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Band Structure Degeneracies and Edge States in Square Lattice Media and Their Deformations
Band Structure Degeneracies and Edge States in Square Lattice Media and Their Deformations
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202105252
- ISBN
- 9798297646735
- DDC
- 510
- 저자명
- Chaban, Jonah.
- 서명/저자
- Band Structure Degeneracies and Edge States in Square Lattice Media and Their Deformations
- 발행사항
- [Sl] : Columbia University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 231 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 87-04, Section: B.
- 주기사항
- Advisor: Weinstein, Michael I.
- 학위논문주기
- Thesis (Ph.D.)--Columbia University, 2025.
- 초록/해제
- 요약We study the spectral properties of conservative linear wave systems modeling two-dimensional periodic "bulk" media, as well as "edge" media formed by the junction of two distinct bulk media. In both scenarios, we consider the cases of (1) periodic bulk media possessing the symmetries of a square lattice, and (2) linear deformations of such media. The dissertation is divided in two parts:In part one, we begin with a spatially infinite bulk medium modeled by the Schrodinger operator H = -Δ + V, where the potential V is real-valued, periodic with respect to a square lattice, and invariant under the symmetry group of the square. The band structure of H is known to contain quadratic band degeneracy points; see also [8]. We prove that, under typical small linear deformations of V, each quadratic band degeneracy splits into a pair of nearby (tilted, elliptical) conical degeneracies, or Dirac points. Further, we show that both types of degeneracies are lifted upon the introduction of appropriate symmetry-breaking perturbations, and discuss the topology of the now non-degenerate bands.In part two, we consider a class of Schrodinger operators, modeling either an undeformed or deformed square lattice bulk medium, perturbed by a spatially non-compact line defect commensurate with the underlying periodic bulk medium. For both cases (1) and (2), we construct edge states, which propagate parallel to the interface but are localized transverse to it. The edge states bifurcate from the band structure degeneracies associated with the unperturbed bulk medium; the bifurcation is controlled by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis. In case (1), the bifurcation is governed by a matrix Schrodinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results as well as numerical simulations, all consistent with the bulk-edge correspondence principle of topological physics.
- 일반주제명
- Mathematics
- 일반주제명
- Quantum physics
- 일반주제명
- Materials science
- 일반주제명
- Engineering
- 키워드
- Band structure
- 키워드
- Dirac points
- 키워드
- Edge states
- 기타저자
- Columbia University Applied Mathematics
- 기본자료저록
- Dissertations Abstracts International. 87-04B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
008260126s2025 us c eng d■001000017360024
■00520260202105252
■006m o d
■007cr#unu||||||||
■020 ▼a9798297646735
■035 ▼a(MiAaPQ)AAI32277228
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aChaban, Jonah.
■24510▼aBand Structure Degeneracies and Edge States in Square Lattice Media and Their Deformations
■260 ▼a[Sl]▼bColumbia University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a231 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 87-04, Section: B.
■500 ▼aAdvisor: Weinstein, Michael I.
■5021 ▼aThesis (Ph.D.)--Columbia University, 2025.
■520 ▼aWe study the spectral properties of conservative linear wave systems modeling two-dimensional periodic "bulk" media, as well as "edge" media formed by the junction of two distinct bulk media. In both scenarios, we consider the cases of (1) periodic bulk media possessing the symmetries of a square lattice, and (2) linear deformations of such media. The dissertation is divided in two parts:In part one, we begin with a spatially infinite bulk medium modeled by the Schrodinger operator H = -Δ + V, where the potential V is real-valued, periodic with respect to a square lattice, and invariant under the symmetry group of the square. The band structure of H is known to contain quadratic band degeneracy points; see also [8]. We prove that, under typical small linear deformations of V, each quadratic band degeneracy splits into a pair of nearby (tilted, elliptical) conical degeneracies, or Dirac points. Further, we show that both types of degeneracies are lifted upon the introduction of appropriate symmetry-breaking perturbations, and discuss the topology of the now non-degenerate bands.In part two, we consider a class of Schrodinger operators, modeling either an undeformed or deformed square lattice bulk medium, perturbed by a spatially non-compact line defect commensurate with the underlying periodic bulk medium. For both cases (1) and (2), we construct edge states, which propagate parallel to the interface but are localized transverse to it. The edge states bifurcate from the band structure degeneracies associated with the unperturbed bulk medium; the bifurcation is controlled by effective (homogenized) edge Hamiltonians derived via multiple-scale analysis. In case (1), the bifurcation is governed by a matrix Schrodinger operator; in case (2), it is governed by a pair of Dirac operators. We present analytical results as well as numerical simulations, all consistent with the bulk-edge correspondence principle of topological physics.
■590 ▼aSchool code: 0054.
■650 4▼aMathematics
■650 4▼aQuantum physics
■650 4▼aMaterials science
■650 4▼aEngineering
■653 ▼aBand structure
■653 ▼aDirac points
■653 ▼aEdge states
■653 ▼aFloquet-Bloch theory
■653 ▼aTopological materials
■690 ▼a0405
■690 ▼a0599
■690 ▼a0794
■690 ▼a0537
■71020▼aColumbia University▼bApplied Mathematics.
■7730 ▼tDissertations Abstracts International▼g87-04B.
■790 ▼a0054
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17360024▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
