One of the fundamentally nonlinear problems we have not yet discussed is the problem of interacting quantum spins. The nonlinearity is embedded in the commutation relations of spin operators; in contrast with Bose or Fermi creation and annihilation operators, commutators of spin operators are not c-numbers, but are still operators. Therefore even apparently very simple spin models, such as the Heisenberg model, for which the Hamiltonian is quadratic in spins, may exhibit complicated dynamics. In fact the Heisenberg model describes the majority of phenomena occurring in magnets, such as various types of magnetic ordering, spin-glass transitions, etc. In the traditional approaches spins are treated as almost classical arrows weakly fluctuating around some fixed reference frame. This reference frame is defined by the existing global magnetic order. For example, in ferromagnets or antiferromagnets, where the global order specifies only one preferential direction (the direction of average magnetization or staggered magnetization), it is supposed that spins fluctuate weakly around this direction. In helimagnets there are two preferential directions describing a spiral; their spins fluctuate around the spiral configuration. When deviations of spins from average positions are small, the spin operators can be approximated by Bose creation and annihilation operators. This approach is called the spin-wave approximation. As I have already said, it is based on two assumptions: existence of a global reference frame and smallness of fluctuations. Difficulties arise when fluctuations become strong and destroy the global order.

In this part of the book I will primarily discuss disordered magnetic systems. However, I shall not touch on such complicated sources of disorder as quenched randomness.

In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise.

In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems.

The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated.

There are several important review articles which one can read to familiarize oneself with the subject.

The bosonization procedure works equally well for fermions with spin, thus providing us with an essentially nonperturbative approach to one-dimensional metals. The foundations of this approach were laid in the late 1970s and early 1980s by various authors (see the review article by Brazovsky and Kirova (1984) and references therein).

The bosonization approach is based on the fact that coherent excitations in onedimensional interacting systems are not renormalized electrons, but collective excitations – bosons. For spinless fermions there is only one branch of collective excitations – charge density waves. For fermions with spin, another branch appears which represents spin density waves. Elementary excitations in the charge sector carry charge ±e and spin 0; excitations in the spin sector are neutral and carry spin 1/2. Since the different branches have different symmetry properties, one can expect them to have quite different spectra. This can even go to such extremes that one branch has a gap and the other does not. It is only natural under such circumstances that electrons, which carry quantum numbers from both the spin and charge sectors, cannot propagate coherently. Roughly speaking, the parts of the electron containing different degrees try to tear it in pieces; it is customary to call this phenomenon spin-charge separation. Empirically this loss of coherence is revealed as an absence of a quasi-particle pole in the single-electron Green's function, an effect whose existence is supported by experimental observations (see discussion at the end of the chapter).

All phenomenona described above (the spin-charge separation and existence of excitations with fractions of quantum numbers) can be described by the model considered below.

The possiblity of a violation of Fermi liquid theory remains one of the most intriguing problems in condensed matter theory. One loophole in the Landau reasoning is easy to find – in the derivation of decay rate given in the previous chapter I assumed that the effective interaction U is weakly momentum dependent. If this is not the case and U(k) is singular at |k| → 0, the decay rate will not be that small at the Fermi surface. Strong momentum dependence corresponds to long distance interactions; hence the presence of such interactions makes the Landau theory inapplicable. The problem is where to get such interactions. It appears that the electrostatic (Coulomb) interaction will do the job. This is a vain hope, however, because the Coulomb force is long distance only in vacuum and in insulators; in metals conduction electrons screen the Coulomb interaction and make it effectively short range. The interaction which remains long distance in metals is a much weaker interaction of electronic currents via the exchange of transverse photons. The fact that this interaction may lead to violations of the Fermi liquid theory was discovered by Holstein et al. (1973), but the real interest in this fact arose after the papers by Reizer (1989, 1991). Since photons are everywhere, this interaction exists in any metal. However, since the current–current interaction is proportional to the square of the ratio of the Fermi velocity to the speed of light, the corresponding bare coupling constant is small. As we shall see, deviations from the Fermi liquid theory may become noticeable only at very low temperatures and only provided the metal is very pure.

The spin S = 1 Heisenberg chain is an interesting object. First of all, as we have already discussed in Part III, spin chains with integer and half-integer spins have very different low energy properties. The results of Chapter 16 suggest that chains with integer spin have a spectral gap which is generated dynamically. The fact that there are many experimental realizations of quasi-one-dimensional antiferromagnets with S = 1 makes the situation even more interesting. The most well studied compound is Ni(C2H8N2)2NO2(ClO4), abbreviated as NENP. Localized spins belong to magnetic Ni ions and all the other ingredients are necessary just to arrange them into well separated chains. The experiments show that there is indeed a spectral gap (Aijro et al., 1989; Renard et al., 1987; see Fig. 36.1). This gap is not very small in comparison with the bandwidth, but one can still hope that the continuum description is good enough. The Monte Carlo simulations give for the spin S = 1 Heisenberg chain the ratio of the correlation length to the lattice spacing ξ/a = 6.2 (Nomura, 1989) which can still be treated as a large number.

There is another intriguing fact about the S =1 Heisenberg chain which makes it attractive for a theorist. It is the presence of a hidden, so-called topological order. This feature puts this theory in a much broader context of spin liquids, magnetically disordered systems with a hidden order.

Spin ladder

The spin S = 1 Heisenberg chain can be described as a limiting case of two ferromagnetically coupled S = 1/2 chains. The problem of coupled chains (so-called spin ladders) is interesting in its own right.