Generalised Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalised Dyson boson-fermion mapping is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that represents a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that it will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this text are to review the theory of generalized Dyson boson-fermion mappings and to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. The Dyson mapping uncovers non-trivial properties of the system that aid the construction of time-independent as well as time-dependent perturbation expansions. The time-independent expansions agree with results that other authors have obtained through methods other than boson expansions. The time-dependent expansions might in future prove useful in understanding aspects of the dynamics of ultra-cold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.
Generalised Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalised Dyson boson-fermion mapping is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that represents a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that it will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this text are to review the theory of generalized Dyson boson-fermion mappings and to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. The Dyson mapping uncovers non-trivial properties of the system that aid the construction of time-independent as well as time-dependent perturbation expansions. The time-independent expansions agree with results that other authors have obtained through methods other than boson expansions. The time-dependent expansions might in future prove useful in understanding aspects of the dynamics of ultra-cold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.
The Richardson Hamiltonian describes superconducting correlations in a metallic nanograin. We do a perturbative analysis of this and related Hamiltonians, around the strong pairing limit, without having to invoke the Bethe-ansatz solvability. Rather, we make use of a boson expansion method known as the Dyson mapping. Thus, we uncover a selection rule that facilitates both time-independent and time-dependent perturbation expansions. In principle, the model we analyze is realized in a very small metallic grain of a very regular shape. The results we obtain point to subtleties sometimes neglected when thinking of the superconducting state as a Bose-Einstein condensate. An appendix contains a general presentation of time-independent perturbation theory for operators with degenerate spectra, with recursive formulas for corrections of arbitrarily high orders.
Recently, the mathematical formalism of the Dyson boson mapping has been extended to a system of 3n fermions, leading to the boson–fermion and the baryon mapping. In the present paper, the case of a restriction to a subset of three-fermion quantum numbers, the collective indices, is discussed. A theory is developed for the representation of fermionic states and operators in a truncated ideal space where only collective boson–fermion pairs or collective ideal baryons are allowed. An exact reproduction of physical properties is proved to be possible provided that the original fermionic problem can be solved in a subspace where all three-fermion subsystems carry collective indices. Examples of simple applications are presented in the two subsequent papers of this series.
Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, aˊla Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.
Given a time-independent Hamiltonian H , one can construct a time-dependent Hamiltonian Ht by means of the gauge transformation Ht=UtHUt†−iUt∂tUt† . Here Ut is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on Ut ensuring that Ht is physical as long as H is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians Ht and H such that the driven many-body dynamics governed by Ht can be reduced to the quench dynamics due to the time-independent H . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H=Ht is time-dependent but dynamically integrable.
Bosonisation is a technique for studying one-dimensional fermion systems by the means of transforming them to a boson system. S. Coleman showed that the correlation functions for the Sine Gordon field, a massless boson with a cosine interaction term, coincide with those of the massive Thirring model, which is a Dirac fermion with self-interactions. An explicit map can be found between these two systems, and it can in fact be generalised to an entire class of fermions in which the boson-fermion correspondence is a well-defined operator equality in Fock space. More generally, it is a widely used technique to transform a model into another whose nature simplifies the solution or allows for obtaining more information that appears more clearly in the new representation. Another example is the Jordan Wigner transformation, which associates quantum spin chains to one-dimensional fermions. Some aspects about the statistical properties of the chain can be inferred from the fermionic representation, and more properties can be studied by applying bosonisation and examine the system in this new representation.