This document describes a numerical renormalization group (NRG) computation of nuclear magnetic relaxation rates using a two-center basis approach. The NRG and Lanczos methods are used to calculate the spin-lattice relaxation rate 1/T1 as a function of temperature T and distance r from the impurity. Results show that 1/T1 dependence on T changes as the probe crosses the Kondo screening cloud radius rK, and the phase of low-energy Friedel oscillations also changes, indicating the Kondo screening cloud radius can be measured via NMR.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 4

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. 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. 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. 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. 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. 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