#### From Introduction to "Bosonization Approach to
Strongly Correlated Systems"
by A. Gogolin, A. Nersesyan and A. Tsvelik.
Cambridge University Press, 1999.
*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. |