Numerical Renormalization Group Calculations for Impurity ...

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24 3. Numerical Renormalization Group Approach The truncated Hamiltonian gives a valid result (low-lying spectrums {E (N) n }) only if the off-diagonal elements for |n − n ′ | > N s are negligibly small compared to the diagonal ones: for |n − n ′ | > N s . Equivalently, t N−1 〈φ (N−1) n ′ 〈ψ (N) n ′ m ′ |H N |ψ (N) nm 〉 ≪ 〈ψ(N) nm |H N|ψ (N) nm 〉 (3.40) |f † N−1σ |φ(N−1) n 〉〈m ′ |f Nσ |m〉 ≪ E n (N−1) + ε N 〈m|f † N f N|m〉 (3.41) for |n − n ′ | > N s . Now, we check the order of magnitude of each element in Eq. (3.41). The first result in Eq. (3.32) tells us, for a given N, t N−1 and ε N are same in order of magnitude: t N , ε N ∼ Λ −N/2 . (3.42) Two new elements, 〈m|f † Nσ f Nσ|m〉 and 〈m ′ |f (†) Nσ |m〉, are order of unity: 〈m|f † Nσ f Nσ|m〉 ∈ {0, 1, 2}, (3.43) 〈m ′ |f Nσ |m〉 ∈ {0, 1}. To make simple explanation, we replace E n (N−1) to E (N−1) 1 and check the inequality (3.41). Since E (N−1) 1 corresponds to the first excitation-energy of the Hamiltonian H 4 N−1 , its energy-scale has to be similar to ε N−1 and t N−1 in order of magnitude. E (N−1) 1 ∼ Λ −N/2 (3.44) From Eq. (3.42), Eq. (3.44) and Eq. (3.44), we can conclude that the inequality (3.41) is satisfied (or truncation is allowed) if we can find an integer N s smaller than M such that 〈φ (N−1) n |f † ′ N−1,σ |φ(N−1) n 〉 ≪ 1, (3.45) for N s < |n − n ′ | < M. To find a proper N s , we use the result in Eq. (3.33). The finite summation in Eq. (3.33) begins with m = i and stops at m = f, which means f N−1,σ involves the single-particle(hole) operators, a mσ (b mσ ), with energy ξ m smaller than ξ i and larger than ξ f . Thus, the transition amplitude 〈φ (N−1) n ′ for ξ f < ξ m < ξ i for i < m < f (3.46) |E (N−1) n ′ |f † N−1,σ |φ(N−1) n 〉 becomes effectively zero − E n (N−1) | > ξ (N−1) i (3.47) 4 To be precise, E (N−1) 1 is the energy-difference between the ground state and the first excited one.
3.3. Flow diagrams and Fixed points 25 and the subspace of the full Hamiltonian H N with energy 0 ≤ E (N−1) n ≤ ξ (N−1) i can be effectively described by a truncated Hamiltonian H N (N s l ×N s l) where N s is defined to satisfy E (N−1) N s ≈ 2ξ i . (3.48) The energy cutoff ξ (N−1) i for large N(> 10), of the operator f N−1,σ shows exponential decrease and ξ (N−1) i ∝ Λ −N/2 . 5 (3.49) If Λ is very close to 1, the cut-off does not change so much with iterations but stays at the initial cut-off (band width) D and there is very little room for truncation. A large Λ(≫ 1) makes computation easy but we lose too much information with logarithmic discretization (or p = 0 approximation). 6 Optimal values of Λ can be different according to the kind of models and also to the type of physical properties to be calculated. For examples, physics at the ground-state are usually obtained with a large value of Λ(∼ 5) whereas relatively small Λ(∼ 2) is demanded to investigate temperature-dependence of (thermo)dynamical quantities. 3.3 Flow diagrams and Fixed points In analytic RG approach, the renormalization group is a mapping R of a Hamiltonian H(K), which is specified by a set of interaction parameters or couplings K = (K 1 , K 2 , ...) into another Hamiltonian of the same form with a new set of coupling parameters K ′ = (K 1 ′, K′ 2 , ...). This is expressed formally by R{H(K)} = H(K ′ ), (3.50) or equivalently, R{K} = K ′ . (3.51) In applications to critical phenomena the new Hamiltonian is obtained by removing short range fluctuations to generate an effective Hamiltonian valid over larger length scales. The transformation is usually characterized by a parameter, say α, which specifies the ratio of the new length or energy scale to the old one. A sequence of transformations, K ′ = R α (K), K ′′ = R α (K ′ ), K ′′′ = R α (K ′′ ), etc. (3.52) generates a sequence of points or, where α is a continuous variable, a trajectory in the parameter space K. 5 In actual calculations, truncation is controlled by keeping the number of states constant. 6 see Section 3.2.1
Page 1 and 2: Numerical Renormalization Group Cal
Page 3: i ACKNOWLEDGMENTS I would like to e
Page 6 and 7: iv Contents 6.2.3 Quantum critical
Page 8 and 9: 2 1. Overview in Chapter 6 indicate
Page 10 and 11: 4 2. Introduction to Quantum Phase
Page 22 and 23: 16 3. Numerical Renormalization Gro
Page 38 and 39: 32 4. Soft-Gap Anderson Model 0.06
Page 40 and 41: 34 4. Soft-Gap Anderson Model model
Page 42 and 43: 36 4. Soft-Gap Anderson Model 4.0 3
Page 44 and 45: 38 4. Soft-Gap Anderson Model 0 1 2
Page 46 and 47: 40 4. Soft-Gap Anderson Model r and
Page 48 and 49: 42 4. Soft-Gap Anderson Model ; see
Page 50 and 51: 44 4. Soft-Gap Anderson Model 1.5 1
Page 52 and 53: 46 4. Soft-Gap Anderson Model Eq. (
Page 54 and 55: 48 4. Soft-Gap Anderson Model
Page 56 and 57: 50 5. Spin-Boson Model compared to
Page 58 and 59: 52 5. Spin-Boson Model α c 1.2 0.8
Page 60 and 61: 54 5. Spin-Boson Model 4 α=0.01 α
Page 62 and 63: 56 5. Spin-Boson Model s=0.8. ∆=0
Page 64 and 65: 58 5. Spin-Boson Model 1 QCP D E N
Page 66 and 67: 60 5. Spin-Boson Model
Page 68 and 69: 62 6. Bosonic Single-Impurity Ander
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74 6. Bosonic Single-Impurity Ander
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76 6. Bosonic Single-Impurity Ander
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78 7. Summary as the most important
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81 A. FIXED-POINTS ANALYSIS: SOFT-G
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A.1. Local moment fixed points 83
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A.1. Local moment fixed points 85 a
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A.1. Local moment fixed points 87 3
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A.1. Local moment fixed points 89 C
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A.2. Details of the Perturbative An
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97 B. THERMODYNAMICS IN THE OHMIC S
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99 0.8 0.6 a S imp (T) 0.4 0.2 α=1
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1/2. The agreement with the exact r
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103 C. BEC OF AN IDEAL BOSONIC GAS
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105 D. DETAILS ABOUT THE ITERATIVE
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Bibliography 107 BIBLIOGRAPHY Affle
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Bibliography 109 Georges, A., Kotli
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Bibliography 111 Vojta, M. and Frit
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CURRICULUM VITAE Name: Hyun-Jung Le
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LIST OF PUBLICATIONS [1] H.-J. Lee
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