This book is written with the graduate student in mind. I had in mind to write a text that would introduce my students to the basic ideas and concepts behind many-body physics. At the same time, I felt very strongly that I would like to share my excitement about this field, for without feeling the thrill of entering uncharted territory I do not think one has the motivation to learn and to make the passage from learning to research.

Traditionally, as physicists we ask “what are the microscopic laws of nature?”, often proceeding with the brash certainty that, once revealed, these laws will have such profound beauty and symmetry that the properties of the universe at large will be self-evident. This basic philosophy can be traced from the earliest atomistic philosophy of Democritus to the most modern quests to unify quantum mechanics and gravity.

The dreams and aspirations of many-body physics interwine the reductionist approach with a complementary philosophy: that of emergent phenomena. In this view, fundamentally new kinds of phenomena emerge within complex assemblies of particles which cannot be anticipated from an a priori knowledge of the microscopic laws of nature. Many-body physics aspires to synthesize, from the microscopic laws, new principles that govern the macroscopic realm, asking:

This is a comparatively modern and far less familiar scientific philosophy. Charles Darwin was perhaps the first to seek an understanding of emergent laws of nature. Following in his footsteps, Ludwig Boltzmann and James Clerk Maxwell were among the first physicists to appreciate the need to understand how emergent principles are linked to microscopic physics. From Boltzmann's biography [1], we learn that he was strongly influenced and inspired by Charles Darwin. In more modern times, a strong advocate of this philosophy has been Philip W. Anderson, who first introduced the phrase “emergent phenomenon” into physics. In an influential article entitled “More is different,” written in 1967 [2], he captured the philosophy of emergence, writing:

On October 19, 2014, as I finished this book, a group of physicists convened at the University of Illinois Urbana-Champaign, home of BCS theory [1], to mark 60 years of progress in strongly correlated electron systems (SCES) and to celebrate the 90th birthday of David Pines. This occasion provided a great opportunity for us to reflect on the dizzying progress of the past 60 years of discovery in many-body physics: a period that spans from the Bohm–Pines plasmon theory of metals [2–4], the discovery of Landau Fermi-liquid theory [5], and BCS theory [1], to the modern era of topological order, strange quantum-critical metals, and high-temperature-superconductivity. This book has only scratched the surface of this period, leaving out some key aspects of the field that I perhaps will go into in a future edition. Yet still, almost everything in this book was unknown 60 years ago. The pioneers of early many-body physics in the 1950s simply could not have imagined the revolution of discovery that has since taken place. David Pines recalls thinking, as a graduate student in the early 1950s, that the physicists 20 years earlier had had all the luck. Yet Pines' thesis work with Bohm marked a beginning of 60 years of tremendous discovery.

You, the reader, might be tempted to think as David Pines did. However, at the “60 years of SCES” meeting in Urbana, many expressed remarkable optimism that the revolution is decidedly unfinished, and that the next 60 years has much in store. Like earthquakes, big scientific discoveries are unpredictable and come at varying intervals. Today, in the early twenty-first century, a large number of unsolved mysteries and problems in condensed matter physics provide the tremors that suggest that there is much yet to discover. I thought it would be interesting, if only as a historical record, to list a few of the challenges we all face:

1 Wanted: a broader understanding of the classes of emergent order in condensed Matter. Between the many decades of complexity that separate simple elements from primitive life surely lie many layers of emergent material behavior. The possibility of new forms of order, as challenging and surprising as superconductivity, cannot be ruled out.

This chapter continues our discussion of superconductivity, considering the effects of repulsive interactions and the physics of anisotropic Cooper pairing. According to an apocryphal story, Landau is reputed to have said that “nobody has yet repealed Coulomb's law” [1]. In the BCS theory of superconductors, there is no explicit appearance of the the repulsive Coulomb interaction between paired electrons. How then do real-world superconductors produce electron pairs, despite the presence of the strong interaction between them?

This chapter we will examine two routes by which Nature is able to satisfy the Coulomb interaction. In conventional superconductors, the attraction between electrons develops because the positive screening charge created by the ionic lattice around an electron remains in place long after the electron has moved away. This process that gives rise to a short-time repulsion between electrons is followed by a retarded attraction which drives s-wave pairing. However, since the 1980s physicists have been increasingly fascinated by anisotropic superconductors. In these systems, it is the repulsive interaction between the fermions that drives the pairing. The mechanism by which this takes place is through the development of nodes in the pair wavefunction – often by forming a higher angular momentum Cooper pair. The two classic examples of this physics are the p-wave pairs of superfluid 3He and the d-wave pairs of cuprate high-temperature superconductors.

In truth, the physics community is still trying to understand the full interplay of superconductivity and the Coulomb force. The discovery of room-temperature superconductivity will surely involve finding a quantum material where strong correlations within the electron fluid lead to a large reduction in the sum total of kinetic and Coulomb energy.

BCS theory with momentum-dependent coupling

We now illustrate these two different ways in which superconductors “overcome” the Coulomb interaction, by returning to the more generalized version of BCS theory with a momentum-dependent interaction:

Notice how we have deliberately included a + sign in front of the interaction HI, to emphasize its predominantly repulsive character.