Concepts of Many-Body Physics

Concepts of Many-Body Physics Jean-SébastienCaux ii Contents Introduction Intro-1 1 Collective phenomena 1-1 2 The operator formalism 2-1 3 Path integrals 3-1 4 Functional integrals 4-1 5 Perturbation theory 5-1 6 Effective theories 6-1 7 Response functions 7-1 A Prerequisites A-1 These lecture notes serve as accompaniment to the Master's course Statistical Physics and Condensed Matter Theory 1 given at the University of Amsterdam. Many parts are directly derived from the book \Condensed Matter Field Theory" by A. Altland and B. Simons, their notation being closely followed in many instances; the equation labels \CMFT(#.#)" in the margins directly refer to equation numbering in this book, which should anyway be consulted for a more complete coverage of the field. iii iv CONTENTS Introduction The great adventure that is physics consists of three different but complementary quests. The starting point is surely the cataloguing of all fundamental constituents of matter. As far as everyday life is concerned, the most important of these are atomic nuclei, electrons and photons. This list is however not exhaustive. For example, we know that nuclei are in fact composite objects made of protons and neutrons. In turn, these are composites of quarks. Whether this chain of ascendency ever terminates remains a quite inspirational mystery. In any case, our catalogue of fundamental constituents is exhaustive enough for most practical applications. Second comes the characterization of the basic interactions between these constituents. The list of fundamental forces currently stands at four: the gravitational force on the one hand, and the electromagnetic, weak and strong nuclear forces on the other. Again here the classification into four forces is not set in stone. Electromagnetic and weak nuclear forces are unified in electroweak theory; grand unified theory then attempts to also incorporate the strong nuclear force. Whether there are indeed only four forces, and whether these are in the end manifestations of a single all-encompassing force again remains a mystery. Nonetheless, as for fundamental constituents, we can be quite satisfied with our catalogue of fundamental forces. The third quest is more subtle, but naturally emerges as one tries to blend fundamental constituents interacting with each other in larger and larger numbers. Put simply, one immediately faces a wall of complexity when one tries to translate detailed information about constituents and fundamental interactions into firm predictions for physical behaviour of many-body systems. The minimal illustration of this is obtained by considering (Newtonian) gravitationally-interacting spherically-symmetric bodies. The two-body case (Kepler problem) can be solved exactly, allow- ing to predict positions and velocities at any time (past or future) from a set of initial conditions. Strikingly, this feature of exact solvability is immediately lost (except for fine-tuned initial conditions) when the problem is complicated by the addition of more bodies. As shown by Bruns and Poincaré,there cannot be an analytical solution to the 3-body problem for arbitrary initial conditions. Motion is then not periodic in general, but rather chaotic. This is also the case for the n-body problem. If we cannot even solve such an already-oversimplified problem, what hope can we possibly have of predicting anything about systems of very (thermodynamically!) large numbers of different types of particles interacting with each other with all available forces? Can one understand barred spiral galaxies from Newtonian gravity? Is the crystal structure of piperidinium copper bromide (C5H12N)2CuBr4 somehow hidden in the Coulomb force? Facing the cliff face of the many-body problem, the fatalistic researcher will either close up shop or refocus towards simply pushing the first two above-mentioned quests further. A more adventurous thinker will however view this third quest, which one usually refers to as the physics of emergence, as carrying more potential for astounding discoveries, and welcome every opportunity to àdd a bit more junk' as an additional chance of generating something unexpected. This contrast was perhaps best expressed by Primo Levi in his book Ìl sistema periodico', when he describes how the element zinc reacts differently to acid as a function of its level of purity: Intro-1 Intro-2 INTRODUCTION `... l'elogio della purezza, che protegge dal male come un usbergo; l'elogio dell'impurezza, che dà adito ai mutamenti, cioèalla vita. Scartai la prima, disgustosamente moralistica, e mi attardai a considerare la seconda, che mi era piúcongeniale.'1 This more congenial path is the one followed by the modern many-body physicist, and is the one which I will do my best to expose in this book. In fact, one can easily get carried away with optimism when thinking about the current state of physics. After all, looking at the world around us with the eyes of a knowledgeable physicist2, 2000_Laughlin_PNAS_97 one can sympathise with the statement from Laughlin and Pines[1] that almost every thing we are confronted with in daily life can be explained by simple nonrelativistic quantum mechanics based on the Schrödingerequation @ i jΨi = HjΨi; (1) eq:MBP:NRSE ~@t where ~ is Planck's constant. The Hamiltonian here describes Ne electrons of fundamental charge −e and mass m, together with Ni atomic nuclei of mass Mα and charge Zαe (α = 1; :::; Ni), interacting following Coulomb's law: Ne 2 Ni 2 Ne Ni 2 Ne 2 Ni 2 X −~ 2 X −~ 2 X X Zαe X e X ZαZβe H = rj + rα − + + (2) eq:MBP:HNRSE 2m 2Mα jrj − Rαj jrj − rkj jRα − Rβj j α j α j<k α<β in which rj is the spatial coordinate of electron j and Rα that of ion α. Missing from this theory are of course any interactions associated to the nuclear forces, and gravity. We should also include spin, to be able to handle problems in magnetism. Coupling to light is easily included; we can even build in special relativity, we'd then simply call this whole edifice QED (quantum electrodynamics). eq:MBP:NRSEeq:MBP:HNRSE Without even going that far, reasonings based on (1, 2) give us an explanation of the sizes of atoms, the strength and scale of chemical bonds, basic properties of bulk matter such as sound waves, why certain materials conduct electricity while others don't, why some are transparent to light of certain frequencies and others not. The accuracy to which this `Theory of Everything' thus describes basic physical properties and processes is simply astounding, and we can easily get ahead of ourselves and repeat the common mantra that it captures all the essential features for explaining essentially all that we see around us. There exist however more complicated phenomena which we cannot reasonably expect to explain from our Theory of Everything. Basic life forms, the human brain, the nonsense of stock market fluctuations, even some very-typical-looking ceramics which happen to superconduct at eq:MBP:NRSE relatively high temperature cannot be modelled in any practically feasible way starting from (1, eq:MBP:HNRSE 2). Besides our model itself being at best only distantly related to what we want to describe, the rules themselves are subject to being questioned. For example, many phenomena observed at the `human' scale can be much better described starting from classical mechanics; it would then be at least less than economical, perhaps at most even nonsensical, to start from a microscopic quantum theory. Citing Laughlin and Pines again, `So the triumph of the reductionism of the Greeks is a pyrrhic victory: We have succeeded in reducing all of ordinary physical behavior to a simple, correct Theory of Everything only to discover that it has revealed exactly nothing about many things of great importance.' 1`... the praise of purity, which protects from evil like a coat of mail; the praise of impurity, which gives rise to changes, in other words to life. I discarded the first, disgustingly moralistic, and I lingered to consider the second, which I found more congenial.' 2To qualify for this, you have to master classical and quantum mechanics, thermodynamics and statistical physics, electromagnetism and all the necessary mathematics (Fourier transforms, calculus, linear algebra, perhaps a bit of complex analysis), and ideally be at ease with a few more subsidiary subjects such as condensed matter physics... Intro-3 In order to make progress, one has to make some sacrifices. First of all, perhaps not all eq:MBP:HNRSE degrees of freedom in (2) are relevant to a specific problem one might think about. For example, if we are interested in the motion of electrons in a conductor, we might reasonably assume that the nuclei are simply sitting at fixed positions and ignore their motion. Writing an effective model involving this reduced number of degrees of freedom then provides a more economical starting point. Once a reasonable effective model has been written down, explicit calculations can be at- tempted. In most cases however, our solution capabilities fall far short of our desires. We might like to know what the frequency-dependent conductivity of the two-dimensional Hubbard model is as a function of interaction strength, filling and temperature, but getting there is no easy task. The dream of finding exact solutions to strongly-correlated many-body systems has only really been realized in the world of one-dimensional quantum (or equivalently, two-dimensional classical) physics, but these firmly-grounded results, though extremely instructive, do not extend to higher dimensions. Over the last few decades, physicists have thus built extensive frameworks to handle such unsolvable problems as best as possible. In a first instance, perturbation theory is the natural starting point. The strategy is simple: from an exactly-solved simple base system (most com- monly: a free (noninteracting) theory, all of whose correlation functions are readily computable), try to approach the desired system by systematically computing the effects of the perturbations required to bring you from the base system to the desired one.

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