學術產出-Theses
Article View/Open
Publication Export
-
題名 非晶形價鍵固態中的隨機單態：強無序重整化群法之研究
Random singlets in an amorphous valence-bond solid: a strong-disorder renormalization group study作者 柯志緯
Ke, Chih Wei貢獻者 林瑜琤
Lin, Yu Cheng
柯志緯
Ke, Chih Wei關鍵詞 非晶形價鍵固態
旋子
強無序重整化群法
價鍵態
amorphous valence-bond solid
spinon
strong-disorder renormalization group
valence-bond states日期 2016 上傳時間 21-Jul-2016 10:03:34 (UTC+8) 摘要 處於絕對零溫時，自旋 1/2 海森堡反鐵磁鍊加上額外的多自旋耦合（稱為 J-Q 鍊）可自發性地失去晶格對稱性而形成價鍵固態，此形成條件為強且均質的多自旋Q 耦合。價鍵固態性質與標準海森堡反鐵磁鍊（又稱 J 鏈）之基態性質截然不同，後者屬臨界態且具晶格對稱性。根據強無序重整化群法分析結果，具非均質無序耦合的海森堡反鐵磁鏈基態為一所謂的「隨機單態」，可視為一組具任意長度的價鍵（雙自旋單態）之組合，此價鍵結構造成無序自旋鏈獨特的低溫性質，包含非尋常的能量－長度關係，以及平均自旋關聯函數之冪次下降行為。藉價鍵態的概念，我們於本論文推導適用於無序 J-Q 鍊的強無序重整化群法則。針對零 J 極限的計算結果顯示：完美價鍵固態在非均質耦合環境下將破裂成無序交錯的二聚化區域,兩相鄰區域間的域壁為一帶自旋 1/2 的旋子，且跨越二 聚化區域的旋子兩兩以微弱價鍵連結。此種「非晶形價鍵固態」於長距離範疇亦雷同無序海森堡反鐵磁鏈之基態屬隨機單態，也就是說，原呈現於均質系統的價 鍵固態相變不存在於無序 J-Q 鍊。此外，我們發現平均四自旋關聯函數在隨機單態中亦呈現冪次下降的形式。
The ground state of the antiferromagnetic spin-1/2 Heisenberg chain with additional multi-spin couplings (the so-called J-Q chain) in the absence of disorder and in the limit of strong multi-spin (Q) couplings is a valence-bond solid (VBS) with spontaneous dimerization; this VBS ground state is different from the critical ground state of the standard Heisenberg chain with nearest-neighbor antiferromagnetic (J) couplings. In the presence of bond randomness, the ground state of the standard Heisenberg chain solved by a strong-disorder renormalization group (SDRG) method was suggested to be in the random-singlet phase consisting of a set of singlets (valence bonds) in a random fashion. This valence-bond (VB) structure leads to unique low-energy properties of the disordered chain, including unconventional energy-length scaling and a power-law decay of the mean spin correlation function. In this thesis we introduce an SDRG scheme using the concepts of the valence-bond basis to study the J-Q chain with random couplings. Our results show that the VBS state breaks into alternating dimerized domains with random singlets formed between spinons localized at domain walls. This amorphous valence-bond solid at long distances is also asymptotically a random-singlet state. Thus, in the random J-Q chain, we do not expect any dimerization phase transition as in the clean system. In addition, we find that the mean dimer correlation function decays algebraically in the random-singlet phase.參考文獻 [1] D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).[2] D. S. Fisher, Phys. Rev. B. 50, 3799 (1994).[3] L. Pauling, The Journal of Chemical Physics 1, 280 (1933).[4] P. W. Anderson, Mat. Res. Bull. 8 (1973).[5] P. Fazekas and P. W. Anderson, Philos. Mag. 30, 423 (1974).[6] P. W. Anderson, Science 235 (1987).[7] F. Alet, S. Capponi, N. Laflorencie, and M. Mambrini, Phys. Rev. Lett. 99, 117204 (2007).[8] Y.-C. Lin and A. W. Sandvik, Phys. Rev. B 82, 224414 (2010).[9] K. S. D. Beach and A. W. Sandvik, Nucl. Phys. B. 750, 142 (2006).[10] E. Merzbacher, Quantum Mechanics, Wiley, 1998.[11] Wikimedia commons, https://commons.wikimedia.org/wiki/File: Spin_branching_diagram.png.[12] T. Oguchi and H. Kitatani, Journal of the Physical Society of Japan 58, 1403 (1989).[13] A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010).[14] S. Liang, B. Doucot, and P. W. Anderson, Phys. Rev. Lett. 61, 365 (1988).[15] J. Lou and A. W. Sandvik, Phys. Rev. B 76, 104432 (2007).[16] E. Manousakis, Rev. Mod. Phys. 63 (1991).[17] K. S. D. Beach, Phys. Rev. B 79, 224431 (2009).[18] C. G. Shull and J. S. Smart, Phys. Rev. 76, 1256 (1949).[19] B.Bernu, P.Lecheminant, C.Lhuillier, and L. Pierre, Phys. Rev. B 50, 10048 (1994).[20] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007).[21] T.-H. Han et al., Nature 492, 406 (2012).[22] Y.-C. Lin, Y. Tang, J. Lou, and A. W. Sandvik, Phys. Rev. B 86, 144405 (2012).[23] B. Sutherland, Phys. Rev. B 37, 3786 (1988).[24] S. Liang, Phys. Rev. B 42, 6555 (1990).[25] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).[26] A. W. Sandvik and H. G. Evertz, Phys. Rev. B 82, 024407 (2010).[27] R. R. P. Singh, M. E. Fisher, and R. Shankar, Phys. Rev. B 39, 2562 (1989).[28] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, Journal of Physics A: Mathematical and General 22, 511 (1989).[29] T. Giamarchi and H. J. Schulz, Phys. Rev. B. 39, 4620 (1989).[30] Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011).[31] S. Sanyal, A. Banerjee, and K. Damle, Phys. Rev. B 84, 235129 (2011).[32] A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).[33] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004).[34] C. Dasgupta and S.-k. Ma, Phys. Rev. B 22, 1305 (1980).[35] S.-k. Ma, C. Dasgupta, and C.-k. Hu, Phys. Rev. Lett. 43, 1434 (1979).[36] D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992).[37] D. S. Fisher, Phys. Rev. B 51, 6411 (1995).[38] J. Hooyberghs, F. Iglói, and C. Vanderzande, Phys. Rev. E 69, 066140 (2004).[39] G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).[40] F. Iglói and C. Monthusc, Phys. Rep 412, 277 (2005).[41] R. N. Bhatt and P. A. Lee, Phys. Rev. Lett. 48, 344 (1982).[42] D. S. Fisher, Physica A: Statistical Mechanics and its Applications 263, 222 (1999).[43] O. Motrunich, S.-C. Mau, D. A. Huse, and D. S. Fisher, Phys. Rev. B 61, 1160 (2000).[44] Y.-C. Lin, R. Mélin, H. Rieger, and F. Iglói, Phys. Rev. B 68, 024424 (2003).[45] Y.-C. Lin, H. Rieger, N. Laflorencie, and F. Iglói, Phys. Rev. B 74, 024427 (2006).[46] R. Mélin, Y.-C. Lin, P. Lajkó, H. Rieger, and F. Iglói, Phys. Rev. B 65, 104415 (2002).[47] Y. Tang, A. W. Sandvik, and C. L. Henley, Phys. Rev. B. 84, 174427 (2011).[48] Y.-R. Shu, D.-X. Yao, C.-W. Ke, Y.-C. Lin, and A. W. Sandvik, arXiv 1603.04362v1 (2016).[49] A. Lavarélo and G. Roux, Phys. Rev. Lett. 110, 087204 (2013).[50] E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee, Phys. Rev. B. 55, 12578 (1997).[51] A. Banerjee and K. Damle, J. Stat. Mech Theor. Exp. (2010).[52] A. W. Sandvik and D. K. Campbell, Phys. Rev. Lett. 83, 195 (1999).[53] G. S. Uhrig and H. J. Schulz, Phys. Rev. B. 54, R9624 (1996).[54] H. Suwa and S. Todo, Phys. Rev. Lett. 115, 080601 (2015).[55] G. S. Uhrig, Phys. Rev. B 57, R14004 (1998).[56] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, 3651 (1993).[57] J. Zhang et al., Phys. Rev. B. 90, 014415 (2014).[58] T. Shiroka et al., Phys. Rev. Lett. 106, 137202 (2011).[59] T. Shiroka et al., Phys. Rev. B 88, 054422 (2013). 描述 碩士
國立政治大學
應用物理研究所
102755008資料來源 http://thesis.lib.nccu.edu.tw/record/#G0102755008 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) 柯志緯 zh_TW dc.contributor.author (Authors) Ke, Chih Wei en_US dc.creator (作者) 柯志緯 zh_TW dc.creator (作者) Ke, Chih Wei en_US dc.date (日期) 2016 en_US dc.date.accessioned 21-Jul-2016 10:03:34 (UTC+8) - dc.date.available 21-Jul-2016 10:03:34 (UTC+8) - dc.date.issued (上傳時間) 21-Jul-2016 10:03:34 (UTC+8) - dc.identifier (Other Identifiers) G0102755008 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/99421 - 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) 102755008 zh_TW dc.description.abstract (摘要) 處於絕對零溫時，自旋 1/2 海森堡反鐵磁鍊加上額外的多自旋耦合（稱為 J-Q 鍊）可自發性地失去晶格對稱性而形成價鍵固態，此形成條件為強且均質的多自旋Q 耦合。價鍵固態性質與標準海森堡反鐵磁鍊（又稱 J 鏈）之基態性質截然不同，後者屬臨界態且具晶格對稱性。根據強無序重整化群法分析結果，具非均質無序耦合的海森堡反鐵磁鏈基態為一所謂的「隨機單態」，可視為一組具任意長度的價鍵（雙自旋單態）之組合，此價鍵結構造成無序自旋鏈獨特的低溫性質，包含非尋常的能量－長度關係，以及平均自旋關聯函數之冪次下降行為。藉價鍵態的概念，我們於本論文推導適用於無序 J-Q 鍊的強無序重整化群法則。針對零 J 極限的計算結果顯示：完美價鍵固態在非均質耦合環境下將破裂成無序交錯的二聚化區域,兩相鄰區域間的域壁為一帶自旋 1/2 的旋子，且跨越二 聚化區域的旋子兩兩以微弱價鍵連結。此種「非晶形價鍵固態」於長距離範疇亦雷同無序海森堡反鐵磁鏈之基態屬隨機單態，也就是說，原呈現於均質系統的價 鍵固態相變不存在於無序 J-Q 鍊。此外，我們發現平均四自旋關聯函數在隨機單態中亦呈現冪次下降的形式。 zh_TW dc.description.abstract (摘要) The ground state of the antiferromagnetic spin-1/2 Heisenberg chain with additional multi-spin couplings (the so-called J-Q chain) in the absence of disorder and in the limit of strong multi-spin (Q) couplings is a valence-bond solid (VBS) with spontaneous dimerization; this VBS ground state is different from the critical ground state of the standard Heisenberg chain with nearest-neighbor antiferromagnetic (J) couplings. In the presence of bond randomness, the ground state of the standard Heisenberg chain solved by a strong-disorder renormalization group (SDRG) method was suggested to be in the random-singlet phase consisting of a set of singlets (valence bonds) in a random fashion. This valence-bond (VB) structure leads to unique low-energy properties of the disordered chain, including unconventional energy-length scaling and a power-law decay of the mean spin correlation function. In this thesis we introduce an SDRG scheme using the concepts of the valence-bond basis to study the J-Q chain with random couplings. Our results show that the VBS state breaks into alternating dimerized domains with random singlets formed between spinons localized at domain walls. This amorphous valence-bond solid at long distances is also asymptotically a random-singlet state. Thus, in the random J-Q chain, we do not expect any dimerization phase transition as in the clean system. In addition, we find that the mean dimer correlation function decays algebraically in the random-singlet phase. en_US dc.description.tableofcontents 第一章 引言 1第二章 價鍵態 3 2.1 自旋 1/2 系統之價鍵基底 3 2.2 價鍵態交疊圖與矩陣元素 11 2.2.1 交疊及交疊圖 11 2.2.2 單態投影算符 12 2.2.3 價鍵態表象中的矩陣元 15第三章 模型概述 17 3.1 一維反鐵磁海森堡模型 17 3.2 一維 Q3 模型 18第四章 強無序重整化群法 19 4.1 一維無序反鐵磁海森堡鍊的重整化群法 19 4.2 一維無序 Q3 鍊的重整化群法 22 4.2.1 耦合項之縮減 23 4.2.2 等效耦合之建立 25 4.3 無序重整化之執行 35第五章 計算結果 37 5.1 強無序重整化群法中的能量標度 37 5.2 自旋關聯函數 39 5.2.1 雙自旋關聯函數 39 5.2.2 四自旋關聯函數 39 5.3 非晶形價鍵固態 45第六章 總結與討論 47參考文獻 49 zh_TW dc.format.extent 1340895 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0102755008 en_US dc.subject (關鍵詞) 非晶形價鍵固態 zh_TW dc.subject (關鍵詞) 旋子 zh_TW dc.subject (關鍵詞) 強無序重整化群法 zh_TW dc.subject (關鍵詞) 價鍵態 zh_TW dc.subject (關鍵詞) amorphous valence-bond solid en_US dc.subject (關鍵詞) spinon en_US dc.subject (關鍵詞) strong-disorder renormalization group en_US dc.subject (關鍵詞) valence-bond states en_US dc.title (題名) 非晶形價鍵固態中的隨機單態：強無序重整化群法之研究 zh_TW dc.title (題名) Random singlets in an amorphous valence-bond solid: a strong-disorder renormalization group study en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).[2] D. S. Fisher, Phys. Rev. B. 50, 3799 (1994).[3] L. Pauling, The Journal of Chemical Physics 1, 280 (1933).[4] P. W. Anderson, Mat. Res. Bull. 8 (1973).[5] P. Fazekas and P. W. Anderson, Philos. Mag. 30, 423 (1974).[6] P. W. Anderson, Science 235 (1987).[7] F. Alet, S. Capponi, N. Laflorencie, and M. Mambrini, Phys. Rev. Lett. 99, 117204 (2007).[8] Y.-C. Lin and A. W. Sandvik, Phys. Rev. B 82, 224414 (2010).[9] K. S. D. Beach and A. W. Sandvik, Nucl. Phys. B. 750, 142 (2006).[10] E. Merzbacher, Quantum Mechanics, Wiley, 1998.[11] Wikimedia commons, https://commons.wikimedia.org/wiki/File: Spin_branching_diagram.png.[12] T. Oguchi and H. Kitatani, Journal of the Physical Society of Japan 58, 1403 (1989).[13] A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010).[14] S. Liang, B. Doucot, and P. W. Anderson, Phys. Rev. Lett. 61, 365 (1988).[15] J. Lou and A. W. Sandvik, Phys. Rev. B 76, 104432 (2007).[16] E. Manousakis, Rev. Mod. Phys. 63 (1991).[17] K. S. D. Beach, Phys. Rev. B 79, 224431 (2009).[18] C. G. Shull and J. S. Smart, Phys. Rev. 76, 1256 (1949).[19] B.Bernu, P.Lecheminant, C.Lhuillier, and L. Pierre, Phys. Rev. B 50, 10048 (1994).[20] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007).[21] T.-H. Han et al., Nature 492, 406 (2012).[22] Y.-C. Lin, Y. Tang, J. Lou, and A. W. Sandvik, Phys. Rev. B 86, 144405 (2012).[23] B. Sutherland, Phys. Rev. B 37, 3786 (1988).[24] S. Liang, Phys. Rev. B 42, 6555 (1990).[25] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).[26] A. W. Sandvik and H. G. Evertz, Phys. Rev. B 82, 024407 (2010).[27] R. R. P. Singh, M. E. Fisher, and R. Shankar, Phys. Rev. B 39, 2562 (1989).[28] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, Journal of Physics A: Mathematical and General 22, 511 (1989).[29] T. Giamarchi and H. J. Schulz, Phys. Rev. B. 39, 4620 (1989).[30] Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011).[31] S. Sanyal, A. Banerjee, and K. Damle, Phys. Rev. B 84, 235129 (2011).[32] A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007).[33] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004).[34] C. Dasgupta and S.-k. Ma, Phys. Rev. B 22, 1305 (1980).[35] S.-k. Ma, C. Dasgupta, and C.-k. Hu, Phys. Rev. Lett. 43, 1434 (1979).[36] D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992).[37] D. S. Fisher, Phys. Rev. B 51, 6411 (1995).[38] J. Hooyberghs, F. Iglói, and C. Vanderzande, Phys. Rev. E 69, 066140 (2004).[39] G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).[40] F. Iglói and C. Monthusc, Phys. Rep 412, 277 (2005).[41] R. N. Bhatt and P. A. Lee, Phys. Rev. Lett. 48, 344 (1982).[42] D. S. Fisher, Physica A: Statistical Mechanics and its Applications 263, 222 (1999).[43] O. Motrunich, S.-C. Mau, D. A. Huse, and D. S. Fisher, Phys. Rev. B 61, 1160 (2000).[44] Y.-C. Lin, R. Mélin, H. Rieger, and F. Iglói, Phys. Rev. B 68, 024424 (2003).[45] Y.-C. Lin, H. Rieger, N. Laflorencie, and F. Iglói, Phys. Rev. B 74, 024427 (2006).[46] R. Mélin, Y.-C. Lin, P. Lajkó, H. Rieger, and F. Iglói, Phys. Rev. B 65, 104415 (2002).[47] Y. Tang, A. W. Sandvik, and C. L. Henley, Phys. Rev. B. 84, 174427 (2011).[48] Y.-R. Shu, D.-X. Yao, C.-W. Ke, Y.-C. Lin, and A. W. Sandvik, arXiv 1603.04362v1 (2016).[49] A. Lavarélo and G. Roux, Phys. Rev. Lett. 110, 087204 (2013).[50] E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee, Phys. Rev. B. 55, 12578 (1997).[51] A. Banerjee and K. Damle, J. Stat. Mech Theor. Exp. (2010).[52] A. W. Sandvik and D. K. Campbell, Phys. Rev. Lett. 83, 195 (1999).[53] G. S. Uhrig and H. J. Schulz, Phys. Rev. B. 54, R9624 (1996).[54] H. Suwa and S. Todo, Phys. Rev. Lett. 115, 080601 (2015).[55] G. S. Uhrig, Phys. Rev. B 57, R14004 (1998).[56] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, 3651 (1993).[57] J. Zhang et al., Phys. Rev. B. 90, 014415 (2014).[58] T. Shiroka et al., Phys. Rev. Lett. 106, 137202 (2011).[59] T. Shiroka et al., Phys. Rev. B 88, 054422 (2013). zh_TW
