Derivação de interações efetivas de elétrons em membrana bidimensional (grafeno) utilizando transformações de Hubbard- Stratonovich
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2015-03-20
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Universidade Federal de Goiás
Resumo
In this paper we investigate two-dimensional membranes, such as graphene, using
Quantum Field Theory, more specifically by the path integral formalism and using Hubbard-
Stratonovich transformations. With graphene as a motivation, we use models for electrons in a
graphene layer, when they interact with phonons, Kekule deformation and gauge fields. We
start by considering a theory in which bosons and fermions interact via a Yukawa type
coupling. We present a method already known to eliminate the degrees of freedom of the
fermionic system and use the Hubbard-Stratonovich transformations to derive an effective
bosonic theory. For this, we introduce an auxiliary field in the model and show that the
effective theory for this field is equivalent to the effective theory for the physical bosonic
field. Thus, we calculated and obtained the gap equations for this system in 1+1, 2+1 and 3+1
dimensions and then compared with the Gross-Neveu model for quartic interactions between
fermions in 1+1 dimensions. We see that for a particular coupling constant the massive
electrons lose all their mass by interacting with bosons, an effect caused by symmetry
breaking. We then present the chiral gauge model of Jackiw-Pi for graphene, where Yukawa
type interactions are present. However, this theory is a particular case of a more general
model proposed by Frederico et al. where bosonic self-interactions at higher orders and
bosons/fermions with more general interactions are considered. Again, we use the Hubbard-
Stratonovich transformations to derive effective models for fermions and the gap equations.
We identified chiral invariance transformations of the group U(1) for the limiting case of
Jackiw-Pi model. Finally, we investigated a model for phonons in the graphene background,
more specifically building on the papers by Katsnelson et al. Guinea et al.. At this point,
anharmonic terms are included in the Lagrangian of the system in an attempt to describe
changes in the background structure. We eliminate the degrees of freedom of the scalar
system, which are responsible for describing power modes on and off plane and thus obtained
an effective theory for the electronic part. As in the previous model, we see a change in the
effective potential and derive their gap equations. Finally, we present the Coulomb potential
to derive an effective theory for fermions when they interact with a gauge field. Thus, we
compare this result with the studied models and analyzed the effective fermionic interactions
present in the obtained Lagrangians.
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FREIRE, L. E. S. Derivação de interações efetivas de elétrons em membrana bidimensional (grafeno) utilizando transformações de Hubbard- Stratonovich. 2015. 76 f. Dissertação (Mestrado em Fisica) - Universidade Federal de Goiás, Goiânia, 2015.