초록

Abstract ▼

The goal of this project was the development of the applications of AdS/CFT correspondence and other string theory methods to various, not necessarily string-related systems. In past years string theory developed an impressive mathematical apparatus among which the AdS/CFT correspondence is probably

The goal of this project was the development of the applications of AdS/CFT correspondence and other string theory methods to various, not necessarily string-related systems. In past years string theory developed an impressive mathematical apparatus among which the AdS/CFT correspondence is probably a most remarkable tool. So there is a natural motivation to try to use these tools in other fields. In the attempts to adapt the string theory methods to other fields and namely to the problems of condensed matter theory there are several obstacles which one has to overcome. One problem is that string theory is by definition a relativistic model and its properties strongly depend on this. On the other hand, many of interesting problems are related to non-relativistic systems. There are at least two approaches to deal with this situation. One is to consider the geometries with the symmetries appropriate for a problem. The other approach is to consider the non-relativistic limits of massive relativistic systems. We followed the last approach. In this context we studied the metamorphosis of the supersymmetry in the nonrelativistic limit starting from the example of ABJM model. The developed methods were applied to the matrix model. We found that a limit similar to non-relativistic limit can be found for a massive Yang-Mills type matrix model. In this limit the model becomes Large N integrable, because it is equivalent to an ensemble of integrable spin chains which are decoupled when N is large. We succeeded to show that non-relativistic corrections are equivalent to higher loop corrections in perturbation theory and we were able to evaluate them without doing, in fact, any perturbative calculation. Another problem we approached was the study of the electronic structure of graphene. In particular we solved the Landau problem for graphene bilayer in tilted magnetic field with arbitrary angle of tilt. We found the Landau level spectrum in terms of spheroidal functions and their eigenvalues. Our exact result can be used to interpolate between the known before result for perpendicular field and the new result for fully in-plane magnetic field. An interesting observation was that in the limit of purely in-plane field the electronic properties of graphene undergoes a serious change, with an additional non-abelian SU(2) symmetry emergnce. Unfortunately, this observation can not be experimentally tested with the present day available techniques, as the effect is expected to be visible at the value of magnetic field of few hundred tesla, not yet reachable in the terrestrial experiment. Further study of graphene lattice revealed that the deformations of the graphene lattice lead in the low energy limit to the emergence of gauge and Yukawa type interactions with the internal group U(2). We were able to classify and describe these degrees of freedom. This approach was particularly useful for the understanding of the Stone-Wales defect important for the mechanical properties of graphene. As it appears a fermi system on a lattice like graphene can be a source of a low energy relativistic system with a multitude of gauge interactions. In a sequel we tried to research if similar systems can be found, described and possibly classified in 3+1 dimensions. We performed an analysis of a wide class of graphs described by their adjacency matrices and found the condition that in the low energy limit they describe a relativistic system with Dirac fermions. Such systems we called Dirac lattices and the respective condition was expressed as an algebraic equation on the adjacency matrix. We were able to find the general solution to this condition in the case of graphs with two-dimensional and three-dimensional translation symmetries. For the case of general dimension we found a class of solutions, which we conjectured to be generic up to the internal re-definitions of the graph structure. In present we develope this approach further by considering the graph models which generate not only the spacial geometry, but also the time evolution.

목차 Contents

• 일반연구자지원사업 최종(결과)보고서 ... 1
• 목 차 ... 3
• Ⅰ. 연구결과 요약문 ... 4
• Ⅱ. 연구내용 및 결과 ... 6
• 1. 연구과제의 개요 ... 6
• 2. 국내외 기술개발 현황 ... 8
• 3. 연구수행 내용 및 결과 ... 10
• 4. 목표 달성도 및 관련 분야에의 기여도 ... 11
• 5. 연구결과의 활용계획 ... 12
• 6. 연구과정에서 수집한 해외과학기술정보 ... 13
• Ⅲ. 연구성과 ... 14
• 끝페이지 ... 15