Numerical Renormalization Group computation of magnetic relaxation rates
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Numerical Renormalization-Group computation of magnetic relaxation rates Krissia de Zawadzki, Luiz Nunes de Oliveira, Jos´ e 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
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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.
Transcript of Numerical Renormalization Group computation of magnetic relaxation rates
Numerical Renormalization-Group computationof magnetic relaxation rates
Krissia de Zawadzki, Luiz Nunes de Oliveira, Jose Wilson M. Pinto
Instituto de Fısica de Sao Carlos - Universidade de Sao Paulo
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
𝑅𝑘
𝑅𝐾 ∝ 𝑇−1𝐾
General consensus
𝑅𝐾 = ~𝑣𝐹 /𝑘𝐵𝑇𝐾
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: 𝑅 from the impurity
NRG computation of the spin
lattice relaxation rate 1/(𝑇1𝑇 ) as
function of 𝑇 and 𝑅
Can we measure 𝑅𝐾 via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses 𝑅𝐾
Phase of low-𝑇 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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
𝑅𝑘
𝑅𝐾 ∝ 𝑇−1𝐾
General consensus
𝑅𝐾 = ~𝑣𝐹 /𝑘𝐵𝑇𝐾
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: 𝑅 from the impurity
NRG computation of the spin
lattice relaxation rate 1/(𝑇1𝑇 ) as
function of 𝑇 and 𝑅
Can we measure 𝑅𝐾 via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses 𝑅𝐾
Phase of low-𝑇 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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
𝑅𝑘
𝑅𝐾 ∝ 𝑇−1𝐾
General consensus
𝑅𝐾 = ~𝑣𝐹 /𝑘𝐵𝑇𝐾
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: 𝑅 from the impurity
NRG computation of the spin
lattice relaxation rate 1/(𝑇1𝑇 ) as
function of 𝑇 and 𝑅
Can we measure 𝑅𝐾 via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses 𝑅𝐾
Phase of low-𝑇 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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
𝑅𝑘
𝑅𝐾 ∝ 𝑇−1𝐾
General consensus
𝑅𝐾 = ~𝑣𝐹 /𝑘𝐵𝑇𝐾
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: 𝑅 from the impurity
NRG computation of the spin
lattice relaxation rate 1/(𝑇1𝑇 ) as
function of 𝑇 and 𝑅
Can we measure 𝑅𝐾 via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses 𝑅𝐾
Phase of low-𝑇 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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
𝑅𝑘
𝑅𝐾 ∝ 𝑇−1𝐾
General consensus
𝑅𝐾 = ~𝑣𝐹 /𝑘𝐵𝑇𝐾
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: 𝑅 from the impurity
NRG computation of the spin
lattice relaxation rate 1/(𝑇1𝑇 ) as
function of 𝑇 and 𝑅
Can we measure 𝑅𝐾 via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses 𝑅𝐾
Phase of low-𝑇 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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
𝐻 =
𝐻𝑐𝑜𝑛𝑑⏞ ⏟ ∑k
𝜀k𝑐†k𝑐k
+
𝐻𝑑⏞ ⏟ 𝜀𝑑𝑐
†𝑑𝑐𝑑 + 𝑈𝑛𝑑↑𝑛𝑑↓ +
𝐻𝑖𝑛𝑡⏞ ⏟ √Γ
𝜋(𝑓†
0𝑐𝑑 +𝐻.𝑐.)
𝜀 = 𝑣𝐹𝐷
(𝑘 − 𝑘𝐹 )
+𝐷
−𝐷
𝑘𝐹
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴[Ψ†
↑(��)Ψ↓(��)𝐼− +𝐻.𝑐.]
Ψ𝜇 =∑k
𝑒𝑖k.R𝑐k
1
𝑇1=
4𝜋
~
∑𝐼,𝐹
𝑒−𝛽𝐸𝐼 |⟨𝐼|𝐻𝑝𝑟𝑜𝑏𝑒|𝐹 ⟩|2𝛿(𝐸𝐼 − 𝐸𝐹 )
𝑓0 =1√𝜌
∑k
𝑐k
𝑅
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
𝐻 =
𝐻𝑐𝑜𝑛𝑑⏞ ⏟ ∑k
𝜀k𝑐†k𝑐k +
𝐻𝑑⏞ ⏟ 𝜀𝑑𝑐
†𝑑𝑐𝑑 + 𝑈𝑛𝑑↑𝑛𝑑↓
+
𝐻𝑖𝑛𝑡⏞ ⏟ √Γ
𝜋(𝑓†
0𝑐𝑑 +𝐻.𝑐.)
𝜀 = 𝑣𝐹𝐷
(𝑘 − 𝑘𝐹 )
+𝐷
−𝐷
𝑘𝐹
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴[Ψ†
↑(��)Ψ↓(��)𝐼− +𝐻.𝑐.]
Ψ𝜇 =∑k
𝑒𝑖k.R𝑐k
1
𝑇1=
4𝜋
~
∑𝐼,𝐹
𝑒−𝛽𝐸𝐼 |⟨𝐼|𝐻𝑝𝑟𝑜𝑏𝑒|𝐹 ⟩|2𝛿(𝐸𝐼 − 𝐸𝐹 )
𝑓0 =1√𝜌
∑k
𝑐k
𝑅
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
𝐻 =
𝐻𝑐𝑜𝑛𝑑⏞ ⏟ ∑k
𝜀k𝑐†k𝑐k +
𝐻𝑑⏞ ⏟ 𝜀𝑑𝑐
†𝑑𝑐𝑑 + 𝑈𝑛𝑑↑𝑛𝑑↓ +
𝐻𝑖𝑛𝑡⏞ ⏟ √Γ
𝜋(𝑓†
0𝑐𝑑 +𝐻.𝑐.)
𝜀 = 𝑣𝐹𝐷
(𝑘 − 𝑘𝐹 )
+𝐷
−𝐷
𝑘𝐹
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴[Ψ†
↑(��)Ψ↓(��)𝐼− +𝐻.𝑐.]
Ψ𝜇 =∑k
𝑒𝑖k.R𝑐k
1
𝑇1=
4𝜋
~
∑𝐼,𝐹
𝑒−𝛽𝐸𝐼 |⟨𝐼|𝐻𝑝𝑟𝑜𝑏𝑒|𝐹 ⟩|2𝛿(𝐸𝐼 − 𝐸𝐹 )
𝑓0 =1√𝜌
∑k
𝑐k
𝑅
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
𝐻 =
𝐻𝑐𝑜𝑛𝑑⏞ ⏟ ∑k
𝜀k𝑐†k𝑐k +
𝐻𝑑⏞ ⏟ 𝜀𝑑𝑐
†𝑑𝑐𝑑 + 𝑈𝑛𝑑↑𝑛𝑑↓ +
𝐻𝑖𝑛𝑡⏞ ⏟ √Γ
𝜋(𝑓†
0𝑐𝑑 +𝐻.𝑐.)
𝜀 = 𝑣𝐹𝐷
(𝑘 − 𝑘𝐹 )
+𝐷
−𝐷
𝑘𝐹
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴[Ψ†
↑(��)Ψ↓(��)𝐼− +𝐻.𝑐.]
Ψ𝜇 =∑k
𝑒𝑖k.R𝑐k
1
𝑇1=
4𝜋
~
∑𝐼,𝐹
𝑒−𝛽𝐸𝐼 |⟨𝐼|𝐻𝑝𝑟𝑜𝑏𝑒|𝐹 ⟩|2𝛿(𝐸𝐼 − 𝐸𝐹 )
𝑓0 =1√𝜌
∑k
𝑐k
𝑅
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
𝑐𝜀 =∑k
𝑐 k 𝛿(𝜀− 𝜀k) (around impurity)
𝑑𝜀 =∑k
𝑐 k 𝑒𝑖k.R𝛿(𝜀− 𝜀k) (around probe)
𝑐 𝜀
𝑑 𝜀
NRG
analytical
{𝑐†𝜀, 𝑑𝜀′} = sin(𝑘𝑅)𝑘𝑅 𝛿(𝜀− 𝜀′)
Gram-Schmidt construction
𝑐𝜀𝜇 = 1√1−𝑊 2
(𝑑𝜀𝜇 −𝑊𝑐𝜀𝜇)
𝑊 = 𝑊 (𝜀,𝑅) = sin(𝑘𝑅)𝑘𝑅
𝑘𝑅 = 𝑘𝐹𝑅(1 + 𝜀
𝐷
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
𝑐𝜀 =∑k
𝑐 k 𝛿(𝜀− 𝜀k) (around impurity)
𝑑𝜀 =∑k
𝑐 k 𝑒𝑖k.R𝛿(𝜀− 𝜀k) (around probe)
𝑐 𝜀
𝑑 𝜀
NRG
analytical
{𝑐†𝜀, 𝑑𝜀′} = sin(𝑘𝑅)𝑘𝑅 𝛿(𝜀− 𝜀′)
Gram-Schmidt construction
𝑐𝜀𝜇 = 1√1−𝑊 2
(𝑑𝜀𝜇 −𝑊𝑐𝜀𝜇)
𝑊 = 𝑊 (𝜀,𝑅) = sin(𝑘𝑅)𝑘𝑅
𝑘𝑅 = 𝑘𝐹𝑅(1 + 𝜀
𝐷
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
𝑐𝜀 =∑k
𝑐 k 𝛿(𝜀− 𝜀k) (around impurity)
𝑑𝜀 =∑k
𝑐 k 𝑒𝑖k.R𝛿(𝜀− 𝜀k) (around probe)
𝑐 𝜀
𝑑 𝜀
NRG
analytical
{𝑐†𝜀, 𝑑𝜀′} = sin(𝑘𝑅)𝑘𝑅 𝛿(𝜀− 𝜀′)
Gram-Schmidt construction
𝑐𝜀𝜇 = 1√1−𝑊 2
(𝑑𝜀𝜇 −𝑊𝑐𝜀𝜇)
𝑊 = 𝑊 (𝜀,𝑅) = sin(𝑘𝑅)𝑘𝑅
𝑘𝑅 = 𝑘𝐹𝑅(1 + 𝜀
𝐷
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
𝐻𝑁 =1
𝒟𝑁
(𝑁−1∑𝑛=0
𝑡𝑛(𝑓†𝑛𝑓𝑛+1 +𝐻.𝑐.) +
√Γ
𝜋(𝑐†𝑑𝑓0 +𝐻.𝑐.) +𝐻𝑑
)NRG[4]
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴 [ 𝜙†↑𝜙↓ +Φ†
↑Φ↓ + (𝜙†↑Φ↓ +Φ†
↑𝜙↓) ] I− +𝐻.𝑐.
𝜙𝜇(𝑅) ≡∫ 𝐷
−𝐷
𝑑𝜀√
1 − 𝑊 (𝜀,𝑅)𝑐𝜀𝜇 Φ𝜇(𝑅) ≡∑𝑛
𝛾𝑛𝑓𝑛
analytically
numerically
1
𝑇1=
(1
𝑇1
)𝜙𝜙⏟ ⏞
1−𝑊 2𝐹
+
(1
𝑇1
)ΦΦ⏟ ⏞
𝑊 2𝐹
+
(1
𝑇1
)Φ𝜙⏟ ⏞
(1−𝑊𝐹 )𝑊𝐹
cte
𝑘𝐹𝑅≪
1
𝑘𝐹𝑅
≫1
DULL
SMALL
𝑊𝐹 =sin(𝑘𝐹𝑅)
𝑘𝐹𝑅
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
𝐻𝑁 =1
𝒟𝑁
(𝑁−1∑𝑛=0
𝑡𝑛(𝑓†𝑛𝑓𝑛+1 +𝐻.𝑐.) +
√Γ
𝜋(𝑐†𝑑𝑓0 +𝐻.𝑐.) +𝐻𝑑
)NRG[4]
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴 [ 𝜙†↑𝜙↓ +Φ†
↑Φ↓ + (𝜙†↑Φ↓ +Φ†
↑𝜙↓) ] I− +𝐻.𝑐.
𝜙𝜇(𝑅) ≡∫ 𝐷
−𝐷
𝑑𝜀√
1 − 𝑊 (𝜀,𝑅)𝑐𝜀𝜇 Φ𝜇(𝑅) ≡∑𝑛
𝛾𝑛𝑓𝑛
analytically
numerically
1
𝑇1=
(1
𝑇1
)𝜙𝜙⏟ ⏞
1−𝑊 2𝐹
+
(1
𝑇1
)ΦΦ⏟ ⏞
𝑊 2𝐹
+
(1
𝑇1
)Φ𝜙⏟ ⏞
(1−𝑊𝐹 )𝑊𝐹
cte
𝑘𝐹𝑅≪
1
𝑘𝐹𝑅
≫1
DULL
SMALL
𝑊𝐹 =sin(𝑘𝐹𝑅)
𝑘𝐹𝑅
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
𝐻𝑁 =1
𝒟𝑁
(𝑁−1∑𝑛=0
𝑡𝑛(𝑓†𝑛𝑓𝑛+1 +𝐻.𝑐.) +
√Γ
𝜋(𝑐†𝑑𝑓0 +𝐻.𝑐.) +𝐻𝑑
)NRG[4]
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴 [ 𝜙†↑𝜙↓ +Φ†
↑Φ↓ + (𝜙†↑Φ↓ +Φ†
↑𝜙↓) ] I− +𝐻.𝑐.
𝜙𝜇(𝑅) ≡∫ 𝐷
−𝐷
𝑑𝜀√
1 − 𝑊 (𝜀,𝑅)𝑐𝜀𝜇 Φ𝜇(𝑅) ≡∑𝑛
𝛾𝑛𝑓𝑛
analytically
numerically
1
𝑇1=
(1
𝑇1
)𝜙𝜙⏟ ⏞
1−𝑊 2𝐹
+
(1
𝑇1
)ΦΦ⏟ ⏞
𝑊 2𝐹
+
(1
𝑇1
)Φ𝜙⏟ ⏞
(1−𝑊𝐹 )𝑊𝐹
cte
𝑘𝐹𝑅≪
1
𝑘𝐹𝑅
≫1
DULL
SMALL
𝑊𝐹 =sin(𝑘𝐹𝑅)
𝑘𝐹𝑅
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
𝐻𝑁 =1
𝒟𝑁
(𝑁−1∑𝑛=0
𝑡𝑛(𝑓†𝑛𝑓𝑛+1 +𝐻.𝑐.) +
√Γ
𝜋(𝑐†𝑑𝑓0 +𝐻.𝑐.) +𝐻𝑑
)NRG[4]
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴 [ 𝜙†↑𝜙↓ +Φ†
↑Φ↓ + (𝜙†↑Φ↓ +Φ†
↑𝜙↓) ] I− +𝐻.𝑐.
𝜙𝜇(𝑅) ≡∫ 𝐷
−𝐷
𝑑𝜀√
1 − 𝑊 (𝜀,𝑅)𝑐𝜀𝜇 Φ𝜇(𝑅) ≡∑𝑛
𝛾𝑛𝑓𝑛
analytically
numerically
1
𝑇1=
(1
𝑇1
)𝜙𝜙⏟ ⏞
1−𝑊 2𝐹
+
(1
𝑇1
)ΦΦ⏟ ⏞
𝑊 2𝐹
+
(1
𝑇1
)Φ𝜙⏟ ⏞
(1−𝑊𝐹 )𝑊𝐹
cte
𝑘𝐹𝑅≪
1
𝑘𝐹𝑅
≫1
DULL
SMALL
𝑊𝐹 =sin(𝑘𝐹𝑅)
𝑘𝐹𝑅
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
𝐻𝑁 =1
𝒟𝑁
(𝑁−1∑𝑛=0
𝑡𝑛(𝑓†𝑛𝑓𝑛+1 +𝐻.𝑐.) +
√Γ
𝜋(𝑐†𝑑𝑓0 +𝐻.𝑐.) +𝐻𝑑
)NRG[4]
𝐻𝑝𝑟𝑜𝑏𝑒 = −𝐴 [ 𝜙†↑𝜙↓ +Φ†
↑Φ↓ + (𝜙†↑Φ↓ +Φ†
↑𝜙↓) ] I− +𝐻.𝑐.
𝜙𝜇(𝑅) ≡∫ 𝐷
−𝐷
𝑑𝜀√
1 − 𝑊 (𝜀,𝑅)𝑐𝜀𝜇 Φ𝜇(𝑅) ≡∑𝑛
𝛾𝑛𝑓𝑛
analytically
numerically
1
𝑇1=
(1
𝑇1
)𝜙𝜙⏟ ⏞
1−𝑊 2𝐹
+
(1
𝑇1
)ΦΦ⏟ ⏞
𝑊 2𝐹
+
(1
𝑇1
)Φ𝜙⏟ ⏞
(1−𝑊𝐹 )𝑊𝐹
cte
𝑘𝐹𝑅≪
1
𝑘𝐹𝑅
≫1
DULL
SMALL
𝑊𝐹 =sin(𝑘𝐹𝑅)
𝑘𝐹𝑅
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Friedel oscillations
Friedel oscillations
9.8 10.0 10.2 10.4 10.6 10.8 11.0 11.2kFRπ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
10.5π
10π 10.25π
T=1.7569e−4
T=9.8711e−8
10.0 10.5 11.0
0.16
0.17
1T
1T(kFR
)2
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 6 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Relaxation rate - temperature dependence
𝑘𝐵𝑇𝐾 = 1.25× 10−5
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
(101+0.25)π
(102+0.25)π
(103+0.25)π
(104+0.25)π
(105+0.25)π
(106+0.25)π
(107+0.25)π
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 7 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Interference relaxation rate
Interference relaxation rate
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-20
1
2
3
4
5
6
7
R≈RK
outside
inside
(kFR
)2
kBT
( 1 T1
)
kB T
kFR=(n+14)π
λB =2πvFkB T
de Broglie
n=102
n=105
n=107
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
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Friedel oscillations
Friedel oscillations
101 102 103 104 105 106 107 108
kFRπ
1.58
1.60
1.62
1.64
1.66
1.68
1.70
1.72
RK →TK =1.25e−05
nπ
(n+1/2)π
104 105 106
0.00010
0.00015
+1.6131
1T
1T(kFR
)2
T≈5.62e−11
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 9 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Conclusions
NMR to measure 𝑅𝐾 : OK!
Inside cloud, 𝑇 -dependent rate follows universal curve
Outside cloud, rate follows different curve
Phase of Friedel oscillations reverses around 𝑅 = 𝑅𝐾
Future prospects:
Other geometries
P-h symmetric case differs from assymetric ?
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 10 / 11
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
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 4
Additional results
Relaxation rate - 𝒢𝑠𝑖𝑑𝑒(𝑇 ) profile
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
(101+0.5)π
(102+0.5)π
(103+0.5)π
(104+0.5)π
(105+0.5)π
(106+0.5)π
(107+0.5)π
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 4
Additional results
Relaxation rate - 𝒢𝑆𝐸𝑇 (𝑇 ) profile
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
101 π
102 π
103 π
104 π
105 π
106 π
107 π
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/𝑇1 as function of 𝑇
𝑘𝐹𝑅 = 𝑛𝜋 and 𝑘𝐹𝑅 = (𝑛+ 12 )𝜋, 𝑛 = 10
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-9 10-8 10-71.4
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/𝑇1 as function of 𝑇
𝑘𝐹𝑅 = 𝑛𝜋 and 𝑘𝐹𝑅 = (𝑛+ 12 )𝜋, 𝑛 = 103
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-8 10-71.4
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/𝑇1 as function of 𝑇
𝑘𝐹𝑅 = 𝑛𝜋 and 𝑘𝐹𝑅 = (𝑛+ 12 )𝜋, 𝑛 = 105
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-7
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/𝑇1 as function of 𝑇
𝑘𝐹𝑅 = 𝑛𝜋 and 𝑘𝐹𝑅 = (𝑛+ 12 )𝜋, 𝑛 = 107
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-91
2
3
4
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
