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ago [25–28], the consistent theory with account of quan- tum fluctuations has been developed only recently [29]. Generalization of the spiral theory to bilayer YBCO has been discussed in Ref. 30. The theory is based on ex- pansion in powers of doping x , so it is parametrically justified at x ≪ 1. The theory is formulated in terms of the bosonic ⃗ n -field ( n 2 = 1) that describes the stag- gered component of the copper spins, and in terms of fermionic holons ψ . The holon field ψ carries electric charge and it has a pseudospin that originates from two sublattices. Minima of the holon dispersion are at the nodal points q$_{0}$ = ( ± π/ 2 , ± π/ 2), so there are holons of two types ψ$_{α}$ , α = 1 , 2, corresponding to two nodal directions. The dispersion in a pocket is somewhat anisotropic, but for simplicity we use here the isotropic approximation, ϵ ( p ) ≈ β p $^{2}$/ 2, where p = q − q$_{0}$ . Lattice spacing of the square lattice is set to be equal to unity a = 3 . 81 ˚ A → 1. The effective Lagrangian for a single layer system reads [29] 

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First two terms in the Lagrangian represent the usu al nonlinear σ model, with magnetic susceptibility χ$_{⊥}$ ≈ 0 . 066 J and spin stiffness ρ$_{s}$ ≈ 0 . 175 J , where J 130 meV is the antiferromagnetic exchange in the pa r- ent Mott insulator. The extended t - J model predicts the following values of the coupling constant and the invers e effective mass, g ≈ 1 . 0 J , β ≈ 2 . 5 J ( m ∗ = 1 . 6 m$_{e}$ ). B e low we will not use this value of β , instead we will trea t β as a fitting parameter. The pseudospin operator is 1 $_{2}$⃗ σ , and e$_{α}$ = (1 $_{/}$√$_{2}$$_{,}$ ± 1 $_{/}$√$_{2) is a unit vector orthogonal}$ to the face of the magnetic Brillouin zone (MBZ) where th e holon is located. Note that usage of the MBZ notatio ns does not imply that there is a long range magnetic o r- der. This is just a convenient way to avoid double coun t- ing of degrees of freedom, and the pseudospin accoun ts for doubling of the area of MBZ. Due to the MBZ n o- tations one should consider two full pockets located at q$_{0}$ = (+ π/ 2 , − π/ 2) and q$_{0}$ = (+ π/ 2 , + π/ 2), hence the index α takes two values. We stress that it does not matter if the ground state expectation value of the n- field is nonzero, 〈 ⃗ n 〉 ̸ = 0 (magnetic ordering), or zero, 〈 ⃗ n 〉 = 0. The only condition for validity of Eq.(2) is that dynamic fluctuations of the ⃗ n -field are sufficiently slow. Typical energy scale of the ⃗ n -field quantum fluctuations is E$_{cross}$ ∝ x 3 / 2 (position of the neck of the “hour glass” spin wave dispersion), as discussed in Ref. 29, and it must be small compared to the holon Fermi energy ϵ$_{F}$ ∝ x . The inequality E$_{cross}$ ≪ ϵ$_{F}$ is valid up to optimal doping, 

x ≈ 0 . 15, below which Eq.(2) is parametrically justified. To account for the interaction with magnetic field, we make the following modifications in Eq.(2) [29]: A mag- netic vector potential A is included in the long deriva- tive P → P − e $_{c}$A , kinetic energy of n − field is modi- fied ˙ ⃗ n 2 → ( ˙ ⃗ n − [ ⃗ n × ⃗ B ] ) 2 , and an extra term δ L B = 1 $_{2}$( ⃗ B · ⃗ n ) ψ † $_{α}$( ⃗ σ · ⃗ n ) ψ$_{α}$ is included. Here we include the Bohr magneton in definition of the magnetic field, 2 µ$_{B}$B → B . The precise meaning of spin-charge separation is clear under this context: In a normal Fermi liquid the spin in- teraction is δ L NFL B = 1 $_{2}$ψ † α ⃗ B · ⃗ σψ$_{α}$ . In contrast, for a spin- spiral state the g -term in Eq.(2) enforces ⃗ n ⊥ ⃗ σ , hence the expectation value of δ L B goes to zero, 〈 ( ⃗ B · ⃗ n )( ⃗ σ · ⃗ n ) 〉 = 0. Thus the pseudospin does not interact with magnetic field in the first order in B and this is the meaning of the spinless nature of the holon. 

In the effective action (2) the ρ$_{s}$ -term, the ϵ ( P )-term and the g -term are important in semiclassical approxi- mation. All other terms contain time derivatives, which are important only for quantum fluctuations and we dis- regard them for now. For the single layer case, we choose the plane of the coplanar spin spiral to be the xy -plane in the spin space, ⃗ n = (cos Q · r , sin Q · r , 0). Due to the g -term in Eq.(2) the psedospin of a holon is quantized along z-axis. Energy of the holon is ϵ ( p ) = ± gQ + β p $^{2}$/ 2, where ± gQ is the pseudospin splitting. The splitting be- tween two pseudospin branches is shown in the left panel of Fig.1. Minimization of the semiclassical energy shows 

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that Q is directed along the CuO bond [ Q ∝ (1 , 0) or Q ∝ (0 , 1)] and has value Q = g ρ s x . The upper pseu- dospin branch of the holon dispersion is always empty, as shown in the left panel, Fig.1, hence the area of the holon pocket is A k A$_{BZ}$ = 1 $_{2}$x . Quantum fluctuations reduce the static value of spin, |〈 ⃗ n 〉| < 1, moreover, at x > x$_{QCP}$ ≈ 0 . 1 the expectation value of ⃗ n vanishes, 〈 ⃗ n 〉 = 0, and the spiral becomes fully dynamic [29]. However, this does 

