical and chemical methods, which often use harmful chemicals that induce health hazards and pollute the environment. To overcome these issues, green synthesis techniques have evolved as alternative and eco-friendly approaches to the synthesis of environmentally safe and less-expensive nanoparticles using naturally available metabolites from plants and microorganisms such as bacteria, fungi, and algae. This review provides an overview of the advances in the synthesis of AuNPs using different biological resources with examples, and their profound applications in biomedicine. A special focus on the biosynthesis of AuNPs using different medicinal plants and their multifunctional applications in antibacterial, anti-inflammatory, and immune responses are featured. Additionally, the applications of AuNPs in cancer theranostics, including contrast imaging, drug delivery, hyperthermia, and cancer therapeutics, are comprehensively discussed. Moreover, this review will shed light on the importance of the green synthesis approach, and discuss the advantages, challenges, and prospects in this field.

The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\varphi$ and no coupling to other slow variables (“model $A$”) is investigated in film geometries for the cases of periodic and free boundary conditions. The Hamiltonian $\mathcal{H}$ governing the stationary equilibrium distribution is taken to be $O\left(n\right)$ symmetric and to involve, in the case of free boundary conditions, the boundary terms ${\int }_{{\mathfrak{B}}_j}{\mathring{c}}_j{\varphi }^2/2$ associated with the two confining surface planes ${\mathfrak{B}}_j$, $j=1,2$, at $z=0$ and $z=L$. Both enhancement variables ${\mathring{c}}_j$ are presumed to be subcritical or critical, so that no long-range surface order can occur above the bulk critical temperature $T_{c,\infty }$. A field-theoretic renormalization-group study of the dynamic critical behavior at $d=4-ϵ$ bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at $T_{c,\infty }$. This applies when either both ${\mathring{c}}_j$ take the critical value ${\mathring{c}}_{\text{sp}}$ associated with the special surface transition or else periodic boundary conditions are imposed. Owing to the zero modes, the $ϵ$ expansion becomes ill-defined at $T_{c,\infty }$. Analogously to the static case, the field theory can be reorganized to obtain a well-defined small-$ϵ$ expansion involving half-integer powers of $ϵ$, modulated by powers of $\text{ln}ϵ$. This is achieved through the construction of an effective $\left(d-1\right)$-dimensional action for the zero-mode component of the order parameter by integrating out its orthogonal component via renormalization-group improved perturbation theory. Explicit results for the scaling functions of temperature-dependent finite-size susceptibilities at temperatures $T\ge T_{c,\infty }$ and of layer and surface susceptibilities at the bulk critical point are given to orders $ϵ$ and ${ϵ}^{3/2}$, respectively. They show that $L$ dependent shifts of the multicritical special point occur along the temperature and enhancement axes. For the case of periodic boundary conditions, the consistency of the expansions to $O\left({ϵ}^{3/2}\right)$ with exact large-$n$ results is shown. We also discuss briefly the effects of weak anisotropy, relating theories whose Hamiltonian involves a generalized square gradient term $B^{kl}{\partial }_k\mathbf{\varphi }\cdot {\partial }_l\mathbf{\varphi }$ to those with a conventional ${\left(\nabla \mathbf{\varphi }\right)}^2$ term.

The difficulties arising in the investigation of finite-size scaling in $d$--dimensional O(n) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance $r$ as $r^{-d-\sigma}$ ($0<\sigma\leq2$), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag-Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.

The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a $d$-dimensional model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime }}\times {\infty }^{d^{\prime }}\times L_{\tau }^z$ $(L$-linear space size and $L_{\tau }$-temporal size) and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1<d<\mathrm{}3\mathrm{}$ and $0<~d^{\prime }<~d$ a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.