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2D QUANTUM ANTIFERROMAGNET 191 ∫ I ( f ) = f (M)δ(Ω · M) dM, (14) where f (M) is an arbitrary function of M. Let us insert in the right hand side of (14) the identity 1 = Z ga ∫ ( exp − 1 ) dϕ 2 ϕ ˆB ga ϕ √ , Zga 2 = det( ˆB ga ), (15) 2π where ˆB ga is a positive number or a positive definite operator for some multidimensional generalization. After changing the orders of integration over M and ϕ we can make the change of variables M: M → M − Ωϕ. After that, due to the delta function, we have ϕ = (Ω · M) so that the delta function disappears from integral (14), I ( f ) = Z ga ∫ f (M tr ) exp(A ga ) dM/ √ 2π, −A ga = L ga = 1 2 (Ω · M) ˆB ga (Ω · M), (16) where M tr ≡ M − Ω(Ω·M). With the help of identity (16) we can remove the delta function δ(Ω·M) from measure (11). As a result we must substitute M → M tr in the Lagrangian L ΩM (10) and add the gauge fixing Lagrangian L ga due to the additional exponential in (16). It is convenient to choose the Lagrangian L ga in the form L ga = Js ∑ 2 [(Ω · M) 2 + (Ω · M)(Ω ′ · M ′ )]. (17) l ′ ∈〈l〉 Such choice removes the major dependence on Ω from the Lagrangian (12) which appears due to substitution M → M tr . We can also replace M tr → M in the first term of the Lagrangian L quad (12) due to the identity (Ω · ˙Ω) = 0. In this way, the expression (12) for L quad is valid in the leading order in the small parameter 1/2s. The final expression for the GPF of the QAF is ∫ Z = ∫ ··· exp(A(Ω, M,λ))Z ga Dµ((Ω, M)Dµ(λ), Dµ(Ω, M) = ∏ τl Dµ(λ) = ∏ τl (2s + 1) 3 (1 − M 2 /4) (2π) 3 (1 + M 2 /4) 3 dΩ(τ,l) dM(τ,l), (18) 2 dλ(τ,l)/ [ (2s + 1)(2π) 1/2] . Here Dµ(Ω, M,λ) = Z ga Dµ(Ω, M)Dµ(λ) is the measure of integration, and the action A(Ω, M,λ) is determined by the total Lagrangian L tot = L ΩM + L ga + L λ . 2.4. Properties of the Basic Approximation One can change to the q = (ω,k) momentum representation (ω is the frequency, k is the wave vector) and write out the total quadratic part of the Lagrangian in the matrix form

192 BELINICHER AND PROVIDÊNCIA L quad (q) = sXq ∗ ˆ(q)X q , Xq ∗ = (Ω∗ q , M∗ q ), ( )( ) P ′ ˆ(q)X q = k , ω Ωq , P −ω, Q k M k ′ = P k + c µ , q c µ = µ 2 0 /2J , Q k = J (1 + γ k ), P k = J (1 − γ k ), J = jsz, γ k = (1/2)(cos(k x a) + cos(k y a)). (19) Here X q is a two component vector field which combines the vector fields Ω q and M q ; the constant c µ (13) is expressed through the constant µ 0 which is the mass of the Ω field in the lowest order of perturbation theory. One can invert the 2 × 2 matrix ˆ(q) and get the bare Green function (GF) Ĝ q of the Ω q and M q fields Ĝ q = 1 2s ( ˆ(q)) −1 = 1 ( ) Qk , −ω 2sL q ω, P k ′ L q = ω 2 + ω 2 0k , ω2 0k = ( 1 − γ 2 k ) J 2 + (1 + γ k )µ 2 0/ 2, where ω 0k is the primary magnon frequency in the paramagnetic phase. Below we shall use the notations Gq Ω = G11 q , Gd q = G21 q , Gu q = G12 q , and G q M = G22 q for the matrix elements of the matrix GF Ĝ q . First let us discuss the parameter of perturbation theory. One can see from the explicit form of the Lagrangian (10) that the spin wave nonlinearity of the theory is due to the term M tr and its modifications. Its average value 〈M 2 tr 〉 is (N − 1) ∑ q G M q = (N − 1)T 2s ∑ ω=2πnT,k (20) P k ω 2 + ω0k 2 , (21) where N = 3 is the number of components of the Ω field. The summation over ω is obtained by standard methods [15] and we have 〈 M 2 tr 〉 = (N − 1) 2s ∑ k P k (N − 1) (1 + 2n 0k ) = 2ω 0k 4s { CM0 , T ≪ J (T/J )C M∞ , T ≫ J . (22) Here, n 0k = (exp(ω 0k /T )−1) −1 is the Planck function, and summation over k means the normalized integration over the AF Brillouin band. The constants C M0 and C M∞ are defined by the relations C M0 = ∑ √ 1 − γ k = c 0 − c 1 = 0.65075, 1 + γ k k C M∞ = ∑ k 2 1 + γ k = 1.48491, where all sums of type (23) are calculated by the following method (23) c n = ∑ k γ n k √ 1 − γ 2 k ρ(ɛ) = ∑ k , ∑ f (γ k ) = k ∫ 1 δ(ɛ − γ k ) = 4 π 2 K (1 − ɛ2 ). 0 f (ɛ)ρ(ɛ) dɛ, (24)

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