Title Page
Contents
Abstract 12
요약 13
Chapter Ⅰ. Introduction 14
Chapter Ⅱ. Backgrounds 18
1. Transverse-field Ising chain 18
1.1. Basic properties of TFIC 19
1.2. Kramers-Wannier duality 22
1.3. Jordan-Wigner transformation 24
1.4. Universality of TFIC 28
2. Random transverse-field Ising chain 28
2.1. SDRG transformation 30
2.2. Flow equation 35
2.3. Infinite randomness fixed point(IRFP) 37
3. Previous studies of the two-dimensional random models 40
Chapter Ⅲ. Methods 44
1. Quantum-to-classical mapping 44
1.1. Suzuki-Trotter approximation 44
1.2. Equivalence between d-dimensional TFIMand (d+1)-dimensional anisotropic Ising model 49
2. Continuous-time cluster algorithm 50
2.1. Description of algorithm 50
2.2. Observables 52
3. Finite-size scaling analysis 55
Chapter Ⅳ. Results 62
1. 2D RTFIF 62
2. 2D McCoy-Wu model 69
Chapter Ⅴ. Discussion 76
Bibliography 80
Table Ⅱ-1. Summary of previous studies on the 2D RTFIF. The last column has been calculated from β=νη/2 [14]. This table contains for same distribution shown in... 41
Table Ⅱ-2. Critical exponents from the 2D random contact process [40], which is believed to belong to the 2D RTFIF universality class. ν⊥ represents the spatial correlation...[이미지참조] 42
Table Ⅱ-3. Earlier results on the 2D McCoy-Wu model. In the first row, I write the same value for Ψ as in Table Ⅱ-1, as well as for η, because these exponents have not... 43
Table Ⅱ-4. The QMC studies in this table are parametrized with K ≈ (1/2) ln coth(△τΓ), which becomes exact in the limit of infinite imaginary-time slices, i.e., △τ → 0,... 43
Table Ⅳ-1. Summary of estimated values for the 2D RTFIF. 69
Table Ⅳ-2. Summary of the results for the 2D McCoy-Wu model. 74
Figure Ⅱ-1. Schematic diagram of the dual lattice. Each dual variable τi is located between spin i and i + 1.[이미지참조] 22
Figure Ⅱ-2. Domain wall configuration with dual variable. τix=−1 implies there is a kink between spin i and i + 1.[이미지참조] 23
Figure Ⅱ-3. Physical meaning of the τiz. It creates or annihilates a kink between spin i and i + 1.[이미지참조] 23
Figure Ⅱ-4. Energy spectrum of the TFIC in momentum space in (a) g=0.5, (b) g=gc=1, (c) g=1.5.[이미지참조] 29
Figure Ⅱ-5. schematic diagram of site-decimation process of the strong-disorder renormalization group transformation for RTFIC. 32
Figure Ⅱ-6. schematic diagram of bond-decimation process of the strong-disorder renormalization group transformation for RTFIC. 35
Figure Ⅲ-1. Schematic representation of a single quantum spin in the continuous-time cluster algorithm. Solid line represents a σi=1 segment, vice versa.[이미지참조] 50
Figure Ⅲ-2. Magnetization and single-time magnetization for the two-dimensional TFIM with L=8 at Γ=3.0440. 53
Figure Ⅲ-3. Comparison of magnetization between quantum Monte Carlo and exact diagonalization of two-dimensional TFIM with L=4 at b=300. 54
Figure Ⅲ-4. Binder's cumulant of the two-dimensional TFIM for the different system size L at the off-critical region (a) Γ=3.00 and (b) Γ=3.08. 58
Figure Ⅲ-5. Binder's cumulant of the two-dimensional TFIM for the different system size L at the Γc=3.0440. Dynamic critical exponent z=1 is used. Dashed...[이미지참조] 59
Figure Ⅲ-6. Scaling collapse of the (a) single-time magnetization and (b) magnetization of the two-dimensional TFIM for different system size L at the Γc=3.0440....[이미지참조] 61
Figure Ⅳ-1. Binder's cumulant of 2d RTFIF at (a) Γ=7.51, (b) Γ=7.52, and (c) Γ=7.54. Gray line indicates U* ≈ 0.33 predicted in Ref. 64
Figure Ⅳ-2. Scaling plot of the single-time magnetization at b ~ 4 × 10³ with respect to system size. 65
Figure Ⅳ-3. Estimation of sub-leading correction exponent with magnetization. Solid line indicates the fitting curve with Equation (IV.4) with A=0.38(6), B=0.67(32),... 66
Figure Ⅳ-4. Estimation of correlation length exponent ν=1.6(3). 67
Figure Ⅳ-5. Test for dynamic scaling. In (a), I employ conventional scaling form from Equation (IV.6) and estimate z=3.3(3). (b) And with activated scaling form from... 68
Figure Ⅳ-6. Scaling collapse for the single-time magnetizaiton. Same exponents were used with Figure Ⅳ-5. 70
Figure Ⅳ-7. Binder's cumulant of the 2D McCoy-Wu model for different system size at (a) Γ=1.56, (b) Γ=1.57, and (c) Γ=1.58 in units of J. 72
Figure Ⅳ-8. Estimation of ν for 2D McCoy-Wu model. ν=0.95(2) is used in this plot. 73
Figure Ⅳ-9. Estimation of β/ν using Equation (IV.1). 73
Figure Ⅳ-10. (a) Conventional and (b) activated dynamic scaling for the 2D McCoy-Wu model for 2D McCoy-Wu model. 75
Figure Ⅴ-1. Comparison of b~L³·³ and b~exp(L⁰·⁵) within the range of system sizes used in this work. 77