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| ???org.dspace.app.webui.jsptag.ItemTag.dcfield??? | Value | Language |
|---|---|---|
| dc.contributor.advisor | 高英哲 | zh_TW |
| dc.contributor.advisor | Ying-Jer Kao | en |
| dc.contributor.author | 林昶滕 | zh_TW |
| dc.contributor.author | Chang-Teng Lin | en |
| dc.date.accessioned | 2024-08-14T16:09:07Z | - |
| dc.date.available | 2024-08-15 | - |
| dc.date.copyright | 2024-08-13 | - |
| dc.date.issued | 2024 | - |
| dc.date.submitted | 2024-08-09 | - |
| dc.identifier.citation | [1] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2–111, 2006.
[2] Shi Feng, Adhip Agarwala, Subhro Bhattacharjee, and Nandini Trivedi. Anyon dynamics in field-driven phases of the anisotropic kitaev model. Physical Review B, 108(3):035149, July 2023. [3] Zheng Zhu, Itamar Kimchi, D. N. Sheng, and Liang Fu. Robust non-abelian spin liquid and a possible intermediate phase in the antiferromagnetic kitaev model with magnetic field. Physical Review B, 97(24):241110, June 2018. [4] Matthias Gohlke, Roderich Moessner, and Frank Pollmann. Dynamical and topological properties of the kitaev model in a [111] magnetic field. Physical Review B, 98(1):014418, July 2018. arXiv:1804.06811 [cond-mat]. [5] Wei-Lin Tu, Laurens Vanderstraeten, Norbert Schuch, Hyun-Yong Lee, Naoki Kawashima, and Ji-Yao Chen. Generating function for projected entangled-pair states. PRX Quantum, 5(1):010335, March 2024. [6] Boris Ponsioen, Juraj Hasik, and Philippe Corboz. Improved summations of n-point correlation functions of projected entangled-pair states. Physical Review B, 108(19):195111, November 2023. [7] Roman Orus. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349:117–158, October 2014. arXiv: 1306.2164. [8] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326(1):96–192, 2011. [9] Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio Cirac. Entropy scaling and simulability by matrix product states. Physical Review Letters, 100(3):030504, January 2008. arXiv:0705.0292 [cond-mat, physics:quant-ph]. [10] M B Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08024–P08024, August 2007. [11] Ramis Movassagh and Peter W. Shor. Supercritical entanglement in local systems: Counterexample to the area law for quantum matter. Proceedings of the National Academy of Sciences, 113(47):13278–13282, November 2016. [12] M. B. Hastings. Lieb-schultz-mattis in higher dimensions. Physical Review B, 69(10):104431, March 2004. [13] M. B. Hastings. Solving gapped hamiltonians locally. Physical Review B, 73(8):085115, February 2006. [14] Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio Cirac. Computational complexity of projected entangled pair states. Physical Review Letters, 98(14):140506, April 2007. [15] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. (arXiv:quant-ph/0608197), May 2007. arXiv:quant-ph/0608197. [16] David Perez-Garcia, Frank Verstraete, J. Ignacio Cirac, and Michael M. Wolf. Peps as unique ground states of local hamiltonians. (arXiv:0707.2260), July 2007. arXiv:0707.2260 [cond-mat, physics:math-ph, physics:quant-ph]. [17] Norbert Schuch, Ignacio Cirac, and David Perez-Garcia. Peps as ground states: degeneracy and topology. Annals of Physics, 325(10):2153–2192, October 2010. arXiv:1001.3807 [cond-mat, physics:quant-ph]. [18] Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck, and Claudius Hubig. Time-evolution methods for matrix-product states. Annals of Physics, 411:167998, December 2019. [19] Michael P. Zaletel, Roger S. K. Mong, Christoph Karrasch, Joel E. Moore, and Frank Pollmann. Time-evolving a matrix product state with long-ranged interactions. Physical Review B, 91(16):165112, April 2015. [20] Guifré Vidal. Efficient simulation of one-dimensional quantum many-body systems. Physical Review Letters, 93(4):040502, July 2004. [21] J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac. Classical simulation of infinite-size quantum lattice systems in two spatial dimensions. Physical Review Letters, 101(25):250602, December 2008. [22] Hai-Jun Liao, Jin-Guo Liu, Lei Wang, and Tao Xiang. Differentiable programming tensor networks. Physical Review X, 9(3):031041, September 2019. [23] Boris Ponsioen, Fakher Assaad, and Philippe Corboz. Automatic differentiation applied to excitations with projected entangled pair states. SciPost Physics, 12(1):006, January 2022. [24] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. [25] Yu-Hsueh Chen, Ke Hsu, Wei-Lin Tu, Hyun-Yong Lee, and Ying-Jer Kao. Variational tensor network operator. (arXiv:2207.01819), July 2022. arXiv:2207.01819 [cond-mat, physics:hep-lat]. [26] S. R. White and R. M. Noack. Real-space quantum renormalization groups. Physical Review Letters, 68(24):3487–3490, June 1992. [27] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69:2863–2866, Nov 1992. [28] U. Schollwöck. The density-matrix renormalization group. Reviews of Modern Physics, 77(1):259–315, April 2005. [29] R. J. Baxter. Dimers on a rectangular lattice. Journal of Mathematical Physics, 9(4):650–654, April 1968. [30] R. J. Baxter. Variational approximations for square lattice models in statistical mechanics. Journal of Statistical Physics, 19(5):461–478, November 1978. [31] Rodney J. Baxter. Exactly solved models in statistical mechanics. Academic Press, London; New York, 1982. [32] Tomotoshi Nishino and Kouichi Okunishi. Corner transfer matrix renormalization group method. Journal of the Physical Society of Japan, 65(4):891–894, April 1996. [33] Michael Levin and Cody P. Nave. Tensor renormalization group approach to two-dimensional classical lattice models. Physical Review Letters, 99(12):120601, September 2007. [34] G. Evenbly and G. Vidal. Tensor network renormalization. Physical Review Letters, 115(18):180405, October 2015. [35] J. Hasik, G. B. Mbeng, W.-L. Tu, and S.-S. Diop. A tensor network library for twodimensional lattice models. https://github.com/jurajHasik/peps-torch, 2020. [36] Philippe Corboz, Steven R. White, Guifré Vidal, and Matthias Troyer. Stripes in the two-dimensional t-j model with infinite projected entangled-pair states. Physical Review B, 84(4):041108, July 2011. [37] Philippe Corboz, T. M. Rice, and Matthias Troyer. Competing states in the t-j model: uniform d-wave state versus stripe state. Physical Review Letters, 113(4):046402, July 2014. arXiv:1402.2859 [cond-mat]. [38] M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete. Faster methods for contracting infinite 2d tensor networks. Physical Review B, 98(23):235148, December 2018. arXiv:1711.05881 [cond-mat, physics:quantph]. [39] I. V. Lukin and A. G. Sotnikov. Corner transfer matrix renormalization group approach in the zoo of archimedean lattices. Physical Review E, 109(4):045305, April 2024. arXiv:2401.07274 [cond-mat]. [40] Atılım Gunes Baydin, Barak A Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey. [41] Soeren Laue. On the equivalence of automatic and symbolic differentiation. (arXiv:1904.02990), December 2022. arXiv:1904.02990 [cs]. [42] Andreas Griewank and Andrea Walther. Evaluating Derivatives. Society for Industrial and Applied Mathematics, second edition, 2008. [43] [Overview of PyTorch Autograd Engine] https:// pytorch.org/ blog/ overview-ofpytorch-autograd-engine/. [44] [Autograd Mechanics - PyTorch 2.3 Documentation] https://pytorch.org/docs/stable/notes/autograd.html. [45] James Townsend. J. townsend, differentiating the singular value decomposition, technical report 2016, . [46] Jutho Haegeman, Spyridon Michalakis, Bruno Nachtergaele, Tobias J. Osborne, Norbert Schuch, and Frank Verstraete. Elementary excitations in gapped quantum spin systems. Physical Review Letters, 111(8):080401, August 2013. arXiv:1305.2176 [cond-mat, physics:math-ph, physics:quant-ph]. [47] Wei-Lin Tu, Huan-Kuang Wu, Norbert Schuch, Naoki Kawashima, and Ji-Yao Chen. Generating function for tensor network diagrammatic summation. Physical Review B, 103(20):205155, May 2021. [48] Benyamin Ghojogh, Fakhri Karray, and Mark Crowley. Eigenvalue and generalized eigenvalue problems: Tutorial. (arXiv:1903.11240), May 2023. arXiv:1903.11240 [cs, stat]. [49] Michiel E. Hochstenbach, Christian Mehl, and Bor Plestenjak. Solving singular generalized eigenvalue problems by a rank-completing perturbation. SIAM Journal on Matrix Analysis and Applications, 40(3):1022–1046, 2019. [50] Run-Ze Chi, Yang Liu, Yuan Wan, Hai-Jun Liao, and T. Xiang. Spin excitation spectra of anisotropic spin-1/2 triangular lattice heisenberg antiferromagnets. Physical Review Letters, 129(22):227201, November 2022. arXiv:2201.12121 [cond-mat]. [51] Hyun-Yong Lee, Ryui Kaneko, Tsuyoshi Okubo, and Naoki Kawashima. Gapless kitaev spin liquid to classical string gas through tensor networks. Physical Review Letters, 123(8):087203, August 2019. [52] Lucile Savary and Leon Balents. Quantum spin liquids: a review. Reports on Progress in Physics, 80(1):016502, January 2016. [53] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner. Dynamics of a twodimensional quantum spin liquid: Signatures of emergent majorana fermions and fluxes. Physical Review Letters, 112(20):207203, May 2014. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/93994 | - |
| dc.description.abstract | 在強相關系統的領域中，計算二維系統基態的2點函數是一項具有挑戰性的任務。這尤其適用於蜂巢結構的Kitaev模型，其中較大的鍵結維度和漢密爾頓量的複雜性，使得該模型不能僅使用實數進行計算。我們選擇在投影糾纏對態(PEPS)上的生成函數來計算有限晶格2點函數收縮，而無需手動進行容易出錯的張量網絡收縮，並在進行收縮時減小中間張量的大小。在這項工作中，我們將原始生成函數方法從實張量擴展到複張量微分，並計算蜂巢結構Kitaev模型的激發譜，靜態結構因子，和譜函數。對於激發譜，確定截斷Norm矩陣的正確維度一直是一個問題。我們提出利用譜函數和靜態結構因子之間的關係，找到符合總和規則的最佳截斷維度，並成功確定了蜂巢結構Kitaev模型等向點的截斷維度。 | zh_TW |
| dc.description.abstract | In the realm of strongly correlated systems, the calculation of the 2-point function of the underlying state in two-dimensional systems is a challenging task. This is especially true for the Honeycomb Kitaev Model, where the larger bond dimension and the fact that the model cannot be calculated using only real numbers due to the complexity of the Hamiltonian. Without approximating the two-dimensional system on the one-dimensional cylinder, we choose the generating function on Projected Entangled-Pair State(PEPS) to calculate the finite lattice 2-point function contraction without the need to conduct error-prone tensor network contraction by hand and reduce the size of intermediate tensor when doing the contraction. In this work, we extend the original generating function method from real tensor to complex tensor differentiation and calculate the excitation spectrum, static structure factor, and spectral function of the Honeycomb Kitaev model. For the excitation spectrum, it has been a problem to determine the correct dimension to truncate the Norm matrix. We purposed to exploit the relation between spectral function and static structure factor, finding the best truncation dimension that meets the sum rule, and we succeeded in determining the truncation dimension in the isotropic point of the Honeycomb Kitaev model. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2024-08-14T16:09:06Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2024-08-14T16:09:07Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | 口試委員審定書 i
致謝 iii 摘要 v Abstract vii Contents ix Chapter 1 Introduction 1 Chapter 2 Tensor Network State 3 2.1 Matrix Product State . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Projected Entangled Pair State . . . . . . . . . . . . . . . . . . . . . 10 Chapter 3 Ground State Calculation 15 3.1 Imaginary Time Evolution . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Density Matrix Renormalization Group . . . . . . . . . . . . . . . . 18 3.4 Corner Transfer Matrix Renormalization Group . . . . . . . . . . . . 19 Chapter 4 Automatic-Differentiation 25 4.1 Forward Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Backward Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Complex Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 28 4.4 Singular Value Decomposition Differentiation . . . . . . . . . . . . . 30 Chapter 5 Generating Function 31 5.1 Single Mode Approximation . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Two-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . 34 5.4 Finite Size Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 Static Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.6 Dynamical Spectral Function . . . . . . . . . . . . . . . . . . . . . . 39 5.7 Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.8 Projector Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 6 Honeycomb Kitaev Model 43 6.1 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Chapter 7 Results 48 7.1 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.1.1 Transverse Field Ising Model . . . . . . . . . . . . . . . . . . . . . 48 7.1.2 Fixed CTM Projector . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.1.3 Projector Differentiation . . . . . . . . . . . . . . . . . . . . . . . 50 7.2 Honeycomb Kitaev Model . . . . . . . . . . . . . . . . . . . . . . . 50 7.2.1 Fixed CTM Projector . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2.2 Projector Differentiation . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter 8 Summary and Outlook 73 References 74 | - |
| dc.language.iso | en | - |
| dc.subject | 廣義特徵值問題 | zh_TW |
| dc.subject | 靜態結構因子 | zh_TW |
| dc.subject | 激發譜 | zh_TW |
| dc.subject | 自動微分 | zh_TW |
| dc.subject | 維度截斷 | zh_TW |
| dc.subject | 蜂巢結構Kitaev模型 | zh_TW |
| dc.subject | 範數矩陣 | zh_TW |
| dc.subject | 轉角轉移矩陣 | zh_TW |
| dc.subject | 投影糾纏對態 | zh_TW |
| dc.subject | 譜函數 | zh_TW |
| dc.subject | 有效漢密爾頓矩陣 | zh_TW |
| dc.subject | 生成函數 | zh_TW |
| dc.subject | Dimension Truncation | en |
| dc.subject | Honeycomb Kitaev Model | en |
| dc.subject | Generating Function | en |
| dc.subject | Spectral Function | en |
| dc.subject | Projected Entangled-Pair States | en |
| dc.subject | Corner Transfer Matrix | en |
| dc.subject | Norm Matrix | en |
| dc.subject | Effective Hamiltonian Matrix | en |
| dc.subject | Generalized Eigenvalue Problem | en |
| dc.subject | Static Structure Factor | en |
| dc.subject | Excitation Spectrum | en |
| dc.subject | Automatic Differentiation | en |
| dc.title | 生成函數的Kitaev模型譜函數研究 | zh_TW |
| dc.title | Generating Function Study of Kitaev Model Spectral Function | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 112-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 陳柏中;鐘佳民 | zh_TW |
| dc.contributor.oralexamcommittee | Po-Chung Chen;Chia-Min Chung | en |
| dc.subject.keyword | 蜂巢結構Kitaev模型,生成函數,譜函數,投影糾纏對態,轉角轉移矩陣,範數矩陣,有效漢密爾頓矩陣,廣義特徵值問題,靜態結構因子,激發譜,自動微分,維度截斷, | zh_TW |
| dc.subject.keyword | Honeycomb Kitaev Model,Generating Function,Spectral Function,Projected Entangled-Pair States,Corner Transfer Matrix,Norm Matrix,Effective Hamiltonian Matrix,Generalized Eigenvalue Problem,Static Structure Factor,Excitation Spectrum,Automatic Differentiation,Dimension Truncation, | en |
| dc.relation.page | 79 | - |
| dc.identifier.doi | 10.6342/NTU202403459 | - |
| dc.rights.note | 同意授權(全球公開) | - |
| dc.date.accepted | 2024-08-12 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 物理學系 | - |
| Appears in Collections: | 物理學系 | |
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| ntu-112-2.pdf | 7.21 MB | Adobe PDF | View/Open |
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