The total number of states of any system is produced by all the possible interactions of the particles in that system. It follows therefore that there is need to obtain the states of the system before a model can be applied to it. Recently we developed a simple method to obtain the electronic states in the two electron lattice systems in all the three dimensions. This method encompasses rules formulated from a detailed study of each dimension. In the current study, the method will be extended to bosons which do not obey the Pauli exclusion principle like the electrons so that their interactions are different from those of the latter. The need for the extension of the method to bosons is also discussed in the work. Keywords: lattices, bosons, fermions, interactions, bosonic states Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 129-135
The total number of states of any system is produced by all the possible interactions of the particles in that system. It follows therefore that there is need to obtain the states of the system before a model can be applied to it. Recently we developed a simple method to obtain the electronic states in the two electron lattice systems in all the three dimensions. This method encompasses rules formulated from a detailed study of each dimension. In the current study, the method will be extended to bosons which do not obey the Pauli exclusion principle like the electrons so that their interactions are different from those of the latter. The need for the extension of the method to bosons is also discussed in the work. Keywords: lattices, bosons, fermions, interactions, bosonic states Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 129-135
The degrees of freedom studied so far have been either real or complex variables. These variables commute under multiplication, in the sense that two numbers a, b satisfy ab = ba ; commuting variables are generically called bosonic variables, or bosonic degrees of freedom. Typical of the bosonic case are the degrees of freedom for a collection of quantum mechanical particles. Interactions of fundamental particles are generally mediated by bosonic fields such as the Maxwell electromagnetic field, whereas mass is usually carried by particles that are fermions, the most familiar being the electron. Two key features distinguish fermions from bosons: Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is the reason the concept of intensity does not apply to a fermion. A high intensity electric field is a reflection of the presence of a large number of photons, which are bosons, in the same quantum state; for photons, any number of photons can be in the same quantum state. In contrast, an electron is either in a quantum state or it is not; in particular, ignoring spin, an electron exists at a point or there is no electron there. The state function of a multi-bosonic system is totally symmetric in that the exchange of any two bosonic degrees of freedom yields the same state function. In contrast, a multi-fermion system is totally anti-symmetric: the exchange of any two fermion degrees of freedom gives the same state – but with a negative sign .
“Bosonization” refers to the possibility of describing a theory of relativistic Dirac fermions obeying standard anticommutation rules by a boson field theory. While this may be possible in all dimensions, it has so far proved most useful in d = 1, where the bosonic version of the given fermionic theory is local and simple, and often simpler than the Fermi theory. This chapter should be viewed as a stepping stone toward a more thorough approach, for which references are given at the end. In this chapter I will set up the bosonization machine, explaining its basic logic and the dictionary for transcribing a fermionic theory to a bosonic theory. The next chapter will be devoted to applications. To my knowledge, bosonization, as described here, was first carried out by Lieb and Mattis [1] in their exact solution of the Luttinger model [2]. Later, Luther and Peschel [3] showed how to use it to find asymptotic (low momentum and energy) correlation functions for more generic interacting Fermi systems. It was independently discovered in particle physics by Coleman [4], and further developed by Mandelstam [5]. Much of what I know and use is inspired by the work of Luther and Peschel. Preamble Before getting into any details, I would first like to answer two questions. First, if bosonization applies only to relativistic Dirac fermions, why is it of any interest to condensed matter theory where relativity is not essential? Second, what is the magic by which bosonization helps us tame interacting field theories? As for the first question, there are two ways in which Dirac fermions enter condensed matter physics. The first is in the study of two-dimensional Ising models, where we have already encountered them. Recall that if we use the transfer matrix approach and convert the classical problem on an N × N lattice to a quantum problem in one dimension we end up with a 2 N -dimensional Hilbert space, with a Pauli matrix at each of N sites. The two dimensions at each site represent the twofold choice of values open to the Ising spins. Consider now a spinless fermion degree of freedom at each site. Here too we have two choices: the fermion state is occupied or empty.
For any purely fermionic system on a lattice in arbitrary space dimension we prove that there is a different but well-defined system on the same lattice, consisting of both bona fide fermions and bosons with an interaction depending on a parameter \ensuremath{\lambda}, which characterizes a same-site repulsion between particles. The energy spectrum and the scattering matrix of the former are identical to those in the finite energy sector of the latter, in the limit \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} (i.e., hard-core repulsion). This general theorem is applied to a resonance-boson model in a lattice fermion system, whose solution may have relevance to high-Tc superconductivity.
There is growing belief that both the conventional Bose Einstein condensation (CBEC) and non-conventional Bose Einstein condensation (NBEC) can be obtained from repulsive and attractive interactions of bosons respectively. However, there is still no generally accepted model to obtain both condensates. Since results in the literature show that the Hubbard model two electron interactions has both repulsive and attractive regions depending on the interaction strength, we are considering its extension to charged bosons to investigate the possibility of obtaining the condensates from it. As a preliminary investigation, we have compared in this present study, the two electron variational Hubbard Hamiltonian with that modified for charged bosons in N = 2, N x N = 4 x 4 and N x N x N = 5 x 5 x 5 lattices. The results show only slight differences between the cases of the electrons and bosons. The implication that there is possibility of obtaining both condensates from the Hubbard model is then discussed in relation to superconductivity. Keywords: Bose Einstein condensation, superconductivity, Hubbard model, electrons, bosons Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 119-123
The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities.