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# Theory of strongly correlated systems

#### We used to think that if we know one, we knew two, because one and one are two. We are finding that we must learn a great deal more about and'.

Sir Arthur Eddington, from The Harvest of a Quiet Eye, by A. Mackay

The behaviour of large and complex aggregations of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understanding of the new behaviours requires research which I think is as fundamental in its nature as any other.

### P. W. Anderson, from More is different (1972)

My subject is condensed matter theory. Since most students do not consider it as sexy  as particle physics I feel a need for an explanation. Without any doubt the particle theory is a fascinating subject and I do not want to belittle my colleagues involved in research in that area. However, its public perception is distorted by a poor philosophy. People inside and outside of science believe that particle physics is somehow more fundamental' than other subjects. Sometimes this belief goes to such extremes that people start talking about Theory of Everything which is expected to come from particle physics as some sort of messiah.  Consciously or unconsciously those who have such expectations adopt an image of the Universe as a jig-saw puzzle where large and complicated things are composed of things small and simple. Of course, the very term elementary particle' invokes that sort of image. Despite the fact that sometimes such reductionist description works well, it has its limitations. Surely, when one says that hydrogen atom consists of one proton and one electron, this is a rather accurate description. This is because a hydrogen atom is formed by electromagnetic forces and the binding energy of the electron and proton is small compared to their masses: $E \sim - \alpha^2 m_ec^2$, where $\alpha = e^2/hc \approx 1/137$ is the fine structure constant and $m_e$ is the electron mass. The smallness of the dimensionless coupling constant $\alpha$ obscures the quantum character of electromagnetic forces yielding a very small cross section for processes of transformation of photons into electron--positron pairs. In the first order in $\alpha$ one can consider the hydrogen atom as a two-body problem and forget about the fact the electromagnetic force binding the system together is quantum in nature consisting' itself of all photons in the Universe. However, when we go further up in energy and ask what are constituent parts of proton things change dramatically. Can we say that proton consists of three quarks?   Yes, if you mean that it has the same quantum numbers as a certain three- particles bound state. No, if you mean that to describe it one needs to solve a three-body problem of quantum mechanics. This is because the fine structure constant for quark-quark interactions is not small and gluons are constantly born and destroyed in the process of interaction. Thus to describe one proton one needs to solve a problem of infinite number of particles!

Here particle physics merges with condensed matter theory. Both disciplines study problems of infinite number of particles using for these purposes statistical description. In both disciplines elementary excitations' or particles' emerge not like independent jig-saw pieces, but as waves on a surface of the sea called vacuum. The only difference is that in the particle theory studies the Ocean - the vacuum of all interactions and condensed matter one deals with various small vacua - ferromagnetic, superconducting, spin liquid etc. Therefore it is not surprising that particle physics and condensed matter like to borrow concepts and models from each other. For example, the Anderson--Higgs phenomenon of particle theory (screening of the weak interactions) appears in condensed matter as the Meissner effect (screening of the magnetic field in superconductors); the concept of inflation' in cosmology is taken from the physics of first order phase transitions; the hypothetical cosmic strings' are similar to magnetic field vortex lines in type II superconductors; the Ginzburg--Landau theory of superfluid He$^3$ has many features common with the theory of hadron-meson interaction etc. When you realize the existence of this astonishing parallelism, it is very difficult not to think that there is something very deep about it, that here you come across a general principle of Nature according to which same ideas are realized on different space-time scales, on different hierarchical layers', as a Platonist would put it. So instead of being a jig-saw the Universe appears as a simphony where the same tune is played parts and in different modifications. This view puts things in a new perspective where truth is no longer out there', but may be seen equally well in a grain of sand' as in an elementary particle.

There is an area in theoretical physics where the parallels between high energy and condensed matter physics are especially strong. This area is the theory of strongly correlated systems. One outstanding problem in this area is the problem of quark confinement - the fact that individual quarks are non-observable but always exist only inside of other particles. There many other problems however, a lot of them in condensed matter physics. It turns out that reduction of dimensionality may be of a great help in solving models of strongly correlated systems. Most nonperturbative solutions presently known (and only nonperturbative ones are needed in physics of strongly correlated systems) are related to (1 + 1)-dimensional quantum or two-dimensional classical models. There are two ways to relate such solutions to reality. One way is that you imagine that reality on some level is also two dimensional. If you believe in this you are a string theorist. Another way is to study systems where the dimensionality is artificially reduced. Such systems are known in condensed matter physics; these are mostly materials consisting of well separated chains, but there are other examples of effectively one-dimensional problems such as problems of solitary magnetic impurities in metallic host (Kondo effect) or of edge states in the Quantum Hall effect. So if you are a theorist who is interested in seeing your predictions fulfilled during your life time, condensed matter physics gives you a chance. Curiously enough the Kondo problem has turned out to be intimately related with the notorious problem of the Schroedinger cat. Its solution has greatly helped to resolve the corresponding paradoxes. At present, there are two approaches to strongly correlated systems. One approach operates with exact solutions of many-body theories. Needless to say not every model can be solved exactly, but fortunately many interesting ones can. Some of my research is related to exactly solvable models.

The other approach is to try to reformulate complicated interacting models in such a way that they become weakly interacting. This is the idea of bosonization which was pioneered by Jordan and Wigner in 1928 when they established equivalence between the spin S = 1/2 anisotropic Heisenberg chain and the model of interacting fermions. Thus in just two years after introduction of the exclusion principle by Pauli it was established that in many-body systems the wall separating bosons from fermions might become penetrable. The example of the spin-1/2 Heisenberg chain has also made it clear that a way to describe a many-body system is not unique, but is a matter of convenience. The spin S = 1/2 Heisenberg chain has provided the first example of particles transmutation'. Here the many-body system can be equally well described with bosons and with fermions. The low-energy excitations in this model differ drastically from the constituent particles. Of course, there are elementary cases when constituent particles are not observable at low energies, for example, in crystalline bodies atoms do not propagate and at low energies we observe propagating sound waves -- phonons; in the same way in magnetically ordered materials instead of individual spins we see magnons etc. These examples, however, are related to the situation where the symmetry is spontaneously broken, and the spectrum of the constituent particles is separated from the ground state by a gap. Despite the fact that continuous symmetry cannot be broken spontaneously in (1 + 1)-dimensions and therefore there is no finite order parameter even at T = 0, spectral gaps may form. This nontrivial fact, known as dynamical mass generation, was discovered by Vaks and Larkin in 1961. However, one does not need spectral gaps to remove single electron excitations since they can be suppressed by overdamping occuring when T = 0 is a critical point. In this case propagation of a single particle causes a massive emission of soft critical fluctuations. The fact that soft critical fluctuations may play an important role in (1 + 1)-dimensions became clear as soon as theorists started to work with such systems. It also became clear that the conventional methods would not work. Bychkov, Gor'kov and Dzyaloshinskii (1966) were the first who pointed out that instabilities of one-dimensional metals cannot be treated in a mean-field-like approximation. They applied to such metals an improved perturbation series summation scheme called parquet' approximation (see also Dzyaloshinskii and Larkin (1972)). Originally this method was developed for meson scattering by Diatlov, Sudakov and Ter-Martirosyan (1957) and Sudakov (1957). It was found that such instabilities are governed by quantum interference of two competing channels of interaction -- the Cooper and the Peierls ones. Summing up all leading logarithmic singularities in both channels (the parquet approximation) Dzyaloshinskii and Larkin obtained differential equations for the coupling constants which later have been identified as Renormalization Group equations (Solyom (1979)). From the flow of the coupling constants one can single out the leading instabilities of the system and thus conclude about the symmetry of the ground state. It turned out that even in the absence of a spectral gap a coherent propagation of single electrons is blocked.

The charge--spin separation -- one of the most striking features of one dimensional liquid of interacting electrons -- had already been captured by this approach. The problem the diagrammatic perturbation theory could not tackle was that of the strong coupling limit. Since phase transition is not an option in (1 + 1)-dimensions, it was unclear what happens when the renormalized interaction becomes strong (the same problem arises for the models of quantum impurities as the Kondo problem where similar singularities had also been discovered by Abrikosov (1965)). The failure of the conventional perturbation theory was sealed by P. W. Anderson (1971) who demonstrated that it originates from what he called orthogonality catastrophy': the fact that the ground state wave function of an electron gas perturbed by a local potential becomes orthogonal to the unperturbed ground state when the number of particles goes to infinity. (Particle transmutation includes orthogonality catastrophy as a particular case.) That was an indication that the problems in question cannot be solved by a partial summation of perturbation series. This does not prevent one from trying to sum the entire series which was brilliantly achieved by Dzyaloshinskii and Larkin (1974) for the Tomonaga--Luttinger model using the Ward identities. In fact, the subsequent development followed the spirit of this work, but the change in formalism was almost as dramatic as between the systems of Ptolemeus and Kopernicus.

As it almost always happens, the breakthrough came from a change of the point of view. When Kopernicus put the Sun in the centre of the coordinate frame, the immensely complicated host of epicycles was transformed into an easily intelligeble system of concentric orbits. In a similar way a transformation from fermions to bosons (hence the term {\it bosonization}) has provided a new convenient basis and lead to a radical simplification of the theory of strong interactions in (1 + 1)-dimensions. The bosonization method was conceived in 1975 independently by two particle and two condensed matter physicists -- Sidney Coleman and Sidney Mandelstam, and Daniel Mattis and Alan Luther respectively. (The first example of bosonization was considered earlier by Schotte and Schotte (1969).) The focal point of their approach was the property of Dirac fermions in (1 + 1)-dimensions. They established that correlation functions of such fermions can be expressed in terms of correlation functions of a free bosonic field. In bosonic representation the fermion forward scattering became trivial which made a solution of the Tomonaga--Luttinger model a simple exercise. The new approach had been immediately applied to previously untreatable problems. The results by Dzyaloshinskii and Larkin were rederived for short range interactions and generalized to include effects of spin. It was then understood that low-energy sector in one-dimensional metallic systems might be described by a universal effective theory later christened Luttinger-' or Tomonaga--Luttinger liquid'. The microscopic description of such a state was obtained by Haldane (1981), the original idea, however, was suggested by Efetov and Larkin (1975). Many interesting applications of bosonization to realistic quasi-one-dimensional metals had been considered in the 1970s by many researches. Another quite fascinating discovery was also made in the 1970s and concerns particles with fractional quantum numbers. Such particles appear as elementary excitations in a number of one-dimensional systems, with typical example being spinons in the antiferromagnetic Heisenberg chain with half-integer spin. Imagine that you have a magnet and wish to study its excitation spectrum. You do it by flipping individual spins and looking at propagating waves. Naturally, since the minimal change of the total spin projection is $|\Delta S^z| = 1$ you expect that a single flip generates a particle of spin-1. In measurements of dynamical spin susceptibility $\chi''(\omega, q)$ an emission of this particle is seen as a sharp peak. This is exactly what we see in conventional magnets with spin-1 particles beeing magnons. However, in many one dimensional systems instead of a sharp peak in $\chi''(\omega, q)$, we see a continuum. This means that by flipping one spin we create at least two particles with spin-1/2. Hence fractional quantum numbers. However, excitations with fractional spin are subject of topological restriction -- in the given example this restriction tells us that the particles can be produced only in pairs. Therefore one can say that the elementary excitations with fractional spin (spin-1/2 in the given example) experience topological confinement'. Topological confinement puts restriction only on the overall number of particles leaving their spectrum unchanged. Therefore it should be distinguished from dynamical confinement which occurs, for instance, in a system of two coupled spin-1/2 Heisenberg chains. There the interchain exchange confines the spinons back to form S = 1 magnons giving rise to a sharp single-magnon peak in the neutron cross section which spreads into the incoherent two-spinon tail at high energies. An important discovery of non-Abelian bosonization was made in 1983--4 by Polyakov and Wiegmann (1983), Witten (1984), Wiegmann (1984) and Knizhnik and Zamolodchikov (1984). The non-Abelian approach is very convenient when there are spin degrees of freedom in the problem. Its application to the Kondo model done by Affleck and Ludwig in the series of papers (see references in Part III) has drastically simplified our understanding of the strong coupling fixed point. The year 1984 witnessed another revolution in low-dimensional physics. In this year Belavin, Polyakov and Zamolodchikov published their fundamental paper on conformal field theory (CFT). CFT provides a unified approach to all models with gapless linear spectrum in (1 + 1)-dimensions. It was established that if the action of a (1 + 1)-dimensional theory is quantizable, that is its action does not contain higher time derivatives, the linearity of the spectrum garantees that the system has an infinite dimensional symmetry (conformal symmetry). The intimate relation between CFT and the conventional bosonization had became manifest when Dotsenko and Fateev represented the CFT correlation functions in terms of correlators of bosonic exponents (1984). In the same year Cardy (1984) and Bl\"ote, Cardy and Nightingale (1984) found the important connection between finite size scaling effects and conformal invariance. Both non-Abelian bosonization and CFT are steps from the initial simplicity of the bosonization approach towards complexity of the theory of integrable systems. Despite the fact that correlation functions can in principle be represented in terms of correlators of bosonic exponents, the Hilbert space of such theories is not equivalent to the Hilbert space of free bosons. In order to make use of the bosonic representation one must exclude certain states from the bosonic Hilbert space. It is not always convenient to do this directly; instead one can calculate the correlation functions using the Ward identities. It is the most important result of CFT that correlation functions of critical systems obey an infinite number of the Ward identities which have a form of differential equations. Solving these equations one can uniquely determine all multi-point correlation functions. This approach is a substitute for the Hamiltonian formalism, since the Hamiltonian is effectively replaced by Ward identities for correlation functions. Conformally invariant systems being systems with infinite number of conservation laws constitute a subclass of exactly solvable (integrable) models. After many years of intensive development the theory of strongly correlated systems became a vast and complicated area with many distingushed researchers working in it. Different people have different styles and different interests -- some are concerned with applications and some with technical developments. There is still a gap between those who develop new methods and those who apply them, but it is closing fast.

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