Quantum condensates classified in superconductivity with topology in the Minkowski space-time

Abstract

A substantial problem in the macroscopic theory of pure superconductivity has been left forgotten for a long time since London and London in 1935. An impression survived that the Meissner effect is more substantial than the zero-resistivity. But, the London equation [I], the Newtonian equation of motion, was abandoned, whereas the London equation [II], derived from the Maxwell equations, was postulated. The London equation [II] included the logical gap [ α ] in real time, whereas the London equation [I] has been ignored without even noting the logical gap [ β ] in space. Microscopically, after the publication of F. London's book and the discovery of the isotope effect in 1950, the success of the Bardeen--Cooper--Schrieffer (BCS) theory in 1957 was likely to have finally given the definitive explanation on superconductivity by proving only the London equation [II] that claimed the coherent condensation of Cooper pairs in the momentum space. Since then, these arguments have been regarded to be a standard among various preceding theories. Meanwhile, the London equation [I] has faded away and has been long-forgotten. But we must not abandon the London equation [I], and, rather, retrieve it. We later recognized also that the DC-component of a persistent current can never be determined by using the Fourier transform analysis, because of its singularity at ω = 0 and q  = 0 with huge differences of space-time domain. Quite recently, in 2003, we first recognized a proper and harmonious view to simultaneously account for (i) the zero-resistivity in an open system with (i-c) the resultant persistent current in a closed system, and (ii) the perfect diamagnetism at T ≅ 0 K in the space-time aspects in terms of the gauge field theory. Here, we further clarify where and how we have lost and found a properly perspective view of the superconductivity. Here, we eliminate two logical gaps [ α ] and [ β ] by using the gauge field theory for further clarifying a position of the previous and present works. We especially classify superconductors with topology which eventually leads us such as (ii-2D) magnetic flux quantization in a ring. By projecting the 3-dimensional BCS-theory with the concept of ‘coherence’ among an enormous number of Bosons like Cooper pairs onto the (1 + 3)-dimensional Minkowski space-time [β = (v/c) = 0], we clarify responses of the ground state Ψ _macro at T ≅ 0 K in a set of the basic equations, for (i) the zero-resistivity, [E _K  − qφ( R )] = 0 at ω = 0 and (ii) the perfect diamagnetism [ℏK  − qA ( R )] = 0 at q  = 0 as an inevitable consequence at the gauge fields in the proper theory of superconductivity.

Keywords:

• Pure superconductivity
• London equations
• Quantum coherent condensation
• Minkowski space-time
• Macroscopic quantum numbers
• Gauge field theory
• Topology

Acknowledgements

Naturally, the author respects all of our great predecessors, not all noted here but like Kamerlingh Onnes, Meissner, Ochsenfeld, Londons, Fritz and Heinz, Landau, Ginzburg, Abrikosov, Fröhlich, Pippard, Bardeen, Cooper, Schrieffer, Gorkov, Josephson, Feynman, Ashcroft, Mermin, contributed at various stages in the historical developments of the physics of superconductivity. Professor Frederick Seitz has encouraged the author to present his viewpoints. It will be recalled that Seitz was one of John Bardeen's lifetime friends, having first encountered him as a fellow graduate student at Princeton University in the 1930s. He also helped arrange an appointment for Bardeen at the University of Illinois to undertake his work on superconductivity. The author would also express his sincere thanks to one of the his best friends Dr Tsuyoshi Uda, the Advance Soft-Project, for his kind and long-sustained friendly advice to publish the more detailed and comprehensive concept of the space-time aspect of the superconductivity as well as stimulating discussions in details in the present manuscript. Dr Jaw-Shen Tsai, Fundamental Research Laboratories, NEC Corporation, also kindly gave the author his friendly advice in an earlier stage of the present work. Professor Dr Giyuu Kido, Director, the High Magnetic Field Centre, the Nano-Materials Laboratory and Dr Tadashi Takamasu, the group leader of the Nano-Materials Physics, together with their group members, now at the Quantum Dot Research Centre, have been extremely encouraging on continuation of the author's research work and have offered long-sustaining support for it, to which the author is sincerely grateful, and also the recent generous support by the CROSS Tukuba. No doubt this work has never been completed without Dr Giyuu Kido's warm friendship and long-sustaining support with the COE (Centre Of Excellence)-Project at the National Research Institute of Metals later developed as the National Institute for Materials Science.

Notes

¹ We try to adopt the SI Units as possible as we can throughout the text of this manuscript. Nevertheless, in the cases where several of predecessors utilized the CGS units before 1960, we leave their expressions as they described, especially in equations and figures etc., as far as we have no possibility of confusion.

² It has been known that there exist a few recognitions on this point for fermion systems. For example, there is the Bloch theorem 12, 13 (p. 594) based on the free electron-like model with taking account of inter-electronic interactions. This will be discussed here newly with the concept of Boson-like Cooper's pairs later in section 4. Also, the Aharanov--Bohm effect 41 will be discussed in different aspect as remarked later in section 4. These issues were treated mainly only within an entirely different space-time domain. Here, we consider the issue topologically both for an open system and a closed system in space-time at ω = 0 and at q  = 0, when and where ∂φ/∂t = 0 and div  A  = 0, the London gauge holds and the issue is gauge invariant.

³ Here, the former is the particular solution among for the general solutions of the differential equations in (1) time or (2) space, respectively. We must be careful also of an implicit introduction of ‘Topology’.

⁴ de Brolie 45 and Weinberger 46. The famous article by de Broglie 45, was developed on the basis of his idea of a ‘wild speculation’ according to Weinberger in 2006 45. However, Weinberger defends de Broglie in the comment 45 ‘this paper proves that speculations are an essential part of physics; without them no new ideas and theories are born’, We eventually obtained the Schrödinger equation later proved experimentally further later by Davison and Germer. But we indicate that our argument here is not even the speculation but only a naïve result for β = 0 in the Minkowski space-time for ‘Quantum condensate physics’.

⁵ Here, since the mass of an enormous number of coherent Boson-like Cooper pairs is huge but with no relative motion so that β = 0 due to their coherence in superconductors, we are taking helm forwards neither to the Klein-Gordon equation nor the Dirac equation.

⁶ The notations of the macroscopic quantum numbers [(K, N), L, M] here may look parallel with those used in the standard style of textbooks of quantum mechanics. There one usually starts, after the one dimensional motion, harmonic oscillator, from the issue associated with the angular momentum towards a central force, with [l, m, n] etc. whereas we here start directly from the issue of the translational and angular momenta of an enormous number of coherent ensemble of the Cooper pairs in superconductors, which is an example of quantum condensate of Bosons. Thus, there exists a substantial difference. More in general, E_K instead of K.

⁷ “Geometrical phase factor” such as exp[iγ _n (t)] (Berry 47) never comes in here for an open system not in the Hilbert space. Besides, γ_n(C) is insignificant or γ _n (C) = 0 for the Ψ _macro of Bosons at T ≅ 0 K in a long open wire with their coherent zero-point vibration, in particular, with no approximation presumed.

⁸ Here, we consider the issue topologically both for an open system and a closed system further in space-time at ω = 0 and q  = 0, when and where ∂φ/∂t = 0 and div  A  = 0, the London gauge holds.

⁹ In general, E _K may be taken at any stably fixed level as the reference point in energy, depending on phenomena. The energy gap equation may be simplified due to coherence in ▵ _K at T ≅ 0 K.

¹⁰ This is possible only in the case of the [quasi-] Bose--Einstein condensation of a coherent ensemble of large number of Cooper pairs because of their coherent zero-point vibrations.

¹¹ The author apologizes in advance for the “patchwork” style of the discussions below.

¹² Historically, [I] the zero-resistivity of superconductor was discovered by Kamerlingh Onnes in 1911 prior to [II] the perfect diamagnetism–the Meissner--Ochsenfeld effect in 1933. If our history of superconductivity were to have developed chronologically in an inverse order such as [II] the perfect diamagnetism at first and later [I] the zero-resistivity, as noted before 10, then London and London 9 must have suffered from a different kind of torment in obtaining E ( r ) = 0 from curl  E ( r ) = 0 with the logical gap [ β ] in space r , instead of in obtaining B ( r ) = 0 from d B /dt = 0 with the logical gap [ α ] in time t. They may not have reached their famous equation [II] J _s( r ) = −[n _s( r )e ²/m] A ( r ). They must not have abandoned the London equation [I] d J _s/dt = [n _s e ²/m] E . But, in any case, we must retrieve it even only in the framework of classical electrodynamics.

¹³ Kittel 12 left an interesting piece of statement on superconductivity in the earlier version of his famous textbook; “Introduction to Solid State Physics” by Charles Kittel 12, 13 published in 1956. In his acknowledgement of this version, we can find the names of Professor Frederick Seitz for reviewing the chapters dealing with imperfections in solids and the name of Professor John Bardeen for reviewing the chapters on superconductivity and semiconductors. Kittel stated on superconductivity in his earlier editions of text book 12, 13 that “the Meissner effect ( B  = 0) contradicts this results [−d B /dt = curl  E  = 0 derived from E  = 0] and suggests that perfect diamagnetism and zero resistivity are two independent essential properties of the superconducting state”. But, in his later editions at latest the 3rd edition in 1966 14 down to even the 8th edition in 2004 17, he has been puzzled and states only that “ ‘This argument’ [based on d B /dt = 0] ‘is not entirely transparent’ [to obtain B  = 0,] ‘but the result predicts that the flux through the metal cannot change on cooling through the transition. ‘The Meissner effect contradicts this result and suggests that perfect diamagnetism is an essential property of the superconducting state’ ”. “Perhaps, this is the period when we lost something”. What happened during the period between 1956 and 1966? Highly probable, the BCS-theory in 1957 27, Landau--Lifschitz in 1960 18 and Josephson 28–30 had given an enormous amounts of deep impacts and left influences for scientists specialized in superconductivity.

¹⁴ In a sense, the expressions in the CGS units may directly reveal us a clearer significance in physics such as in [ℏK  − (q/c) A ( R )] = 0 in the case of (3.11).

¹⁵ This is significant even at a well-defined finite E _K , only if the Bose condensation of material particles such as all Cooper pairs, may occur also say, of an ensemble of coherent excitons in semiconductors, if any, in their zero-point vibrations still remaining even at excited states effectively at T _ex ≅ 0 K.

¹⁶ Here, we consider the issue topologically both for an open system and a closed system further in space-time at ω = 0 and q  = 0, when and where ∂φ/∂t = 0 and div  A  = 0, both the law of charge conservation and the London gauge hold, respectively.

¹⁷ “Geometrical phase factor” in the Hilbert space such as exp[iγ_n (t)] (Berry 47) never comes in here for an open system. Besides, γ_n (C) = 0 is insignificant for the Ψ _macro of Bosons at T ≅ 0 K in a long open wire with their coherent zero-point vibration, particularly, with no approximation presumed.

¹⁸ There may exists a small rotational energy self-induced, as noted Silsbee's hypothesis 8, 12. But, as we have been working at H ≪ H _c, we can handle it here to be the secondary effect. Later, for A.5[C] Closed system, an externally applied magnetic field only attracts substantial attention of primary importance.

¹⁹ The notations of the macroscopic quantum numbers [( K ,  N ),  L ,  M ] here may look parallel with those used in the standard style of textbooks of quantum mechanics. There one usually starts, after the one dimensional motion, harmonic oscillator, from the issue associated with the angular momentum towards a central force, with [l, m, n] etc. whereas we here start directly from the issue of the translational and angular moment a of an enormous number of coherent ensemble of the Cooper pairs in superconductors, which is an example of quantum condensate of Bosons. Thus, there exists a substantial difference.

²⁰ Of course, the situations are entirely different from the Type II superconductors 32, 33.

²¹ “Geometrical phase factor” in the Hilbert space such as exp[iγ _n(t)] (Berry 47) never comes in either here even for a closed system in real space. In particular, γ_n (C) = 0 is insignificant for the Ψ _macro,G of Bosons at T ≅ 0 K in a closed disk or rod with their coherent zero-point vibration, particularly, when no approximation presumed.

²² Naturally, the situations are entirely different from the Type II superconductors 32, 33.

²³There definitely exists a small or a large rotational energy induced at the side surface by externally applying an intense magnetic field. But, as far as we work at H ≪ H _c, we can take into accounts of it which never violates the conservation of energy as a whole. That is the perfect diamagnetism-the Meissner effect for 2 closed system as explained in the text on the basis of ‘coherence’ of an enormous numbers of Boson-like Cooper's pair, which is substantially of primary importance at magnetic floating.

²⁴ Normally, we express  = −qA _μ ( R ) rather than  = −[qA _μ ( R ) ×  R ], whereas the latter includes even a more significance.