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effectively the states unrelated to the Andreev levels. 

To calculate the ”phase” Φ ( T, H) from Eq. (2), we should know the distribution of the vector potential field inside the normal metal. Zaikin [1] has shown that the proximity effect caused the Meissner effect leads to an inhomogeneous distribution of the vector potential field over the N layer of the structure: 

where µ$_{0}$ is the permeability of free space ( the SI sys- tem of units is employed, the geometry of the proxim- ity model system is the same as in [19] Fig 1). This expression can be obtained from the Maxwell equa- tion rot ( rot ( ⃗ A )) = µ$_{0}$ ⃗ j = (0 , µ$_{0}$j, 0) assuming that the current density is uniform over the cross-section of the conductor and the boundary condition ⃗ A ∣ x =0 = 0, rot ( ⃗ A )∣ ∣ x = d = ⃗ B (0 , 0 , µ$_{0}$ H) is met. The fact that the current density is constant in the N-layer follows from spatial homogeneity of the density of Andreev levels over the whole thickness of the N-layer. In cylindrical geome- tries, if the N-layer thickness is not thin compared to the radius ( d ≳ R ), the current density is not constant in space [20]. 

The magnetic moment per unit length of the N-layer M (H) = − 1 µ$_{0}$ d Ω( T, Φ(H)) d H ( z-component ) and the current density ⃗ j are related via a relation 

where V$_{N}$ is a volume of the N-layer unity height, Ω ( T, Φ) is the free energy per unit length. Then, the current appears from Eq. (6) is a function of the magnetic flux Φ and temperature: 

We can write down the self-consistent equation for Φ ( T, H) using Eqs. (2), (5) and (7): 

where h = H $_{H$_{0}$}$, H$_{0}$ = Φ$_{0}$ πd $^{2}$· $_{µ$_{0}$}$, η = d 2 3 R $^{2}$Φ$_{0}$ $_{H$_{0}$}$, M $^{∗}$(Φ) = − d Ω d Φ [21]. To describe the field effect on the magnetic moment 

of a NS structure, it is necessary to find the dependence Φ ( T, h) from Eq. (8). After calculating the free energy 

## B. Self-consistent equation 

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Ω ( T, Φ) from the spectrum of Eq. (4), we can estimate the magnetic moment Eq. (9) using a solution of the dif- ferential equation (8): M ( T, H) = 1 η · µ$_{0}$ H$_{0}$ (Φ ( T, h) − h). In this article, we used the ”thermodynamic” approach Eq. (7) which leads to the first order differential equation (8) for the function Φ ( T, h). However, another approach based on of the Eilenberger formalism ( [1], [18] ) yields an algebraic self-consistent equation ( Eq. 22 in ([18]) ) for the ”phase” Φ ( T, h): 

( in notation Eq., (8) ). In this equation the function j (Φ) is described by the expression Eq., 13 in [1]. Clearly, both approaches (8), (10) will lead to quite different de- pendencies of M ( T, H) on magnetic field and tempera- ture. In our point of view, the self-consistent equation (10) cannot be applied to the cylindrical NS structures. 

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