The Theoretical And Numerical Study On The Holographic Vortex Lattice For Lifshitz Geometry With Hyperscaling Violation

Over the past few years, it has been shown that various properties of condensed matter systems can be reproduced using general relativity. This surprising result is motivated by the remarkable gauge/gravity duality which relates a theory of gravity to a strongly coupled quantum field theory. The duality is called “holographic” since the quantum field theory lives in a lower dimensional space. The holographic approach has been applied to many interesting phenomena, including superconductivity and superfluidity, Fermi surfaces and non-Fermi liquids, particularly the lattice(one key ingredient of condensed matter systems).In this thesis, by making use of the holography, we propose a holographic model of vortex lattice for Lifshitz theories with hyperscaling violation. The thermodynamic properties of this model and the possible configurations of the lattice are also investigated. For this sake, we firstly construct a Lifshitz background with hyperscaling violation. Fermionic field as a test field is introduced into this spacetime. We then analytically consider the spontaneous formation of a fermionic crystalline geometry in a gravity background with Lifshitz scaling and/or hyperscaling violation. Fermionic vortex lattice solution sourced by the lowest Landau level has been obtained. In order to obtain the lattice configurations, we then consider the backreactions of the lattice to the metric and the gauge field. Based on this, free energy of the system is obtained. Thermodynamic analysis shows that the fermionic vortex lattice favors a triangular configuration, regardless of the value of the Lifshitz scaling z and the hyperscaling violation exponent ?. Our results also show that the larger z or lower ? leads to more stable lattices thermodynamically...