Duality of boson and fermion: New intermediate-statistics

Lian-Ao Wu, Zhaoyan Wu, Jing Sun

Subir Sachdev A

V. S. Shchesnovich

Eduardo C. Marino

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.

G. Kaniadakis

In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.

Jingyuan Chen, Jun Ho Son, Chao Wang +1 more

The idea of statistical transmutation plays a crucial role in descriptions of the fractional quantum Hall effect. However, a recently conjectured duality between a critical boson and a massless two-component Dirac fermion extends this notion to gapless systems. This duality sheds light on highly nontrivial problems such as the half-filled Landau level, the superconductor-insulator transition, and surface states of strongly coupled topological insulators. Although this boson-fermion duality has undergone many consistency checks, it has remained unproven. We describe the duality in a nonperturbative fashion using an exact UV mapping of partition functions on a 3D Euclidean lattice.