Quantum Phase Transitions and Dynamics in Perturbed Flatbands = 섭동된 플랫밴드에서의 양자상전이 및 동역학

국문 초록 (Abstract)

최근 거시적 겹침에너지를 갖는 플랫밴드 시스템이 많은 관심을 받고 있다. 이러한 계는 다양한 섭동항과 다체상호작용에 아주 예민하여 다양한 물리적 현상을 관찰할 수 있는 수학적뼈대를 제공한다. 또한 이들과 관련된 실험적 실현 dl 해당분야의 경계를 확장시키는데에 기여하고 있다. 이 학위논문은 박사과정 동안 얻은 결과를 모두 정리하고 설명한다. 첫번째로, 두 개의 평평한 에너지띠를 갖는 일차원 플랫밴드 격자에 준주기적인 섭동이 미치는 영향을 심층적으로 탐구하였다. 주요 성과로, 이전의 섭동된 플랫밴드 시스템에서 찾지 못한 임계-절연체전이(critical-to-insulator transition) 및 에너지 의존 임계-절연체전이인 프랙탈 경계면(fractality edges)을 발견한 것이다. 다음으로, 플랫밴드가 존재하는 일차원 및 이차원 십자뜨기-격자사슬에서 딱딱핵 보손간 상호작용이 어떤 효과를 가져오는지 면밀히 조사하였다. 중요한 발견 중 하나는바닥상태 에너지가 옹골국소고유상태(compact localized state)로부터 결정된다는 것이다. 주요 성과로는, 십자뜨기-격자사슬에 옹골국소고유상태로 이루어진 장벽이 존재하고 그 안에 딱딱핵 보손이 위치해 있을 때, 그 입자는 장벽을 빠져나오지 못하는 비에르고딕(non-ergodic) 성질을 발견하였고 최근활발히 연구되는 힐베르트 공간분할 (Hilbert space fragmentation)을 야기한다는 것을 발견하였다. 더 나아가,전기회로로 플랫밴드가 존재하는 일차원 다이아몬드-격자사슬을 구성한 후, 사슬 일부분에 국소적으로 드라이빙을 입력하여 옹골국소고유상태를 성공적으로 생성하였다. 이는 전기회로를 이용하여 기존 방법보다 쉽게 플랫밴드를 만들고 조정할 수 있는 발판을 이루었다. 또한, 양자정보 분야에서 잠재적인 활용·응용가능성이 있을 것으로생각된다. 마지막으로, 이러한 노력이 해당분야의 경계를 넓혀 과학계에 의미있는 기여를 했기 바란다.

다국어 초록 (Multilingual Abstract)

In recent years, there has been a growing interest in flatband systems which exhibit macroscopic degeneracies. These systems offer a valuable mathematical framework for the extreme sensitivity to perturbations and interactions. This sensitivity unveils a wide variety of exotic and unconventional physical phenomena. Moreover, the progress in their experimental realization contributes to the expanding landscape of exploration in this field. This thesis aims to summarize all the works throughout the Ph.D. program. Firstly, an in-depth exploration was conducted on the impact of weak quasiperiodic perturbations on one- dimensional two-band all-bands-flat lattices. These tight-binding Hamiltonian are diagonalized through a sequence of local unitary transformations. By adjusting the quasiperiodic potential parameters, the key achievement involves finding a critical-to-insulator transition and identifying fractality edges in the flatband systems with quasiperiodic perturbations. Next, the investigation delved into the effects of on-site interactions among hard-core bosons in one- and two- dimensional cross-stitch lattices. One key finding is that groundstate energy primarily depends on compact localized states. Moreover, the presence of barriers of compact localized states trap bosons, leading to the emergence of non-ergodic excitation and Hilbert space fragmentation. Lastly, a compact localized eigenstate of the one-dimensional diamond chain using an electric circuit was successfully generated via local (linear and non-linear) driving. This achievement opens up a versatile circuit platform for the generation of flatbands and holds promise for potential applications in the field of quantum information. I hope these collective efforts have expanded the frontiers of the field and made a meaningful contribution to the scientific community.

목차 (Table of Contents)

• 1 Introduction
• 1.1 Flatband systems and compact localization
• 1.1.1 Flatbands
• 1.1.2 Compact localized states
• 1.1.3 Construction of flatband systems
• 1.2 Experimental realizations
• 1.3 Effect of perturbation and interaction
• 1.4 Contributed work and outline of the thesis
• 2 Generating Critical States from Quasiperturbed Flatbands
• 2.1 Introduction
• 2.2 Model
• 2.2.1 Local unitary transformation and ABF lattice
• 2.2.2 Geometric symmetry of ABF lattice
• 2.3 Numerical methods
• 2.3.1 Inverse participation ratio
• 2.3.2 Defining a linear model for localized states
• 2.4 Weak perturbation
• 2.4.1 Extended Harper model
• 2.4.2 Off-diagonal Harper model
• 2.4.3 Lyapunov exponent and localization length
• 2.4.4 Case one - β = 0 and λ1 ≠ λ2 at θ1,2 = π/4
• 2.4.5 Case two - λ2/λ1 = 0 for any phase difference β
• 2.4.6 Case three - λ2 = λ1 and β = π
• 2.4.7 Remark - Near zero angles θ1,2 → 0
• 2.5 Wavepacket spreading in projected model
• 2.5.1 Lower bound of diffusion exponent
• 2.5.2 Upper bound of diffusion exponent
• 2.5.3 Numerical results
• 2.6 Finite perturbation
• 2.6.1 Representation in semi-detangled basis
• 2.6.2 Critical-to-insulator transition in finite perturbation
• 2.6.3 Non-degenerate compact localized states
• 2.6.4 Non-simple fractality edges
• 2.6.5 Perturbation independent fractality edges for λ2 = 0
• 2.7 Conclusion
• 3 Trapped hard-core bosons in flatband systems
• 3.1 Introduction
• 3.2 Models
• 3.2.1 One-dimensional and two-dimensional cross-stitch lattices
• 3.2.2 Properties of hard-core bosons
• 3.3 CLS groundstates
• 3.3.1 ν ≤ 1/2 - filling of the flatband CLS
• 3.3.2 ν ≤ 1 - filling of the an, bn dimers
• 3.4 Non-ergodic excitation in cross-stitch lattices
• 3.4.1 One hard-core boson between two CLSs
• 3.4.2 Mapping to spin-one chain
• 3.4.3 Localized excitations in two-dimensional lattices
• 3.5 One-dimensional diamond lattice and non-ergodic excitation
• 3.6 Fluxed one-dimensional diamond lattice and ergodicity
• 3.7 Conclusion
• 4 Flatband Electric Circuits
• 4.1 Introduction
• 4.2 Models
• 4.2.1 One-dimensional diamond electric lattice
• 4.2.2 One-dimensional stub electric lattice
• 4.3 Experimental and numerical results
• 4.3.1 Results on the one-dimensional diamond lattice
• 4.3.2 Nonlinear driving and robustness of diamond CLS
• 4.3.3 Results on the one-dimensional stub lattice
• 4.4 Conclusion
• 5 Final remarks
• 5.1 Scientific contribution and summary
• 5.2 Possible future research
• Bibliography