Hubbard Model on the Honeycomb Lattice: A renormalization group approach

Dasgupta, Sohail (2019) Hubbard Model on the Honeycomb Lattice: A renormalization group approach. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

That a lifetime of man is insufficient to appreciate the beauty and magnanimity of nature is hard to fathom, yet nature hides its treasures in the most unlikely of places only to be found by the lucky few who happen to stumble upon them which inevitably changes their lives forever. Such has been the story of low dimensional materials. From the isolation of graphene by pencil and duct tape, it has been an obsession over the last couple of decades for the scientific community. In this work, the focus is on the theoretical aspects of the honeycomb lattice in two dimensions with only nearest neighbor hopping and on-site Hubbard interaction at half- filling. Many electronic systems have been found to exhibit this lattice structure, the most well-known among them being graphene a two-dimensional allotrope of carbon. This model is studied under a recently developed non-perturbative renormalization group method which avoids projecting out the doublon subspace and a priori taking the thermodynamic limit. In fact, no continuum limit is taken and the system is studied on a finite lattice. The RG method has been discussed in great details in this thesis. The various mathematical and physical ideas of the past that went into its creation has been worked out for the reader and its shortfall and relevance to the RG discussed. The method uses the ideas of many-body quantum mechanics and Gauss-Jordan diagonalization process in a new avatar to achieve its goal. The procedure of the RG has been shown by a prototype calculation on the 1D Hubbard model at half-filling leading to the phase diagram. Finally, it has been used on the 2D honeycomb lattice, which has a dispersion graph with two inequivalent Dirac points which at half filling form the Fermi surface. Hubbard interactions added to this gives a rich phase diagram. The effective Hamiltonian of the phases and the ground state wave-functions at the erstwhile Dirac points have been computed. That the phases are topological in nature is explained using Volovik's invariant.

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