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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. 

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. 

## 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 

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The moment M$_{q}$ is given by the exact expression 

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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 

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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 

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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 

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 

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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 

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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. c 

To estimate the zero-temperature transition point J 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. 

Since we will compare the RGM data with ED data for finite lattices, see Sec. III, we have to adopt the RGM to 

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