Quantum criticality in
Quantum impurity systems andLuttinger-Friedel sum rule
~ Luttinger integral as a topological invariant ~Curtin O. J., Nishikawa Y., Hewson A. C., Crow D. J. G. JOURNAL OF PHYSICS COMMUNICATIONS 2 ( 031001 ) 2018：https://arxiv.org/abs/1703.01807
Nishikawa Y., Curtin O. J., Hewson A. C., Crow D. J. G.,PHYSICAL REVIEW B 98 ( 104419 ) 2018,：
https://arxiv.org/abs/1712.08771

International Symposium in Honor of Professor Nambu
for the 10th Anniversary of his Nobel Prize in Physics

Yunori Nishikawa(Osaka City University, NITEP)Collaborators( Imperial College London )A.C.Hewson（Former PhD students）
O.J.Curtin, D.J.G. Crow

Outline of Presentation
1. Model and Systems
General Background
Model and Some Quantum Critical Points

2. Method and Some Results
Renormalized Perturbation Theory and Numerical Renormalization Group
Application to Single Impurity Anderson Model
Application to Two Impurity Anderson Model

3. Results and Discussion
Luttinger –Friedel sum rule
Violation of Luttinger-Friedel sum rule
Discussion

4. Summary and Conclusion

Model and Systems

General BackgroundSingle Impurity Anderson Model ( SIAM )continuous energy levels(non-interacting system)

discrete energy level + interaction

hybridization

High-Tc superconductor(Cuprate: YBa2Cu3O7-δ)Fig.1

Scanning tunneling microscopy
(STM)

CuO
BaO

Quantum dot

CuO2
YCuO2
BaO

Dilute magnetic alloys(Kondo effect)
Co

Au
Fig.2

Mn

CuO

Co
Fig.3

Au

Fig.3

Fig.4

Model
Two-Impurity Anderson Model (2IAM)

AM 1

AM 2

Some Quantum Critical Points
AF J
Fig.1

QCP1: J
QCP2:U12

Kondo effect

QCP3:U12
Fig.3

Fig.2

Positive U12 > U

Negative U12 < -|U|

Method and Some ResultsRenormalized Perturbation Theory and Numerical Renormalization Group
General assumption for low energy states (Fermi liquid)
Fermi liquid state : quasiparticles with residual renormalized interactions

Renormalized Perturbation Theory (RPT) : A. C. Hewson, Phys. Rev. Lett. 70, 4007 (1993)
Perturbation expansion in powers of the renormalized interaction parameters (overcounting are avoided)
Exact expressions of some physical quantities in terms of the renormalized parameters

Numerical Renormalization Group (NRG)

Fig.1

Logarithmic discretization+Transformation to linear chains

NRG flow and NRG fixed point
NRG flow

NRG flowApplication to SIAM
Bethe ansatz (Exact) : special parameter
Electron-hole symmetry, D→∞

Parameter space

Our method : all parameters (in principle)

U/πΔ = 2

Wavefunction renomarization
factor z = Δ~/Δ

Renormalized U~

Wilson ratio R = χ/γExact (Bethe ansatz)

0.23920.2301

1.9620

NRG Λ=2.00.2389

0.22981.9620

NRG Λ=2.50.2390

0.23001.9621

NRG Λ=3.00.2390

0.22991.9619

NRG Λ=3.50.2390

0.22991.9620A. C. Hewson, A. Oguri, and D. Meyer. Eur. Phys. J. B B40 177 2004

Quite good agreement

Application to 2IAM
For 2IAM, no exact results such a Bethe ansatz solution (so far).

Exact expressions of some susceptibilities ( RPT )Spin susceptibility

Charge susceptibility
Staggered spin susceptibility
Staggered charge susceptibility

Quasiparticle density of states

Dimensionless renormalized interaction parameters( Nishikawa Y., Crow D. J. G., Hewson A. C., PHYSICAL REVIEW LETTERS 108 ( 056402 ) 2012, PHYSICAL REVIEW B 86 ( 125134 ) 2012 )@QCP
At least, one susceptibility diverges because(assumption)
At a particular QCP, only one susceptibility associated with the QCP diverges.
(necessary condition for the assumption)⇒ one η → 4 and the other η s → 0.
It enables us to predict the values of the renormalized interaction parametersat the QCP using the following equations.

Kondo effectCO[ Predicted values ]AF[ Predicted values ]1.999

0.99960.000160-0.9995

0.99900.000279

QCP2(Kondo-CO)

QCP1(Kondo-AF)

Kondo effectAP[ Predicted values ]10^-9-0.9999-0.9999

QCP3(Kondo-AP)

Our method is reliable for 2IAM.

Results and Discussion

Luttinger – Friedel sum rule
● Luttinger’s theorem
The volume inside the Fermi surface is invariant by the interaction.
(If the number N of particles is held fixed.)(1)● Friedel sum rule
(2)
(4)

(3)

Violation of Luttinger –Fridel sum rule 1
QCP1’

local AF (electron-hole symmetry)

0.5

Fig.1

Magnetic field

Violation of Luttinger –Fridel sum rule 2
QCP1’’

Magnetic field

local AF (electron-hole asymmetry)

Nishikawa Y., Curtin O. J., Hewson A. C., Crow D. J. G.,PHYSICAL REVIEW B 98 ( 104419 ) 2018

Violation of Luttinger –Fridel sum rule 3
QCP1
e-h asym
Kondo effect(electron-hole asymmetry)

local AF(electron-hole asymmetry)

Curtin O. J., Nishikawa Y., Hewson A. C., Crow D. J. G.
JOURNAL OF PHYSICS COMMUNICATIONS 2 ( 031001 ) 2018

The local AF QCP (QCP1) is robust against parameter perturbations( magnetic field, electron-hole asymmetry, ( geometric asymmetry ), and some of them )Violation of Luttinger –Fridel sum rule 4
QCP
magnetic fieldmagnetic fieldDiscussion(1)

1. Long-lived quasiparticle excitation
at the Fermi level

2. Adiabatic continuous[ Luttinger integral ](2)Constant in a phase

Quantized

Robustness of the QCPagainst some perturbations

Luttinger integral as a Topological invariant
・ characterizing the two Fermi liquid phases
・ induced by electron correlations

Luttinger’s theorem…. something topological…
For example,[1] M.Oshikawa, PHYSICAL REVIEW LETTERS 84 (3370) 2000： https://arxiv.org/abs/cond-mat/0002392

Q1. “ a Fermi liquid which violates Luttinger’s theorem can exist? ”⇒● (Local) Fermi liquid states violating Luttinger’s theorem ( Luttinger-Friedel sum rule).
● Deviations from Luttinger-Friedel sum rule are quantized.
[2]Curtin O. J., Nishikawa Y., Hewson A. C., Crow D. J. G. JOURNAL OF PHYSICS COMMUNICATIONS
2 ( 031001 ) 2018：https://arxiv.org/abs/1703.01807
Nishikawa Y., Curtin O. J., Hewson A. C., Crow D. J. G.,PHYSICAL REVIEW B 98 ( 104419 ) 2018,：https://arxiv.org/abs/1712.08771
Other example :[3]

G. G. Blesio, L. O. Manuel, P. Roura-Bas, A. A. Aligia, PHYSICAL REVIEW B 98 (195435) 2018 ：https://arxiv.org/abs/1805.11518

Q2. the topological understanding⇒ ●Winding number[4] K. Seki, S. Yunoki, PHYSICAL REVIEW B 96 (085124)2017 ：https://arxiv.org/abs/1708.00986

Related experiment : heavy fermion system
Fig.1
Fig.2

http://www2.cpfs.mpg.de/~wirth/rec/stm4.html

Philipp Gegenwart, Qimiao Si, Frank Steglich,
Nature Physics volume 4, 186–197 (2008)

Summary and Conclusion• Quantum criticality in two impurity Anderson model• Method : RPT and NRG• Violation of Luttinger-Friedel sum rule in some Fermi liquid states• Luttinger integral as a topological invariant
Characterizing the two fermi liquid phases separated by the QCP
Induced by electron correlations• Related experiment on a heavy fermion metal, YbRh2Si2Thank you for your attention.