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2 . The physical quantities in the presence of the logarithmic singularity near the Fermi level are investigated. We calculate the magnetic susceptibility and specific heat, in particular at low temperatures, discuss the problem if universal behavior and calculate the Wilson ratio. II.
(the energy is referred to the Fermi level, the constants A, B and ∆ are determined by the band spectrum). The logarithmic divergence in ρ(E) is typical for the two-dimensional case (in particular, for the layered ruthenates). However, similar strong Van Hove singularities can occur also in some three-dimensional systems like Pd alloys and weak itinerant ferromagnets ZrZn2 and TiBr2 [28, 29]. First we consider perturbation expansion for the resistivity, following to the original approach by Kondo . We write down the inverse transport relaxation time with the Kondo correction τ −1 (E) = τ0−1 (E)[1 + 4J g(E, 0)].
is the Pauli susceptibility of non-interacting conduction electrons with the singular density of states, which is shown in Fig.1. For ∆ 6= 0 this quantity has a maximum at T ≃ ∆/2. Such a behavior is typical for the case where a density-of-states peak is present near the Fermi level .
However, the NRG calculations  confirms the perturbation expression (8) rather than (18) (see the discussion below). The corresponding problems of the parquet approximation in the Hubbard model are discussed in the works . III.
The details of NRG calculations are discussed in Appendix.
For the standard flat-band case we have T χ(T ) = φ(T /TK ), so that the curves T χ(T ) are universal: a change in J results in a change of TK only. In our situation, such a simple universality does not hold. In particular, for t′ = 0 this fact (illustrated by Fig.4) was demonstrated in Ref..
These asymptotics are independent of J (cf. Fig.4) since at sufficiently low temperatures any value of |J| manyfold exceeds both the temperature and the width of the infinitely thin logarithmic peak in ρ(E). According to Ref., the NFL behavior with C ∝ 1/ ln4 (TK /T ) takes place in the case of underscreened (S > 1/2) Kondo problem . Fig. 8 shows the temperature dependence of impurity specific heat Cimp = Ctot − Cband for different t′ . As a rule, this dependence demonstrates two peaks. At not too small ∆, the high-temperature maximum occurs at the temperature, determined by the distance from VHS to EF . This is owing to the non-monotonous dependence of χ(0) (T ), see Eqs.(10), (13). When decreasing temperature and passing this maximum, Cimp (T ) acquires a minimum and even can become negative. The low-temperature peak is owing to the Kondo effect and takes place in the standard flat-band situation too (see Ref. ). One can see that its position corresponds roughly to the Kondo temperature. For small ∆ < TK , the order of positions of the maxima becomes interchanged. The corresponding magnetic entropy Simp is shown in Fig. 9. One can see that this quantity tends to the value ln 2 = ln(2S + 1) at high temperatures and demonstrates the Kondo compensation at low temperatures. The behavior turns out to be non-monotonous due to the maximum in χ(0) (T ) (see Eq.(12)).
A similar problem which occurs at calculating the slope of specific heat γimp is solved by the same way. The method was tested for the flat-band case to obtain the values R = 2.008, 2.016 and w = 0.416, 0.417 for J = −0.1, −0.2 respectively (cf. Table 1). Our method differs from  by that we calculate χ and γ directly rather than by constructing an effective Hamiltonian explicitly near the fixed point J = −∞. Although resulting in a slight decrease of accuracy, such an approach can be applied more widely, in particular to obtain a NFL behavior (see Figs.4–6).
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