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Superconductivity by Berry Connection from Many-body Wave Functions: Revisit to Andreev−Saint-James Reflection and Josephson Effect

Abstract

Although the standard theory of superconductivity based on the BCS theory is a successful one, several experimental results indicate the necessity for a fundamental revision. We argue that the revision is on the origin of the phase variable for superconductivity; this phase appears as a consequence of the electron-pairing in the standard theory, but its origin is a Berry connection arising from many-body wave functions. When this Berry connection is non-trivial, it gives rise to a collective mode that generates supercurrent; this collective mode creates number-changing operators for particles participating in this mode, and these number-changing operators stabilize the superconducting state by exploiting the Cooper instability. In the new theory, the role of the electron-pairing is to stabilize the nontrivial Berry connection; it is not the cause of superconductivity. In BCS superconductors, however, the simultaneous appearance of the nontrivial Berry connection and the electron-pairing occurs. Therefore, the electron-pairing amplitude can be used as an order parameter for the superconducting state. We revisit the Andreev−Saint-James reflection and the Josephson effect. They are explained as consequences of the presence of the Berry connection. Bogoliubov quasiparticles are replaced by the particle-number conserving Bogoliubov excitations that describe the transfer of electrons between the collective and single-particle modes. There are two distinct cases for the Josephson effect; one of them contains the common Bogoliubov excitations for the two superconductors in the junction, and the other does different Bogoliubov excitations for different superconductors. The latter case is the one considered in the standard theory; in this case, the Cooper pairs tunnel through without Bogoliubov excitations, creating an impression that the supercurrent is a flow of Cooper pairs; however, it does not explain the observed ac Josephson effect under the experimental boundary condition. On the other hand, the former case explains the ac Josephson effect under the experimental boundary condition. In this case, it is clearly shown that the supercurrent is a flow of electrons brought about by the non-trivial Berry connection which provides an additional U(1) gauge field besides the electromagnetic one.

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References

  1. 1.

    Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175 (1957)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Bednorz, J.G., Müller, K.A.: Possible high t\(_c\) superconductivity in the ba-la-cu-o system. Z. Phys. B 64, 189 (1986)

    ADS  Article  Google Scholar 

  3. 3.

    Emery, V.J., Kivelson, S.A.: Importance of phase fluctuation in superconductors with small superfluid density. Nature 374, 434 (1995)

    ADS  Article  Google Scholar 

  4. 4.

    Hirsch, J.: Bcs theory of superconductivity: it is time to question its validity. Physica Scripta 80, 035702 (2009)

    ADS  Article  Google Scholar 

  5. 5.

    Koizumi, H.: Spin-vortex superconductivity. J. Supercond. Nov. Magn. 24, 1997 (2011)

    Article  Google Scholar 

  6. 6.

    Hirsch, J.E.: Momentum of superconducting electrons and the explanation of the Meissner effect. Phys Rev. B 95, 014503 (2017)

    ADS  Article  Google Scholar 

  7. 7.

    Hirsch, J.E.: Entropy generation and momentum transfer in the superconductor-normal and normal-superconductor phase transitions and the consistency of the conventional theory of superconductivity. Int. J. Mod. Phys. B 32, 1850158 (2018)

    ADS  Article  Google Scholar 

  8. 8.

    Hirsch, J.E.: Inconsistency of the conventional theory of superconductivity. EPL 130, 17006 (2020)

    ADS  Article  Google Scholar 

  9. 9.

    Koizumi, H.: Reversible superconducting-normal phase transition in a magnetic field and the existence of topologically-protected loop currents that appear and disappear without Joule heating. EPL 131(3), 37001 (2020)

    ADS  Article  Google Scholar 

  10. 10.

    Keesom, W., Kok, J.: Measurements of the latent heat of thallium connected with the transition, in a constant external magnetic field, from the supraconductive to the non-supraconductive state. Physica 1(1), 503–512 (1934). https://doi.org/10.1016/S0031-8914(34)90059-8

    ADS  Article  Google Scholar 

  11. 11.

    Keesom, W., Van Laer, P.: Measurements of the latent heat of tin in passing from the supraconductive to the non-supraconductive state. Physica 3(6), 371–384 (1936). https://doi.org/10.1016/S0031-8914(36)80002-0.

    ADS  Article  Google Scholar 

  12. 12.

    Keesom, W., van Laer, P.: Measurements of the latent heat of tin while passing from the superconductive to the non-superconductive state at constant temperature. Physica 4(6), 487–493 (1937). https://doi.org/10.1016/S0031-8914(37)80081-6

    ADS  Article  Google Scholar 

  13. 13.

    van Laer, P.H., Keesom, W.H.: On the reversibility of the transition processs between the superconductive and the normal state. Physica 5, 993 (1938)

    ADS  Article  Google Scholar 

  14. 14.

    Hirsch, J.E.: The london moment: what a rotating superconductor reveals about superconductivity. Physica Scripta 89(1), 015806 (2013). https://doi.org/10.1088/0031-8949/89/01/015806

    ADS  Article  Google Scholar 

  15. 15.

    Koizumi, H.: London moment, london’s superpotential, nambu-goldstone mode, and berry connection from many-body wave functions. J. Supercond. Nov. Magn. (2020). https://doi.org/10.1007/s10948-021-05827-9

    Article  Google Scholar 

  16. 16.

    London, F.: Superfluids, vol. 1. Wiley, New York (1950)

    MATH  Google Scholar 

  17. 17.

    Hildebrandt, A.F.: Magnetic field of a rotating superconductor. Phys. Rev. Lett. 12, 190–191 (1964). https://doi.org/10.1103/PhysRevLett.

    ADS  Article  Google Scholar 

  18. 18.

    Zimmerman, J.E., Mercereau, J.E.: Compton wavelength of superconducting electrons. Phys. Rev. Lett. 14, 887–888 (1965). https://doi.org/10.1103/PhysRevLett.14.887

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Brickman, N.F.: Rotating superconductors. Phys. Rev. 184, 460–465 (1969). https://doi.org/10.1103/PhysRev.184.460

    ADS  Article  Google Scholar 

  20. 20.

    Tate, J., Cabrera, B., Felch, S.B., Anderson, J.T.: Precise determination of the Cooper-pair mass. Phys. Rev. Lett. 62, 845–848 (1989). https://doi.org/10.1103/PhysRevLett.62.845

    ADS  Article  Google Scholar 

  21. 21.

    Tate, J., Felch, S.B., Cabrera, B.: Determination of the Cooper-pair mass in niobium. Phys. Rev. B 42, 7885–7893 (1990). https://doi.org/10.1016/S0921-4526(09)80176-2

    ADS  Article  Google Scholar 

  22. 22.

    Verheijen, A., van Ruitenbeek, J., de Bruyn Ouboter, R., de Jongh, L.: The London moment for high temperature superconductors. Physica B: Condensed Matter 165-166, 1181–1182. LT-19 (1990)

  23. 23.

    Verheijen, A.A., van Ruitenbeek, J.M., de Bruyn Ouboter, R., de Jongh, L.J.: Measurement of the London moment in two high-temperature superconductors. Nature 345(6274), 418–419 (1990). https://doi.org/10.1038/345418a0

    ADS  Article  Google Scholar 

  24. 24.

    Sanzari, M.A., Cui, H.L., Karwacki, F.: London moment for heavy-fermion superconductors. Applied Physics Letters 68(26), 3802–3804 (1996). https://doi.org/10.1063/1.116622

    ADS  Article  Google Scholar 

  25. 25.

    Koizumi, H.: Explanation of superfluidity using the Berry connection for many-body wave functions. J. Supercond. Nov. Magn. 33, 1697–1707 (2020)

    Article  Google Scholar 

  26. 26.

    Andreev, A.F.: Thermal conductivity of the intermediate state of superconductors. Sov. Phys. JETP. 19(19), 1228 (1964)

    Google Scholar 

  27. 27.

    Saint-James, D.: Excitations élémentaires au voisinage de la surface de séparation d’un métal normal et d’un métal superconducteur. J. Phys. France 25(10), 899–905 (1964)

    Article  Google Scholar 

  28. 28.

    Josephson, B.D.: Possible new effects in superconductive tunneling. Phys. Lett. 1, 251 (1962)

    ADS  Article  Google Scholar 

  29. 29.

    Aharonov, Y., Bohm, D.: Significance of elecromagnetic potentials in the quantum theory. Phys. Rev. 115, 167 (1959)

    Article  Google Scholar 

  30. 30.

    Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. Zh. Exsp. Teor. Fiz. 20, 1064 (1950)

    Google Scholar 

  31. 31.

    Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174 (1957)

    Google Scholar 

  32. 32.

    de Gennes, P.G.: Superconductivity of Metals and Alloys. Benjamin Inc, W. A (1966)

    MATH  Google Scholar 

  33. 33.

    Koizumi, H., Ishikawa, A.: Theory of supercurrent in superconductors. Int. J. Mod. Phys. B 34(31), 2030001 (2020). https://doi.org/10.1142/S0217979220300017

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Zhu, JX.: Bogoliubov-de Gennes Method and Its Applications. Springer. (2016)

  35. 35.

    Koizumi, H.: Possible occurrence of superconductivity by the \(\pi\)-flux Dirac string formation due to spin-twisting itinerant motion of electrons. Symmetry 12, 776 (2020)

    Article  Google Scholar 

  36. 36.

    Koizumi, H., Tachiki, M.: Supercurrent generation by spin-twisting itinerant motion of electrons: Re-derivation of the ac josephson effect including the current flow through the leads connected to josephson junction. J. Supercond. Nov. Magn. 28, 61–69 (2015)

    Article  Google Scholar 

  37. 37.

    Bocquillon, E., Deacon, R.S., Wiedenmann, J., Leubner, P., Klapwijk, T.M., Brüne, C., Ishibashi, K., Buhmann, H., Molenkamp, L.W.: Gapless andreev bound states in the quantum spin hall insulator hgte. Nature Nanotechnology 12(2), 137–143 (2017). https://doi.org/10.1038/nnano.2016.159

    ADS  Article  Google Scholar 

  38. 38.

    Ueda, K., Matsuo, S., Kamata, H., Sato, Y., Takeshige, Y., Li, K., Samuelson, L., Xu, H., Tarucha, S.: Evidence of half-integer shapiro steps originated from nonsinusoidal current phase relation in a short ballistic inas nanowire josephson junction. Phys. Rev. Research 2, 033435 (2020). https://doi.org/10.1103/PhysRevResearch.2.033435

    ADS  Article  Google Scholar 

  39. 39.

    Zhang, H., de Moor, M.W.A., Bommer, J.D.S., Xu, D., Wang, G., van Loo, N., Liu, C.X., Gazibegovic, S., Logan, J.A., Car, D., het Veld, R.L.M.O., van Veldhoven, P.J., Koelling, S., Verheijen, M.A., Pendharkar, M., Pennachio, D.J., Shojaei, B., Lee, J.S., Palmstrøm, C.J., Bakkers, E.P.A.M., Sarma, S.D., Kouwenhoven, L.P.: Large zero-bias peaks in insb-al hybrid semiconductor-superconductor nanowire devices. arXiv:2101.11456 (2021)

  40. 40.

    Ambegaokar, V., Baratoff, A.: Tunneling between superconductors. Phys. Rev. Lett. 10, 486–489 (1963)

    ADS  Article  Google Scholar 

  41. 41.

    Shapiro, S.: Josephson currents in superconducting tunneling: the effect of microwaves and other observations. Phys. Rev. Lett. 11, 80 (1963)

    ADS  Article  Google Scholar 

  42. 42.

    Peierls, R.: Spontaneously broken symmetries. J. Phys. A 24, 5273 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  43. 43.

    Feynman, R.P.: Quantum mechanics and path integrals. McGraw-Hill companies, Inc. (1965)

  44. 44.

    Dirac, P.: Quantised singularities in the electromagnetic field. Proc. Roy. Soc. London 133, 60 (1931)

    ADS  MATH  Google Scholar 

  45. 45.

    Gor’kov, L.P.: Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity. Sov. Phys. JETP 36(9), 1364–1367 (1959)

    MATH  Google Scholar 

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Appendices

Appendix I : Modification of Maxwell’s Equations in the Presence of the Berry Connection from Many-body Wave Functions

From the view point of the Feynman path integral formalism of quantum mechanics [43], a wave function is a sum of contributions from all paths each contributes an exponential whose phase is the classical action divided by \(\hbar\) for the path in question.

For the system of charged particles and electromagnetic field, the classical action S are composed of the following three terms,

$$\begin{aligned} S=S_1+S_2+S_3 \end{aligned}$$
(62)

where

$$\begin{aligned} S_1=\sum _i { m \over 2} \int dt \ \dot{\mathbf{r}}_i^2 \end{aligned}$$
(63)

is the action for the particles,

$$\begin{aligned} S_2&=-\int d^3r dt \ \left[ \rho \phi ^\mathrm{em}(\mathbf{r},t) -{1 \over c} \mathbf{j} \cdot \mathbf{A}^\mathrm{em}(\mathbf{r},t) \right]\\& =-q\sum _i \int dt \ \left[ \phi ^\mathrm{em}(\mathbf{r}_i,t) -{1 \over c} \dot{\mathbf{r}}_i \cdot \mathbf{A}^\mathrm{em}(\mathbf{r}_i,t) \right] \nonumber \\ \end{aligned}$$
(64)

is the action for the interaction between the field and particles, and

$$\begin{aligned} S_3&={1 \over {8 \pi }}\int d^3r dt \ \left[ (\mathbf{E}^\mathrm{em})^2- (\mathbf{B}^\mathrm{em})^2 \right] \\&={1 \over {8 \pi }}\int d^3r dt \ \left[ \left( -\nabla \phi ^\mathrm{em} -{1 \over c} {{\partial \mathbf{A}^\mathrm{em}} \over {\partial t}} \right) ^2- \left( \nabla \times \mathbf{A}^\mathrm{em}\right) ^2 \right] \nonumber \end{aligned}$$
(65)

is the action for the field. Here c is the speed of light in vacuum, and \(\phi ^\mathrm{em}\) and \(\mathbf{A}^\mathrm{em}\) are the scalar and vector potentials for the electromagnetic field, respectively; \(\rho\) and \(\mathbf{j}\) are the electric charge and current densities, respectively; q and m are the charge and mass of the particle, respectively.

In the following, we consider the case where the electric field \(\mathbf{E}^\mathrm{em}\) is absent, and only the magnetic field \(\mathbf{B}^\mathrm{em}\) is present.

In our previous work [15, 25, 33], it is shown that the Berry connection arising from many-body wave functions modifies the momentum operator \(-i \hbar \nabla\) in the Schrödinger representation of quantum mechanics as follows

$$\begin{aligned} -i \hbar \nabla \longrightarrow -i \hbar \nabla + \hbar \mathbf{A}_{\Phi }^\mathrm{MB} \end{aligned}$$
(66)

where \(\mathbf{A}_{\Phi }^\mathrm{MB}\) is the Berry connection defined by

$$\begin{aligned} \mathbf{A}^\mathrm{MB}_{\Phi }(\mathbf{r},t)=-i \langle n_{\Phi }(\mathbf{r},t) |\nabla |n_{\Phi }(\mathbf{r},t) \rangle \end{aligned}$$
(67)

and \(|n_{\Phi }(\mathbf{r}) \rangle\) is the parameterized wave function with the parameter \(\mathbf{r}\) and integration coordinates \(\mathbf{r}_2, \cdots \mathbf{r}_N\) given by

$$\begin{aligned} \langle \mathbf{r}_{2}, \cdots , \mathbf{r}_{N} |n_{\Phi }(\mathbf{r},t) \rangle = { {\Phi (\mathbf{r}, \mathbf{r}_{2}, \cdots , \mathbf{r}_{N},t)} \over {|C_{\Phi }(\mathbf{r} ,t)|^{{1 \over 2}}}} \end{aligned}$$
(68)

with \(|C_{\Phi }(\mathbf{r},t)|\) being the normalization constant given by

$$\begin{aligned} |C_{\Phi }(\mathbf{r},t)|=\int d\mathbf{r}_{2} \cdots d\mathbf{r}_{N}\Phi (\mathbf{r}, \mathbf{r}_{2}, \cdots )\Phi ^{*}(\mathbf{r}, \mathbf{r}_{2}, \cdots ) \end{aligned}$$
(69)

Inclusion of \(\hbar \mathbf{A}_{\Phi }^\mathrm{MB}\) means the inclusion of the gauge field that describes the interaction between particles through the wave function they share.

As a consequence, the effective vector potential in the system becomes

$$\begin{aligned} \mathbf{A}^\mathrm{eff}=\mathbf{A}^\mathrm{em}+\mathbf{A}^\mathrm{fic}, \quad \mathbf{A}^\mathrm{fic}= \hbar \mathbf{A}_{\Phi }^\mathrm{MB} \end{aligned}$$
(70)

due to the presence of the “fictitious” vector potential \(\mathbf{A}^\mathrm{fic}\) that arises as the Berry connection.

Actually, \(\mathbf{A}^\mathrm{fic}\) is given by

$$\begin{aligned} \mathbf{A}^\mathrm{fic}=-{{\hbar c} \over {2e}} \nabla \chi \end{aligned}$$
(71)

where \(\chi\) is an angular variable with period \(2 \pi\) [35]. This appears through the spin-twisting itinerant motion of electrons, and the spin-twisting is caused by the Rashba spin-orbit interaction. Although the energy gain by the spin-twisting itself is very small, the Berry connection creates the number changing operators that make the energy gain by exploiting the Cooper instability possible. In other words, as far as the energy gain by the electron-pair formation exceeds other energy deficits, the spin-twisting itinerant motion of electrons occurs and non-trivial Berry connection is generated.

By including the Berry connection, \(S_2\) becomes,

$$\begin{aligned} S_2'={1 \over c}\int d^3r dt \ \mathbf{j} \cdot \mathbf{A}^\mathrm{eff}(\mathbf{r},t) \end{aligned}$$
(72)

where we only retain the term with vector potential assuming that electric field is absent. This gives rise to “Lorentz force” in the classical electromagnetic dynamics. The Lorenz force from \(\mathbf{A}^\mathrm{fic}\) is zero in classical mechanics; however, \(\mathbf{A}^\mathrm{fic}\) may affect the dynamics of charged particles through the Aharonov-Bohm effect [29] in quantum mechanics. Actually, this effect is the main concern of the present work. We call this term, the Lorentz interaction term, instead of the Lorentz force term, to emphasize it contains the Aharonov-Bohm effect.

Since the electromagnetic field energy is the energy stored in the space through the Lorentz interaction term \(S_2'\), \(S_3\) should be modified using \(\mathbf{B}^\mathrm{eff}=\mathbf{B}^\mathrm{em}+\mathbf{B}^\mathrm{fic}\),

$$\begin{aligned} S_3'=-{1 \over {8 \pi }}\int d^3r dt \ (\mathbf{B}^\mathrm{eff})^2 \end{aligned}$$
(73)

where

$$\begin{aligned} \mathbf{B}^\mathrm{fic}=\nabla \times \mathbf{A}^\mathrm{fic}={{\hbar c} \over {2e}} \nabla \times \nabla \chi \end{aligned}$$
(74)

\(\mathbf{B}^\mathrm{fic}\) may not be zero due to the fact that \(\chi\) may be multi-valued.

Using \(S_2'\) and \(S_3'\), two of the Maxwell’s equations are modified as

$$\begin{aligned} \nabla \cdot \mathbf{B}^\mathrm{eff}= & {} 0 \\ \nabla \times \mathbf{B}^\mathrm{eff}= & {} {{4 \pi } \over c}{} \mathbf{j} \end{aligned}$$
(75)

The first one gives rise to a Dirac monopole as shown below. It is written as

$$\begin{aligned} \nabla \cdot \mathbf{B}^\mathrm{em}=-\nabla \cdot (\nabla \times \mathbf{A}^\mathrm{fic}) \end{aligned}$$
(76)

When the both sides of the above equation are integrated for a closed region with surface \(\mathrm{Sf}\), we have

$$\begin{aligned} \int _\mathrm{Sf} d\mathbf{S} \cdot \mathbf{B}^\mathrm{em}=-\int _\mathrm{Sf} d\mathbf{S} \cdot (\nabla \times \mathbf{A}^\mathrm{fic}) \end{aligned}$$
(77)

We split \(\mathrm{Sf}\) into two surfaces \(\mathrm{Sf}_1\) and \(\mathrm{Sf}_2\) with common boundary loop, \(C=\partial (\mathrm{Sf}_1)=-\partial (\mathrm{Sf}_2)\). Then, we have

$$\begin{aligned}& \int _{\mathrm{Sf}_1} d\mathbf{S} \cdot (\nabla \times \mathbf{A}^\mathrm{fic})+\int _{\mathrm{Sf}_2} d\mathbf{S} \cdot (\nabla \times \mathbf{A}^\mathrm{fic})\\& \quad =\int _{\partial ( \mathrm{Sf}_1)} d\mathbf{r} \cdot \mathbf{A}^\mathrm{fic}+\int _{\partial ( \mathrm{Sf}_1)} d\mathbf{r} \cdot \mathbf{A}^\mathrm{fic} \nonumber \\ \end{aligned}$$
(78)

We examine the case in which singularities exist in \(\mathbf{A}^\mathrm{fic}\). Let us consider a closed surface S with boundary \(C=\partial S\), and \(\mathbf{A}^\mathrm{fic}\) has a singularity in S. Then, we have

$$\begin{aligned} \int _{C} d\mathbf{r} \cdot {{\hbar c} \over {2e}} \nabla \chi ={{h c} \over {2e}} n \end{aligned}$$
(79)

where n is an integer.

If we have \(n=0\) for \(\partial ( \mathrm{Sf}_1)\) term, and \(n=1\) for \(\partial ( \mathrm{Sf}_2)\) term, we have

$$\begin{aligned} \int _\mathrm{Sf} d\mathbf{S} \cdot \mathbf{B}^\mathrm{em}={{h c} \over {2e}} \end{aligned}$$
(80)

This shows that a monopole with magnetic charge \({{h c} \over {2e}}\) exists in the region enclosed by \(\mathrm{Sf}\). This corresponds to the monopole considered by Dirac [44].

The second one in Eq. (75) is equal to

$$\begin{aligned} \nabla \times \mathbf{B}^\mathrm{em}= & {} {{4 \pi } \over c}{} \mathbf{j} \end{aligned}$$
(81)

since \(\nabla \times \mathbf{B}^\mathrm{fic}=0\) is satisfied as shown below; it is well-known that \(\nabla \chi\) in \(\mathbf{A}^\mathrm{fic}\) can be decomposed as

$$\begin{aligned} \nabla \chi =\nabla \chi _0 +\nabla f, \quad \nabla ^2 \chi _0=0 \end{aligned}$$
(82)

where f is a single-valued, and \(\chi _0\) may be multi-valued. Thus, we have

$$\begin{aligned} \nabla \times \mathbf{B}^\mathrm{fic}=\nabla \times (\nabla \times \nabla \chi _0)= \nabla (\nabla ^2 \chi _0)-\nabla ^2 \nabla \chi _0=0 \end{aligned}$$
(83)

As a consequence, Eq. (75) is reduced to the original one in Eq. (81).

Appendix II : Modification of the Ginzburg–Landau Theory Including the Berry Connection from Many-body Wave Functions and its Consequences

The Ginzburg–Landau theory [30] is based on the London theory [16]. In the London theory, the velocity field for electrons in superconductors is given by

$$\begin{aligned} \mathbf{v}=-{ q \over {mc}} \left( \mathbf{A}^\mathrm{em} - { {c \hbar } \over q} \nabla \chi ^\mathrm{super} \right) \end{aligned}$$
(84)

where \(\chi ^\mathrm{super}\) is the superpotential assumed to exist in superconductors.

The Ginzburg–Landau theory uses a free energy consists of the material part and the magnetic field part. It assumes the presence of the effective wave function of superconducting electrons \(\Psi _\mathrm{GL}\) in the superconducting phase.

Using \(\Psi _\mathrm{GL}\), the material part of the free energy for a superconductor is given by

$$\begin{aligned} F_\mathrm{mat}= &F_\mathrm{normal}+ \int d^3 r {1 \over {2m}} \left| \left( { \hbar \over i} \nabla -{q \over c}{} \mathbf{A}^\mathrm{em} \right) \Psi _\mathrm{GL} \right| ^2 \\&+ \int d^3 r \left( \alpha |\Psi _\mathrm{GL}|^2 +{ \beta \over 2}|\Psi _\mathrm{GL}|^4 \right) \nonumber \\ \end{aligned}$$
(85)

where \(\alpha\) is a negative real parameter, and \(\beta\) is a positive real parameter.

We can express \(\Psi _\mathrm{GL}\) using the supercurrent carrier density \(n_s\) and the superpotential \(\chi ^\mathrm{super}\) as

$$\begin{aligned} \Psi _\mathrm{GL}=n_s^{1/2}e^{ i \chi ^\mathrm{super}} \end{aligned}$$
(86)

Then, the kinetic term becomes

$$\begin{aligned} \int d^3 r {1 \over {2m}} \left| \left( { \hbar \over i} \nabla -{q \over c}{} \mathbf{A}^\mathrm{em} \right) \Psi _\mathrm{GL} \right| ^2 =F_k + \int d^3 r {{\hbar ^2 (\nabla n_s)^2} \over {8m \ n_s}} \end{aligned}$$
(87)

where the supercurrent kinetic energy is given by

$$\begin{aligned} F_k=\int d^3 r {m \over {2}}n_s\mathbf{v}^2=\int d^3 r {{q^2 n_s} \over {2mc^2}} \left( \mathbf{A}^\mathrm{eff} \right) ^2 \end{aligned}$$
(88)

Here, \(\mathbf{A}^\mathrm{eff}\) in Eq. (69) is used by identifying

$$\begin{aligned} \mathbf{A}^\mathrm{fic} =- { {c \hbar } \over q} \nabla \chi ^\mathrm{super}=- { {c \hbar } \over {2e}} \nabla \chi , \end{aligned}$$
(89)

assuming that \(\chi ^\mathrm{super}\) arises from the Berry connection.

In the standard theory, the Ginzburg–Landau theory is derived from the BCS theory, yielding \(q=-2e\) [45]. In this case the mass of the charge carriers becomes \(m=2m^{*}\), thus, \({ m \over q}\) in the London moment becomes \(-{ m^{*} \over e}\), disagrees with the experimental value \(-{ m_e \over e}\). This indicates that the accepted derivation of the Ginzburg–Landau theory from the standard theory is incorrect.

We will use \(q=-e\) here, and identify

$$\begin{aligned} \nabla \chi ^\mathrm{super}= -{1 \over 2 }\nabla \chi \end{aligned}$$
(90)

with \(m=m_e\). This relation, \(m=m_e\), can be explained in the new theory [15].

Using \(F_k\), the current density \(\mathbf{j}\) is calculated as

$$\begin{aligned} \mathbf{j}=-c {{\delta F_k} \over {\delta {\mathbf{A}^\mathrm{em}}}}=-{{e^2 n_s} \over {m_e c}} \mathbf{A}^\mathrm{eff} \end{aligned}$$
(91)

Since

$$\begin{aligned} \mathbf{j}=-e n_s\mathbf{v} \end{aligned}$$
(92)

Eq. (91) is equivalent to the London equation in Eq. (84). Actually, the above relation should be regarded as the definition of \(n_s\) through \(\mathbf{j}\) and \(\mathbf{v}\).

Now, consider the magnetic field part of the GL free energy,

$$\begin{aligned} F_m=\int d^3 r {1 \over {8\pi }} \left( \mathbf{B}^\mathrm{eff} \right) ^2 \end{aligned}$$
(93)

This is different from the one employed by the original GL work due to the use of \(\mathbf{B}^\mathrm{eff}\) in place of \(\mathbf{B}^\mathrm{em}\).

The stationary condition of \(F_k+F_m\) with respect to the variation of \(\mathbf{A}^\mathrm{fic}\) yields,

$$\begin{aligned} -{1 \over c}{} \mathbf{j}+{ 1 \over {4 \pi }}\nabla \times \mathbf{B}^\mathrm{eff}= -{1 \over c}{} \mathbf{j}+{ 1 \over {4 \pi }}\nabla \times \mathbf{B}^\mathrm{em}=0 \end{aligned}$$
(94)

where Eq. (83) is used. This is one of the Maxwell’s equations.

Using Eq. (91) and neglecting the spatial variation of \(n_s\), we have

$$\begin{aligned} \nabla \times \mathbf{j}=-{{ n_s e^2} \over {m_e}}\left[ \nabla \times \mathbf{A}^\mathrm{em} +\nabla \times \mathbf{A}^\mathrm{fic} \right] = -{{ n_s e^2} \over {m_e}}\left[ \mathbf{B}^\mathrm{em} +\nabla \times \mathbf{A}^\mathrm{fic} \right] \end{aligned}$$
(95)

From Eq. (94), the following relation is obtained,

$$\begin{aligned} \nabla \times \mathbf{j}={{ c} \over {4 \pi }}\nabla \times (\nabla \times \mathbf{B}^\mathrm{em})= -{{ c} \over {4 \pi }}\nabla ^2 \mathbf{B}^\mathrm{em} \end{aligned}$$
(96)

Here, \(\nabla \cdot \mathbf{B}^\mathrm{em}=0\) is assumed.

Combining Eqs. (95) and (96), the following is obtained,

$$\begin{aligned} \nabla ^2 \mathbf{B}^\mathrm{em}-{{1} \over {\lambda ^2}}{} \mathbf{B}^\mathrm{em}={{1} \over {\lambda ^2}}\nabla \times \mathbf{A}^\mathrm{fic} ={{1} \over {\lambda ^2}}{} \mathbf{B}^\mathrm{fic} \end{aligned}$$
(97)

where \(\lambda\) is the London penetration depth

$$\begin{aligned} \lambda = \sqrt{ {m_e c} \over { 4 \pi n_s e^2}} \end{aligned}$$
(98)

Now we consider the loop current formation by following Abrikosov [31]. Actually, the presence of \(\mathbf{B}^\mathrm{fic}=\nabla \times \mathbf{A}^\mathrm{fic}\) in Eq. (97) naturally gives rise to vortices. The characteristic length scale for the spatial variation of \(n_s\) in the Ginzburg–Landau theory is

$$\begin{aligned} \xi _\mathrm{GL}=\sqrt{ \hbar ^2 \over {2m_e|\alpha |}} \end{aligned}$$
(99)

Abrikosov argued that if \(\lambda \gg \xi _\mathrm{GL}\) is satisfied and the singularity of \(\nabla \chi\) is along the z-axis, Eq. (97) can be approximated as

$$\begin{aligned} \nabla ^2 \mathbf{B}^\mathrm{em}-{{1} \over {\lambda ^2}}{} \mathbf{B}^\mathrm{em}={{1} \over {\lambda ^2}} \Phi ^\mathrm{fic} \mathbf{e}_z \delta ^{(2)}(\mathbf{r}) \end{aligned}$$
(100)

in the region away from the core, where \(\Phi ^\mathrm{fic}\) is given by

$$\begin{aligned} \Phi ^\mathrm{fic} =\int _{S} \mathbf{B}^\mathrm{fic} \cdot d{ \mathbf{S}} = \int _{C} \mathbf{A}^\mathrm{fic} \cdot d{ \mathbf{r}}= { {c \hbar } \over {2e}} \int _{C} \nabla \chi \cdot d{ \mathbf{r}} \end{aligned}$$
(101)

and \(\delta ^{(2)}(\mathbf{r})\) is the delta function in two-dimension with singularities along the z-axis.

The solution is known to be \(\mathbf{B}^\mathrm{em}={ B}^\mathrm{em}(r)\mathbf{e}_z\), where \({ B}^\mathrm{em}(r)\) is given by

$$\begin{aligned} { B}^\mathrm{em}(r)= -2\pi \Phi ^\mathrm{fic}\lambda ^{-2}K_0(r/\lambda ) \end{aligned}$$
(102)

Here \(K_0\) is the modified Bessel function of the 2nd kind, and r is the distance from the z-axis. This expresses a vortex along the z-axis with core size \(\xi _\mathrm{GL}\), accompanied by loop current around it. Thus, \(\xi _\mathrm{GL}\) can be identified as the core size of the loop current that exits in a superconductor.

In the BCS theory, a different coherence length

$$\begin{aligned} \xi _\mathrm{BCS}= {{ \hbar v_\mathrm{Fermi}} \over {\pi \Delta }} . \end{aligned}$$
(103)

is defined, where \(v_\mathrm{Fermi}\) is the velocity of the electron at the Fermi energy.

It is known that \(\xi _\mathrm{GL}\) and \(\xi _\mathrm{BCS}\) are similar in size at very low temperatures for BCS superconductors. However, \(\xi _\mathrm{BCS}\) is regarded as the size of the Cooper pair. In our previous work, it has been argued that the Cooper pair formation is accompanied by the loop current formation that encircles a section of the Fermi surface. This loop current gives rise to \(\mathbf{A}^\mathrm{fic}\) given in Eq. (70) [33, 35]. Actually, we can relate \(\xi _\mathrm{BCS}\) to the core size of this loop current as explained below.

First, we associate \(\xi _\mathrm{BCS}\) to the wave number \(q_c\) that has an excitation energy equal to the gap energy \(\Delta\),

$$\begin{aligned} \Delta =\hbar q_c v_\mathrm{Fermi} \end{aligned}$$
(104)

If we identify

$$\begin{aligned} \xi _\mathrm{BCS} ={ 1 \over {\pi q_c}} \end{aligned}$$
(105)

we obtain Eq. (103). This may be interpreted that \(\xi _\mathrm{BCS}\) is an estimate of the size of the loop current whose excitation energy is equal to the gap energy.

The presence of the loop current by \(\mathbf{A}^\mathrm{fic}\) is plausible from the experimental fact that the superconducting - normal metal phase transitions in a magnetic field are reversible [9] since it explains the the reversible energy transfer between the kinetic energy of the supercurrent and the magnetic field energy. We will explain this point below: by taking into account only the change of \(\mathbf{A}^\mathrm{fic}\) in the time interval \(\Delta t\) during the phase transition, the change of the kinetic energy is given by

$$\begin{aligned} \Delta F_k= & {} \int d^3 r {{e^2 n_s} \over {m_e c^2}} \mathbf{A}^\mathrm{eff} \cdot \int _t^{t+\Delta t} \partial _t \mathbf{A}^\mathrm{fic}dt \nonumber \\= & {} -{1 \over c} \int d^3 r \mathbf{j}\cdot \int _t^{t+\Delta t} \partial _t \mathbf{A}^\mathrm{fic}dt \end{aligned}$$
(106)

and the change of the magnetic field energy is given by

$$\begin{aligned} \Delta F_m= & {} \int d^3 r {1 \over {4 \pi }} \mathbf{B}^\mathrm{eff} \cdot \int _t^{t+\Delta t} \partial _t \mathbf{B}^\mathrm{fic}dt \nonumber \\= & {} \int d^3 r {1 \over {4 \pi }}\nabla \times \mathbf{B}^\mathrm{eff} \cdot \int _t^{t+\Delta t} \partial _t \mathbf{A}^\mathrm{fic}dt \nonumber \\= & {} {1 \over c} \int d^3 r \mathbf{j} \cdot \int _t^{t+\Delta t} \partial _t \mathbf{A}^\mathrm{fic}dt \end{aligned}$$
(107)

Thus, the free energy conservation, \(\Delta F_m+ \Delta F_k=0\), is satisfied [9].

The transition with \(\Delta F_\mathrm{mag}+ \Delta F_\mathrm{kin}=0\) in the time interval \(\Delta t\) occurs via

$$\begin{aligned} \int _t^{t+\Delta t} dt \ \partial _t \mathbf{A}^\mathrm{fic} \end{aligned}$$
(108)

This gives rise to a quantized change without Joule heating brought about by the modification of the winding numbers for \(\chi\). This is the key step to realize the reversible superconducting-normal metal phase transition in a magnetic field. Other changes occur among \(\mathbf{j}_s\), \(\mathbf{B}^\mathrm{em}\), \(n_s\), to satisfy \(\nabla \mathbf{B}^\mathrm{em}={ {4\pi } \over c}{} \mathbf{j}_s\) and \(\partial _t n_s +\nabla \cdot \mathbf{j}_s=0\). However, those changes can proceed without Joule heating [9].

If we assume that the situation considered above is smoothly connected to the \(\mathbf{B}^\mathrm{em}=0\) case, the existence of the loop currents with net zero macroscopic current is expected. In other words, the phase transition between the superconducting and normal phases occurs through the creation and annihilation of the loop currents generated by \(\mathbf{A}^\mathrm{em}\) in general.

Let us consider the free energy balance for the \(\mathbf{B}^\mathrm{em}=0\) case. In this case \(\mathbf{A}^\mathrm{em}\) is a pure gauge, and it cancels \(\mathbf{A}^\mathrm{fic}\) except in the core region of the size \(\xi\).

For simplicity, we assume that singularities of \(\chi\) form vortices along the z-direction. Let us pick up one of them, and take the z-axis along it. Then, the supercurrent carrier density is given by

$$\begin{aligned} n_s=n_0 e^{ -{{2 r} \over {\xi }}} \end{aligned}$$
(109)

near the vortex, where \(n_0\) is a constant.

The sum of the energies from the spatial variation of \(n_s\) and the term linear to \(n_s\) in Eqs. (85) and (87) is given by

$$\begin{aligned} \int d^3 r {{\hbar ^2 (\nabla n_s)^2} \over {8m_e \ n_s}} +\int d^3 r \alpha |\Psi _\mathrm{GL}|^2 = \int d^3 r {{\hbar ^2 (\nabla n_s)^2} \over {8m_e \ n_s}} +\int d^3 r \alpha n_s \end{aligned}$$
(110)

It becomes zero when \(\xi\) satisfies

$$\begin{aligned} \xi =\xi _\mathrm{GL} \end{aligned}$$
(111)

If \(\xi > \xi _\mathrm{GL}\), it is negative, indicating that the vortex formation may be possible if the energy gain from it is more than the energy deficit from the core formation. In other words, if the energy deficit by the creation of vortex cores is compensated by the energy gain by the generation of the non-trivial Berry connection, the loop currents generation by the Berry connection will be realized.

The energy gain here comes from the electron-pair formation in the BCS superconductors. In the new theory, the non-trivial Berry connection \(\mathbf{A}^\mathrm{fic}\) appears when the spin-twisting itinerant motion of the core size \(\xi _\mathrm{BCS}\) is realized, leading to the relation \(\xi _\mathrm{GL} \approx \xi _\mathrm{BCS}\). The non-trivial Berry connection \(\mathbf{A}^\mathrm{fic}\) and the pairing energy gap appear, simultaneously; thus, the pairing energy gap can be used as the superconducting order parameter.

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Koizumi, H. Superconductivity by Berry Connection from Many-body Wave Functions: Revisit to Andreev−Saint-James Reflection and Josephson Effect. J Supercond Nov Magn 34, 2017–2029 (2021). https://doi.org/10.1007/s10948-021-05905-y

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Keywords

  • Andreev reflection
  • Josephson effect
  • Berry connection