Numerical Renormalization-Group computation 
of magnetic relaxation rates 
Krissia de Zawadzki, Luiz Nunes de Oliveira, Jo...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Radius of Kondo screening cloud 
Radius of Kon...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Radius of Kondo screening cloud 
Radius of Kon...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Radius of Kondo screening cloud 
Radius of Kon...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Radius of Kondo screening cloud 
Radius of Kon...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Radius of Kondo screening cloud 
Radius of Kon...
ndings: 
Yes, we can! 
T dependence changes as probe 
crosses 푅퐾 
Phase of low-푇 Friedel oscillations 
also changes 
LASZL...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
The quantum system 
NRG Probe 
Single-impurity...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
The quantum system 
NRG Probe 
Single-impurity...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
The quantum system 
NRG Probe 
Single-impurity...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
The quantum system 
NRG Probe 
Single-impurity...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Two-center basis 
Two-center basis 
Sphericall...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Two-center basis 
Two-center basis 
Sphericall...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Two-center basis 
Two-center basis 
Sphericall...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
NRG and Lanczos basis 
NRG and Lanczos basis 
...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
NRG and Lanczos basis 
NRG and Lanczos basis 
...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
NRG and Lanczos basis 
NRG and Lanczos basis 
...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
NRG and Lanczos basis 
NRG and Lanczos basis 
...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
NRG and Lanczos basis 
NRG and Lanczos basis 
...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Friedel oscillations 
Friedel oscillations 
0....
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Relaxation rate - temperature dependence 
푘퐵푇퐾...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Interference relaxation rate 
Interference rel...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Friedel oscillations 
Friedel oscillations 
1....
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Conclusions 
NMR to measure 푅퐾 : OK! 
Inside c...
Introduction NRG calculations Numerical results Conclusions Acknowledgment 
Acknowledgment 
Thank you! 
Zawadzki, K. de; O...
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 4
Additional results 
Relaxation rate - 풢푠푖푑푒(푇) profile 
7 
6 
5 
T1 ´ 
kB T ³1 
4 
3 
2 
1 
10-10 10-9 10-8 10-7 10-6 10-5...
Additional results 
Relaxation rate - 풢푆퐸푇 (푇) profile 
7 
6 
5 
T1 ´ 
kB T ³1 
4 
3 
2 
1 
10-10 10-9 10-8 10-7 10-6 10-5...
Additional results 
Particle-hole symmetric case 
Particle-hole symmetric case: 1/푇1 as function of 푇 
푘퐹푅 = 푛휋 and 푘퐹푅 = ...
Additional results 
Particle-hole symmetric case 
Particle-hole symmetric case: 1/푇1 as function of 푇 
푘퐹푅 = 푛휋 and 푘퐹푅 = ...
Additional results 
Particle-hole symmetric case 
Particle-hole symmetric case: 1/푇1 as function of 푇 
푘퐹푅 = 푛휋 and 푘퐹푅 = ...
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Numerical Renormalization Group computation of magnetic relaxation rates

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We report an essentially exact numerical renormalization-group (NRG)
computation of the temperature-dependent NMR rate $1/T_1$ of a probe at a
distance $R$ from a magnetic impurity in a metallic host. We split the metallic
states into two subsets, A and B. The former comprises electrons $a_k$ in
$s$-wave states about the magnetic-impurity site. The coupling between the
$a_k$ band and the impurity is described by the Anderson Hamiltonian,
diagonalizable by the NRG procedure. Each state $b_k$ in the B subset is a
linear combination of an $s$-wave state about the probe site with the
degenerate $a_k$, constructed to be orthogonal to all the $a_k$'s. The
$b_k$ band hence decouples from the impurity and is analytically treatable. We
show that the relaxation rate has three components: (i) a constant associated
with the $b_k$'s; (ii) a $T$-dependent term associated with the
$a_k$'s, which decays in proportion to $1/(k_FR)^2$, where $k_F$ is the
Fermi momentum; and (iii) another $T$-dependent term due to the interference
between the $a_k$'s and the $b_k$'s. The interference term shows
Friedel oscillations whose amplitude, proportional to $1/k_FR$, can be mapped
onto the universal function of $T/T_K$ describing the Kondo resistivity. We
compare our findings with results in the literature.

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Numerical Renormalization Group computation of magnetic relaxation rates

  1. 1. Numerical Renormalization-Group computation of magnetic relaxation rates Krissia de Zawadzki, Luiz Nunes de Oliveira, Jose Wilson M. Pinto Instituto de F´ısica de S˜ao Carlos - Universidade de S˜ao Paulo Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 11
  2. 2. Introduction NRG calculations Numerical results Conclusions Acknowledgment Radius of Kondo screening cloud Radius of Kondo screening cloud 푅푘 LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
  3. 3. Introduction NRG calculations Numerical results Conclusions Acknowledgment Radius of Kondo screening cloud Radius of Kondo screening cloud 푅푘 푅퐾 ∝ 푇−1 퐾 General consensus 푅퐾 = ~푣퐹 /푘퐵푇퐾 Boyce Slichter NMR: LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
  4. 4. Introduction NRG calculations Numerical results Conclusions Acknowledgment Radius of Kondo screening cloud Radius of Kondo screening cloud 푅푘 푅퐾 ∝ 푇−1 퐾 General consensus 푅퐾 = ~푣퐹 /푘퐵푇퐾 Boyce Slichter NMR: LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
  5. 5. Introduction NRG calculations Numerical results Conclusions Acknowledgment Radius of Kondo screening cloud Radius of Kondo screening cloud 푅푘 푅퐾 ∝ 푇−1 퐾 General consensus 푅퐾 = ~푣퐹 /푘퐵푇퐾 Boyce Slichter NMR: Experimental arrangement: NMR probe: 푅 from the impurity NRG computation of the spin lattice relaxation rate 1/(푇1푇) as function of 푇 and 푅 LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
  6. 6. Introduction NRG calculations Numerical results Conclusions Acknowledgment Radius of Kondo screening cloud Radius of Kondo screening cloud 푅푘 푅퐾 ∝ 푇−1 퐾 General consensus 푅퐾 = ~푣퐹 /푘퐵푇퐾 Boyce Slichter NMR: Experimental arrangement: NMR probe: 푅 from the impurity NRG computation of the spin lattice relaxation rate 1/(푇1푇) as function of 푇 and 푅 Can we measure 푅퐾 via NMR? Our
  7. 7. ndings: Yes, we can! T dependence changes as probe crosses 푅퐾 Phase of low-푇 Friedel oscillations also changes LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
  8. 8. Introduction NRG calculations Numerical results Conclusions Acknowledgment The quantum system NRG Probe Single-impurity Anderson model 퐻 = 퐻푐표푛푑 ⏞Σ︁ ⏟ k 휀k푐† k푐k 휀 = 푣퐹 퐷 (푘 − 푘퐹 ) +퐷 −퐷 푘퐹 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
  9. 9. Introduction NRG calculations Numerical results Conclusions Acknowledgment The quantum system NRG Probe Single-impurity Anderson model 퐻 = 퐻푐표푛푑 ⏞Σ︁ ⏟ k 휀k푐† k푐k + 퐻푑 ⏞ ⏟ 휀푑푐† 푑푐푑 + 푈푛푑↑푛푑↓ 휀 = 푣퐹 퐷 (푘 − 푘퐹 ) +퐷 −퐷 푘퐹 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
  10. 10. Introduction NRG calculations Numerical results Conclusions Acknowledgment The quantum system NRG Probe Single-impurity Anderson model 퐻 = 퐻푐표푛푑 ⏞Σ︁ ⏟ k 휀k푐† k푐k + 퐻푑 ⏞ ⏟ 휀푑푐† 푑푐푑 + 푈푛푑↑푛푑↓ + 퐻푖푛푡 ⏞√︂ ⏟ Γ 휋 (푓† 0 푐푑 + 퐻.푐.) 휀 = 푣퐹 퐷 (푘 − 푘퐹 ) +퐷 −퐷 푘퐹 푓0 = 1 √ 휌 Σ︁ k 푐k Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
  11. 11. Introduction NRG calculations Numerical results Conclusions Acknowledgment The quantum system NRG Probe Single-impurity Anderson model 퐻 = 퐻푐표푛푑 ⏞Σ︁ ⏟ k 휀k푐† k푐k + 퐻푑 ⏞ ⏟ 휀푑푐† 푑푐푑 + 푈푛푑↑푛푑↓ + 퐻푖푛푡 ⏞√︂ ⏟ Γ 휋 (푓† 0 푐푑 + 퐻.푐.) 퐻푝푟표푏푒 = −퐴 [︁ Ψ† ↑(⃗푅 )Ψ↓(⃗푅)퐼− + 퐻.푐. ]︁ Ψ휇 = Σ︁ k 푒푖k.R푐k 1 푇1 = 4휋 ~ Σ︁ 퐼,퐹 푒−훽퐸퐼 |⟨퐼|퐻푝푟표푏푒|퐹⟩|2훿(퐸퐼 − 퐸퐹 ) 푓0 = 1 √ 휌 Σ︁ k 푐k 푅 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
  12. 12. Introduction NRG calculations Numerical results Conclusions Acknowledgment Two-center basis Two-center basis Spherically symmetric operators 푐휀 = Σ︁ k 푐 k 훿(휀 − 휀k) (around impurity) 푑휀 = Σ︁ k 푐 k 푒푖k.R훿(휀 − 휀k) (around probe) 푐휀 푑휀 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
  13. 13. Introduction NRG calculations Numerical results Conclusions Acknowledgment Two-center basis Two-center basis Spherically symmetric operators 푐휀 = Σ︁ k 푐 k 훿(휀 − 휀k) (around impurity) 푑휀 = Σ︁ k 푐 k 푒푖k.R훿(휀 − 휀k) (around probe) 푐휀 푑휀 휀, 푑휀′} = sin(푘푅) {푐† 푘푅 훿(휀 − 휀′) Gram-Schmidt construction 푐¯휀휇 = √ 1 1−푊2 (푑휀휇 −푊푐휀휇) 푊 = 푊(휀,푅) = sin(푘푅) 푘푅 푘푅 = 푘퐹푅 (︀ 1 + 휀 퐷 )︀ Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
  14. 14. Introduction NRG calculations Numerical results Conclusions Acknowledgment Two-center basis Two-center basis Spherically symmetric operators 푐휀 = Σ︁ k 푐 k 훿(휀 − 휀k) (around impurity) 푑휀 = Σ︁ k 푐 k 푒푖k.R훿(휀 − 휀k) (around probe) 푐휀 푑휀 NRG analytical 휀, 푑휀′} = sin(푘푅) {푐† 푘푅 훿(휀 − 휀′) Gram-Schmidt construction 푐¯휀휇 = √ 1 1−푊2 (푑휀휇 −푊푐휀휇) 푊 = 푊(휀,푅) = sin(푘푅) 푘푅 푘푅 = 푘퐹푅 (︀ 1 + 휀 퐷 )︀ Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
  15. 15. Introduction NRG calculations Numerical results Conclusions Acknowledgment NRG and Lanczos basis NRG and Lanczos basis 퐻푁 = 1 풟푁 (︃ 푁Σ︁−1 푛=0 푡푛(푓†푛 푓푛+1 + 퐻.푐.) + √︂ Γ 휋 (푐† 푑푓0 + 퐻.푐.) + 퐻푑 )︃ NRG[4] 퐻푝푟표푏푒 = −퐴[ 휙† ↑휙↓ + Φ† ↑Φ↓ + (휙† ↑Φ↓ + Φ† ↑휙↓) ] I− + 퐻.푐. 휙휇(푅) ≡ ∫︁ 퐷 −퐷 푑휀 √︁ 1 −푊(휀,푅)¯푐휀휇 Φ휇(푅) ≡ Σ︁ 푛 훾푛푓푛 analytically numerically WILSON, K. Rev Mod Phys, 47, 773 (1975). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
  16. 16. Introduction NRG calculations Numerical results Conclusions Acknowledgment NRG and Lanczos basis NRG and Lanczos basis 퐻푁 = 1 풟푁 (︃ 푁Σ︁−1 푛=0 푡푛(푓†푛 푓푛+1 + 퐻.푐.) + √︂ Γ 휋 (푐† 푑푓0 + 퐻.푐.) + 퐻푑 )︃ NRG[4] 퐻푝푟표푏푒 = −퐴[ 휙† ↑휙↓ + Φ† ↑Φ↓ + (휙† ↑Φ↓ + Φ† ↑휙↓) ] I− + 퐻.푐. 휙휇(푅) ≡ ∫︁ 퐷 −퐷 푑휀 √︁ 1 −푊(휀,푅)¯푐휀휇 Φ휇(푅) ≡ Σ︁ 푛 훾푛푓푛 analytically numerically 1 푇1 = + + WILSON, K. Rev Mod Phys, 47, 773 (1975). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
  17. 17. Introduction NRG calculations Numerical results Conclusions Acknowledgment NRG and Lanczos basis NRG and Lanczos basis 퐻푁 = 1 풟푁 (︃ 푁Σ︁−1 푛=0 푡푛(푓†푛 푓푛+1 + 퐻.푐.) + √︂ Γ 휋 (푐† 푑푓0 + 퐻.푐.) + 퐻푑 )︃ NRG[4] 퐻푝푟표푏푒 = −퐴[ 휙† ↑휙↓ + Φ† ↑Φ↓ + (휙† ↑Φ↓ + Φ† ↑휙↓) ] I− + 퐻.푐. 휙휇(푅) ≡ ∫︁ 퐷 −퐷 푑휀 √︁ 1 −푊(휀,푅)¯푐휀휇 Φ휇(푅) ≡ Σ︁ 푛 훾푛푓푛 analytically numerically 1 푇1 = (︂ 1 푇1 )︂ 휙휙 ⏟ ⏞ 1−푊2 퐹 + + cte 푊퐹 = sin(푘퐹푅) 푘퐹푅 WILSON, K. Rev Mod Phys, 47, 773 (1975). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
  18. 18. Introduction NRG calculations Numerical results Conclusions Acknowledgment NRG and Lanczos basis NRG and Lanczos basis 퐻푁 = 1 풟푁 (︃ 푁Σ︁−1 푛=0 푡푛(푓†푛 푓푛+1 + 퐻.푐.) + √︂ Γ 휋 (푐† 푑푓0 + 퐻.푐.) + 퐻푑 )︃ NRG[4] 퐻푝푟표푏푒 = −퐴[ 휙† ↑휙↓ + Φ† ↑Φ↓ + (휙† ↑Φ↓ + Φ† ↑휙↓) ] I− + 퐻.푐. 휙휇(푅) ≡ ∫︁ 퐷 −퐷 푑휀 √︁ 1 −푊(휀,푅)¯푐휀휇 Φ휇(푅) ≡ Σ︁ 푛 훾푛푓푛 analytically numerically 1 푇1 = (︂ 1 푇1 )︂ 휙휙 ⏟ ⏞ 1−푊2 퐹 + (︂ 1 푇1 )︂ ⏟ ⏞ ΦΦ 푊2 퐹 + cte 푘퐹 푅 ≪ 1 푊퐹 = sin(푘퐹푅) 푘퐹푅 WILSON, K. Rev Mod Phys, 47, 773 (1975). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
  19. 19. Introduction NRG calculations Numerical results Conclusions Acknowledgment NRG and Lanczos basis NRG and Lanczos basis 퐻푁 = 1 풟푁 (︃ 푁Σ︁−1 푛=0 푡푛(푓†푛 푓푛+1 + 퐻.푐.) + √︂ Γ 휋 (푐† 푑푓0 + 퐻.푐.) + 퐻푑 )︃ NRG[4] 퐻푝푟표푏푒 = −퐴[ 휙† ↑휙↓ + Φ† ↑Φ↓ + (휙† ↑Φ↓ + Φ† ↑휙↓) ] I− + 퐻.푐. 휙휇(푅) ≡ ∫︁ 퐷 −퐷 푑휀 √︁ 1 −푊(휀,푅)¯푐휀휇 Φ휇(푅) ≡ Σ︁ 푛 훾푛푓푛 analytically numerically 1 푇1 = (︂ 1 푇1 )︂ 휙휙 ⏟ ⏞ 1−푊2 퐹 + (︂ 1 푇1 )︂ ⏟ ⏞ ΦΦ 푊2 퐹 + (︂ 1 푇1 )︂ Φ휙 ⏟ ⏞ (1−푊퐹 )푊퐹 cte 푘퐹 푅 ≪ 1 푘퐹푅 ≫ 1 DULL SMALL 푊퐹 = sin(푘퐹푅) 푘퐹푅 WILSON, K. Rev Mod Phys, 47, 773 (1975). Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
  20. 20. Introduction NRG calculations Numerical results Conclusions Acknowledgment Friedel oscillations Friedel oscillations 0.7 0.6 0.5 0.4 0.3 0.2 0.1 T=1.7569e−4 T=9.8711e−8 0.17 9.8 10.0 10.2 10.4 10.6 10.8 11.0 11.2 kF R ¼ 0.0 10.5¼ 10¼ 10.25¼ 10.0 10.5 11.0 0.16 1 T1 T (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 6 / 11
  21. 21. Introduction NRG calculations Numerical results Conclusions Acknowledgment Relaxation rate - temperature dependence 푘퐵푇퐾 = 1.25 × 10−5 7 6 5 T1 ´ kB T ³1 4 3 2 1 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 (101+0.25)¼ (102+0.25)¼ (103+0.25)¼ (104+0.25)¼ (105+0.25)¼ (106+0.25)¼ (107+0.25)¼ (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 7 / 11
  22. 22. Introduction NRG calculations Numerical results Conclusions Acknowledgment Interference relaxation rate Interference relaxation rate 7 6 5 1 kB T ³T1 ´ kB T 4 3 2 1 R¼RK outside inside 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 0 (kF R)2 kF R=(n+1 4 )¼ ¸B =2¼ vF kB T de Broglie n=102 n=105 n=107 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 8 / 11
  23. 23. Introduction NRG calculations Numerical results Conclusions Acknowledgment Friedel oscillations Friedel oscillations 1.72 1.70 1.68 1.66 1.64 1.62 1.60 0.00015 101 102 103 104 105 106 107 108 kF R ¼ 1.58 RK !TK =1.25e−05 n¼ (n+1/2)¼ 104 105 106 0.00010 +1.6131 1 T1 T (kF R)2 T¼5.62e−11 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 9 / 11
  24. 24. Introduction NRG calculations Numerical results Conclusions Acknowledgment Conclusions NMR to measure 푅퐾 : OK! Inside cloud, 푇-dependent rate follows universal curve Outside cloud, rate follows dierent curve Phase of Friedel oscillations reverses around 푅 = 푅퐾 Future prospects: Other geometries P-h symmetric case diers from assymetric ? Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 10 / 11
  25. 25. Introduction NRG calculations Numerical results Conclusions Acknowledgment Acknowledgment Thank you! Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 11 / 11
  26. 26. Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 4
  27. 27. Additional results Relaxation rate - 풢푠푖푑푒(푇) profile 7 6 5 T1 ´ kB T ³1 4 3 2 1 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 (101+0.5)¼ (102+0.5)¼ (103+0.5)¼ (104+0.5)¼ (105+0.5)¼ (106+0.5)¼ (107+0.5)¼ (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 4
  28. 28. Additional results Relaxation rate - 풢푆퐸푇 (푇) profile 7 6 5 T1 ´ kB T ³1 4 3 2 1 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 101 ¼ 102 ¼ 103 ¼ 104 ¼ 105 ¼ 106 ¼ 107 ¼ (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 4
  29. 29. Additional results Particle-hole symmetric case Particle-hole symmetric case: 1/푇1 as function of 푇 푘퐹푅 = 푛휋 and 푘퐹푅 = (푛 + 1 2 )휋, 푛 = 10 7 6 5 T1 ´ kB T ³1 4 3 2 1 2.0 1.8 1.6 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 10-9 10-8 10-7 1.4 (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
  30. 30. Additional results Particle-hole symmetric case Particle-hole symmetric case: 1/푇1 as function of 푇 푘퐹푅 = 푛휋 and 푘퐹푅 = (푛 + 1 2 )휋, 푛 = 103 7 6 5 T1 ´ kB T ³1 4 3 2 1 2.0 1.8 1.6 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 10-8 10-7 1.4 (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
  31. 31. Additional results Particle-hole symmetric case Particle-hole symmetric case: 1/푇1 as function of 푇 푘퐹푅 = 푛휋 and 푘퐹푅 = (푛 + 1 2 )휋, 푛 = 105 7 6 5 T1 ´ kB T ³1 4 3 2 1 2.0 1.8 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 10-7 1.6 (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
  32. 32. Additional results Particle-hole symmetric case Particle-hole symmetric case: 1/푇1 as function of 푇 푘퐹푅 = 푛휋 and 푘퐹푅 = (푛 + 1 2 )휋, 푛 = 107 7 6 5 T1 ´ kB T ³1 4 3 2 1 4 3 2 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 kB T 0 10-9 1 (kF R)2 Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4

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