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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 高英哲 | |
| dc.contributor.author | Hsin-Chih Hsiao | en |
| dc.contributor.author | 蕭信智 | zh_TW |
| dc.date.accessioned | 2021-06-15T04:15:01Z | - |
| dc.date.available | 2010-01-21 | |
| dc.date.copyright | 2010-01-21 | |
| dc.date.issued | 2010 | |
| dc.date.submitted | 2010-01-13 | |
| dc.identifier.citation | [1] O. F. Sylju˚asen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).
[2] E. Y. Loh Jr. et. al., Phys. Rev. B 41, 9301 (1990). [3] P. Henelius and A. W. Sandvik, Phys. Rev. B 62, 1102 2000). [4] S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [5] S. R. White, Phys. Rev. B 48, 10345 (1993). [6] U. Schollw¨ock, Rev. Mod. Phys. 77, 259 (2005). [7] S. ¨Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995). [8] S. Rommer and S. ¨ Ostlund, Phys. Rev. B 55, 2164 (1997). [9] F. Verstraete, D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004). [10] F. Verstraete and J. I. Cirac, cond-mat/0407066. [11] A. W. Sandvik, Phys. Rev. Lett. 101, 140603 (2008). [12] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 040501 (2008). [13] A. W. Sandvik and G. Vidal, Phys. Rev. Lett. 99, 220602 (2007). [14] H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett.99, 220602 (2007). [15] A. Isacsson and Olav F. Sylju˚asen, Phys. Rev. E 74, 026701 (2006). [16] Z.-C. Gu, M. Levin, and X.-G. Wen, Phys. Rev. B 78, 205116 (2008). [17] L. Wang, Y.-J. Kao, and A. W. Sandvik, arXiv: 0901.0214 [18] R. P. Brent, Algorithms for Minimizations without Derivatives (Englewood Cliffs, NJ: Prentice-Hall, 1973). [19] R. P. Feynman, Statistical Mechanics: A Set of Lectures (Benjamin, Reading, MA, 1972). [20] L. Wang (private communication). [21] A. W. Sandvik, Phys. Rev. E 68, 056701 (2003). [22] A. W. Sandvik, Phys. Rev. B 56, 11678 (1997). [23] H. J. Schulz, T. A. Ziman and D. Poilblanc, J. Phys. I 6 (1996) 675-703. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/45339 | - |
| dc.description.abstract | 張量網狀態 (tensor network states)可作為研究量子自旋系統基態的試驗波函數。在這一篇論文裡，我們介紹兩種方法，為張量糾纏重整化群 (tensor entanglement renormalization group)及方磚重整法 (plaquette renormalization scheme)。張量糾纏重整化群使用張量乘積態 (tensor product states)做為試驗波函數且為了有效率的計算做了估計。方磚重整法使用方磚重整態 (plaquette renormalized states)做為試驗波函數而不需要做估計。我們使用二維的伊辛模型 (Ising model)來驗證這些方法我們發現張量糾纏重整化群在相對直接對角化 (exact diagonalization)的大尺寸下可使用。藉由使用方磚重整態，我們能夠有信心地找出量子自旋系統的基態。 | zh_TW |
| dc.description.abstract | Tensor network states can be used as a trial wave function to study ground states for quantum spin systems. In this thesis, we introduce two methods, tensor entanglement renormalization group (TERG) method and plaquette renormalization scheme. TERG uses tensor product states as a trial wave function and does an approximation for efficient calculations. Plaquette renormalization scheme uses plaquette renormalized states as a trial wave function without approximate procedures. We demonstrate these methods
with two dimensional transverse Ising model. It’s shown that TERG works in a large size compared to exact diagonalization. By using plaquette renormalized states, we can represent ground states faithfully for frustrated spin systems. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T04:15:01Z (GMT). No. of bitstreams: 1 ntu-99-R95222019-1.pdf: 641690 bytes, checksum: fe84a19ff0f48e6cf977757a8d9010be (MD5) Previous issue date: 2010 | en |
| dc.description.tableofcontents | Contents
Abstract i 1 Introduction 1 1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Variational principle . . . . . . . . . . . . . . . . . . . . . . . 2 2 Tensor network states 4 2.1 Key idea of DMRG andmatrix product states . . . . . . . . . 4 2.2 Tensor product states and plaquette renormalized states . . . 8 3 Tensor renormalization for two dimensional systems 12 3.1 TERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.1 Transverse field Ising model . . . . . . . . . . . . . . . 21 3.2.2 Fully frustrated XYmodel . . . . . . . . . . . . . . . . 23 4 Plaquette renormalized trial wave function for two dimensional systems 26 4.1 Basic formulation . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Transverse Ising model . . . . . . . . . . . . . . . . . . 30 4.2.2 Frustrated Heisenberg J1 − J2 model . . . . . . . . . . 33 5 Conclusions 39 Bibliography 41 List of Figures 2.1 A universe block is devided into a system block and an environment block. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 We add a single site labeled by sL to the smaller block representing L − 1 sites. The basis states are represented by {|sL ⊗ |βL−1}. We keep only D states to form a new basis states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A graphic representation of matrix product states on a periodic spin chain. The links represent index summations in the MPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 A graphic representation of rank four tensor. . . . . . . . . . . 8 2.5 A graphic representation of tensor product states on a square lattice. The links represent index summations in the TPS. . . 8 2.6 The indices of the tensors have a range D. After summing over the internal indices, the four-linked tensor in (a) can be viewed as a single tensor whose indices have a range D2. In (b), the indices of the four-linked tensor have a range D. . . . 9 2.7 A graphic representation of rank-three renormalization tensor. 10 2.8 The graphic representation of plaquette renormalized states on an 8×8 lattice. There are two kinds of 3-index renormalization tensors, smaller squares and larger squares. . . . . . . . . . . . 11 3.1 A graphic representation of the translational invariant tensor product state on a square lattice. The matrices at each site are the same. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 A graphic representation of the normalization factor of tensor product states with periodic boundary conditions. . . . . . . . 15 3.3 A graphic representation of the energy term of tensor product states with periodic boundary conditions. . . . . . . . . . . . . 15 3.4 (a)On a square lattice, we decompose the tensors in two different ways, purple and green. (b)Contracting the internal indices of the reduced tensors in a square produces a new tensor.( reprinted fromRef. [16]) . . . . . . . . . . . . . . . . . . . 16 3.5 There are four impurity tensors in the center. First, we decompose the lattice, and then we perform the renormalization to reach the next level.(reprinted fromRef. [16]) . . . . . . . . 18 3.6 The decomposition and renormalization procedure on an 8×8 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.7 The variational ground state energy and magnizations v.s. external field on a 16 ×16 lattice. . . . . . . . . . . . . . . . . . 22 3.8 The ferromagnetic interaction is indicated by a blue bond and other interactions are antiferromagnetic. The pair of the up spins tend to lie parallel, but the other pairs all tend to lie anti-parallel. There is a competition between them. . . . . . . 23 3.9 A graphic representation of the tensor pattern for fully frustrated XY model. The ferromagnetic interactions are indicated by blue bonds and other interactions are antiferromagnetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.10 The distributions of singular values of transverse Ising model and fully frustrated XY model. We normalize λ1 to 1. . . . . . 25 4.1 A graphic representation of plaquette renormalized states on a 4 ×4 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 A graphic representation of the normalization factor for the trial wave function. . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 A graphic representation of the expectation value of the twobody interaction for the trial wave function. The impurity tensors which have non-identity local operators are indicated by color circles. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 We can decompose a plaquette renormalization into a sequence of summations and the scaling of each step is shown in the figure.(reprinted fromRef. [17]) . . . . . . . . . . . . . . . . . 29 4.5 A graphic representation of the tensor product state which has no renormalization tensors and has translational invariance symmetry. The tensors at each site are the same. . . . . . . . 30 4.6 A tensor network for the plaquette renormalized wave function. Spins at different lattice sites inside the plaquette have different environments. . . . . . . . . . . . . . . . . . . . . . . 31 4.7 We set the tensors inside the plaquette to be different because the spins inside the plaquette have different environments. . . 32 4.8 The pairs of the nearest-neighbor spins tend to lie antiparallel and the pairs of the next-nearest-neighbor spins also tend to lie antiparallel. There is a competition between them. . . . . . 33 4.9 The relative error v.s. J2/J1 with D = 2 and D = 3 on a 4×4 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.10 A graphric representation of the f(j) in Eq. (4.12) associated with (a) Q = (π, π), (b) Q = (π, 0) , and (c) Q = (0, π). . . . 36 4.11 Energy of the variational ground state with D = 2 for a 4× 4 system, D = 2 for a 8×8 system and D = 3 for a 4×4 system v.s. J2/J1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.12 The squares of the staggered magnetization M12 with D = 2 (black circles), M22 with D = 2 (blue squares), exact M12 (maroon diamonds) and exact M22 (red triangles) v.s. J2/J1 on a 4 × 4 lattice. The exact results for comparison are from Ref.[23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.13 The squares of the staggered magnetization M12 with D = 3 (black circles), M22 with D = 3 (blue squares), exact M12 (maroon diamonds) and exact M22 (red triangles) v.s. J2/J1 on a 4 × 4 lattice. The exact results for comparison are from Ref. [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.14 The squares of the staggered magnetization M12 and M22 v.s. J2/J1 with D = 2 on an 8 × 8 lattice. M12 with QMC at J2/J1 = 0 is 0.177839 [22]. . . . . . . . . . . . . . . . . . . . . 38 List of Tables 4.1 Relative error is defined as ΔE = (E − Evar)/E. The exact energy at h = 3 for a 4 × 4 system is from Ref. [11]. The result at h = 3 for a 8 × 8 system with QMC is from Ref. [20] 31 4.2 The results and error on a 4×4 lattice with D = 2 and D = 3. 32 4.3 The results for a 8 × 8 system near the quantum critical point. 33 4.4 The energy and error for different sizes at J1 = 1 and J2 = 0. The results for comparison with QMC are from Ref. [22]. . . . 35 | |
| dc.language.iso | en | |
| dc.subject | 張量網狀態 | zh_TW |
| dc.subject | 調變原理 | zh_TW |
| dc.subject | 最佳化 | zh_TW |
| dc.subject | 重整化群 | zh_TW |
| dc.subject | 挫折性系統 | zh_TW |
| dc.subject | variational principle | en |
| dc.subject | frustrated systems | en |
| dc.subject | renormalization group | en |
| dc.subject | tensor network states | en |
| dc.subject | optimization | en |
| dc.title | 張量網狀態在二維量子自旋系統的應用 | zh_TW |
| dc.title | Tensor Network States in
Two Dimensional Quantum Spin Systems | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 98-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 管希聖,陳柏中 | |
| dc.subject.keyword | 張量網狀態,調變原理,最佳化,重整化群,挫折性系統, | zh_TW |
| dc.subject.keyword | tensor network states,variational principle,optimization,renormalization group,frustrated systems, | en |
| dc.relation.page | 42 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2010-01-13 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 物理研究所 | zh_TW |
| 顯示於系所單位： | 物理學系 | |
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