Quantum impurity systems and Luttinger-Friedel sum rule ~ Luttinger - - PowerPoint PPT Presentation
Quantum criticality in Quantum impurity systems and Luttinger-Friedel sum rule ~ Luttinger integral as a topological invariant ~ Internat ationa onal Symp mpos osiu ium in H Honor of Profe fesso sor Nambu for the 10 10th A Annivers
~ Luttinger integral as a topological invariant ~
Internat ationa
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ium in H Honor of Profe fesso sor Nambu for the 10 10th A Annivers rsar ary y of hi his N Nobel Prize in P Physics
Yunori Nishikawa (Osaka City University)
General Background Model and Some Quantum Critical Points
Renormalized Perturbation Theory and Numerical Renormalization Group Application to Single Impurity Anderson Model Application to Two Impurity Anderson Model
Luttinger –Friedel sum rule Violation of Luttinger-Friedel sum rule Discussion
Quantum dot High-Tc superconductor (Cuprate: YBa2Cu3O7-δ)
BaO BaO CuO2 CuO2 CuO CuO Y
Dilute magnetic alloys (Kondo effect) Au Mn Scanning tunneling microscopy (STM) Au Co Co
Single Impurity Anderson Model ( SIAM )
continuous energy levels (non-interacting system) hybridization discrete energy level + interaction
AM 1 AM 2
Two-Impurity Anderson Model (2IAM)
AF J Kondo effect
Positive U12 > U Negative U12 < -|U|
Logarithmic discretization + Transformation to linear chains
Perturbation expansion in powers of the renormalized interaction parameters (overcounting are avoided) Exact expressions of some physical quantities in terms of the renormalized parameters Fermi liquid state : quasiparticles with residual renormalized interactions
NRG flow NRG flow Numerical Renormalization Group (NRG)
Renormalized Perturbation Theory and Numerical Renormalization Group
Renormalized Perturbation Theory (RPT) General assumption for low energy states (Fermi liquid) NRG flow and NRG fixed point
: A. C. Hewson, Phys. Rev. Lett. 70, 4007 (1993)
U/πΔ = 2 Wavefunction renomarization factor z = Δ~/Δ Renormalized U~ Wilson ratio R = χ/γ Exact (Bethe ansatz) 0.2392 0.2301 1.9620 NRG Λ=2.0 0.2389 0.2298 1.9620 NRG Λ=2.5 0.2390 0.2300 1.9621 NRG Λ=3.0 0.2390 0.2299 1.9619 NRG Λ=3.5 0.2390 0.2299 1.9620 Our method : all parameters (in principle) Parameter space Electron-hole symmetry, D→∞ Bethe ansatz (Exact) : special parameter
Quite good agreement
Exact expressions of some susceptibilities ( RPT )
Spin susceptibility Charge susceptibility Staggered spin susceptibility Staggered charge susceptibility
For 2IAM, no exact results such a Bethe ansatz solution (so far).
Dimensionless renormalized interaction parameters Quasiparticle density of states
( Nishikawa Y., Crow D. J. G., Hewson A. C., PHYSICAL REVIEW LETTERS 108 ( 056402 ) 2012, PHYSICAL REVIEW B 86 ( 125134 ) 2012 )
(necessary condition for the assumption) ⇒ one η → 4 and the other η s → 0. At least, one susceptibility diverges because (assumption) At a particular QCP, only one susceptibility associated with the QCP diverges. It enables us to predict the values of the renormalized interaction parameters at the QCP using the following equations.
0.000160 1.999 0.9996 0.000279
0.9990
[ Predicted values ]
QCP1(Kondo-AF) QCP2(Kondo-CO)
[ Predicted values ]
AF CO Kondo effect
Our method is reliable for 2IAM.
10^-9
[ Predicted values ]
QCP3(Kondo-AP) AP Kondo effect
Luttinger’s theorem
The volume inside the Fermi surface is invariant by the interaction. (If the number of particles is held fixed.)
Friedel sum rule
local AF (electron-hole symmetry)
Magnetic field QCP1’
0.5
Unpublished result
Nishikawa Y., Curtin O. J., Hewson A. C., Crow D. J. G.,PHYSICAL REVIEW B 98 ( 104419 ) 2018
local AF (electron-hole asymmetry)
Magnetic field QCP1’’
Curtin O. J., Nishikawa Y., Hewson A. C., Crow D. J. G. JOURNAL OF PHYSICS COMMUNICATIONS 2 ( 031001 ) 2018
Kondo effect(electron-hole asymmetry) local AF(electron-hole asymmetry) QCP1 e-h asym
The local AF QCP (QCP1) is robust against parameter perturbations ( magnetic field, electron-hole asymmetry, ( geometric asymmetry ), and some of them )
QCP magnetic field magnetic field Unpublished result
1. Long-lived quasiparticle excitation at the Fermi level
Constant in a phase Quantized [ Luttinger integral ] Luttinger integral as a Topological invariant ・ characterizing the two Fermi liquid phases ・ induced by electron correlations Robustness of the QCP against some perturbations
For example, in the paper by M. Oshikawa ( PHYSICAL REVIEW LETTERS 84 (3370) 2000 ) ,
“ Luttinger’s theorem might be actually the first example of the topological quantization discovered in quantum many-body problem… ” “Luttinger's theorem perhaps does not look like a quantization, because the volume of the Fermi sea takes continuous values depending on the particle density. However, the insensitivity to the interaction resembles other quantization phenomena, …” “ In fact, it is not clear whether a Fermi liquid which violates Luttinger’s theorem can exist. ” “… the topological understanding has been missing for a long time. ”
⇒ ● (Local) Fermi liquid states violating Luttinger’s theorem ( Luttinger-Friedel sum rule).
(Nishikawa Y., Curtin O. J., Hewson A. C., Crow D. J. G.,PHYSICAL REVIEW B 98 ( 104419 ) 2018, Curtin O. J., Nishikawa Y., Hewson
⇒ Winding number for finite systems ( K. Seki, S. Yunoki, PHYSICAL REVIEW B 96 (085124)2017 ) → consistent with our results and proposal
(See also, G. G. Blesio, L. O. Manuel, P. Roura-Bas, A. A. Aligia, PHYSICAL REVIEW B 98 (195435) 2018)
Luttinger’s theorem…. something topological…
Related experiment : heavy fermion system
Philipp Gegenwart, Qimiao Si, Frank Steglich, Nature Physics volume 4, 186–197 (2008)
http://www2.cpfs.mpg.de/~wirth/rec/stm4.html
Characterizing the two fermi liquid phases separated by the QCP Induced by electron correlations
Thank you for your attention.