自旋系統的量子糾纏量子相變與拓樸序

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(1)Department of Physics National Taiwan Normal University PhD Thesis. Numerical study on quantum entanement, quantum phase transition, and topological order in spin system. Ching-Yu Huang. Advisor: Feng-Li Lin, Ph.D. August.4.2011.

(2) c Copyright by Ching-Yu Huang 2011. ii.

(3) to my PARENTS and my TEACHERS with love. iii.

(4) Acknowledgements I would like to express my deep-felt gratitude to my advisor, Pro. Feng-Li Lin of the Department of Physics at National Taiwan Normal University, for his advice, encouragement, enduring patience and constant support. Without his consistent and illuminating instruction, this thesis could not have reached its present form. I would like to express my heartfelt gratitude to Pro. Hong-Yi Chen, who help me a lot on the computer. I am also greatly indebted to Pro. Ming-Che Chang, Pro. Po-Chung Chen, Pro. Ying-Jer Kao, and Pr. Yu-Cheng Lin. They introduce and teach me the numerical methods. I enjoy the discussion with them. Additionally, I want to thank the university of department Physics professors and staff for all their hard work and dedication, providing me the means to complete my degree. I also owe my sincere gratitude to my friends, I-Ching, and my fellow classmates who gave me their help and time in listening to me and helping me work out my problems during the difficult course of the thesis. And finally, I want to thank my family, my mom, Yu-Cheng, Chun-Hui, Yu Hung, and my puppy DiDi. I love you very much. Without your encouragement and support, I cannot finish my PhD degree. I must thank my dear mother specially for putting up with me during the development of this work with continuing, loving support and no complaint. I do not have the words to express all my feelings here, only that I love you, Mom!. iv.

(5) Abstract This thesis concerns the ground state property and the quantum entanglement in the oneand two-dimensional quantum spin systems. We use the Iterative Projection method to find the ground states numerically in the form of the tensor product states, and then evaluate their expectation values and their entanglement measures such as the geometric entanglement by the method of tensor renormalization group. We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points. We also study the scaling behaviors of the entanglement measures near the quantum critical point by the ideas of quantum-state renormalization group transformations. We find some universal features for one-dimensional spin system. However, we fails to capture the area-law for two-dimensional spin system. We then study the connection between topological order and the degeneracy of the singular value spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of tensor product state ansatz. By evaluating the topological entanglement entropy, we identify a new phase with topological order in this model, in which the singular value spectrum is doubly degenerate. Degeneracy of the singular value spectrum is robust against various types of perturbations, in accordance with our expectation for topological order.. KEY WORD: Quantum entanglement, Quantum phase transition, Topological order, Spin system, MPS, TPS.. v.

(6) Table of Contents Page Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vi. List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. 1.2. 1.3. 1.4. 1. Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.1. The basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.2. The properties of quantum phase transition . . . . . . . . . . . . .. 2. 1.1.3. Topological order and spin liquid . . . . . . . . . . . . . . . . . . .. 3. Entanglement measure and quantum phase transition. . . . . . . . . . . .. 7. 1.2.1. Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.2.2. Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.2.3. Block entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 10. 1.2.4. Entanglement spectrum . . . . . . . . . . . . . . . . . . . . . . . . 12. 1.2.5. Topological entanglement entropy . . . . . . . . . . . . . . . . . . . 13. 1.2.6. Geometric entanglement . . . . . . . . . . . . . . . . . . . . . . . . 15. 1.2.7. Scaling of the geometric entanglement in one-dimensional system . 16. Matrix product state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1. The basic concepts of the matrix product state. . . . . . . . . . . . 17. 1.3.2. Valence-bond picture . . . . . . . . . . . . . . . . . . . . . . . . . . 17. 1.3.3. Vidal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 1.3.4. The canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 1.3.5. Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 1.3.6. Time-evolving block-decimation . . . . . . . . . . . . . . . . . . . . 23. 1.3.7. Expectation value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 1.3.8. Quantum state renormalization group in the matrix product state . 26. Tensor product state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 vi.

(7) 1.5. 1.4.1. The basic concepts of the tensor product state . . . . . . . . . . . . 28. 1.4.2. The imaginary time evolution with the tensor product state . . . . 30. 1.4.3. Tensor reorganization group . . . . . . . . . . . . . . . . . . . . . . 31. 1.4.4. Quantum state renormalization group in the tensor product state . 34. Universal Quantum Resource via TPS . . . . . . . . . . . . . . . . . . . . 36 1.5.1. Teleportation quantum computation model . . . . . . . . . . . . . 37. 1.5.2. Measurement based quantum computation . . . . . . . . . . . . . . 38. 1.5.3. The cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 1.5.4. Quantum state as computation resource . . . . . . . . . . . . . . . 40. 2 Multipartite Entanglement Measures and Quantum Criticality from MPS and TPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 2.2. Entanglement measure in one-dimensional spin system . . . . . . . . . . . 43. 2.3. 2.4. 2.2.1. Spin 1/2 chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 2.2.2. Spin 1 chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. Entanglement measure in two-dimensional spin systems . . . . . . . . . . . 47 2.3.1. The two-dimensional transverse Ising model . . . . . . . . . . . . . 48. 2.3.2. The two-dimensional XYX model . . . . . . . . . . . . . . . . . . . 49. 2.3.3. The two-dimensional XXZ model . . . . . . . . . . . . . . . . . . . 51. Comment on scaling behavior of entanglement measure . . . . . . . . . . . 52 2.4.1. One-dimensional scaling via matrix product states . . . . . . . . . . 53. 2.4.2. Two-dimensional scaling via tensor product states? . . . . . . . . . 55. 3 Topological Order and Degenerate Singular Value Spectrum in two-Dimensional Dimerized Quantum Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . 58 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58. 3.2. The Model and order parameters . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.1. Order parameters for N´eel and dimer phases . . . . . . . . . . . . . 61. 3.2.2. Characterizing topological phase by degeneracy of singular value spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 3.3. Numerical results for the phase diagram of J-J’ model . . . . . . . . . . . . 62. 3.4. Topological Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1. J-J’ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 vii.

(8) 3.4.2 3.5. Toric code like state . . . . . . . . . . . . . . . . . . . . . . . . . . 69. Degeneracy of Singular Value Spectrum and Topological Order . . . . . . . 70 3.5.1. Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. 3.5.2. Topological order of J-J’ model by singular value spectrum and its robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. 3.5.3 4. Remove short range entanglement . . . . . . . . . . . . . . . . . . . 75. The projection symmetry group analysis of J-J’ model . . . . . . . . . . . . . . 78 4.1. General conditions on PSG on the square lattice . . . . . . . . . . . . . . . 78. 4.2. Classification of Z2 PSG on the J-J’ model . . . . . . . . . . . . . . . . . . 80. 4.3. A summary of Z2 PSGs on the J-J’ model . . . . . . . . . . . . . . . . . . 82. 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 A Schmidt decomposition and Singular value decomposition . . . . . . . . . . . . 88 B The properties of entanglement measure . . . . . . . . . . . . . . . . . . . . . . 90 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97. viii.

(9) List of Figures 1.1. Two schemes for evaluating the topological entanglement entropy. . . . . . 14. 1.2. The MPS representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. 1.3. Update the MPS after the gate U has been applied. . . . . . . . . . . . . . 24. 1.4. The calculation of expectation value of the MPS representation. . . . . . . 26. 1.5. The quantum state renormalization group od the MPS . . . . . . . . . . . 27. 1.6. The TPS representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. 1.7. Schematic reforestation of the tensor product state on the square lattice. . 30. 1.8. The TRG method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. 1.9. This figure shows how the lattice is changed after performing the TRG. . . 33. 1.10 Renormalization procedure on square lattice part 1. . . . . . . . . . . . . . 35 1.11 Renormalization procedure on square lattice part 2. . . . . . . . . . . . . . 36 1.12 The gate in the teleportation quantum computation . . . . . . . . . . . . . 38 1.13 The gate in the one-way quantum computer. . . . . . . . . . . . . . . . . . 38 1.14 The reduction of 1D AKLT state. . . . . . . . . . . . . . . . . . . . . . . . 41 2.1. The magnetizations v.s. transverse magnetic field of XY spin chain using MPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 2.2. Entanglement measures v.s. transverse magnetic field of XY spin chain using MPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 2.3. Staggered magnetization of spin 1 XXZ chain. . . . . . . . . . . . . . . . . 46. 2.4. The entanglement measures of spin 1 XXZ chain. . . . . . . . . . . . . . . 47. 2.5. The phase diagram of the 2D Ising model. . . . . . . . . . . . . . . . . . . 49. 2.6. The phase diagram of the 2D XYX model. . . . . . . . . . . . . . . . . . . 50. 2.7. The phase diagram of the 2D XXZ model. . . . . . . . . . . . . . . . . . . 51. 2.8. The block entanglement entropy SL for spin 1 XXZ chain. . . . . . . . . . 54. 2.9. The GE of block size L for spin 1 XXZ chain. . . . . . . . . . . . . . . . . 55. 2.10 The scaling behavior of bipartite entanglement for spin 1 XXZ chain. . . . 56 2.11 Spectra of singular values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. ix.

(10) 3.1. J-J’ model on the 2D square lattice. . . . . . . . . . . . . . . . . . . . . . . 60. 3.2. Magnetization hMsz i v.s. J 0 /J. . . . . . . . . . . . . . . . . . . . . . . . . . 63. 3.3. hMsz i vs J 0 /J for different bond dimensions. . . . . . . . . . . . . . . . . . 64. 3.4. The ground state energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65. 3.5. Dimerization Dx and Dy vs. J 0 /J. . . . . . . . . . . . . . . . . . . . . . . 65. 3.6. Visualization of Spin-Spin correlation of neighboring sites. . . . . . . . . . 66. 3.7. J,J The singular value measures SBP and S1 v.s. J 0 /J. . . . . . . . . . . . . . 67. 3.8. Topological entanglement entropy in the J-J’ model. . . . . . . . . . . . . . 68. 3.9. The toric code is defined on a 2D square lattice. . . . . . . . . . . . . . . . 70. 0. 3.10 Topological entanglement entropy from Kitaev-Preskill’s scheme. . . . . . . 71 3.11 The singular value spectrum per bond ofthe J-J’ model.. . . . . . . . . . . 73. 3.12 Singular value spectrum per bond of the J-J’ model with the perturbation.. 74. 3.13 The singular value spectrum of J-J’ model under quantum state RG flow. . 75 3.14 The renormalization flow in the toric code like model. 4.1. . . . . . . . . . . . 76. The lattice structure of the J-J’ model. . . . . . . . . . . . . . . . . . . . . 78. x.

(11) Chapter 1. Introduction This thesis presents the studies of one- and two-dimensional (2D) spin system at and near the critical points by using tensor product state representation. In addition to the symmetry breaking phase, we also consider the topological phase that can not be characterized by the local order parameter. From quantum information perspective, we attempt to understand these phases from the qualitative behavior of entanglement and their fixed point tensor states. The outline of the thesis is as follows. In this chapter, we give the background of our work and expand on three topics: quantum phases, quantum entanglement, and tensor network state. In the second chapter, we explore the one- and two-dimensional quantum criticality numerically by the entanglement measure such as geometric entanglement. In the third chapter, we investigate the connection between topologically ordered phase and degeneracy of the singular value spectrum by studying the two-dimensional dimerized quantum Heisenberg model. In the fourth chapter, we summarize all the possible Z2 spin liquids on the J-J’ lattice based on the projective symmetry group (PSG) analysis of the slave-boson approach. Some technical details are summarized in Appendices.. 1.1 1.1.1. Quantum phase transitions The basic concepts. Phase transition is an interesting phenomenon in nature. It includes the boiling of water, the melting of ice, and the more complicated such as the transition of a metal into the superconducting state when cooling below the critical temperature. A phase transition takes place at some values of the well-defined parameters that determine the phase. Phase transitions are classified into first-order and continuous transition depending on the lowest derivative of the free energy that is discontinuous or not at the transition point. For the first-order phase transition the first derivative of the free energy is discontinuous. For example, when the pressure reaches the gas-liquid point, there will be a discontinuous 1.

(12) jump in the volume as all the gas changes to liquid at this pressure. In contrast, the continuous transitions are continuous in the first derivative of the free energy. For example, there is the order-disorder transitions in the Ising ferromagnet by tuning the temperature. The ferromagnetic order will go to zero continuously at Tc . Furthermore, an observable that can distinguish different phases is called ”order parameter”. The order parameters for different phases may not be the same. The liquid and the gas phases may be distinguished by the density, but the order parameter between the liquid and the solid is the shear modulus. The continuous transitions can be characterized by the critical exponents. The important one is the critical exponent ν which could describe the divergence of the correlation length ξ at the critical point, ξ ∝ t−ν = (. T − Tc −ν ) , Tc. (1.1). where Tc is critical temperature. Similar behaviors also appear for other physical quantities such as specific heat, susceptibility, and order parameter. These exponents are related by scaling relations and are universal, that is to say, the critical exponents could be the same for different physical systems. 1.1.2. The properties of quantum phase transition. The quantum phase transition (QPT) [1] is the phase transition between different quantum phases at zero temperature. The transition describes a drastic change in the ground state of a many-body system due to the quantum fluctuations. It is to tune some nontemperature parameters such as the magnetic field to obtain different phases. The quantum phase transition can also be first-order or continuous transition depending on the behavior of the derivative of order parameter. The quantum phase transition is at zero temperature, so we are always seeking the quantum state with the lowest total energy, i.e., the ground state. The simple example is the Ising chain model with transverse field, H=. X. z −σiz σi+1 + gσix ,. (1.2). i=1. where g is the coupling constant. In the thermodynamic limit, the ground state shows a second-order transition by tuning the coupling constant g. When g 1, the ground. 2.

(13) states are two degenerate ferromagnetic states, | ↑i| ↑i| ↑i...| ↑i;. | ↓i| ↓i| ↓i...| ↓i. (1.3). where | ↑i, | ↓i are the eigenstates of Pauli matrix σ z . In the thermodynamical limit, the quantum tunneling is suppressed so that the ground state should be in either of the above two states. This is a simple example of a common phenomenon in physics: the spontaneously breaking of the Z2 symmetry. On the other hand, when g 1, the ground state is the paramagnetic state | →i| →i| →i...| →i. For this reason, there must be a critical value g = gc . It could be characterized by the magnetization that is an order parameter of this transition. On the other hand, at g = gc , the ground state will be in a very complicated and entangled qubit arrangement. This is one of our motivations to probe the criticality from the quantum entanglement. In any practical experiment, it is essential to describe the physics at finite temperature. It need more complicated structure like its excited states to describe this system. The phase diagram will be more involved. 1.1.3. Topological order and spin liquid. Landau symmetry-breaking theory is regarded as a very successful theory to describes all possible orders, and all possible phase transitions. The key assumption is that all the orders are associated with symmetries. However, in 1982, one discovered a new kind of states, namely Fractional Quantum Hall liquid that cannot be described by the Landau symmetry breaking theory. Quantum Hall liquid means the electrons in some special cases will be in liquid phase. These phases have many special properties: for instance, different quantum Hall states which have the same symmetry and are beyond the Landau symmetry-breaking description. The existence of these states tells us we need new way to characterize the different orders distinguishing them. The new orders are named quantum orders. The quantum order of Fractional Quantum Hall liquids has a special property that all excitations above the ground state have energy gaps. This kind of quantum orders is also called topological order[2, 3]. However, topological orders can be described by some other means, such as ground state degeneracy [2], quasiparticle fractional statistics, edge states, topological entropy [4, 5], etc. Roughly speaking, topological order is a pattern of long-range quantum entanglement in quantum states [6]. 3.

(14) Nontrivial topological orders are not only appearing in Fractional Quantum Hall liquid but also in spin liquids at zero temperature. The spin liquid phase approach is based on the Mott insulator phase at the high-Tc superconductors. The Mott insulator means there are odd number of electrons per unit cell. Furthermore, many important properties of high-Tc superconductors may be directly linked to the Mott insulator at half filling. At half filling with large Coulomb repulsion, the charge excitations can be ignored, so one can use pure spin model to describe the system. By analyzing the spin liquid phases, one can study the Mott insulator phase of high-Tc superconductors. On the other hand, the numerical and experimental results support the existence of quantum spin liquid in some frustrated systems. In 1991, superconductors which have a Z2 topological order were constructed theoretically [7, 8]. The topologically ordered states are also regarded as the resource of quantum computing [9, 10]. A topologically ordered state with non-local quantum entanglement is robust. This significantly reduces the effect of decoherence. In other words, topologically ordered states may provide a natural tool for quantum memory and quantum computation. Slave-boson approach. We illustrate the slave-boson approach in the spin liquid phase. Besides this approach, spin liquid has also been studied using σ-model approach [11], quantum dimer model [12], and others numerical method [13]. In third chapter, we employ a numerical method, called tensor product states (TPSs), to study the spin liquid phase. In the slave-boson approach, a spin 1/2 operator S~i can be represented as 1 † ~σαβ fiβ , S~i = fiα 2. (1.4). with fermionic operator fiα , α = 1, 2, which carries spin 1/2 and no charge. We explain P the spin liquid phase of a Heidelberg spin-1/2 model. Its Hamiltonian H = hiji Jij S~i S~j can be rewritten as H=. X hiji. 1 − Jij 2. . † † fiα fjα fjβ fiβ. 1 † † + fiα fiα fjβ fjβ 2. .. (1.5). Because of the enlarged Hilbert space, we need to impose some constraints on the physical. 4.

(15) Hilbert space, i.e., † fiα fiα = 1,. fiα fiβ αβ = 0.. (1.6). This constraints can be relaxed in the mean field treatment so that they are satisfied by the ground state average, namely, † hfiα fiα i = 1,. hfiα fiβ αβ i = 0,. (1.7). where αβ is the antisymmetric tensor. Such constraints can be realized by adding a site-dependent and time-independent Lagrangian multiplier al0 (i), l = 1, 2, 3 in the Hamiltonian. Then, we introduce the mean field parameters by the ground state average as follows, ηij αβ = −2hfiα fjβ i, † χij δαβ = 2hfiα fjβ i,. ηij = ηji χij = χ†ji .. (1.8). After some mainpulations, we can obtain the zero-order mean-field Hamiltonian i X 3 h † † † Hmean = − Jij χij fiα fjα + ηij fiα fjβ αβ + H.c. − |χij |2 − |ηij |2 8 hiji i Xh † + a30 (fiα fiα − 1) + (a10 + ia20 )fiα fiβ αβ + H.c. , (1.9) i. where χij and ηij must satisfy the self-consistency condition in Eq.(1.8), and al0 (i) must be chosen to satisfy the constraints in the ground state average. We regard such the χij , ηij , and al0 (i) as the mean-field solution. The Hamiltonian and the constraints have a local SU (2) symmetry. To make this SU (2) symmetry manifest, we introduce the two T † component fermion ψi = fi,↑ , fi,↓ and the matrix of mean-field solution amplitudes,   † ηij χ . Uij =  †ij (1.10) ηij −χij In terms of ψi , the Hamiltonian of the slave-boson representation is invariant under a local SU (2) transformation W (i). This transformation can modify the mean-field solution as following: ψi → W (i)ψi , Uij → W (i)Uij W † (j), W (i) ∈ SU (2). 5. (1.11).

(16) This SU (2) gauge structure is originated from Eq.(1.4), and the local SU (2) transformation leaves the physical spin operator unchanged. Then, we can rewrite the constraints and the Hamiltonian into the following hψi† τ l ψi i = 0, X † X Hmean = − ψi uij ψj + al0 ψi† τ l ψi ,. (1.12) (1.13). i. hiji. where τ l , l = 1, 2, 3 are the Pauli matrices. uij =. 3 8 Jij Uij. and al0 are the mean-field. ansatz. By imposing some symmetries, different ansatz may describe the different spin liquid phases. In other words, different gauge transformations can distinguish different spin liquid phases with the same symmetry. The mathematical tool to classify different spin liquid phases is the projection symmetry group (PSG) [14]. Projection symmetry group. The projection symmetry group (PSG) which is the property of the ansatz (uij , al0 ) can be used to classify the spin liquid phases that have the same symmetries. We can group the ansatzs that share the same PSG to form a class. A PSG contains a symmetry transformation U and a gauge transformation GU . Under its PSG, the GU U , the ansatz is invariant, i.e., GU U (uij ) = uij ,. with. U (uij ) ≡ uU (i),U (j) , GU (uij ) ≡ GU (i)uij G†U (j), GU (i) ∈ SU (2).. (1.14). There is a special subgroup of PSG, called invariant gauge group (IGG). It is formed by the gauge transformation that leaves the ansatz unchanged. IGG = {Wi |Wi uij Wi† = uij , Wi ∈ SU (2)}.. (1.15). In this gauge, IGG can be a Z2 transformation {Wi = ±τ 0 }, U (1) transformation {Wi = 3. eiθτ , θ ∈ [0, 2π]}, or SU (2) transformation {Wi = eiθˆnτ , θ ∈ [0, 2π], n ˆ ∈ S 2 }. This means that the symmetric spin liquid whose low energy gauge structure is Z2 , U (1), or SU (2). The fluctuations of the ansatz need to be take into account, and it will influence the stability of the mean-field state. 6.

(17) If GU U ∈ P SG and G ∈ IGG, then GGU U will also be in the PSG. In other words, for each symmetry transformation U , the different gauge transformations GU have one-to-one mapping by using IGG, SG = P SG/IGG.. (1.16). This relation shows that the PSG is an extension of SG by IGG. A symmetry can describe the classical order. Consequently, we can use the PSG to describe the quantum order. There are many studies where they classify all possible spin liquids in spin systems by using PSG and projection contruction [15, 16, 17, 18]. We will classify the topological orders of J-J’ spin model by using PSG as an example in the chapter four. In addition to the self-consistency Eq.(1.8), we can obtain the mean-field solutions by minimizing the energy. Those different mean-field solutions will be the local minima of the ground state energy of the mean-field ansatz, so we could compare the energies between these states.. 1.2. Entanglement measure and quantum phase transition. Entanglement has been considered as ”spooky” non-locality intrinsically to quantum mechanics. The EPR experiment, in the form as analyzed by Bell, emphasizes that entanglement leads to a degree of correlation beyond what can be explained in terms of local hidden variables. Due to this special property, entanglement is a uniquely quantum resource that plays a key role in many interesting applications of quantum information processing [19, 20]. It is believed that entanglement is one of the main ingredient to speed up the quantum computation and communication. Several quantum protocols have been developed with the help of entangled states, such as quantum dense coding [21], teleportation [22], quantum secret sharing [23], and so on. On the other hand, entanglement also plays an important role to understand the quantum phase transitions and strongly correlated systems. The intuition is that quantum correlation is related to how entanglement is distributed in the system. The notion of entanglement has been applied to many-body systems [24, 25, 26, 27], and especially to systems that are under quantum phase transitions, mainly via bipartite measures such as concurrence [28, 29] and entanglement entropy [30, 31]. Nielsen(2002) [28] and Osterloh(2002) [29] have considered the entanglement in the spin model under the quantum 7.

(18) phase transition. Their analysis focused on single-spin entropies and two-spin quantum correlations, called concurrence [32, 33, 34]. Several other authors have also studied the entanglement of formation in spin system [35, 36, 37, 38]. In this section, we do not attempt to give an detailed review of entanglement. We define some entanglement measures that are being used to quantify entanglement in many-body systems on pure state. 1.2.1. Concurrence. Let us expound the concurrence that we mention above. For a pure state |ΨAB i of two P qubits, if this pure state can be written in the Bell basis |ei i, as |ΨAB i = i αi |ei i, 1 |e1 i = √ (| ↑↑i + | ↓↓i), 2 1 |e2 i = √ (| ↑↑i − | ↓↓i), 2 1 |e3 i = √ (| ↑↓i + | ↓↑i), 2 1 |e4 i = √ (| ↑↓i − | ↓↑i), 2. (1.17) (1.18) (1.19) (1.20). then, the the concurrence [34] is. X. αi2 |,. (1.21). := |hΨ|σ y ⊗ σ y |Ψi|.. (1.22). C[|ΨAB i] := |. i. It is one of the entanglement measures in the bipartite systems. When a bipartite system AB is in a mixed state, it is formulated differently. One way is the entanglement of formation [32, 33] applicable to bipartite mixed states. They thought that the mixed state entanglement is an average of the corresponding pure states that make up the mixed state. If ρ is any density matrix of two qubits, the concurrence of mixed state of two qubits is C[ρ] := max[0, λ1 − λ2 − λ3 − λ4 ], where the λi ’s are the eigenvalues of the Hermitian matrix R ≡. (1.23). p√ √ ρ˜ ρ ρ, and ρ˜ = (σ y ⊗. σ y )ρ∗ (σ y ⊗ σ y ). The range of concurrence C is from 0 (no entanglement) to 1 (maximally entangled state). In Nielsen (2002) [28] and Osterloh(2002) [29], they considered a spin chain model, and connected the theory of critical phenomena with entanglement. They discussed the nearest neighbor concurrence C(1). The concurrence is a smooth function 8.

(19) of the coupling constant, and reaches a maximum near the critical point. The critical properties of the ground state can be captured by the derivatives of the concurrence. 1.2.2. Entanglement entropy. Bipartite entanglement of pure state is conceptually well known. A pure bipartite state is not entangled if and only if it can be written as a product of pure states. In Ref. [40], they showed the entanglement entropy is the unique bipartite entanglement measure for pure state. Here, we define the von Neumann entropy as entanglement measure. A detailed explanation can be found in several reviews [24, 39, 25, 41, 42, 43, 44]. Start with a bipartite system |Ψi belonging to H = HA ⊗ HB . According to the Schmidt decomposition (see appendix A), the pure state can be written as |ψiAB =. χ X. λi |αi i|βi i,. (1.24). i=1. where λi ’s are real and positive and are called Schmidt coefficients, χ := min(dim HA , dim HB ) is the Schmidt rank. The Schmidt decomposition is the characterization of quantum entanglement for pure state: (1) If there is only one non-zero Schmidt coefficient (or Schmidt rank equals one), the state is unentangled. (2) If Schmidt rank is larger than one, then the state is entangled. We can use the Schmidt rank to judge if this state is entangled or not, but cannot quantify the quantum entanglement. Quantitatively, the entanglement of a pure state is conveniently measured by the entanglement entropy. The von Neumann entropy can be written in terms of the reduced density matrices of each part of the system. SA = SρA ≡ −tr(ρA log ρA ),. (1.25). where ρA is the reduced density matrix of part A. Then, the von Neumann’s definition can be re-expressed with the Schmidt coefficients of bipartite system, i.e., SA = SB ≡ −. X. λ2i log λ2i ,. (1.26). i. where λ2i ’s are eigenvalues of the reduced density matrix ρ. We can find that SA = SB because the above eigenvalues for the sub-systems A and B are the same. This is an important property of entropy (others see. Appendix B). For example, a Bell state, √ |φi = 1/ 2(| ↑↑i + | ↓↓i), has Schmidt rank two, and its bipartite entanglement entropy 9.

(20) is SA = SB = 1. This is maximally entangled. A product state, |φi = | ↑↑i, has Schmidt rank one, and its bipartite entanglement entropy is SA = SB = 0, The state is unentangled. A special example is when we trace out all but one spins to obtain the corresponding reduced density matrix ρA , we can then define the one-tangle [45] as S1 = 4 det ρA .. (1.27). In the Ref. [46, 47, 48], they computed the one-tangle S1 and the sum of squared concurrence C2 to identify the critical point in the one-dimensional(1D) and two-dimensional(2D) XYX models. Furthermore, the bipartite entanglement entropy has also been a powerful tool in numerical studies to identify the topological phases [49, 50]. 1.2.3. Block entanglement entropy. Many studies have been proposed to characterize the ground state of the spin models through the spectral properties of the reduce density matrix for a block of spins and its entanglement entropy. A key property of the reduced density matrix is what has been known as the “area law” [51]. This means that entanglement entropy would depend on the size of the surface separating the two regions A and B. One explores the behavior of entanglement entropies at different length scales and captures the universal scaling near the various 1D quantum critical points [30, 31]. If |Gi represents ground state of N -spin chain system, ρL = trN −L |GihG| is the reduced density matrix of the block of L spins. If ρL has many non-zero eigenvalues, this indicates a lot of entanglement between the block and the rest of the chain. The von Neumann entropy is SL ≡ −tr(ρL log ρL ),. (1.28). as a entanglement measure which quantifies the quantum entanglement [52] between the block and the rest of the chain. There are some general properties of the SL . First, SL is positive. Second, the block and the rest of the chain will have the same entanglement entropy. Finally, if the system is translational invariant, SL is a concave function, SL ≥. SL−M + SL+M 2. where L = 0, 1, ..., N , and M = 0, 1, ... min{N − L, L}.. 10. (1.29).

(21) Furthermore, in the Ref. [30], they studied the spin chain model with open boundary conditions by using the Bethe ansatz. They proceed to compute the entanglement entropy for XY model and XXZ model given by 1X z 1X 1+γ x x 1−γ y y σi σi+1 σi σi+1 − λσi , HXY = − 2 2 2 2 i i X y y x x z z z HXXZ = σi σi+1 + σi σi+1 + ∆σi σi+1 − λσi .. (1.30) (1.31). i. Some quantum critical regions occur in the parameter space (γ, λ) for the XY model. The Ising model with γ = 1 has a critical point at λ = 1. The XX model with γ = 0 is critical in the interval λ ∈ [0, 1]. Within the critical regime, the block entanglement entropy displays a logarithmic divergence for the large L. Away from the critical point, the block entanglement will be saturated. For the finite-chain, SL shows the feature combining the critical logarithmic behavior with the finite size effect. In 1D, the surface separating the two regions is constituted of two points. So, the area law in 1D would imply that the block entanglement entropy is independent on the block size. At critical point, a 1D system has a block entropy which diverges logarithmically with the block size. If the block contains L spins, and the system is of N spins with periodic boundary condition, then SL is given by c N π SL = log2 sin( L) + As , 3 πa N. (1.32). where c is the central charge of the underlying conformal field theory and a is the ultraviolet regularization cut-off. As is a non-universal constant. For examples, in the Ising model c equals 1/2, and c = 1 in the Heisenberg model. Away from the critical point, the block entanglement entropy, a growing function with the block size L, will be saturated at the order of the correlation length ξ, i.e., SL =. c ξ log2 3 a. for. L ≥ ξ.. (1.33). In the non-critical regime, the correlation length ξ, taking the distance between the nearest neighboring spins as the unit length, is a finite value. The entanglement entropy between a block and the rest of an infinite chain has a fixed value for the block size larger than the correlation length ξ. The scaling of the block entanglement entropy gives the excellent signature for the critical point as well as the concurrence considered in the previous section. 11.

(22) If we consider a quantum spin system on a n-dimensional lattice, and look at the reduced density operator ρL of a block of spins with linear size L. The ground states of local Hamiltonian seem to have the property that the entanglement entropy only scales as the boundary area of the block, SρL ∝ Ln−1 .. (1.34). This means the most of the entanglement must be concentrated around the boundary. If this state exhibits some locality, we could use a few parameters to describe this phases. It is at present not clear if the logarithmic corrections will occur in the case of higher dimensional systems. 1.2.4. Entanglement spectrum. Another interesting quantity in characterizing the quantum entanglement is the full spectrum of the reduced density matrix. It is called the entanglement spectrum [53]. The entanglement spectrum is a set of eigenvalues of the reduced density matrix. In contrast, the entanglement entropy is just a single number. The entanglement spectrum contains more complete information. One focused on the entanglement spectrum at and near the critical point with conformal invariance [54, 55]. The entanglement spectrum can be obtained from the traces of all powers of the reduced density matrix X Rα ≡ trραA = λαi ,. (1.35). i. for any α. In order to make the entanglement spectrum well-defined, Rα can be written as −c(α−1/α)/6. Rα = cα Lef f. ,. (1.36). where cα is a nonuniversal constant and Lef f is the relevant length scale in the considered regime. Following [54], we introduce the parameter b and rewrite this as Rα = cα exp [−b(α − 1/α)],. b=. c log Lef f . 6. (1.37). For conformally invariant models, the parameter b is related to the maximum eigenvalue, b = − log λmax . Note that the parameter b is simply related to the entanglement entropy S, b = S/2. By using this, an distribution function n(λ) can be represented as Z λmax p n(λ) = dλP (λ) = I0 2 b log(λmax /λ) , λ. 12. (1.38).

(23) where P (λ) is the distribution of eigenvalues. In the Ref. [54, 55], they proved that the entanglement spectrum is described by a universal scaling function depending only on the central charge. Actually, the entanglement spectrum carries highly nonlocal information. The double degeneracy of the entanglement spectrum has a physical consequence. The entanglement spectrum is the Schmidt values of the bi-partition of the whole system, so the existence of the dominant degenerate Schmidt values implies that the wave function is close to the maximally entangled state (Bell state). In this case, the two parts of the system is coherently long-range entangled. On the other hand, the degeneracy of the entanglement spectrum has recently implemented to characterize the topological orders for some 2D quantum Hall states [53] and some 1D symmetry protected topological phases [56]. In the Ref. [56], they showed the Haldane phase of S = 1 chains is characterized by a double degeneracy of the entanglement spectrum. They considered the spin-1 model Hamiltonian, H=. X. ~i+1 + Uzz (S z )2 . ~i S S i. (1.39). i=1. By tuning the coupling constant Uzz , the ground state is in Z2z phase characterized by the nonzero expectation value of hS z i. As Uzz increases, the ground state becomes the Haldance phase, which is a symmetry protected topological phase with the double degeneracy of the entanglement spectrum. As the parameter is tuned to be large enough, the ground state becomes the eigenstate of (Siz )2 , namely the trivial insulating phase (TRI). There are both single and multiply degenerate levels in TRI. They also added a perturbation to break some symmetry. Then, in the Haldance phase, the double degeneracy is robust against the perturbations. Entanglement spectrum is an indicator of topological order. 1.2.5. Topological entanglement entropy. It is more difficult to characterize the topological phase since there is no symmetry breaking order parameter for it. The topological entanglement entropy, usually denoted by γ, is a number characterizing many-body ground states that possess topological order. On typical quantity for the 2D systems is the topological entanglement entropy [4, 5]. It is the sub-leading constant term in the entanglement entropy SL = αL − γ + O(L−ν ) , 13. ν>0,. (1.40).

(24) where L is the boundary size of the block for which the entanglement entropy is evaluated by tracing out the degrees of freedom inside/outside it.. A. A. B. D. C. B. C. D. Figure 1.1: Two schemes for evaluating the topological entanglement entropy. (Left) KitaevPreskill’s scheme for the square block with even number of sites on each side. (Right) Levin-Wen’s scheme for the square block with odd number of sites on each side.. Since the constant term γ is topological and universal, we can extract it by appropriately subtracting the entanglement entropies of different blocks. There are two subtraction schemes, one is proposed by Kitaev and Preskill [4], and the other is by Levin and Wen [5]. For the square lattice it is easier to implement the numerical evaluation of the entanglement entropy for a square or rectangular block. We find that it is convenient to adopt Kitaev-Preskill’s scheme as shown in the left of Fig. 1.1 for the square block with even number of sites on each side of the square block. The extraction for the topological entanglement entropy then goes as follows KP Stopo = SA + SB + SC + SD − SAB − SBC − SCD − SAD + SABCD ,. (1.41). where SAB.. denotes the von Neumann entropy of the density matrix ρAB... in the region AB... ≡ A ∪ B ∪ .... As for the square block with odd number of sites on each side, it is convenient to adopt Levin-Wen’s scheme as shown in the right of Fig. 1.1, and the extraction is LW = SABCD − SABD − SACD + SA + SD . Stopo. (1.42). In both cases, Stopo = −γ < 0, which is related to total quantum dimension D by γ = log D [4, 5]. For example, the simplest fractional quantum Hall states [57], the Laughlin states at filling fraction ν, have γ = 1/2 log(1/ν). The Z2 fractionalized states, such as topologically 14.

(25) ordered states of Z2 spin-liquid [58], quantum dimer models on triangular lattice [59], and the toric code state [60], are characterized by γ = log(2). 1.2.6. Geometric entanglement. The amount of entanglement is generally difficult to define once we are not considering the bipartite system. In the case of systems composed of m > 2 subsystems the definition of separable and entangled states is richer than in the bipartite case. Although, mutipartitle entanglement in many-body system is much less studied, it has already been mentioned that several quantities are useful as the indicators for multipartite entanglement when the whole system is in a pure state [61, 62]. One of the multipartite entanglement is the geometric measure. The geometric measure of entanglement quantifies the entanglement of a pure state |ψi through the minimal distance of the state from the set of pure product states {|φi} [61], namely, d = min k|ψi − |φik, {|φi}. where, |φi ≡. Nn. (i) i=1 |φ i. (1.43). is an separable n-particle pure state. Hence, we define Λmax as. the maximal overlap with the closest separable state, i.e., Λmax (ψ) ≡ |hφ|ψi|.. (1.44). The larger Λmax indicates that the less entangled is |ψi. A well-defined global measure of entanglement is obtained by taking the logarithm of Λ2max , i.e., E(ψ) = − log Λ2max .. (1.45). To characterize the properties of a quantum critical point, one uses the quantity per party En given by En = E(ψ)/n. In fact, the difficult task is the maximization over all possible separable states. To determine the closet separable state is a nonlinear eigenproblem. It can be solved numerically. However, the symmetry of the state can alleviate the difficulty associated with solving the nonlinear eigenproblem. For example, we consider a two-qubit Bell state (1.17) with permutation symmetry. One can assume that the closet state is |ψi = (cos θ| ↑i + sin θ| ↓i)⊗2 ,. 15. (1.46).

(26) √ √ for which the overlap is 1/ 2(cos2 θ + sin2 θ). Then, Λmax = 1/ 2. Otherwise, it is zero for the unentangled states. In the Ref. [63], they quantified the multipartite entanglement in the 1D spin-1/2 XY model (1.30). The energy eigenproblem can be solved by applying the Jordan-Wigner transformation. Similar to the bipartite entanglement, the geometric entanglement is very sensitive to the existence of QPTs. 1.2.7. Scaling of the geometric entanglement in one-dimensional system. In fact, the scaling behavior of the geometric entanglement in several 1D models is similar to the block entanglement entropy [64]. Near criticality and for the 1D quantum systems of finite correlation length ξ, the geometric entanglement will be saturated when increasing the block size L E=. ξ c log2 12 . L ξ ,. (1.47). where is the UV regularization parameter that coincides with the lattice spacing for lattice systems. However, at criticality, the geometric entanglement per block L shows the logarithmic scaling behavior [65] E∼. 1.3. c log2 L. 12. (1.48). Matrix product state. In the case of 1D systems, many analytical methods have been used to find the exact ground states and excited states of some special Hamiltonians. On the other hand, White (1993) developed a very power numerical simulation method known as density matrix renormalization group (DMRG) [67, 68] that could determine the physical properties of generic spin chains. A simple interpretation of the DMRG casts it as a variational method with matrix product states (MPS) [69, 70]. In the two or higher dimensions, almost no model could be solved exactly. However, DMRG is mainly restricted to the 1D systems. A natural generalization of 1D MPS to two and higher dimensions, called tensor product states (TPS) [71, 72], had been presented. In the following, we will provide an overview of two concepts of central importance to this thesis, namely, the matrix-product states and tensor-product states.. 16.

(27) 1.3.1. The basic concepts of the matrix product state. Matrix product state (MPS) appeared in 1995 [69, 70] is a very well known method that enables us to treat quantum many-body systems. It arises from a family of 1D quantum systems whose description is local, in the sense the site spins interact and correlate only locally. Moreover, the total amount of correlations is controlled by a parameter called the bond dimension. Many studies have been performed based on the applications of MPS. In the Ref. [73], it laid the basic understanding of MPS, and introduced the useful techniques characterizing the MPS. Furthermore, MPSs were also explained from the quantum information point of view [74]. They described a quantum state with ”virtual” pairs in the maximally entangled state between adjacent sites. On the other hand, MPSs play an important role also in the context of quantum information processing. Such as the cluster states [75, 76, 77], they can be described by low-dimensional MPSs and are considered as the valuable resources for quantum information and communication tasks. For 1D quantum many-body systems, the ground states could be expressed in terms of a MPS as |ψi =. X. T r[A[1] (s1 )A[2] (s2 )...A[N ] (sN )]|s1 , ..., sn i,. (1.49). s1 ,s2 ...,sN. where si = 1, ..., ds for i = 1, ..., N , and A[i] (si )’s are χi−1 by χi matrices, with ds denoting the physical dimension and χi the dimension of the i-th bond. The parameter χi is the number of the Schmidt values for the bipartition at the site i. If the amount of entanglement of the system is low, we only keep a few of the Schmidt coefficients. Let us give a more precise account of the matrix product states from two different perspectives. First, from the ground state of the AKLT model, MPS may be intuitively given in terms of a valence-bond picture. Second, it is a mathematical scheme through the Schmidt decomposition. 1.3.2. Valence-bond picture. Given a spin-1 antiferromagnetic Heisenberg chain with bi-quadratic interations X ~i S ~i+1 + J(S ~i S ~i+1 )2 , H= S. (1.50). i. where Si are spin-1 operators. The model in the J = 1/3 is an exactly solvable model. Proposed by Affleck, Kennedy, Lieb and Tasaki [78, 79], they proved the existence of a 17.

(28) Haldane gap and exponential decay of the correlation function in the ground state. In order to describe the exact ground state of the AKLT model, One employs the special basis for the state space. One splits the spin-1 space into two virtual spin-1/2 spaces and constructs a state of the virtual particles with a valence bond, | ↑↓i − | ↓↑i, between each pair of adjacent sites i and i + 1. The space of spin-1 may be formed by taking the symmetric part of the product of two spin-1/2 spaces, as shown in the Fig. 1.2 (b).. A[1]. A[2]. A[3]. s2. s1. A[4]. (a). s4. s3. (b). Γ[1] s1. λ[1]. Γ[2]. λ[2]. λ[3] Γ[3]. s2. s3. λ[4] Γ[4]. (c). s4. Figure 1.2: (a)Graphical representation of MPS. (b)In the Valence bond picture. The valence bond state on the chain. Each dot and line represents a singlet pair. The dotted circle represents two spin 1/2 spaces to create spin 1 space. (c)In the Vidal representation.. We label the virtual spins αi and βi at each site i, then the spins βi−1 and αi at the sites i − 1 and i respectively form the valence bond states. The state of the spin at a each site is 1 |ϕαβ i = √ (|αi ⊗ |βi + |βi ⊗ |αi), 2. (1.51). where |αi and |βi denote the eigenstates | ↑i and | ↓i of σ z , thus |ϕαβ i and |ϕβα i denote the same state. These state may be written in terms of a more standard basis for spin 1,. 18.

(29) namely, ϕ↑↑ =. √ 2| + 1i,. ϕ↑↓ = ϕ↓↑ = |0i, √ ϕ↑↓ = ϕ↓↑ = 2| − 1i.. (1.52). We also can rewrite it as the projector that map the virtual system to the physical one. P = | + 1ihϕ↑↑ | + |0i(. hϕ↑↓ | + hϕ↓↑ | √ ) + | − 1ihϕ↓↓ |. 2. (1.53). For each site i,. X. Pi =. i Asα,β |si ihα, β|,. (1.54). si ,α,β. where the elements of the matrix A denote the coefficients in the projector P . We can then determine the MPS form of the AKLT state. In general, we can consider a system of N particles, each of which lives in the ddimensional Hilbert space. Following the valence-bond construction, we can assign two virtual spins of dimension χ, and assume that each pair of neighboring virtual spins is in Pχ the maximally entangled state |φi = α=1 |α, αi. Apply the map (1.54) we can get the MPS form (1.49). 1.3.3. Vidal decomposition. Vidal [81] proposed an efficient protocol for simulation of slightly entangled pure state by using Schmidt decomposition. There, we consider the expansion of a n-qubit state in the computation basis (| ↑i, | ↓i) |Ψi =. 1 X. .... s1 =0. 1 X. cs1 ...sn |s1 i ⊗ . . . ⊗ |sn i.. (1.55). sn =0. The key ingredient of this simulation protocol is a particular decomposition of the coefficient cs1 ...sn cs1 ...sn =. X. [1]s. [1] [2]s. [2] [3]s. [n]s. Γα1 1 λα1 Γα1 α22 λα2 Γα2 α33 . . . Γαn−1n .. (1.56). α1 ,...,αn−1. This decomposition employs n tensors (of rank 3) {Γ[1] , . . . , Γ[n] } and n − 1 vectors {λ[1] , . . . , λ[n] }, whose indices si and αi take values in {0, 1} and {1, . . . , χ}. Here, χ 19.

(30) is the maximal values of the rank of the reduce density matrix in the bipartite splitting of the state Eq. (1.55). In Eq. (1.56), the 2n coefficients cs1 ...sn could be expressed in terms of (2χ2 + χ)n parameters. This is more efficient. This decomposition contains a concatenation of n − 1 Schmidt decompositions. Now, we will introduce this decomposition. First, we compute the Schmidt decomposition of the bipartite splitting: {1|2, . . . , n} of |Ψi so that. X. |Ψi =. [1]. [1]. [2...n]. λα1 |Φα1 i|Φα1. i,. α1. X. =. [1]. [1]s. [2...n]. Γα1 1 λα1 |s1 i|Φα1. i,. (1.57). α1 [1]. where we expand the Schmidt vector |Φα1 i =. P. [1]s. α1. Γα1 1 |s1 i in terms of the computational. basis. Second, we expand the Schmidt vector for qubit 2, [2...n]. |Φα1. i=. X. [3...n]. |s2 i|τα1 s2 i.. (1.58). s2 [3...n]. [3...n]. Third, write |τα1 s2 i in terms of the (at most) χ Schmidt vectors {|Φα2. i}χα2 =1 (the. eigenvectors of ρ[3...n] ). [3...n]. |τα1 s2 i =. X. [2]s. [2]. [3...n]. Γα1 α22 λα2 |Φα2. i.. (1.59). α2. Finally, we combine (1.58) and (1.59) so that |Ψi could be rewritten |Ψi =. X. [1]s. [1] [2]s. [2]. [3...n]. Γα1 1 λα1 Γα1 α22 λα2 |s1 s2 i|Φα2. i.. (1.60). α1. Iterating the above steps one can express state |Ψi of (1.56) as shown in Fig. 1.2(c). By √ √ combining the vectors { λ[i−1] , λ[i] } with the tensors {Γ[i] }, we arrive at the standard representation of the MPS in the (1.49). 1.3.4. The canonical form. In the MPS representation of the translationally invariant state A[i] ’s are uniform χ × χ matrices. The correlation length of the MPS can be determined by the eigenvalue spectrum of the completely positive (CP) map [80] acting on the space of χ × χ matrices ε(X) =. X. A[i] XA[i]† .. i. 20. (1.61).

(31) The CP map has spectral radius equal to 1. That means the largest eigenvalue λ1 of ε for a normalized MPS is always to 1. The second largest eigenvalue λ2 of ε determines the correlation length as follows: 1 . (1.62) log(λ2 /λ1 ) P The CP map and the transfer matrix E = i A[i] ⊗ A¯[i] have the same spectrum owing ξ=−. to the relation hβ|ε(|αihα0 |)|β 0 i = hββ 0 |E|αα0 i.. (1.63). The matrices A[i] can be written as a product of χ × χ matrix Γ[i] and a positive, real, diagonal matrix Λ. The matrices Γ and Λ can be chosen to be in the canonical form, that is, they satisfy the following canonical conditions: P (1) i Γ[i]† Λ2 Γ[i] = Iχ , P (2) i Γ[i] Λ2 Γ[i]† = Iχ , (3) 1 is the only fixed point of the operator ε(X) =. P. [i] 2 [i]† i Γ XΛ Γ .. The conditions can be understood from the transfer matrix Eαα0 ,ββ 0 =. X. [i]. [i]. Γαβ Λβ Λβ 0 (Γα0 β 0 )∗ .. (1.64). i. This should have a right eigenvector δββ 0 with eigenvalue λ = 1. 1.3.5. Simple examples. For a clearer understanding of MPS and this notation, we will give some simple examples of these states showing their corresponding matrices. (1) Product state |00 . . . 00i: Using χ = 1 and the physical dimension d = 2, we hve A[i[ (0) = 1, A[i] (1) = 0.. (1.65). (2) AKLT state: Using χ = 2 and the physical dimension d = 3, its representation is given by.   √  2 0 0 1 0 0  , A[i] (−1) =  √ .  , A[i] (1) =  A[i[ (0) =  − 2 0 0 0 0 −1 . . . 21. (1.66).

(32) (3) Majumdar-Gosh model: This is a paradigmatic example of a frustrated 1-D spin 1/2 chain with next nearest-neighbor interaction. The Hamiltonian is H=. X. 2~σi~σi+1 + ~σi~σi+2 .. (1.67). i. The ground state is composed of singlets between the nearest-neighbor spins. The equal weight supposition of singlets between even-odd spins and singlets between odd-even spins is translationally invariant. The MPS representation using χ = 3 is. . 0 1. 0. . . 0 0 0. .      , A[i] (1) =  1 0 0  . A[i[ (0) =  0 0 −1     0 0 0 0 1 0. (1.68). (4) Greenberger-Horne-Zeilinger (GHZ) state |00 . . . 00i + |11 . . . 11i: GHZ states are important because for many entanglement measures they are maximally entangled. It is viewed as a resource for some quantum information protocols. Using χ = 2 and the physical dimension d = 2, we have.  A[i[ (0) = . 1 0 0 0. . .  , A[i] (1) = . 0 0 0 1.  .. (1.69). (5) Cluster states: They are the unique ground state of the three-body interaction z x z i σi σi+1 σi+2 .. P. They are also revelent to one-way quantum computing. Using χ = 2 and. the physical dimension d = 2, Their representation is given by     1 0 0 1  , A[i] (1) =  . A[i[ (0) =  1 0 0 −1. (1.70). This representation is for the 1D system. An interesting question is how to determine the TPS form of the 2D cluster state. These examples are the exact results. Almost all the quantum states can be approximated by the MPS ansatz and we have methods to determine the elements of MPS. In the following, we will focus our attention on a numerical method, called the time-evolving block decimation (TEBD) [81, 82, 83, 84], by using which an accurate ground state wave function can be projected out. 22.

(33) 1.3.6. Time-evolving block-decimation. Several algorithms have been developed to determine the ground state of a Hamiltonian with short range interaction. Some of the algorithms are to minimize the ground state energy [71, 74, 85]. Others are to find the ground state wave function via an imaginary time evolution [81, 82, 83, 84, 86, 87]. We consider a 1D spin system. Our aim is to simulate the evolution of a system with n spins, whose wave function |Ψ0 i, for a time t according to a two-body interaction Hamiltonian H. In the real time |Ψt i = exp(−iHt)|Ψ0 i,. (1.71). and in the imaginary time |Ψτ i =. exp(−Hτ )|Ψ0 i . k exp(−Hτ )|Ψ0 i|. (1.72). In this sense, we must break the evolution operations exp(−iHt) and exp(−Hτ ) into a sequence of local gates using a Suzuki-Trotter expansion [88], and update the MPS after applying the local gate. First, it is convenient to decompose H =. P. h[i,i+1] as a sequence of small two-site. gates F ≡. X. F [i] ≡. even i. G≡. X odd i. X. h[i,i+1] ,. even i. X. [i. G ≡. h[i,i+1] ,. (1.73). odd i. 0. where [F [i] , F [i ] ]=0 for different even i, i0 . For a small δ, the Suzuki-Trotter expansion of order p for exp(−iHt) is e−i(F +G)t = [e−i(F +G)δ ]t/δ ' [fp (UF δ , UGδ )]t/δ ,. (1.74). where UF δ ≡ e−iF δ , UGδ ≡ e−iGδ , f1 (x, y) = xy is the first order term in the expansion, and f2 (x, y) = x1/2 yx1/2 is the second order one. If we expand to higher order terms, the Trotter error will decrease. By using the Suzuki-Trotter expansion, the evolution operators can be expressed as a product of two-body gate. For this reason, the simulation of evolution is achieved by updating the MPS after a set of gates UF δ and UGδ have been applied. If the spin system is in the thermodynamic limit, the action of gates preserve 23.

(34) the invariance of the state under shifts by two sites. Then, only tensor ΓA , λA , ΓB , and λB need to be updated. This algorithm is called infinite TEBD or iTEBD [83, 84]. Vidal(2004) [82] showed how to update the description of the state |Ψi when one- and two- qubit gates are applied. Given a matrix product state |Ψi ( 1.55). In particular, for a one-qubit operator V acting on the i-th spin, it suffices to update the local matrix by [i],s0. [i],s. Γαi−1i αi → Vss0 i Γαi−1i αi .. (1.75). i. The update rule for an application of a two-qubit operator U acting on the neighboring spin i and i+1 (see Fig. 1.3) is as following. First, after the operation U has been applied, we get a large tensor Θα,i,i+1,γ =. [i−1] λα. X X. s0 ,s0i+1 [i],s0i [i] [i+1],s0i+1 Usii,si+1 Γα,β λβ Γβ,γ. . [i+1]. λγ. .. (1.76). s0i ,s0i+1 β. Then we rewrite the state |Ψi into the following |Ψi =. X. Θα,i,i+1,γ |αi|si i|si+1 i|γi.. (1.77). α,γ,si ,si+1. (a). (b) Γ[i+1]. Γ[i] λ [i]. λ [i-1]. λ [i+1]. U i. (c). Θ SVD. α. Y[i+1]. β β i+1α. i i+1. (e). (d) λ [i-1] λ [i-1]. X[i]. λ [i+1]. (λ [i-1])-1. X[i]. Y[i+1]. λ [i+1]. (λ [i+1])-1. Figure 1.3: Update the MPS after the gate U has been applied. We contract the tensor (a) into a tensor Θα,i,i+1,γ (b). We then compute the singular value decomposition of Θ as in (c). By attaching the inverse of Schmidt values in (d). Finally, we update the MPS in (e).. Using the singular value decomposition (SVD), the matrix can be decomposed into P [i] ˜ [i] [i+1] Θα,i,i+1,γ = β X(α,si )β λβ Y(β,si+1 )γ . Here, the truncation error depends on the bond 24.

(35) ˜ [i] and Γ ˜ [i+1] can be obtained from the tensor X and dimension. The updated tensors Γ Y by attaching the inverse of λ[i] and λ[i+1] , i.e., −1 −1 ˜ [i] = λ[i] ˜ [i+1] = λ[i+1] Γ X [i] , Γ Y [i+1] .. (1.78). ˜ and the new tensors Γ ˜ [i] and Γ ˜ [i+1] The decomposition procedure to get the updated λ requires O(d2 χ2 ) computational cost. Here, we need to take care of the Trotter and truncation error. Using the MPS ansatz, we can use imaginary time evolution (1.72) to look for the ground state of the system with local Hamiltonian. In the limit τ → ∞, exp(−Hτ )|Ψ0 i will converge to the ground state. In Ref. [86], they proposed a projection method, called the Iterative Projection method, similar to Vidal method to calculate the ground state wave function with tensor network in 2D system. In this method, the projection operator obtained by using Suzuki-Trotter expansion is applied to a random-generated initial wave function. This method is efficient and accurate. By using the procedure, an accurate ground state can be projected out. 1.3.7. Expectation value. By using iTEBD method, we get a ground state with MPS representation. Here we will discuss how to calculate the expectation value of the operator in MPS representation. Given the state (1.49), we would like to calculate the expectation value of some operator O which is the product of local operators Oi for each site i, O = O1 ⊗ O2 ⊗ . . . ⊗ On . Then expectation value could be represented as d X. hΨ|O|Ψi =. Tr. s1 ,...sn ,s01 ,...s0n =1. = Tr. n Y. n Y. ! [i]. A (si ) T r. i=1. n Y. ! A¯[i] (s0i ). i=1. n Y hs0i |Oi |si i, i=1. ! E [i] (si ) ,. (1.79). i=1. where [i]. E (si ) =. d X. hs0i |Oi |si iA¯[i] (s0i ) ⊗ A[i] (si ).. (1.80). si ,s0i =1. E [i] (si ) denotes the transfer matrix shown in the Fig. 1.4. Then we have. hΨ|O|Ψi = T r[E [1] . . . E [n] ]. 25. (1.81).

(36) A[1]. A[2]. O[1]. [2]. O. ... .... A[i] [i]. O. .... ... .... A[n] E[1]. [n]. O. E[2]. .... E[i]. .... E[n]. .... Figure 1.4: The matrices associated to each spins are represented by the squares and observables are represented by the circles. One calculate the expectation value of the MPS representation by using the transfer matrix E [i] . This contraction can be done efficiently.. Therefore, the calculation of expectation values of products of local observable is reduced to the multiplication of some transfer matrices. This skill can also be used to calculate the geometric entanglement in 1D system. The only difference is the form of the corresponding transfer matrix. 1.3.8. Quantum state renormalization group in the matrix product state. The renormalization group (RG) refers to a mathematical tool to average over short distance degree of freedom, and can be used to demonstrate the property of the universality at the fixed-point. Extending Wilson’s original idea, a more efficient algorithm was proposed, called the density matrix renormalization group (DMRG) algorithm. It has been successful in solving one-dimensional noncritical system by retaining the relevant degrees of freedom. Here, we will introduce another application, called the quantum state renormalization group (QSRG). The RG transformation on the quantum states was introduced by Verstraete et al. [89]. They employed the entanglement as a qualification to do quantum coarse-graining transformation and classify the fixed point states of QSRG. First, they merge two neighboring sites, says i and i + 1, into one new block, and in term of the matrices As it is (pq) A˜αγ. ≡. χ X. Apαβ Aqβγ .. (1.82). β=1. We further perform the singular value decomposition min(d2 ,χ2 ) (pq) A˜αγ. ≡. X l=1. 26. †(pq). Ul. l λV(αγ) ,. (1.83).

(37) where (pq) and (αγ) mean the combined indices. Here, one keeps the relevant degrees of n. freedom. By performing n coarse-graining steps, each local site is d2 -dimensional and corresponds to 2n original spins. However, for the noncritical spin chains, the block entanglement entropy will be saturated to a finite value. This shows only few degrees of freedom couple to the outer sites. From a quantum information perspective, there will exn. ist a unitary transformation that could transform the d2 -dimensional space into a smaller one. These relevant degrees of freedom will correspond to the eigenvalues of the block reduced density matrix. In MPS representation, the dimension of Hilbert spaces of the block remains bounded above by χ2 as can be seen from the singular value decomposition. In this way, it is possible to perform the coarse-graining without any truncation even at the critical point where the effective correlation length of the MPS is close to χ2 . By considering equivalence classes, one ignores the local unitary operations and select a new representation Ap −→RG A0l = λl V l .. (1.84). In summary, under one-step RG transformation of merging two neighbors sites shown in the Fig. 1.5, the state |ψi is transformed to ˜ = U1,2 ⊗ U3,4 ⊗ ... ⊗ U2i−1,2i ⊗ ...|ψi. |ψi → |ψi. (1.85).  U12. U34. U56. U78 RG transformation. . Figure 1.5: The coarse-graining of the matrix product states. Merge two sites to forms a new site.. However, it can also be used to find the fixed point of the quantum states. The transfer matrix E uniquely determines the MPS up to the local unitary operator in the physical system. It provides a way to classify the MPS under local unitary transformation by studying the E∞ = limn→∞ E n , the largest eigenvalue of E will be non-degenerate and the magnitude is 1 or smaller than 1. That is to say, the second largest eigenvalue 27.

(38) will decay exponentially along the coarse-graining steps. Otherwise, its left and right eigenvectors will have a maximal Schmidt rank. Let us consider the AKLT state. The transfer matrix E has singular values as follows: { √312 , √112 , √112 , √112 } with no degeneracy in the largest eigenvalues. The fixed point will be given by the eigenvector corresponding to the largest singular value. Thus, E∞ = P2 |ΦR ihΦL |, its left and right eigenvector are i=1 |iii which is a maximally entangled states. In the cluster state, the fixed point is obtained after one step of coarse-graining. P2 The transfer matrix is E 2 = i,j=1 12 |iiihjj| and has rank 1.. 1.4. Tensor product state. Novel ideals from quantum information have guided a series of new simulation algorithms based on an efficient representation of the many-body wave function in higher dimensions through a tensor network or tensor product. This class contains states such as projected entangled pair state (PEPS) [71], tree tensor state (TNS) [72], and multiscale renormalization ansatz (MERA) [90, 91]. For example, projected entangled pair state is the generalization of MPS to two or higher dimensions, motivated by the quantum information perspective. Additionally, before the tensor netork was used in the quantum system, the tensor product representations of the partition functions for classical lattice models with local interactions have been used in statistical physics [92]. We should emphasize that the tensor product states is not yet as well established as the one-dimensional MPS or DMRG methods. This is mainly due to its bigger complexity, but there are some ways to improve it. 1.4.1. The basic concepts of the tensor product state. In the past few years, tensor product states (TPS) have been used in many studies on lattice systems, for example, the infinite projected entangled-pair states algorithm (iPEPS) [71, 93, 94]. TPS has been proven to work well for the Heisenberg antiferromagnet [86, 95, 96], hard-core boson in a two-dimensional optical lattice [93], the quantum orbital compass model [97], and some frustrated spin-systems [98] beyond the reach of quantum Monte Carlo methods. Even spin state at finite temperature could be approximated. Some studies also extend PEPS to the fermionic system with local interactions, called fPEPS [99, 100]. Furthermore, PEPS is more complicated than MPS, it is shown 28.

(39) that there exist PEPSs with power law decaying correlations [101]. The PEPSs exhibit rich properties, indeed. It has been shown how to regard 2D cluster state as a PEPS, which can be used in the measurement based quantum computation [102].. Ar ,d ,l ,u ( si , j ). u l. r si,j. d (c). (b). (a). Figure 1.6: (a)The bond represents a pair of maximally entangled state, and the circle denotes projector. (b) Representation of the tensor corresponding to a single site. The letters r,d,l,u are the bond indices, and s is the physical index. (c) Representation of the whole state.. Here, we take a 2D square lattice as an example, and the TPSs in other geometric figures can also be constructed by the same way. Following MPS representation, we can assign four virtual spins, ai,j , bi,j ,ci,j and di,j ,of dimension χ, and assume that each pair of Pχ neighboring virtual spins is in the maximally entangled state |φi = α=1 |α, αi as shown in Fig. 1.6 (a). Then, by applying a projection operator that maps four-virtual-spin space to the space of the physical spin at each site, we can obtain the state |Ψi as follows: |Ψi = P1,1 ⊗ . . . ⊗ PN,N |φi ⊗ . . . ⊗ |φi X = tT r[A[1,1] (s1,1 )A[1,2] (s1,2 ) . . . A[N,N ] (sN,N )]|s1,1 , s1,2 , . . . , sN,N i. s1,1 ,s1,2 ,...,sN,N. (1.86) Here, the map at the site (i,j) is Pi,j =. X. [i,j]. Ar,d,l,u (si,j )|si ihr, d, l, u|,. (1.87). si ,r,d,l,u. where the values of the physical spins si,j = 1, · · · ds for i, j = 1, . . . , N , the notation tT r [i,j]. denotes the sum over all indices of the site tensors, and Ar,d,l,u (si,j )’s (see Fig. 1.6 (b)) 29.

(40) are rank-five tensors with the bond indices r, d, l, u = 1, · · · , χ. We again call χ the bond dimension and ds the physical dimension. This construction can be generalized to any lattice shape and one can see that any state can be written as a TPS if they allow the bond dimension to become very large. In 2D system, the entropy is proportional to the area of the block, which counts the bonds that connect this block with the rest of the system. 1.4.2. The imaginary time evolution with the tensor product state. Let us move to describe how a time-evolution can be simulated on a TPS. We will assume a 2D square lattice and the Hamiltonian only couples nearest neighbors. First, an arbitrary state |Ψ0 i with physical dimension d = 2 and virtual dimension χ is chosen as the starting state. The evolution operators could be the real-time evolution e−iHt or the imaginarytime one e−Ht .. A. λ1. B. λ7. D. λ2. λ3 λ5. λ8. C λ4. λ6 Hl. Hu Hr Hd. Figure 1.7: The lattice is divided into four sublattices, represented by the black, green, blue and red circles. On each bonds, there is an associated diagonal matrix, λi , i = 1, 2, ..., 8.. This is done by performing the Trotter approximation. First, the Hamiltonian H are decomposed along the horizontal and vertical directions. The component Hamiltonians are labelled as even and odd depending on whether the operator is action on eve-odd or odd-even sites. The Hamiltonian can then be decomposed into four components shown in the Fig. 1.7: H = Hr + Hd + Hl + Hu .. (1.88). These terms commute with each other. Using the Trotter approximation, the time-. 30.

(41) evolution operator e−Ht can be written e−Hδt ≈ e−Hr δt e−Hd δt e−Hl δt e−Hu δt + O(δt2 ).. (1.89). This is similar to 1D case. Each evolution operator equals a product of two particle operators acting on neighboring sites. In our example, we assume the tensor product state contains four kinds of tensors and eight bonds. In the first step, the operators e−Hr δt is done by merging two neighbors spins that connected by the horizontal bonds. The χ3 d × χ3 d matrix can be defined by XX 0 p Θd0 l0 u0 s0 ,dlus = hs0 s|e−Hr δλt |p0 piλ7 λ2 λ8 Aprd0 l0 u0 λ1 Bruld λ2 λ3 λ4 . p0 p. (1.90). r. Taking the singular value decomposition for this matrix, this matrix can be written as 0. p ˜ Tp Θd0 l0 u0 s0 ,dlus = Urd 0 l0 u0 λ1 Vruld ,. (1.91). where U and V are two unitary matrices and λ˜1 is a positive diagonal matrix of the dimension χ3 d. Then we truncate this dimension by keeping χ largest singular values of λ˜1 and update the tensor A and B, i.e., 0. 0. −1 −1 p Apr0 d0 l0 u0 = λ−1 2 λ7 λ8 Ur0 d0 l0 u0 ,. (1.92). p −1 −1 p Bruld = λ−1 2 λ3 λ4 Vruld .. (1.93). The remaining steps can be done similarly with e−Hd δt , e−Hl δt , and e−Hu δt . However, as compared with MPS, PEPS is hard to deal with: for example, evaluating the expectation values of local observables. 1.4.3. Tensor reorganization group. We now show how to determine the expectation values of operators in the state |Ψi given in the form of (1.86). This is done by contracting all the bond indices to determine Q hΨ|O|Ψi and hΨ|Ψi. For any operations O = i,j Oi,j its expectation value is formally given by hΨ|O|Ψi = tT r[T [O1,1 ] . . . T [ON,N ] ], where the rank-four tensor T Oi,j is X [i,j] [i,j] [T Oi,j ]rr0 ,dd0 ,ll0 ,uu0 = hsi,j |Oi,j |s0i,j iAr,d,l,u (si,j ) ⊗ Ar0 ,d0 ,l0 ,u0 (s0i,j ). si,j s0i,j. 31. (1.94). (1.95).