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This important technique represents one of the most powerful nonperturbative approaches to many-body systems currently available. The first part of the book examines the technical aspects of bosonization. Topics include one-dimensional fermions, the Gaussian model, the structure of Hilbert space in gosonization theories, Bose-Einstein condensation in two di- mensions, non-Abelian bosonization, and the Systeks and WZNW models. The third part addresses the problems of quantum impurities.

Chapters cover poten- tial scattering, the X-ray edge problem, impurities in Tomonaga-Luttinger liquids and the multi-channel Kondo problem. This book will be an excel- lent reference for researchers and graduate students working in theoretical bosonizatiob, condensed matter physics and field theory.

We are finding bosonizationn we must learn a great deal more about ‘and’. 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.

Anderson, from More is different High energy physics continues to fascinate people inside and outside of science, being percieved as the ‘most fundamental’ area of research. It is believed somehow that the deeper inside the matter we go the closer we get to the truth.

So it is believed that ‘the truth is out there’ – at high energies, small distances, short times. Therefore the ultimate theory. Theory of Ev- erything, must be a theory operating at smallest distances and times possible where there is no difference between gravitational and all other forces the Planck scale.

All this looks extremely revolutionary and complicated, but once a condensed matter physicist has found time and courage to acquiant himself with these ideas and theories, these would not appear to him ut- terly unfamihar. Moreover, despite the fact that the two branches of physics study objects of vastly different sizes, the deeper into details you go, the more parallels you will find between the concepts used. In many cases the only dif- ference is that models are called by different names, but this has more to do with funding than with the essence.

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. 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.

In this book wc arc going to deal with the area of theoretical physics where the parallels between high energy and condensed matter physics arc especially strong. This area is the theory of strongly correlated low-dimensional sys- tems. Below we will briefly go through these paralellisms and discuss the history of this discipline, its main concepts, ideas and also the features which excite interest in different communities of physicists.

The problems of strongly correlated systems are among the most difficult problems of physics we are now aware of. By definition, strongly correlated systems are those ones which cannot be described as a sum of weakly in- teracting parts. So here we encounter a situation when the whole is greater than its parts, which is always difficult to analyse.

The well-known example of such problem in particle physics is the problem of strong interactions – that is a problem of formation and structure of heavy particles – hadrons with proton and neutron being the examples and mesons. In popular lit- erature, which greatly infiuences minds outside physics, one may often read that particles constituting atomic nuclei consist in their turn of ‘smaller’, or ‘more elementary’, particles called quarks, coupled together with gluon fields.

However, invoking images and using language quite inadequate for the essence of the phenomenon in question this description more confuses 4 than explains. The confusion begins with the word ‘consist’ which here does not have the same meaning as when we say that a hydrogen atom consists of a proton and an electron. 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: The smallness of the dimensionless coupling constant a obscures the quantum character of electromagnetic forces yielding a very small cross section for processes of transformation of photons into electron-positron pairs.

Thus a serves as a small parameter in a perturbation scheme where in the first approximation the hydrogen atom is represented as a system of just two particles. Without small a quantum mechanics would be a purely academic discipline. Therefore gluon forces are of essentially quantum nature, in the sense that virtual gluons constantly emerge from vacuum and disappear, so that the problem involves an infinite number of particles and therefore is absolutely non-quantum-mechanical.

It turns out, however, that the proton and neutron have the same quantum numbers as a quantum mechanical bound state of three particles of a certain kind. Only in this sense can one say that ‘proton consists of three quarks’. The reader would probably agree that this is a very nontraditional use of this word. So it is not actually a statement about the material content of a proton as a wave on a surface of the sea, it does not have any permanent material con- tentbut about its symmetry properties, that is to what representation of the corresponding symmetry group it belongs.

It turns out that reduction of dimensionality may be of a great help in solving models of strongly correlated systems. There are two ways to relate such so- lutions 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 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. At present, there are two approaches to strongly correlated systems. One approach, which will be only very briefly discussed in this book, operates with exact solutions of many-body theories. Needless to say not every model can be solved exactly, but fortunately many interesting ones can. So this method can provide a treasury of valuable information.

The other approach is to try to reformulate complicated interacting mod- els in such a way that they become weakly interacting. 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.

## Condensed Matter > Strongly Correlated Electrons

Thus, at least at this point, the excitation spectrum and hence thermodynamics can be 6 easely described.

However, since spins are expressed in terms of the fermionic operators in a nonlinear and nonlocal fashion, the problem of correlation functions remains nontrivial to the extent that it took another 50 years to solve it. The transformation from spins to fermions completes the solution only for the special value of anisotropy; at all other values fermions interact.

It turns out however, that in many cases interactions can be effectively removed by the second transformation – in the given case from the fermions to a scalar massless bosonic field. This transformation is called bosonization and holds in the continuous limit, that is for energies much smaller than the bandwidth.

We use these words to describe a situation when low-energy corelated of a many-body system differ drastically from the con- stituent particles. Of course, there are bosonizatiob cases when constituent particles are not observable at low energies, for example, in crystalline bod- ies 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 spec- trum of correlatedd constituent particles is separated from the ground state by a gap.

This nontrivial fact, known as dynamical mass generation, was discovered by Vaks and Larkin in In this case propagation of a single particle causes a mas- 7 sive emission of soft critical fluctuations.

Both scenarios will be discussed in detail in the text. It also became clear that the conventional methods would not work.

Bychkov, Gor’kov and Dzyaloshinskii were the first who pointed bosonuzation that instabilities of one-dimensional metals cannot be treated in a mean-field-like approximation. They applied to such metals an improved perturbation series wystems scheme called ‘parquet’ approximation see also Dzyaloshinskii and Larkin Originally this method was devel- oped stdongly meson scattering by Diatlov, Sudakov and Ter-Martirosyan and Sudakov It was found that such instabilities are governed by quantum interfer- ence of two competing channels of interaction – the Cooper and the Pcicrls ones.

Summing up all leading logarithmic singularities in both channels the parquet approximation Dzyaloshinskii and Larkin obtained differential equa- tions for the coupling constants which later have been identified as Renor- malization Group equations Solyom From the fiow 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.

### [cond-mat/] Bosonization and Strongly Correlated Systems

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 – – l -dimensions, it was unclear what happens when the renormalized inter- action 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 The failure of the conventional perturba- 8 tion theory was sealed by P. Anderson who demonstrated that it originates from what he called ‘orthogonality catastrophy’: That was an indication that the problems in ques- tion 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 for the Tomonaga-Luttinger model using the Ward identities. In fact, the subsequent development fol- lowed the spirit of this boonization, 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.

The bosonization method was conceived in independently by two particle and two condensed mat- ter physicists – Sidney Coleman and Sidney Mandelstam, and Daniel Mattis and Alan Luther respectively. They established that cor- zystems functions of such fermions can be expressed in terms bosonizaion correlation functions of a free bosonic field.

In the 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 untreat- able problems. 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 Zystemsthe original idea, however, was suggested by Efetov and Larkin Many interesting applications of bosonization to realistic quasi-one-dimensional metals had been considered in the s by many researches.

Another quite fascinating discovery was also made in the s and con- cerns particles with fractional quantum numbers. Such particles appear as elementary excitations in a number of one dimensional systems, with typ- ical example being spinous in the antiferromagnetic Heisenberg chain with half-integer spin. A detailed description of such systems will correpated given in the main text; here we just present in the main text; here we just present an impressionistic picture. Imagine that you have a magnet and wish to study its excitation spec- trum.

You do it by flipping individual spins and looking at propagating waves. In measurements of dynamical spin susceptibility q 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 x” u;, qwe see a continuum. 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.

Topological confinement puts restriction only on the overall 10 number of particles leaving their spectrum unchanged.