An electron interacts electrostatically with the surrounding building blocks of a polarisable crystal lattice. The consequence of the interaction is a deformation formed around the electron. This complex is called a polaron, a quasiparticle of the electron-phonon interaction. Polaron properties are a lot different than the properties of a free electron. Physical properties of the system, in which such interaction is present, can, consequently, also be a lot different from the properties of a system, in which this interaction is negligible. It has been known for quite some time already that the interaction of electrons with phonons enables low-temperature superconductivity. However, it is still not certain whether it also plays an important role in the mechanism of the high-temperature superconductors and colossal magnetoresistance in ferromagnetic oxides.
This work is dedicated to the small polaron model. We restrict the standard Holstein-Hubbard model by replacing phonons with hard-core bosons (HCB). We apply the HCB model to the one-dimensional one-electron and two-electron systems. We analyze the static properties such as the energy dispersion of the lowest-lying states, the expected number of bosons in these states, the kinetic energy, the quasiparticle weight, the polaron's effective mass, and also the spectral function. The behavior of the model is a lot different than the prediction of the standard Holstein-Hubbard model due to the restriction. Nevertheless, the model is quite relevant because its degrees of freedom match the degrees of freedom of the spin-1/2 system. In conclusion, we mention some of the potential directions for further work.
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