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arXiv:0801.1746

Bosonized Quantum Hamiltonian of the Two-Dimensional Derivative-Coupling Model

L. V. Belvedere, Aline Ferreira Rodrigues

Published Jan 11, 2008• 0 citations
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Abstract

Using the operator formulation we discuss the bosonization of the two-dimensional derivative-coupling model. The fully bosonized quantum Hamiltonian is obtained by computing the composite operators as the leading terms in the Wilson short distance expansion for the operator products at the same point. In addition, the quantum Hamiltonian contains topological terms which give trivial contributions to the equations of motion. Taking into account the quantum corrections to the bosonic equations of motion and to the scale dimension of the Fermi field operator, the operator solution is obtained in terms of a generalized Mandelstam soliton operator with continuous Lorentz spin (generalized statistics).

DOI
10.48550/arxiv.0801.1746

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Using the operator formulation we discuss the bosonization of the two-dimensional derivative-coupling model. The fully bosonized quantum Hamiltonian is obtained by computing the composite operators as the leading terms in the Wilson short distance expansion for the operator products at the same point. In addition, the quantum Hamiltonian contains topological terms which give trivial contributions to the equations of motion. Taking into account the quantum corrections to the bosonic equations of motion and to the scale dimension of the Fermi field operator, the operator solution is obtained in terms of a generalized Mandelstam soliton operator with continuous Lorentz spin (generalized statistics).
We study simple two-dimensional models with massless and massive fermions in the Hamiltonian framework. While our ultimate goal is to gain a deeper insight to structural differences between the usual (“spacelike” – SL) and light-front (LF) forms of the relativistic dynamics, an attempt is also made to clarify a few conceptual problems of quantum field theory. We point out that contrary to the assumption of canonical quantization, interacting Heisenberg fields do not always reduce to free fields at t=0. We also show that by incorporating operator solutions of the field equations to the canonical formalism, SL and LF Hamiltonians of the derivative-coupling model as well as of the Federbush model acquire an equivalent structure. In the usual canonical treatment, physical predictions in the two schemes disagree – the SL Hamiltonians contain interaction terms while their LF counterparts do not. Using a Bogoliubov transformation, the physical vacuum of the Thirring model is then derived for the first time. It has a form of a coherent state quadratic in composite boson operators which, after bosonization of the vector current, are present in the (nondiagonal) interaction Hamiltonian. To find the vacuum of the Federbush model by an analogous Bogoliubov transformation, we propose a massive version of Klaiberʼs current bosonization and demonstrate advantages of the LF treatment of the model.
We introduce a concise quantum operator formula for bosonization in which the Lie group structure appears in a natural way. The connection between fermions and bosons is found to be exactly the connection between Lie group elements and the group parameters. Bosonization is an extraordinary way of expressing the equation of motion of a complex fermion field in terms of a real scalar boson in two dimensions. All the properties of the fermion field theory are known to be preserved under this remarkable transformation with substantial simplification and elucidation of the original theory, much like Lie groups can be studied by their Lie algebras.
A weakly interacting Bose–Fermi mixture model was investigated using Wisonian renormalization group (RG). This model includes one boson–boson interaction term and one boson–fermion interaction term. The scaling dimensions of the two interaction coupling constants were calculated as 2-D at tree level and the flow equations were derived at one-loop level. We find that in the flow equations the contributions from the fermion loops go to zero as the length scale approaches infinity. In three-dimensional case two fixed points are calculated. One is the Gaussian fixed point and the other one is Wilson–Fisher fixed point. We observe that the boson–fermion interaction decouples at the Wilson–Fisher fixed point. We also find that under RG transformation the boson–fermion interaction coupling constant runs to negative infinity with a small negative initial value, which indicates a boson–fermion pairing instability. Furthermore, the possibility of emergent supersymmetry in this model was discussed.
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This is a pedagogical introduction to the general technique of bosonization of one-dimensional systems starting from scratch and assuming very little besides basic quantum mechanics and second quantization. The formalism is developed in a self-contained fashion and applied to the spinless and spin- Luttinger models, working out both single and two particle correlation functions. The implications of these results for the specific cases ofthe (anisotropic) Heisenberg and the Hubbard models are discussed. Although everything in these notes can befound in the published literature, detailed and explicit calculations of most of the results are given, which mayprove useful to beginning graduate students or researchers in this area.