Equivalence of O(3) nonlinear sigma model and the CP1 model: A path integral approach
aa r X i v : . [ qu a n t - ph ] O c t Equivalence of O(3) nonlinear σ model and the CP model: A path integral approach Ran Cheng, Qian Niu
Department of Physics, University of Texas at Austin, Austin, TX 78712 USA
A rigorous proof is given on the equivalence of the O(3) nonlinear σ model and the CP model via pathintegral approach. The low energy dynamics of anti-ferromagnetically corre-lated spins can be well described by the O(3) nonlinear σ model (NL σ M), which abandoned the strong requirement ofglobal order and only assumes the local
Neel order [1, 2]. Thecentral idea is to express the local spin field s i by a unimodularNeel field | ˆ n ( x µ ) | = 1 and a small canting field | ˆ m ( x µ ) | ≪ , s i = ( − i ˆ n ( x µ ) q − ˆ m ( x µ ) + ˆ m ( x µ ) where x µ represents the Euclidean space of the continuumbackground lattice. By integrating out the small canting field ˆ m ( x µ ) in the path integral formalism, people obtained the de-sired effective action: Z D ˆ n D ˆ mδ (ˆ n − e − S [ˆ n, ˆ m ] = Z D ˆ nδ (ˆ n − e − S eff [ˆ n ] where the original action S [ˆ n, ˆ m ] is derived from the Heisen-berg model [2] and the effective action takes the form: S eff [ˆ n ] = 14 g Z d x∂ µ ˆ n · ∂ µ ˆ n where summation over repeated indices is assumed and g isthe coupling strength determined by the experiments. Theintegration is taken over d + 1 dimensional Euclidean space R d x = R d τ d x .While O(3) NL σ M manifests great usefulness in the studyof antiferromagnetic systems near their critical points, peopleusually solve the model, however, by transforming it intothe celebrated CP model [3] by the Hopf map ˆ n = z † ˆ σz ( σ a = normalized Pauli matrices) where z is the CP field.The High - T c superconductivity is a good example amongthese situations where the CP model is often taken as astarting point [4]. A striking property of the CP modelis the gauge field minimally coupled to CP field acquiresMaxwell dynamics in the long wave length limit, by whichelectrons with opposite spins become attractive [3, 4]. How-ever, although the equivalence of the two models serves asa crucial foundation in these applications, a proof of theirexact equivalence still bears mathematical restrictions andcomplexities [5, 6].In this notes, we perform a simple but rigid proof of thisequivalence via the path integral approach. To start with, wewrite down explicitly the amplitude for the O(3) NL σ M: Z = Z D ˆ nδ (ˆ n − e − g R d x∂ µ ˆ n · ∂ µ ˆ n (1) and the amplitude for the CP model: Z = Z D z D A µ δ ( | z | − e − g R d x | ( ∂ µ − iA µ ) z | (2)Proof of the equivalence between the two models is nothingbut to show Z is proportional to Z under the Hopf map ˆ n = z † ˆ σz . Let us express the CP field as z = ( z , z ) T =( re iα , se iβ ) T , and it is easy to check that r + s = 1 due tothe constraint ˆ n = | z | = | z | + | z | = 1 . This means the CP field is constrained on the unit complex sphere. In termsof r, s, α, and β , the action in Z can be written as: g Z d x∂ µ ˆ n · ∂ µ ˆ n = 1 g Z d x [ r s ( ∂ µ α − ∂ µ β ) + ( ∂ µ r ) + ( ∂ µ s ) ] (3)Next, we integrate out the gauge field in the amplitude Z which is a Gaussian integral and then express the action alsoin terms of the new variables r, s, α, and β : Z = Z D z D A µ δ ( | z | − e − g R d x | ( ∂ µ − iA µ ) z | = Z D z D A µ δ ( | z | − e − g R d x∂ µ z † ∂ µ z e − g R d x [ A µ + iA µ ( z † ∂ µ z − z∂ µ z † )] =( πg ) Z D zδ ( | z | − e g R d x ( r ∂ µ α + s ∂ µ β ) e − g R d x [ r ( ∂ µ α ) + s ( ∂ µ β ) +( ∂ µ r ) +( ∂ µ s ) ] (4)Considering r = r (1 − s ) and s = s (1 − r ) , it isstraightforward to show that the action in the above path inte-gral just equals the action obtained in Eq. (3). Put it anotherway, while the two path integrals have different variables, theirintegrands (the actions) are equal: Z = Z D ˆ nδ (ˆ n − e − S [ˆ n ] (5) Z = ( πg ) Z D zδ ( | z | − e − S [ˆ n ( z )] (6)with S [ˆ n ] = S [ˆ n ( z )] = S [ r, s, α, β ] To proceed, we are to show that the entire amplitudes ofEq. (5) and Eq. (6) are proportional, i.e., the equality: Z D zδ ( | z | − e − S [ˆ n ( z )] = c Z D ˆ nδ (ˆ n − e − S [ˆ n ] (7)where c is an overall constant that can be eliminated by propernormalization.By virtue of the selection rule of the δ function and the Hopfmap ˆ n = z † ˆ σz we used above, we are able to rewrite the lefthand side of Eq. (7) in the form: Z D zδ ( | z | − e − S [ˆ n ( z )] = Z D zδ ( | z | − Z D ˆ nδ (ˆ n − z † ˆ σz ) e − S [ˆ n ] (8)thus the equality of Eq. (7) would be proved if we can showthe following relation: Z D zδ (ˆ n − z † ˆ σz ) δ ( | z | −
1) = c δ (ˆ n − (9)In other words, the proof of the equivalence between the twomodels is now a matter of demonstrating Eq. (9). To proveEq. (9), we first clarify the meaning of D z by: D z = Y x µ ,j =1 , dRe z j ( x µ )dIm z j ( x µ ) (10)and then we carry out the integral in the r, s, α, β coordinates.Since Re z = r cos α , Im z = r sin α , Re z = s cos β , and Im z = s sin β , the Jacobian of the coordinate transformationreads: J = ∂ (Re z , Im z , Re z , Im z ) ∂ ( r, α, s, β ) = rs (11)Then the left hand side of Eq. (9) becomes:L.H.S. = Z ∞ rdr Z ∞ sds Z π dα Z π dβ δ ( r + s − δ ( n x − rs cos( α − β )) δ ( n y + 2 rs sin( α − β )) δ ( n z − ( r − s ))= 116 Z ∞ dR Z ∞ dS Z π dθ Z π − π dφδ ( R + S − δ ( n x − √ RS cos( φ )) δ ( n y + 2 √ RS sin( φ )) δ ( n z − ( R − S )) (12)where some simple transformations of variables have beenused. Integrating out dR and dθ first and then dS , we obtain:L.H.S. = π Z ∞ dS Z π − π dφδ ( n x − p (1 − S ) S cos( φ )) δ ( n y + 2 p (1 − S ) S sin( φ )) δ ( n z − (1 − S ))= π Z π − π dφ δ ( n x − p − n z cos( φ )) δ ( n y + p − n z sin( φ )) (13)where in the last line we have taken into account the peri-odicity of the integrand so that R π − π = 2 R π − π . The last integration over dφ is somewhat tricky. Define the func-tion f ( φ ) = √ − n z cos φ − n x , it has two zero points at φ = ± arccos n x √ − n z , and the absolute values of its deriva-tive at these points are: | f ′ ( φ ) | = p − n z sin φ = p − n x − n z (14)using the relation δ [ f ( φ )] = P φ δ ( φ − φ ) | f ′ ( φ ) | , we obtain:L.H.S. = π Z π − π dφ δ ( n y + p − n z sin( φ )) δ ( φ − arccos n x √ − n z ) + δ ( φ + arccos n x √ − n z ) p − n x − n z = π δ ( n x + n y + n z − π δ (ˆ n − (15)Therefore, Eq. (9) hence Eq. (7) is proved and the constant c = π . Finally, we are safe to claim the exact equivalenceof the O(3) NL σ M and the CP model in the path integralformalism: Z = Z D z D A µ δ ( | z | − e − g R d x | ( ∂ µ − iA µ ) z | = π g Z D ˆ nδ (ˆ n − e − g R d x∂ µ ˆ n · ∂ µ ˆ n = π g Z (16)where the overall constant in front of Z is a trivial factor thatcan be eliminated by proper normalization.Special thanks are offered to Xiao Li for helpful discussionsand calculations. The authors are also grateful to Y. You, G.Fiete. [1] F. D. M. Haldane, Phys. Lett. , 464 (1983); Phys. Rev. Lett. , 1153 (1983).[2] S. Sachdev, Quantum Phase Transitions , Cambrigde 1999.[3] A. M. Polyakov,
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