We have seen in the previous chapter how the existence of an order-disorder duality structure allows the obtainment of a full quantum theory of topological excitations. In the present chapter, conversely, we will see how the same structure is at the very roots of a method by which one can generate, out of bosonic fields, new composite fields with different statistics, either fermionic or generalized. In the first case, the method is usually called bosonization and allows a full description of fermions within the bosonic theory, whereas in the second, the method provides a complete description of anyons, as the particles with generalized statistics have been called [59], also in the framework of the bosonic theory. From their inception, both bosonization and the construction of fields with generalized statistics are deeply related to the order-disorder duality structure, since both fermion and anyon fields can be expressed as products of order and disorder operators, which respectively carry charge and topological charge [31, 32, 45]. The statistics of the resulting composite field, then, is proportional to the product of the charge and topological charge borne by this field. In this chapter we will expose the basic features of bosonization, as well as its relation with generalized statistics, and apply it to different quantum field theory systems in D = 2, 3 and 4, thereby verifying it is an extremely powerful tool for solving interacting field theories. Indeed, in some cases, this method can lead to the exact solution of highly nonlinear systems.

The Symmetrization Postulate and Its Violation: Bosons, Fermions and Anyons

Physical systems most frequently possess identical objects. Such is the case, for instance, of electrons, photons, phonons, etc. Also billiard balls, perhaps. The description of identical objects, however, changes dramatically according to whether we use a classical or quantum-mechanical approach. In the first case, one can always distinguish identical objects because we can follow their trajectories, whereas in the second case, this is no longer possible. Indeed, within a quantummechanical framework the wave-functions of different identical particles may overlap, thus causing us at the end of a physical process, to completely loose track of which particle corresponds to each of the initial ones.

Classical computation is based on algorithms that manipulate binary variables (bits), which convey the information being processed. Such algorithms are formed by logic units (logic gates), which are basic functions of binary variables yielding an output that will be used in the next step of the algorithm, and so on. Algorithms of classical computation make extensive use of irreversible gates, which can only be used in one direction and, consequently, loose information. Quantum computation, conversely, uses the quantum states of a given Hilbert space (qubits) to store the information [290, 291, 292]. Then, it takes advantage of the fact that these states have a unitary evolution to build data-processing algorithms, which only possess reversible, unitary logic gates. An important point in this scheme is that coherence of the quantum states storing the information is a crucial requirement; otherwise, the information is lost. Nevertheless, we know that quantum systems rapidly loose their coherence as we increase their size. A clever method of coherence protection, therefore, must be devised if we want to have a quantum computer operating on a human scale. For this reason, quantum computation, since its inception, has been always related to the principles of quantum mechanics.

The main method of coherence protection involves excitations having the so-called non-abelian statistics. Such excitations were shown to occur in the ν = 5/2 plateaus of systems presenting the quantum Hall effect, which is described by the Moore–Read Pfaffian wave-function. Such a state is associated to a Landau–Ginzburg type field theory consisting in the level 2 non-abelian SU(2) Chern–Simons theory in the same way the odd denominator plateaus described by the Laughlin wave-function are associated to the Zhang–Hansson–Kivelson abelian Chern–Simons–Landau–Ginzburg theory. We have shown that the vortex excitations of this theory correspond to the (abelian) anyon excitations of Laughlis's wave-function. In an analogous way, the vortex excitations of the non-abelian Chern–Simons theory are associated to the Pfaffian wave-function excitations, which present non-abelian statistics. Such gapped bulk excitations produce gapless Dirac excitations on the edges, which may combine in specific ways to reproduce the non-abelian statistics exhibited by the bulk vortex states.

Carbon has four electrons in its outer electronic shell, which contains one s and three p orbitals. This, however, is the picture for an isolated atom. When the carbon atom is part of a material with a given crystal structure, almost invariably, for energetic reasons imposed by geometric constraints associated with the crystal structure of the material, two or more of these orbitals combine in a hybrid form that is a linear combination of the former. This interesting phenomenon is known as hybridization. When the four orbitals combine in the so-called sp3 hybridization, the hybrid orbitals assemble in the form of a tetrahedron, which is the basic building block of diamond. Conversely, when three orbitals, namely, one s and two p, combine to create hybrid sp2 orbitals, these are now co-planar, pointing to directions that make an angle of 120◦ among themselves. An unexpectedly vast and extremely interesting amount of physical phenomena emerge in materials formed by this form of carbon. Among these we find polyacetylene and graphene, respectively, possessing one- and two-dimensional structures.

Polyacetylene is a polymer presenting a sequence of CH radicals, formed by carbon atoms, each one with three sp2 hybridized orbitals having covalent bonds with two adjacent carbon atoms, thus forming a zig-zag chain (transpolyacetylene). The third hybridized orbital of each carbon atom is covalently bonded to a hydrogen atom. There remains a p-orbital, which does not hybridize, which is occupied by a single electron. Since this orbital admits up to two electrons with opposite spins, the carbon p-electrons in polyacetylene can move all over the chain, being therefore responsible for most of the interesting physics of this polymer. Especially interesting effects derive from the interplay of the carbon p-electrons with the lattice, imparticular with deformations thereof possessing nontrivial topological properties and, for this reason, called topological solitons.

Polyacetylene exhibits remarkable effects, such as the Peierls mechanism, induced by the electron-lattice interaction, by which a gap is generated where otherwise there would be a Fermi surface.

Until 1986, the highest temperature at which superconductivity had been observed was 23.2K in Nb3Ge. There was an issue as to whether BCS theory would impose an upper limit on the critical temperature for the onset of superconductivity. Actually, as remarked before, the exponential factor in (23.24) may be viewed as responsible for the relatively low temperatures of the BCS superconductors. That expression for Tc has in fact a mathematical upper limit, of the order of the Debye temperature, which is typically about a few hundred Kelvin. This, however, would correspond to an infinite value of the product ƛN(EF) of coupling parameter and Fermi level density-of-states. The mathematical upper bound, consequently, is way above the temperatures corresponding to realistic physical values of parameters. In 1986, La2–xx BaxCuO4 and La2–x SrxCuO4 (LSCO) where shown to present a superconducting phase at temperatures up to about 40K. Soon after, Y Ba2Cu3O6+x (YBCO) was shown to have a superconducting phase up to 92K. These were the first members of the cuprate family, which contains materials such as HgBa2Ca2Cu3O8+x, which exhibits a superconducting phase up to a temperature of the order of 130K without pressure and of about 160K under high pressure. Many indications suggest that the mechanism leading to superconductivity in cuprates is not the phonon-mediated BCS mechanism. There is general consensus on this point. Nevertheless, there is so far no agreement about the specific mechanism leading to the formation of Cooper pairs in cuprates, despite the many proposals that have been made. In this chapter, we keep mostly in the region where consensus has been reached, except for Section 24.4, which contains our own contribution to the subject.

Crystal Structure, CuO2 Planes and Phase Diagram

A common feature of all materials belonging to the cuprates family is the presence of CuO2 planes, consisting in a square lattice, which, before doping, contains Cu++ ions on the sites and O−− ions on the links. The latter has a 2p6 electronic configuration, whereas the former is in a 3d9 configuration. The oxygen ions are in a noble gas configuration, while the copper ions have an unpaired electron, which creates a localized magnetic moment at each site.