High-T_c superconductivity: the Erice legacy

1 A non-linear approach

It is a real pleasure to take the opportunity given by the beautiful book dedicated to Karl-Alex Müller for his 90th birthday for an excursion through the events that occurred at the Ettore Majorana Foundation and Centre for Scientific Culture (EMFCSC) in Erice, in connection with Bednorz and Müller’s discovery of high-temperature superconductivity in cuprates. Annette Bussmann-Holder and Hugo Keller edited a similar volume entitled “High T_c Superconductors and Related Transition Metal Oxides” (Springer, 2007) on the occasion of Alex Müller’s 80th birthday, and it is significant that for the present volume Antonio Bianconi joined them as a third editor, in view of his involvement in some of the recent breakthroughs in superconductivity “above the lowest Earth temperature”.

The discovery by Marvin Cohen et al. of superconductivity in doped SrTiO₃ dates back to more than half a century ago, while thirty-one years separate the publication of this book from the stunning discovery by Bednorz and Müller of high T_c superconductivity in cuprates. Actually Alex Müller dates back his great intuition to an Erice workshop, that Heinz Bilz and I organized at the EMFCSC in 1983 (fig. 1), and since then Erice hosted a series of several courses and workshops, from 1989 until today, directed first by Alex Müller, with Georg Bednorz, Annette Bussmann-Holder, etc., then by Antonio Bianconi from the 2003 course organized with Alexander Andreev. The most recent workshop of this series, held in Erice in July 2018 under the direction of Bianconi and Yoshiteru Maeno, was devoted to Majorana Fermions and Topological Materials Science. The subject, actually including several lectures on topological superconductivity, gives an idea of how far the research went about unconventional superconductivity, inaugurated with Müller and Bednorz’s discovery in 1986. The Nobel Prize in Physics to K.-A. Müller and Georg Bednorz in 1987 for “for their important break-through in the discovery of superconductivity in ceramic materials” was a sort of plebiscitarian event, being announced just one year after the publication of their famous Zeitschrift für Physik paper –a rather unique case in the Nobel prize history!

The origin of the 1983 Erice workshop deserves being briefly told, as it illustrates the great merits of Heinz Bilz, at the time Director at the Max Planck Institute for Solid State Science in Stuttgart, and of Annette, who joined Bilz’s group at the time of my sabbatical there and with whom I had the pleasure to start in 1979 a collaboration when she was still just Bussmann. Heinz Bilz had the great intuition that the ferroelectric phase transition in perovskites was driven by the strong electron-phonon interaction associated with a nonlinear polarizability of chalcogenide ions, rather than by anharmonicity as assumed in the classical Cochran’s theory. Annette got fully absorbed in this research, going herself through a first-order phase transition with the addition of Holder to her name. The observation of superconductivity in doped strontium titanate convinced Heinz that nonlinear electron-phonon interaction implied in ferroelectricity could be as well a possible driving mechanism of superconductivity in perovskites. In those years Minko Balkanski, director of the Erice International School of Materials Science and Technology, invited me to organize some courses under the auspices of EPS. Minko Balkanski was a board member of EPS Condensed Matter Division and Jozef Devreese, a star in the theory of polarons, its chairman. Heinz Bilz liked very much the proposal to jointly organize a workshop. We quickly put together a far-reaching program on a wide spectrum of nonlinear phenomena, ranging from ferroelectrics and electron-phonon interaction, polarons and bi-polarons, to solitons in biology and ocean waves: a highly productive melting pot of ideas and models. Alex Müller was invited as the most prominent physicist in ferroelectricity and phase transitions. Roland Zeyher gave an invaluable assistance in the editing of the proceedings, while Annette could not come, busy as she was with a second-order phase transition, the birth of her first daughter. Heinz Bilz passed away prematurely on June 28, 1986, less than six weeks after the celebration of his 60th birthday in mid-May. The famous Bednorz-Müller (BM) paper was submitted to Zeitschrift für Physik on April 17 and was not yet in the libraries: unfortunately Heinz did not learn about the great breakthrough, though he was always optimist and confident that one day or the other it would have arrived! Annette cared to publish posthumously the joint papers with Heinz, and continued her highly valuable research work along those lines, strengthening many productive collaborations with Alan Bishop at Los Alamos, Alex Müller, Hugo Keller, Antonio Bianconi, keeping Erice as one of her favorite workshop places, as explained below.

When the BM paper appeared I was back at the Ettore Majorana Centre in Erice. A colleague of Bell Labs, after receiving by fax a copy of the BM paper, showed the paper to me and asked: “Do you know these guys? It looks a blunder...”. “Surely not –I replied– I know quite well KAM, a great and very serious physicist. They must have had some good reason to publish in Zeitschrift für Physik. Great news, they made it!”. “How do you know that?”. Let me answer with the words of Thomas Gull as reported in Margrit Wyder’s book: “Erice, Sicily, summer 1983: K. Alex Müller is sitting on a bench in the castle grounds and enjoying the view. As he gazes into the distance, his mind buzzes with ideas. He had just listened to a lecture by Harry Thomas which dealt with the possible existence of Jahn-Teller polarons –’quasi-particles’ that occur when electrons move through a crystal lattice. Thomas suspected that these polarons, if paired, could result in superconductivity. “As I sat there, gazing at the sea, the idea came to me that such polarons might occur in oxides” K.-Alex Müller recalls. … As is the case with metallic superconductors, the so-called Cooper pairs –in this case bipolarons– enable the current to flow without resistance.”

2 Superconductivity warms up: birth of a workshop series

In 1987, during my visit at IBM Labs in Rüschlikon, Zürich, at the time Alex Müller and Georg Bednorz were preparing their lecture for Stockholm, Alex came out with the idea of a series of courses and workshops on high-T_c superconductivity at the EMFCSC. As mentioned above, the first course of the series, Earlier and Recent Aspects of Superconductivity, was organized in 1989 by Georg Bednorz and Alex Müller themselves within the International School of Materials Science and Technology, co-directed for the occasion by its permanent director Minko Balkanski and myself. The course was a huge success for the excellence of lecturers (whom the readers may try to identify in fig. 2) and the large participation.

The next year Nino Zichichi, president of the EMFCSC, created and wanted me to run the new International School of Solid State Physics (ISSSP) under the auspices of EPS, so as to continue the series of courses and workshops on high-T_c superconductivity. Alex Müller organized the next Erice 1992 workshop on Phase Separation in Cuprate Superconductors. At this workshop the first evidence of stripes in the CuO₂ plane was presented by Bianconi. More precisely the experimental demonstration was given of 1D charge fluctuations in doped perovskites associated with tilting and bond disproportionation, forming linear arrays assigned to self-organization of pseudo-Jahn Teller polarons. This was followed by a second Erice workshop on Phase Separation, Electronic Inhomogeneities and Related Mechanisms for High-T_c Superconductors, organized in 1995 with Carlo Di Castro and Ernst Sigmund (fig. 3), who handed over the workshop care to Vladimir Kresin (Pair Correlation in Many-Fermion Systems, 1997), and Annette Bussmann-Holder. Annette organized the workshop Electron-Phonon Interaction and Phase Transitions (1997), dedicated to the late Heinz Bilz, with whom we shared the early days of this adventure, and in 1998 a memorable one with Vladimir Kresin on Polarons: Condensation, Pairing, Magnetism. The latter course was not only memorable for the exceptional level of participants, but also for a very sad event: Professor Mihail Teplov passed away suddenly during the workshop excursion. Alexander Andreev pronounced a touching praise of Teplov’s great scientific contributions when the workshop restarted, Boris Kochelaev remembered him in his Proceedings article, and a permanent Teplov Scholarship was created by the World Federation of Scientists for Russian students of the next ISSSP courses. It is important to acknowledge that the first events of the series were in the framework of the agreement between the European Physical Society and the UNESCO Regional Office for Science and Technology in Europe (ROSTE), which in the difficult times of the iron curtain collapse provided a substantial support for the mobility of students and scientists from countries of the FSU (Former Soviet Union). A brief illustration of this agreement, strongly advocated by Renato A. Ricci, at the time EPS President, is found in the Europhysics News section dedicated to the 50th anniversary of EPS foundation.

Next to the 1998 workshop, another relevant event closely related to the superconductivity series took place in ISSSP: the course chaired by Joshua Jortner and Nice Terzi on Jahn-Teller Effect and Electron-Phonon Dynamics. The role of Jahn-Teller polarons in the mechanism of high-T_c cuprates has been much discussed since the early days of the 1983 workshop. The Proceedings, edited by G. Bevilacqua, L. Martinelli and N. Terzi, had a relevant section on high-T_c superconductivity, and a foreword by the late Edward Teller, who, despite his past familiarity with the EMFCSC, could not participate, due to his advanced age. Indeed, as mentioned above, nonlinear electronphonon interaction and dynamics in ferroelectrics as closely connected to superconductivity in perovskites, was a major subject behind the 1983 workshop and the last works of Heinz Bilz. The nonlinearity aspects came back in a 2003 workshop on Intrinsic Localized Modes and Discrete Breathers in Nonlinear Lattices, chaired by Vladimir Hizhnyakov and Al Sievers. In the same year the series on unconventional superconductivity was resumed by Antonio Bianconi and Alexander Andreev on Symmetry and Heterogeneity, where the focus was on the new emerging topics like multigap superconductivity and Lifshitz transitions. Besides the two directors, the course had an exceptional participation, with hot sessions chaired by Alex Müller, Phil Allen, Neil Ashcroft, Alan Bishop, Séamus Davis, Takeshi Egami, Francesco Iachello, Kazumi Maki, Kolja Kristoffel, David Pines, Shin-ichi Uchida, Sudodh Shenoi. In particular Ashcroft outlined the roadmap to room temperature superconductivity in hydrogen-rich hydrides with the key role of polarizability. Takeshi Egami presented evidence for lattice complexity, and Campi et al. proposed the key role of anisotropic electronphonon interaction for the stripe formation in lamellar perovskites. This ground-breaking course was followed by that for the Twenty Years from the Discovery of High T_c Superconductivity (2006) and the courses Stripes in High Temperature Superconductivity (2008), Quantum Phenomena in Complex Matter (2010), and Phase Separation and Superstripes in High-T_c Superconductors (2012), all directed by Antonio Bianconi.

Majorana is back home!

The discovery of iron-based superconductivity, the renewed interest for layered chalcogenides in connection with 2D superconductivity, and particularly the search for Majorana fermions in the vortices of p-wave superconductors have opened new investigation avenues and, quite naturally, new trends for unconventional superconductivity and electronic functionalities driven by complexity in quantum matter at the Ettore Majorana Centre in Erice. Multi-condensates superconductivity and Superstripes were the subjects of a course and a workshop, respectively, that Antonio Bianconi, Carlos Sà de Melo and I directed in July 2014. Finally the course of July 2018, including, as mentioned above, topological superconductivity as one of the leading subjects, was de facto the follow-up of a ground-breaking course on Majorana Fermions in Condensed Matter, held in Erice within the ISSSP in 2013. “Majorana returns” announced Frank Wilczek a decade ago in Nature Physics, as well as in his Erice plenary lecture, and actually Majorana was discovered in one of his choice hideouts, p-wave superconductor vortices: Majorana is back home! All this demonstrates how far during the last twenty years the research in condensed matter physics went, triggered by the momentous discovery of high-T_c superconductivity in cuprates.

The book edited last year by Annette Bussmann-Holder, Hugo Keller and Antonio Bianconi to honour Alex-Müller on the occasion of his 90th birthday collects papers of many leading scientists of the Erice courses, who have greatly contributed to the present understanding of high-T_c as well as of other forms of unconventional superconductivity. Altogether 26 great papers, which would be too long to describe, plus a touching Foreword. Soon after the discovery of high-T_c superconductivity in cuprates Alex told me that we had already 120 different theories, and at least 119 were wrong. Probably all 120, as we do not have yet a fully convincing theory, but nothing goes lost in physics. The confluence of many old and new ideas will finally lead to a breakthrough and solve the problem, perhaps in a time shorter than that required by conventional superconductivity!

A great lesson from high-T_c superconductivity discovery, which should orient any future endeavour, is the one I received from another 90th birthday celebration, that of Frederich Hund in Göttingen. Professor Hund answered my question about the three rules to get at his age in a such a good shape: “The first rule is to have in life the maximum spin, the second rule to follow things of maximum momentum, but the third is to go in the opposite direction!”. Alex could say the same: just three years ago, as Editor-in-Chief of EPL, I wanted to inaugurate the series of Perspectives asking Alex to illustrate with Alexander Shengelaya the state of the art. He made it, illustrating the unconventional role of heterogeneity. With the same spirit as at the origin: looking for high-T_c superconductivity in perovskites was like going against many old tenets of the time! With this lesson in mind let me join the celebration, though a bit late, just good for the next birthday, and ideally present Alex an Erice cake in the shape of Sicily (fig. 3), where we were trying in vain to switch off the inestinguishable fire of Etna. This is actually the inestinguishable spirit of Majorana’s Erice: we have many good reasons to believe that Erice will have once more the previlege to witness that long-awaited (theoretical) breakthrough!