Numerical Renormalization Group Calculations for Impurity ...

More documents

Recommendations

Info

40 4. Soft-Gap Anderson Model r and in (Wilson 1975) for r = 0. Whereas an anti-ferromagnetic exchange coupling is marginally relevant in the gap-less case (r = 0), it turns out to be irrelevant for finite r; see (Gonzalez-Buxton and Ingersent 1998). This is of course consistent with the scaling analysis of Section 4.2: the operator (4.22) shows that it decreases as Λ −Nr/2 Λ −N/2 = Λ −N(r+1)/2 with increasing N. Consequently, we have to choose f(N) = Λ Nr/2 . (4.24) This result also directly follows from dim[J]= −r: as the NRG discretization yields a decrease of the running energy scale of Λ −N/2 , the S ⃗ imp · ⃗s 0 term in Eq. (4.22) scales as Λ −Nr/2 . The function f(N) is now simply chosen to compensate this effect using Eq. (4.24) scales as Λ −Nr/2 . The function f(N) is now simply chosen to compensate this effect; using Eq. (4.24) the operator H N ′ becomes exactly marginal. Now we turn to a discussion of the many-body spectrum. The relevant ground state of the effective model for the LM fixed point consists of the filled impurity level (with one electron with either spin ↑ or ↓) and all the conduction electron states with p < 0 filled with both ↑ and ↓, as shown in Fig. A.1. Let us now focus on excitations with energy ɛ 1 +ɛ 2 measured with respect to the ground state. (For more subspaces with different excitation-energy, refer Appendix A.1.) Fig. 4.6 shows one such excitation; in this case, one electron with spin ↓ is removed from the p = −3 level and one electron with spin ↓ is added to the p = 1 level. The impurity level is assumed to be filled with an electron with spin ↑, so the resulting state has Q = 0 and S z = +1/2. In total, there are 32 states with excitation energy ɛ 1 + ɛ 3 . These states can be classified using the quantum numbers Q, S, and S z . Here we consider only the states with quantum numbers Q = 0, S = 1/2, and S z = 1/2 (with excitation energy ɛ 1 +ɛ 3 ) which form a four-dimensional subspace. As the state shown in figure 4.6 is not an eigenstate of the total spin S, we have to form proper linear combinations to obtain a basis for this subspace; this basis can be written in the form |ψ 3 〉 = 1 √ 2 f † ↑ (ξ† 1↑ ξ −3↑ + ξ † 1↓ ξ −3↓)|ψ 0 〉 |ψ 4 〉 = 1 √ 6 f † ↑ (ξ† 1↑ ξ −3↑ − ξ † 1↓ ξ −3↓)|ψ 0 〉 + 2 √ 6 f † ↓ ξ† 1↑ ξ −3↓|ψ 0 〉 (4.25) |ψ 5 〉 = 1 √ 2 f † ↑ (ξ† 3↑ ξ −1↑ + ξ † 3↓ ξ −1↓)|ψ 0 〉 |ψ 6 〉 = 1 √ 6 f † ↑ (ξ† 3↑ ξ −1↑ − ξ † 3↓ ξ −1↓)|ψ 0 〉 + 2 √ 6 f † ↓ ξ† 3↑ ξ −1↓|ψ 0 〉 where the state |ψ 0 〉 is given by the product of the ground state of the conduction
4.3. Structure of the quantum critical points 41 electron chain and the empty impurity level: |ψ 0 〉 = |0〉 imp ⊗ [ ∏ ξ † p↑ ξ† p↓ |0〉 cond] . (4.26) p
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 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
Page 74 and 75: '& &' '& '& &' '&
Page 84 and 85: 78 7. Summary as the most important
Page 87 and 88: 81 A. FIXED-POINTS ANALYSIS: SOFT-G
Page 89 and 90: A.1. Local moment fixed points 83
Page 91 and 92: A.1. Local moment fixed points 85 a
Page 93 and 94: A.1. Local moment fixed points 87 3
Page 95 and 96: A.1. Local moment fixed points 89 C
Page 97 and 98:
A.2. Details of the Perturbative An
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
97 B. THERMODYNAMICS IN THE OHMIC S
Page 105 and 106:
99 0.8 0.6 a S imp (T) 0.4 0.2 α=1
Page 107 and 108:
1/2. The agreement with the exact r
Page 109 and 110:
103 C. BEC OF AN IDEAL BOSONIC GAS
Page 111 and 112:
105 D. DETAILS ABOUT THE ITERATIVE
Page 113 and 114:
Bibliography 107 BIBLIOGRAPHY Affle
Page 115 and 116:
Bibliography 109 Georges, A., Kotli
Page 117 and 118:
Bibliography 111 Vojta, M. and Frit
Page 119:
CURRICULUM VITAE Name: Hyun-Jung Le
Page 123:
LIST OF PUBLICATIONS [1] H.-J. Lee
show all

Numerical Renormalization Group Calculations for Impurity ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?