3 find solutions of the RGM ground-state equations down to J$_{2}$ = 0 . 46. This value lies certainly above the transi- tion point J c $_{2}$. To fix J c 2 we may extrapolate the ground- state energy obtained for case (ii). The crossing point of both energies at J$_{2}$ = 0 . 44 can be considered as the RGM estimate of J c $_{2}$, see the inset of Fig. 1. This value is in reasonable agreement with values for J c 2 reported in other papers. 14,18,19,21,22 The behavior of the spin-spin correlation functions at T = 0 near the transition point J c 2 is illustrated in Fig. 1. For J$_{2}$ ≳ J c 2 there is a noticeable variation in the cor- relation functions. With increasing J$_{2}$ , at J$_{2}$ ≳ 1, the spin-spin correlation functions approach the correspond- ing values of the limiting case J$_{2}$ ≫ 1. This behavior is in qualitative agreement with data from Lanczos ED and coupled-cluster method. 21 At the critical point the correlation functions jump to the exact results of the fer- romagnetic phase, i.e., 〈 S$_{0}$S$_{R}$ 〉 = 0 . 25. Together with the kink in the energy, this indicates a first-order tran- sition. These results confirm the findings in previous papers. 14,19,22 finite N . In this case the quantity C is not related to the magnetization and stays nonzero in the whole tempera- ture region. In Ref. 45 it was shown that C = 2 Tχ/N . To evaluate the thermodynamic quantities, a coupled system of six (for finite systems: seven) nonlinear alge- braic self-consistency equations, including the sum rule C$_{0}$ = 1 $_{2}$, has to be solved numerically to determine the correlators C$_{1}$$_{,}$$_{0}$ , C$_{1}$$_{,}$$_{1}$ , C$_{2}$$_{,}$$_{0}$ , C$_{2}$$_{,}$$_{1}$ , C$_{2}$$_{,}$$_{2}$, ( C ) and the ver- tex parameter α . To solve this system we use Broyden’s method, 46 which yields the solutions with a relative error of about 10 − 8 on the average. The momentum integrals are done by Gaussian integration. To find the numerical solution of the RGM equations for T > 0, we start at high temperatures and decrease T in small steps. Below a certain (low) temperature T$_{0}$ ( J$_{2}$ ) no solutions of the RGM equations (except at T = 0) could be found, since the quantity Δ( T, J$_{2}$ ) in Eqs. (9) and (10) becomes ex- ponentially small which leads to numerical instabilities. Presenting our results in the next section we put J$_{1}$ = − 1. In a first step we use the RGM as described above to determine the transition point J c $_{2}$, where the ferromag- netic ground state gives way for a ground-state phase with zero magnetization. For that we solve the RGM Now we investigate the thermodynamic properties of the model for J$_{2}$ < J c 2 ≈ 0 . 44, where the ground state is ferromagnetic. First we consider the temperature de- B. Thermodynamic properties III. RESULTS A. Phase transition at T = 0 FIG. 2: NN and NNN spin-spin correlation functions for J 2 = 0, 0 . 15, and 0 . 3, from top to bottom, calculated by RGM for N → ∞ (lines) and ED for N = 20 (filled symbols). FIG. 1: Ground-state spin-spin correlation functions 〈 S 0 S R 〉 in dependence on frustration J 2 . (Note that for J 2 < J c 2 ≈ 0 . 44 all 〈 S 0 S R 〉 coincide.) Inset: Ground-state energy E ver- sus J 2 (solid line: RGM data, dashed line: extrapolated RGM data to fix the transition point, see also text). pendence of the spin-spin correlation functions. The NN and NNN correlators are shown in Fig. 2 for various values of J$_{2}$ . For comparison we show also ED data for N = 20 sites. The RGM results agree qualitatively with the ED data. With increasing frustration, the correla- tion functions decrease more rapidly with temperature. Interestingly, for large frustration the NNN correlation function changes its sign at a certain temperature, e.g., equations at T = 0 (i) starting from J$_{2}$ = 0 with increas- ing J$_{2}$ and (ii) starting from large J$_{2}$ ≫ 1 with decreasing J$_{2}$ . In case (i) the RGM equations at T = 0 can be solved until the classical transition at J$_{2}$ = 0 . 5. In case (ii) we −0.05  0  0.05  0.1  0.15  0.2  0.25  0.3  0  0.5  1  1.5  2 < S$^{0}$ S 1,0(1,1) > T NN NNN −0.3 −0.2 −0.1  0  0.1  0.2  0.3  0  0.5  1  1.5  2  2.5  3 < S$^{0}$ S R > J$_{2}$ R=1,0 R=2,0 R=1,1 R=1,2 R=2,2 −0.35 −0.3 −0.25  0.4 0.45  0.5 J$_{2}$ E