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Fig. 3. Critical extrapolation length Ao of the columnar defect as a function of the magnetic field B where the solid line is for m = 0, dotted line—for m = 1, dashed line—for m = —1, dash-dotted curve—for m = 2, and dash-dot-dotted curve—for m = —2. For comparison, AY of the quantum disk is also shown by the thick solid line.

Figure 3 Critical extrapolation length Ao of the columnar defect as a function of the magnetic field B where the solid line is for m = 0, dotted line—for m = 1, dashed line—for m = —1, dash-dotted curve—for m = 2, and dash-dot-dotted curve—for m = —2. For comparison, AY of the quantum disk is also shown by the thick solid line.

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Fig. 1. Schematic of the infinitely long hollow superconductor wire of the inner radius ( and the outer one (9 + d) subjected to the superposition of the uniform magnetic field B and the AB flux ® parallel to its axis. Channel walls support the Robin boundary condition, Eq. (1), with the complex outer A,,; and inner Aj, extrapolation lengths uniform along the corresponding wire surface. Cartesian (x, y, z) and cylindrical (, y, z) systems of coordinates are also shown with their origins lying on the waveguide axis. The curved arrow shows the azimuthal direction in which the polar angle y grows.
Fig. 1. Schematic of the infinitely long hollow superconductor wire of the inner radius ( and the outer one (9 + d) subjected to the superposition of the uniform magnetic field B and the AB flux ® parallel to its axis. Channel walls support the Robin boundary condition, Eq. (1), with the complex outer A,,; and inner Aj, extrapolation lengths uniform along the corresponding wire surface. Cartesian (x, y, z) and cylindrical (, y, z) systems of coordinates are also shown with their origins lying on the waveguide axis. The curved arrow shows the azimuthal direction in which the polar angle y grows.
In other words, the lossless longitudinal flow takes place for the real transverse energies only, as expected. As it will be shown below, for the multiply connected geometry it takes place when the total 2D radial current through the inner circle J,, is equal to its counterpart through the outer border J,,. Expressions for Re(k,) and Im(k,) were derived and analyzed before [1], so, here we simply rewrit them pointing only to the fact that the imaginary part of the wave vector, which, according to (48, defines the longitudinal alteration of the current, vanishes together with I’: 3. Results and discussion In this section the outcome of the calculations based on the theory developed in Section 2 is presented and their mathematical and physical interpretations are given. If in the results below the index or subscript @ is dropped, it means that its particular value of 6 = 0 is used; otherwise, it takes any arbitrary magnitudes.
In other words, the lossless longitudinal flow takes place for the real transverse energies only, as expected. As it will be shown below, for the multiply connected geometry it takes place when the total 2D radial current through the inner circle J,, is equal to its counterpart through the outer border J,,. Expressions for Re(k,) and Im(k,) were derived and analyzed before [1], so, here we simply rewrit them pointing only to the fact that the imaginary part of the wave vector, which, according to (48, defines the longitudinal alteration of the current, vanishes together with I’: 3. Results and discussion In this section the outcome of the calculations based on the theory developed in Section 2 is presented and their mathematical and physical interpretations are given. If in the results below the index or subscript @ is dropped, it means that its particular value of 6 = 0 is used; otherwise, it takes any arbitrary magnitudes.
Fig. 2. Energy EL (left panels) and corresponding magnetization M, (right panels) of the columnar defect as functions of magnetic field B and positive (upper panels) and negative (lower panels) extrapolation length A. Note the different scales for the upper and lower panels. To better emphasize the obtained dependencies, the A- and B- axes in panel (c) have been reversed as compared to panel (a). For the same reason, the field axis in panel (d) grows in the opposite direction of panel (b).
Fig. 2. Energy EL (left panels) and corresponding magnetization M, (right panels) of the columnar defect as functions of magnetic field B and positive (upper panels) and negative (lower panels) extrapolation length A. Note the different scales for the upper and lower panels. To better emphasize the obtained dependencies, the A- and B- axes in panel (c) have been reversed as compared to panel (a). For the same reason, the field axis in panel (d) grows in the opposite direction of panel (b).
2 We point out other typos in the reference literature; namely, Eqs. (13.5.9) in [86] and (6.8.5) in [87] are wrong. For the correct form, one needs in Eq. (6.8.5) of Ref. [87] to convert the rightmost negative sign into its positive counterpart.
2 We point out other typos in the reference literature; namely, Eqs. (13.5.9) in [86] and (6.8.5) in [87] are wrong. For the correct form, one needs in Eq. (6.8.5) of Ref. [87] to convert the rightmost negative sign into its positive counterpart.
Fig. 4. (a) Real EY” and (b) negative double imaginary I parts of the total transverse energy Ed as functions of A and B fo the columnar defect. Note reversed A- and B-axes directions in panels (a) and (b).
Fig. 4. (a) Real EY” and (b) negative double imaginary I parts of the total transverse energy Ed as functions of A and B fo the columnar defect. Note reversed A- and B-axes directions in panels (a) and (b).
We start our analysis from the study of the asymptotic cases of the large and small |A|; namely, similar to the solid cylinder case [1], the limits of the small, |Ainour| < 1, and large, |Ainjour| >> 1, magnitudes of the purely imaginary de Gennes distances can be derived from Eq. (35) using the addition theorem for the Bessel functions and their asymptotic properties [86]. Expressions for the imaginary part of the energy I'g:nm are provided below for all possible permutations of the asymptotics:
We start our analysis from the study of the asymptotic cases of the large and small |A|; namely, similar to the solid cylinder case [1], the limits of the small, |Ainour| < 1, and large, |Ainjour| >> 1, magnitudes of the purely imaginary de Gennes distances can be derived from Eq. (35) using the addition theorem for the Bessel functions and their asymptotic properties [86]. Expressions for the imaginary part of the energy I'g:nm are provided below for all possible permutations of the asymptotics:
Fig.5. Negative double imaginary part Io of the total transverse energy E¢, as a function of the radius p and equal inner an outer purely imaginary distances A = Ain = Aout.
Fig.5. Negative double imaginary part Io of the total transverse energy E¢, as a function of the radius p and equal inner an outer purely imaginary distances A = Ain = Aout.
Fig. 6. (a) One of the surfaces in (9, Ain, Aour) Space on which the energy Ez is real. (b) Real energy Ed, corresponding to pane (a) as a function of ( and Ain.
Fig. 6. (a) One of the surfaces in (9, Ain, Aour) Space on which the energy Ez is real. (b) Real energy Ed, corresponding to pane (a) as a function of ( and Ain.
Fig. 7. (a) Complex energy EL as a function of the inner purely imaginary de Gennes distance Aj, for o9 = 0.5 and Aoy, = 0.2 where the dotted line depicts the real part of the energy E, o while the solid line is for the double negative imaginary part. The dashed horizontal line denotes zero energy. (b) Real (lower at 0 © po + 1) and imaginary (upper at o © > + 1) parts of the total radial functions Roo (~) for 9 = 0.5, Aoue = 0.2 and several i;, where the thick solid lines are for Aj, = 0.2, dotted lines—for Aj, = 0, dashed lines—for Ajn = —0.2033145, dash-dotted lines—for Aj, = —0.35, dash-dot-dotted lines—for Ain = —0.9285438, and thin solid lines are for Aj, = —1.2. Note that real parts vanish identically at p = pp + 1.
Fig. 7. (a) Complex energy EL as a function of the inner purely imaginary de Gennes distance Aj, for o9 = 0.5 and Aoy, = 0.2 where the dotted line depicts the real part of the energy E, o while the solid line is for the double negative imaginary part. The dashed horizontal line denotes zero energy. (b) Real (lower at 0 © po + 1) and imaginary (upper at o © > + 1) parts of the total radial functions Roo (~) for 9 = 0.5, Aoue = 0.2 and several i;, where the thick solid lines are for Aj, = 0.2, dotted lines—for Aj, = 0, dashed lines—for Ajn = —0.2033145, dash-dotted lines—for Aj, = —0.35, dash-dot-dotted lines—for Ain = —0.9285438, and thin solid lines are for Aj, = —1.2. Note that real parts vanish identically at p = pp + 1.
Fig. 8. Current densities of the state with n m oo) 0 for po 0.5, Ao = 0.21 and (a) Am = 0.21,(b) An = 0, (c) Ain = —0.2033145i, (d) Aj,n = —0.35i, (€) Ain = —0.9285438i, and (f) Aj, = —1.2i. For each of the figures, the currents are normalized with respect to their largest value. The longer arrows denote larger currents. Corresponding energies Ex are written near each pattern.
Fig. 8. Current densities of the state with n m oo) 0 for po 0.5, Ao = 0.21 and (a) Am = 0.21,(b) An = 0, (c) Ain = —0.2033145i, (d) Aj,n = —0.35i, (€) Ain = —0.9285438i, and (f) Aj, = —1.2i. For each of the figures, the currents are normalized with respect to their largest value. The longer arrows denote larger currents. Corresponding energies Ex are written near each pattern.
Fig. 9. Real (panels (a) and (c)) and double negative imaginary (panels (b) and (d)) parts of the total transverse energy Ei, as functions of the purely imaginary Robin parameter A and field B for the annulus with inner radius op = 0.1. For the left panels Xd = hour = —Ain while for the right panels A = Aoy, with Aj, = 0. Note reversed A and B axes in panel (b) as compared to other panels.
Fig. 9. Real (panels (a) and (c)) and double negative imaginary (panels (b) and (d)) parts of the total transverse energy Ei, as functions of the purely imaginary Robin parameter A and field B for the annulus with inner radius op = 0.1. For the left panels Xd = hour = —Ain while for the right panels A = Aoy, with Aj, = 0. Note reversed A and B axes in panel (b) as compared to other panels.
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Applied MathematicsMathematical PhysicsPhysicsTheoretical PhysicsQuantum PhysicsAntennas & Radio Wave PropagationParticle PhysicsAbsorbing Boundary ConditionsMathematical Methods of PhysicsHigh Magnetic Fields
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