2 with where K$_{1}$$_{,}$$_{1}$ = 1 + 2 α (2 C$_{1}$$_{,}$$_{1}$ + C$_{2}$$_{,}$$_{0}$ − 5 C$_{1}$$_{,}$$_{0}$ ), K$_{2}$$_{,}$$_{2}$ = 1 + 2 α (2 C$_{2}$$_{,}$$_{0}$ + C$_{2}$$_{,}$$_{2}$ − 5 C$_{1}$$_{,}$$_{1}$ ), K$_{1}$$_{,}$$_{2}$ = 4 α ( C$_{1}$$_{,}$$_{2}$ − C$_{1}$$_{,}$$_{0}$ ), and K$_{2}$$_{,}$$_{1}$ = 4 α ( C$_{1}$$_{,}$$_{0}$ + C$_{1}$$_{,}$$_{2}$ − 2 C$_{1}$$_{,}$$_{1}$ ). From the Green’s func- tion (6) the correlation functions C$_{R}$ = 1 N ∑ $_{q}$C$_{q}$ e i qR of arbitrary range R are determined by the spectral theorem, 40 ω 2 q = 2 ∑ k,l (=1 , 2) J$_{k}$J$_{l}$ ( 1 − γ ( k ) q ) × [ K$_{k,l}$ + 8 αC$_{1}$$_{,k}$$_{−}$$_{1}$ ( 1 − γ ( l ) q )] , (7) C$_{q}$ = 〈 S + $_{q}$S − q 〉 = M$_{q}$ 2 ω$_{q}$ [1 + 2 n ( ω$_{q}$ )] , (8) where n ( ω$_{q}$ ) = ( e ω$_{q}$/T − 1 ) − 1 is the Bose function. Tak- ing the on-site correlator C$_{R}$$_{=}$$_{0}$ and using the oper- ator identity S + i S − i = 1 2 + S z i we get the sum rule C$_{0}$ = 1 N ∑ $_{q}$C$_{q}$ = 1 $_{2}$. The uniform static spin suscep- tibility χ = lim$_{q}$$_{→}$$_{0}$ χ$_{q}$ , where χ$_{q}$ = χ$_{q}$ ( ω = 0) and χ$_{q}$ ( ω ) = 1 $_{2}$χ + − q ( ω ), is given by χ = − 2 Δ ∑ k =1 , 2 kJ$_{k}$C$_{1}$$_{,k}$$_{−}$$_{1}$, Δ = ∑ k,l =1 , 2 kJ$_{k}$J$_{l}$K$_{k,l}$. (9) The correlation length may be calculated by expanding χ$_{q}$ around q = 0, 28,35,36 χ$_{q}$ = χ/ [ 1 + ξ $^{2}$( q 2 x + q 2 y )] . We find The ferromagnetic long-range order, occurring at T = 0 only, is reflected by a non-vanishing quantity C (see, e.g., Ref. 36) according to C$_{R}$ (0) = 1 N ∑ q $_{=0}$C$_{q}$ (0) e i qR + C , where C , typically called condensation term, 36 is con- nected with the magnetization m by m 2 = 3 C/ 2. Due to the exact result C$_{R}$$_{̸}$$_{=}$$_{0}$ (0) = 1 $_{6}$, C$_{q}$ (0) = M$_{q}$ (0) 2 ω$_{q}$ (0) must be independent of q . Examining Eqs. (4) and (7), this requires K$_{k,l}$ (0) = 0 which leads to α (0) = 3 2 and C$_{q}$ (0) = 1 $_{3}$. Hence, in case of a ferromagnetic ground state we find at zero temperature ω$_{q}$ = 2 ρ$_{s}$ q 2 ( | q | << 1), ρ$_{s}$ = − 1 $_{4}$( J$_{1}$ + 2 J$_{2}$ ) (where ρ$_{s}$ is the spin-stiffness) and C$_{R}$ (0) = 1 $_{3}$δ$_{R}$$_{,}$$_{0}$ + 1 $_{6}$. Note that the sum rule C$_{0}$ = 1 2 is fulfilled. In Eqs. (9) and (10), we have Δ(0) = 0 so that χ and ξ diverge as T → 0 indicating the ferromagnetic phase transition. Since we will compare the RGM data with ED data for finite lattices, see Sec. III, we have to adopt the RGM to ξ 2 = 2 α ( J 2 $_{1}$C$_{1}$$_{,}$$_{0}$ + 2 J$_{1}$J$_{2}$ ( C$_{1}$$_{,}$$_{0}$ + C$_{1}$$_{,}$$_{1}$ ) + 4 J 2 $_{2}$C$_{1}$$_{,}$$_{1}$ ) Δ . (10) II. ROTATION-INVARIANT GREEN’S FUNCTION METHOD To calculate the thermodynamic quantities of the model (1), we use the RGM 28,36,37,43,44 for determin- ing the transverse dynamic spin susceptibility χ + − q ( ω ) = −〈〈 S + $_{q}$; S − $_{q}$〉〉$_{ω}$ , where 〈〈 . . . ; . . . 〉〉$_{ω}$ denotes the two-time commutator Green’s function. 40 Taking the equations of motion up to the second step and supposing spin rota- tional symmetry, i.e., 〈 S z i 〉 = 0, we obtain The paper is organized as follows: In Sec. II the RGM applied to model (1) is presented. The results for the thermodynamic quantities are discussed in Sec. III. Fi- nally, a summary is given in Sec. IV. sites are considered. As we will see below, for these lat- tice sizes the finite-size effects at low temperatures are large. Therefore, methods studying infinite systems are highly desirable. Among others, the Green’s function method is a powerful tool to study magnetic systems at finite temperatures, see e.g. Refs. 40–42 and refer- ences therein. A specific variant appropriate for frus- tated systems is the RGM. In the present paper, we use the RGM to investigate thermodynamic properties of the frustrated model (1), focusing on J$_{2}$ ≤ J c 2 ≈ 0 . 4 | J$_{1}$ | . The RGM has been applied successfully to low-dimensional (frustrated) infinite quantum spin systems. 28,35–39,43–45 In particular, for the 1D spin-1 / 2 Heisenberg ferromag- net it was shown that the RGM reproduces Bethe-ansatz results. 28,37 Since the RGM can be formulated also for finite systems, we use full exact diagonalization (ED) of finite square lattices of N = 16 and N = 20 sites to compare the RGM with ED data for finite lattices. where C$_{n,m}$ ≡ C$_{R}$ = 〈 S + 0 S − $_{R}$〉 = 2 〈 S z $_{0}$S z $_{R}$〉 , R = n e$_{x}$ + m e$_{y}$ , γ (1) q = (cos q$_{x}$ + cos q$_{y}$ ) / 2, and γ (2) q = cos q$_{x}$ cos q$_{y}$ . The second derivative − ¨ S + q is approximated in the spirit of the schemes employed in Refs. 28,35–39,43–45. That is, in − ¨ S + i we adopt the decoupling M$_{q}$ = − 8 ∑ k =1 , 2 J$_{k}$C$_{1}$$_{,k}$$_{−}$$_{1}$ ( 1 − γ ( k ) q ) , (4) ω $^{2}$〈〈 S + q ; S − $_{q}$〉〉$_{ω}$ = M$_{q}$ + 〈〈− ¨ S + $_{q}$; S − $_{q}$〉〉$_{ω}$ , (2) M$_{q}$ = 〈 [[ S + $_{q}$, H ] , S$_{−}$$_{q}$ ] 〉 , − ¨ S + q = [[ S + $_{q}$, H ] , H ] . (3) The moment M$_{q}$ is given by the exact expression Following the arguments of Ref. 28 we put α$_{i,k}$ = α in the whole temperature region. We obtain − ¨ S + q = ω 2 $_{q}$S + q and S + i S + j S − k = α$_{i,k}$ 〈 S + i S − k 〉 S + j + α$_{j,k}$ 〈 S + j S − k 〉 S + i . (5) χ + − q ( ω ) = −〈〈 S + $_{q}$; S − $_{q}$〉〉$_{ω}$ = M$_{q}$ ω 2 q − ω $_{2}$, (6) To estimate the zero-temperature transition point J c 2 (see Sec. III A), we determine the ground-state corre- lation functions and the ground-state energy per site, E = 3( J$_{1}$C$_{1}$$_{,}$$_{0}$ + J$_{2}$C$_{1}$$_{,}$$_{1}$ ), for J$_{2}$ > J c $_{2}$. Thereby, we take into consideration the possible existence of a quantum collinear phase described by the magnetic ordering vector Q = ( π, 0) or (0 , π ) and proceed along the lines indicated in Ref. 44.