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題名 非週期性鍵結海森堡模型之重整化群研究
Renormalization group studies of Heisenberg chains with aperiodic couplings作者 王奕翔
Wang, Yi-Xiang貢獻者 林瑜琤
Lin, Yu-Cheng
王奕翔
Wang, Yi-Xiang關鍵詞 張量網路重整化群
密度矩陣重整化群
非週期性
量子自旋鏈
Tensor network renormalization group method
Density matrix renormalization group
Aperiodicity
Quantum spin chains日期 2023 上傳時間 1-Sep-2023 16:28:07 (UTC+8) 摘要 在海森堡（Heisenberg）反鐵磁鏈中，無序性將導致系統基態呈現出隨機單態，此為強無序重整化群法所推演出的關鍵結果。此方法的設計使之在無序系統低溫下能求得近似精確解。在本論文我們採用一強無序重整化群法的改良方法—樹狀張量網路強無序重整化群法，來探討帶有確定性但非週期耦合之自旋 1/2 鏈基態性質。非周期性效應對低溫性質的影響取決於平均耦合常數的局部波動，若非週期性屬攸關擾動，則該系統與完全無序系統會呈現出一定的相似性。若非週期性屬無關擾動，系統行為會表現如同均質情況，而屬邊際型的非週期性則可能導致非普適性的行為。藉由與密度矩陣重整化群法的結果作比較，我們檢驗了在各種類型的非週期性調變下樹狀張量網路重整化群法的效力，而這些非週期性調變在影響自旋鏈的基態性質中，可能屬攸關、邊際或無關型。
Randomness in Heisenberg antiferromagnetic chains leads to the random-singlet ground state, which is a key analytical result of the strong-disorder renormalization group (SDRG) method. This method is designed to be asymptotically exact at low energies in the presence of disorder. Here we use a tree tensor network renormalization group (RG) method, an adaptation of SDRG, to study the ground state properties of S = 1/2 spin chains with deterministic aperiodic couplings. The effects of aperiodicity on low-temperature properties depend on the local fluctuations of the mean coupling constant. If the aperiodicity is a relevant perturbation, the system may bear some similarities with completely random systems. With irrelevant aperiodicity, the system behaves as in the uniform case. Marginal aperiodicity may lead to non-universal behavior. By comparing with density matrix RG results, we examine the validity of the tree tensor RG method for various types of aperiodic modulations that are relevant, marginal, or irrelevant in affecting the ground state properties of the spin chains.參考文獻 [1] Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).[2] Motrunich,O.,Mau,S.-C.,Huse,D.A.&Fisher,D.S.Infinite-randomness quantum ising critical fixed points. Phys. Rev. B 61, 1160–1172 (2000).[3] Igloi,F. Critical behaviour in aperiodic systems. Journal of PhysicsA:Mathematical and General 26, L703 (1993).[4] Hermisson, J., Grimm, U. & Baake, M. Aperiodic ising quantum chains. Journal of Physics A: Mathematical and General 30, 7315 (1997).[5] Hermisson, J. Aperiodic and correlated disorder in xy chains: exact results. Journal of Physics A: Mathematical and General 33, 57 (2000).[6] Harris, A. B. Effect of random defects on the critical behaviour of ising models. Journal of Physics C: Solid State Physics 7, 1671 (1974).[7] Luck, J. M. A classification of critical phenomena on quasi-crystals and other aperiodic structures. Europhysics Letters 24, 359 (1993).[8] Ma, S.-k., Dasgupta, C. & Hu, C.-k. Random antiferromagnetic chain. Phys. Rev. Lett. 43, 1434–1437 (1979).[9] Dasgupta, C. & Ma, S.-k. Low-temperature properties of the random heisenberg antiferromagnetic chain. Phys. Rev. B 22, 1305–1319 (1980).[10] Fisher, D. S. Random antiferromagnetic quantum spin chains. Physical review b 50, 3799 (1994).[11] Hyman, R. A. & Yang, K. Impurity driven phase transition in the antiferromagnetic spin-1 chain. Phys. Rev. Lett. 78, 1783–1786 (1997).[12] Monthus, C., Golinelli, O. & Jolicœur, T. Percolation transition in the random anti- ferromagnetic spin-1 chain. Phys. Rev. Lett. 79, 3254–3257 (1997).[13] Damle, K. & Huse, D. A. Permutation-symmetric multicritical points in random antiferromagnetic spin chains. Phys. Rev. Lett. 89, 277203 (2002).[14] Vieira, A. P. Low-energy properties of aperiodic quantum spin chains. Phys. Rev. Lett. 94, 077201 (2005).[15] CasaGrande,H.L.,Laflorencie,N.,Alet,F.&Vieira,A.P.Analytical and numerical studies of disordered spin-1 heisenberg chains with aperiodic couplings. Phys. Rev. B 89, 134408 (2014).[16] Goldsborough, A. M. & Römer, R. A. Self-assembling tensor networks and holography in disordered spin chains. Phys. Rev. B 89, 214203 (2014).[17] Lin, Y.-P., Kao, Y.-J., Chen, P. & Lin, Y.-C. Griffiths singularities in the random quantum ising antiferromagnet: A tree tensor network renormalization group study. Phys. Rev. B 96, 064427 (2017).[18] Tsai, Z.-L., Chen, P. & Lin, Y.-C. Tensor network renormalization group study of spin-1 random heisenberg chains. The European Physical Journal B 93, 1–10 (2020).[19] Bethe, H. Zur theorie der metalle: I. eigenwerte und eigenfunktionen der linearen atom kette. Zeitschrift für Physik 71, 205–226 (1931).[20] Des Cloizeaux, J. & Pearson, J. Spin-wave spectrum of the antiferromagnetic linear chain. Physical Review 128, 2131 (1962).[21] Luther, A. & Peschel, I. Calculation of critical exponents in two dimensions from quantum field theory in one dimension. Phys. Rev. B 12, 3908–3917 (1975).[22] Affleck, I., Gepner, D., Schulz, H. & Ziman, T. Critical behaviour of spin-s heisen- berg antiferromagnetic chains: analytic and numerical results. Journal of Physics A: Mathematical and General 22, 511 (1989).[23] Giamarchi, T. & Schulz, H. J. Correlation functions of one-dimensional quantum systems. Phys. Rev. B 39, 4620–4629 (1989).[24] Lieb,E.,Schultz,T.&Mattis,D. Two soluble models of an antiferromagnetic chain. Annals of Physics 16, 407–466 (1961).[25] McCoy,B.M.Spin correlation functions of the x−y model.Phys.Rev.173,531–541 (1968).[26] Sequence A003849 in the OEIS. https://oeis.org/A003849.[27] Sequence A010060 in the OEIS. https://oeis.org/A010060.[28] Turban, L., Iglói, F. & Berche, B. Surface magnetization and critical behavior of aperiodic ising quantum chains. Phys. Rev. B 49, 12695–12702 (1994).[29] White, S. R. Density matrix formulation for quantum renormalization groups. Phys- ical review letters 69, 2863 (1992).[30] Hikihara, T., Furusaki, A. & Sigrist, M. Numerical renormalization-group study of spin correlations in one-dimensional random spin chains. Physical Review B 60, 12116 (1999).[31] Goldsborough,A.M.Tensor networks and geometry for the modelling of disordered quantum many-body systems. Ph.D. thesis, University of Warwick (2015).[32] Orús, R. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of physics 349, 117–158 (2014).[33] tensornetwork.org. https://tensornetwork.org.[34] Schollwöck,U.Thedensity-matrix renormalization group in the age of matrix product states. Annals of physics 326, 96–192 (2011).[35] Hubig, C., McCulloch, I. P. & Schollwöck, U. Generic construction of efficientmatrix product operators. Phys. Rev. B 95, 035129 (2017).[36] Oseledets, I. V. Tensor-train decomposition. SIAM Journal on Scientific Computing33, 2295–2317 (2011).[37] Perez-Garcia, D., Verstraete, F., Wolf, M. & Cirac, J. Matrix product state represen-tations. QUANTUM INFORMATION & COMPUTATION 7, 401–430 (2007).[38] Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys. 77,259–315 (2005).[39] Cytnx. https://github.com/Cytnx-dev/Cytnx.[40] Disordered MERA. https://github.com/AMGoldsborough/dMERA.[41] Goldsborough, A. M. & Evenbly, G. Entanglement renormalization for disordered systems. Physical Review B 96, 155136 (2017).[42] Tree tensor network strong disorder renormalisation group. https://github.com/ AMGoldsborough/tSDRG. 描述 碩士
國立政治大學
應用物理研究所
110755002資料來源 http://thesis.lib.nccu.edu.tw/record/#G0110755002 資料類型 thesis dc.contributor.advisor 林瑜琤 zh_TW dc.contributor.advisor Lin, Yu-Cheng en_US dc.contributor.author (Authors) 王奕翔 zh_TW dc.contributor.author (Authors) Wang, Yi-Xiang en_US dc.creator (作者) 王奕翔 zh_TW dc.creator (作者) Wang, Yi-Xiang en_US dc.date (日期) 2023 en_US dc.date.accessioned 1-Sep-2023 16:28:07 (UTC+8) - dc.date.available 1-Sep-2023 16:28:07 (UTC+8) - dc.date.issued (上傳時間) 1-Sep-2023 16:28:07 (UTC+8) - dc.identifier (Other Identifiers) G0110755002 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/147296 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用物理研究所 zh_TW dc.description (描述) 110755002 zh_TW dc.description.abstract (摘要) 在海森堡（Heisenberg）反鐵磁鏈中，無序性將導致系統基態呈現出隨機單態，此為強無序重整化群法所推演出的關鍵結果。此方法的設計使之在無序系統低溫下能求得近似精確解。在本論文我們採用一強無序重整化群法的改良方法—樹狀張量網路強無序重整化群法，來探討帶有確定性但非週期耦合之自旋 1/2 鏈基態性質。非周期性效應對低溫性質的影響取決於平均耦合常數的局部波動，若非週期性屬攸關擾動，則該系統與完全無序系統會呈現出一定的相似性。若非週期性屬無關擾動，系統行為會表現如同均質情況，而屬邊際型的非週期性則可能導致非普適性的行為。藉由與密度矩陣重整化群法的結果作比較，我們檢驗了在各種類型的非週期性調變下樹狀張量網路重整化群法的效力，而這些非週期性調變在影響自旋鏈的基態性質中，可能屬攸關、邊際或無關型。 zh_TW dc.description.abstract (摘要) Randomness in Heisenberg antiferromagnetic chains leads to the random-singlet ground state, which is a key analytical result of the strong-disorder renormalization group (SDRG) method. This method is designed to be asymptotically exact at low energies in the presence of disorder. Here we use a tree tensor network renormalization group (RG) method, an adaptation of SDRG, to study the ground state properties of S = 1/2 spin chains with deterministic aperiodic couplings. The effects of aperiodicity on low-temperature properties depend on the local fluctuations of the mean coupling constant. If the aperiodicity is a relevant perturbation, the system may bear some similarities with completely random systems. With irrelevant aperiodicity, the system behaves as in the uniform case. Marginal aperiodicity may lead to non-universal behavior. By comparing with density matrix RG results, we examine the validity of the tree tensor RG method for various types of aperiodic modulations that are relevant, marginal, or irrelevant in affecting the ground state properties of the spin chains. en_US dc.description.tableofcontents 謝辭 iAbstract iii摘要 vContents vii1 Introduction 11.1 Ground-state properties of the Heisenberg S=1/2 chain 21.2 Effects of disorder and inhomogeneities 32 Aperiodic sequences 72.1 Substitution rules 72.2 Relevance-irrelevance criteria 93 Methods 133.1 Matrix product operator 133.2 Matrix product state 143.3 Density matrix renormalization group 173.4 Tensor-network strong-disorder renormalization group 194 Results 234.1 Two-point correlations 234.2 Bulk correlations 264.3 Configuration-average correlations 275 Conclusion and outlook 35Reference 39 zh_TW dc.format.extent 1481411 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0110755002 en_US dc.subject (關鍵詞) 張量網路重整化群 zh_TW dc.subject (關鍵詞) 密度矩陣重整化群 zh_TW dc.subject (關鍵詞) 非週期性 zh_TW dc.subject (關鍵詞) 量子自旋鏈 zh_TW dc.subject (關鍵詞) Tensor network renormalization group method en_US dc.subject (關鍵詞) Density matrix renormalization group en_US dc.subject (關鍵詞) Aperiodicity en_US dc.subject (關鍵詞) Quantum spin chains en_US dc.title (題名) 非週期性鍵結海森堡模型之重整化群研究 zh_TW dc.title (題名) Renormalization group studies of Heisenberg chains with aperiodic couplings en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).[2] Motrunich,O.,Mau,S.-C.,Huse,D.A.&Fisher,D.S.Infinite-randomness quantum ising critical fixed points. Phys. Rev. B 61, 1160–1172 (2000).[3] Igloi,F. Critical behaviour in aperiodic systems. Journal of PhysicsA:Mathematical and General 26, L703 (1993).[4] Hermisson, J., Grimm, U. & Baake, M. Aperiodic ising quantum chains. Journal of Physics A: Mathematical and General 30, 7315 (1997).[5] Hermisson, J. Aperiodic and correlated disorder in xy chains: exact results. Journal of Physics A: Mathematical and General 33, 57 (2000).[6] Harris, A. B. Effect of random defects on the critical behaviour of ising models. Journal of Physics C: Solid State Physics 7, 1671 (1974).[7] Luck, J. M. A classification of critical phenomena on quasi-crystals and other aperiodic structures. Europhysics Letters 24, 359 (1993).[8] Ma, S.-k., Dasgupta, C. & Hu, C.-k. Random antiferromagnetic chain. Phys. Rev. Lett. 43, 1434–1437 (1979).[9] Dasgupta, C. & Ma, S.-k. Low-temperature properties of the random heisenberg antiferromagnetic chain. Phys. Rev. B 22, 1305–1319 (1980).[10] Fisher, D. S. Random antiferromagnetic quantum spin chains. Physical review b 50, 3799 (1994).[11] Hyman, R. A. & Yang, K. Impurity driven phase transition in the antiferromagnetic spin-1 chain. Phys. Rev. Lett. 78, 1783–1786 (1997).[12] Monthus, C., Golinelli, O. & Jolicœur, T. Percolation transition in the random anti- ferromagnetic spin-1 chain. Phys. Rev. Lett. 79, 3254–3257 (1997).[13] Damle, K. & Huse, D. A. Permutation-symmetric multicritical points in random antiferromagnetic spin chains. Phys. Rev. Lett. 89, 277203 (2002).[14] Vieira, A. P. Low-energy properties of aperiodic quantum spin chains. Phys. Rev. Lett. 94, 077201 (2005).[15] CasaGrande,H.L.,Laflorencie,N.,Alet,F.&Vieira,A.P.Analytical and numerical studies of disordered spin-1 heisenberg chains with aperiodic couplings. Phys. Rev. B 89, 134408 (2014).[16] Goldsborough, A. M. & Römer, R. A. Self-assembling tensor networks and holography in disordered spin chains. Phys. Rev. B 89, 214203 (2014).[17] Lin, Y.-P., Kao, Y.-J., Chen, P. & Lin, Y.-C. Griffiths singularities in the random quantum ising antiferromagnet: A tree tensor network renormalization group study. Phys. Rev. B 96, 064427 (2017).[18] Tsai, Z.-L., Chen, P. & Lin, Y.-C. Tensor network renormalization group study of spin-1 random heisenberg chains. The European Physical Journal B 93, 1–10 (2020).[19] Bethe, H. Zur theorie der metalle: I. eigenwerte und eigenfunktionen der linearen atom kette. Zeitschrift für Physik 71, 205–226 (1931).[20] Des Cloizeaux, J. & Pearson, J. Spin-wave spectrum of the antiferromagnetic linear chain. Physical Review 128, 2131 (1962).[21] Luther, A. & Peschel, I. Calculation of critical exponents in two dimensions from quantum field theory in one dimension. Phys. Rev. B 12, 3908–3917 (1975).[22] Affleck, I., Gepner, D., Schulz, H. & Ziman, T. Critical behaviour of spin-s heisen- berg antiferromagnetic chains: analytic and numerical results. Journal of Physics A: Mathematical and General 22, 511 (1989).[23] Giamarchi, T. & Schulz, H. J. Correlation functions of one-dimensional quantum systems. Phys. Rev. B 39, 4620–4629 (1989).[24] Lieb,E.,Schultz,T.&Mattis,D. Two soluble models of an antiferromagnetic chain. Annals of Physics 16, 407–466 (1961).[25] McCoy,B.M.Spin correlation functions of the x−y model.Phys.Rev.173,531–541 (1968).[26] Sequence A003849 in the OEIS. https://oeis.org/A003849.[27] Sequence A010060 in the OEIS. https://oeis.org/A010060.[28] Turban, L., Iglói, F. & Berche, B. Surface magnetization and critical behavior of aperiodic ising quantum chains. Phys. Rev. B 49, 12695–12702 (1994).[29] White, S. R. Density matrix formulation for quantum renormalization groups. Phys- ical review letters 69, 2863 (1992).[30] Hikihara, T., Furusaki, A. & Sigrist, M. Numerical renormalization-group study of spin correlations in one-dimensional random spin chains. Physical Review B 60, 12116 (1999).[31] Goldsborough,A.M.Tensor networks and geometry for the modelling of disordered quantum many-body systems. Ph.D. thesis, University of Warwick (2015).[32] Orús, R. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of physics 349, 117–158 (2014).[33] tensornetwork.org. https://tensornetwork.org.[34] Schollwöck,U.Thedensity-matrix renormalization group in the age of matrix product states. Annals of physics 326, 96–192 (2011).[35] Hubig, C., McCulloch, I. P. & Schollwöck, U. Generic construction of efficientmatrix product operators. Phys. Rev. B 95, 035129 (2017).[36] Oseledets, I. V. Tensor-train decomposition. SIAM Journal on Scientific Computing33, 2295–2317 (2011).[37] Perez-Garcia, D., Verstraete, F., Wolf, M. & Cirac, J. Matrix product state represen-tations. QUANTUM INFORMATION & COMPUTATION 7, 401–430 (2007).[38] Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys. 77,259–315 (2005).[39] Cytnx. https://github.com/Cytnx-dev/Cytnx.[40] Disordered MERA. https://github.com/AMGoldsborough/dMERA.[41] Goldsborough, A. M. & Evenbly, G. Entanglement renormalization for disordered systems. Physical Review B 96, 155136 (2017).[42] Tree tensor network strong disorder renormalisation group. https://github.com/ AMGoldsborough/tSDRG. zh_TW
