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The static and dynamic properties of the anisotropic XY-model (s = 1/2) on the inhomogeneous periodic chain, composed of N cells with n different exchange interactions and magnetic moments, in a transverse field h, are determined exactly at arbitrary temperatures.

The static and dynamic properties of the isotropic XY-model (s = 1/2) on the inhomogeneous periodic chain, composed of N segments with n different exchange interactions and magnetic moments, in a transverse field h are obtained exactly at arbitrary temperatures. The properties are determined by introducing the generalized Jordan-Wigner transformation and by reducing the problem to a diagonalization of a finite matrix of n-th order. The diagonalization procedure is discussed in detail and the critical behaviour induced by the transverse field, at T = 0, is presented. The quantum transitions are determined by analyzing the behaviour of the induced magnetization, defined as (1/n) n m=1 µ m < S z j,m > where µ m is the magnetic moment at site m within the segment j, as a function of the field, and the critical fields determined exactly. The dynamic correlations, < S z j,m (t)S z j ′ ,m ′ (0) >, and the dynamic susceptibility χ zz q (ω) are also obtained at arbitrary temperatures. Explicit results are also presented in the limit T = 0, where the critical behaviour occurs, for the static susceptibility χ zz q (0) as a function of the transverse field h, and for the frequency dependency of dynamic susceptibility χ zz q (ω). Also in this limit, the transverse time-correlation < S x j,m (t)S x j ′ ,m ′ (0) >, the dynamic and isothermal susceptibilities, χ xx q (ω) and χ xx T , are obtained for the transverse field greater or equal than the saturation field.

We present a modified diffusive epidemic process (DEP) that has a finite threshold on scale-free graphs, motivated by the COVID-19 pandemic. The DEP describes the epidemic spreading of a disease in a non-sedentary population, which can describe the spreading of a real disease. Our main modification is to use the Gillespie algorithm with a reaction time t max, exponentially distributed with mean inversely proportional to the node population in order to model the individuals’ interactions. Our simulation results of the modified model on Barabasi–Albert networks are compatible with a continuous absorbing-active phase transition when increasing the average concentration. The transition obeys the mean-field critical exponents β = 1, γ′ = 0 and ν ⊥ = 1/2. In addition, the system presents logarithmic corrections with pseudo-exponents β ̂ = γ ̂ ′ = − 3 / 2 on the order parameter and its fluctuations, respectively. The most evident implication of our simulation results is if the individuals avoid social interactions in order to not spread a disease, this leads the system to have a finite threshold in scale-free graphs.

We investigated the susceptible-infected-susceptible model on a square lattice in the presence of a conjugated field based on recently proposed reactivating dynamics. Reactivating dynamics consists of reactivating the infection by adding one infected site, chosen randomly when the infection dies out, avoiding the dynamics being trapped in the absorbing state. We show that the reactivating dynamics can be interpreted as the usual dynamics performed in the presence of an eective conjugated field, named the reactivating field. The reactivating field scales as the inverse of the lattice number of vertices n, which vanishes at the thermodynamic limit and does not aect any scaling properties including ones related to the conjugated field.