## 1 Introduction
The LDA+DMFT calculations suggest that electron correlation effects play an essential role in the formation and the enhancement of orbital moments and magnetic anisotropy in the YCo\({}_{5}\) magnet. ND, XRD measurements, and FPLMTO-DMFT technique have been used [77] to describe an isomorphic (without change of the symmetry) lattice collapse and the electronically topological transition (ETT), which is the first order Lifshitz phase transition in the YCo\({}_{5}\) magnet at pressure about 18 \(\pm\) 2 GPa first reported in refs. [62; 63] based on XRD measurements and FPLO calculations [66]. According to these calculations, the Curie temperature of the YCo\({}_{5}\) magnet, \(T_{c}\)\(\approx\) 987 K, is in the range of experimental measurements. These calculations reveal that, at ambient pressure, the orbital moments are 0.22 \(\mu_{\text{B}}\) and 0.18 \(\mu_{\text{B}}\) for 2c and 3g, correspondingly, and at pressure of 7.2 GPa, the orbital moments are 0.18 \(\mu_{\text{B}}\) and 0.15 \(\mu_{\text{B}}\) for 2c and 3g, correspondingly, which is in reasonable agreement with the experimental observation [28] (at ambient pressure) and also in agreement with results of LDA-DMFT calculations [70]. The atom-resolved Co\({}_{1}\)(2c) and Co\({}_{2}\)(3g) orbital moments reach the maximum values for the Coulomb parameter \(U\)\(\sim\)2 eV. These results for the orbital moments are consistent with previous XMCD measurements [71], inelastic spin flip-neuron scattering experiments [16], and OP calculations [37; 55; 56]. The temperature dependence of MAE and the spontaneous magnetization of the YCo\({}_{5}\) magnet has been calculated by utilizing the relativistic KKRASA-CPA-DLM approach [71; 74; 75; 76; 81; 82]. The calculated total magnetic moment, \(m^{(tot)}\), is 8.50 \(\mu_{\text{B}}\)/f.u. 8.03 \(\mu_{\text{B}}\)/f.u. at 0 K and around \(T\) = 100 K, correspondingly, [71], which is in an fair accord with the experimental data, 8.33 \(\mu_{\text{B}}\)/f.u., at liquid-helium temperature [29], and 7.99 \(\mu_{\text{B}}\)/f.u. at room temperature [28]. Calculated MAE is equal to \(K_{1}\)\(\sim\)4.63 MJ/m\({}^{3}\) at \(T\)\(\approx\)50 K [71], which is larger than reported in the previous calculations, \(K_{1}\)\(\sim\)2.83 MJ/m\({}^{3}\) at \(T\) = 0 K [58; 59; 60; 61] (see Figure 1, [71]), but smaller than the experimental value of \(K_{1}\)\(\sim\)7.38 MJ/m\({}^{3}\) at \(T\) = 4.2 K [24], \(K_{1}\)\(\sim\)6.40 MJ/m\({}^{3}\) at \(T\) = 77 K [15] as well as \(K_{1}\)\(\sim\)6.03 MJ/m\({}^{3}\)[15], \(K_{1}\)\(\sim\)5.50 MJ/m\({}^{3}\)[69], and \(K_{1}\)\(\sim\)5.00 MJ/m\({}^{3}\)[17] at room temperature, \(T\) = 293 K. The calculated MAE decreases as temperature rises, for example, \(K_{1}\)\(\sim\)1.1 MJ/m\({}^{3}\) at 600 K [71]. Calculated Curie temperatures \(T_{c}\)\(\sim\)965 K [71] and \(T_{c}\)\(\sim\)885 K 74] compare well with experimental measurements \(T_{c}\)\(\sim\)920 K [28] and \(T_{c}\)\(\sim\)987 K [15], KKRASA-CPA-DLM formalism has been applied to study temperature dependence of magnetization and the Curie temperature of the YCo\({}_{5}\) magnet doped by Ni and Fe [74]. Both experiments and calculations show an increase or a decrease in magnetization with Fe or Ni replacement, correspondingly. The same KKRASA-CPA-DLM formalism has been applied to study the temperature dependence of magnetization and MAE for the Y(Co\({}_{1\text{-}\text{x}}\))Fe\({}_{\text{x}}\)Cu\({}_{\text{y}}\))\({}_{5}\) magnets [81]. These calculations show that MAE of the YCo\({}_{5}\) magnet could be increased by adding relatively minor amounts of Fe and/or Cu, with the larger MAE field detected for the composition Y(Co\({}_{0.82}\)Fe\({}_{0.09}\)Cu\({}_{0.09}\))\({}_{5}\) with the condition of a preferred replacement of Fe and Cu at 3g and 2c, correspondingly. The recent improvement of KKRASA-CPA-DLM methodology allowed [PERSON] and [PERSON] [82] to reproduce the experimental MAE of the YCo\({}_{5}\) magnet, \(K_{I}\)\(\sim\)5.0 MJ/m\({}^{3}\)[17], at room temperature.