Download Renormalized Parameters for Strong Correlation Impurity and

Renormalized Parameters for
Strong Correlation Impurity and Lattice Models
near Quantum Critical Points
Alex Hewson, Khan Edwards, Daniel Crow, Vassilis Pandis , Imperial College London, U.K
Yunori Nishikawa, Osaka City University, Japan; Johannes Bauer, Harvard, USA
Distinctive features of Strongly Correlated Electron Systems
1. The emergence of a low energy scale T* << TF
----- large effectives masses m*/m ~ 1000 for heavy fermions
----- chi/gamma’ or Wilson ratios of the order of 1
------ strongly renormalized quasiparticles
2 . The energy scale T* is unique in some cases leading to universal behaviour
eg. Kondo model for magnetic impurity
T*=TK (Kondo temperature)
3. New symmetries emerge on a scale T<T* in some cases
Low temperature magnetic transitions and unusual types of superconductivity
4. Breakdown of Fermi Liquid theory at quantum critical points (QCP)
•
scaling eg. YbRh2Si2
Renormalized Parameters
eg. Anderson Impurity model
A model of localized states of an impurity in a metallic host, or more recently as a model of a
quantum dot
impurity level
coupling to magnetic field
This part determines the low
energy/temperature behaviour
hybridization width
on-site interaction
Counter terms important on
on higher energy scales
Exact Relations in terms of renormalized parameters
specific heat coefficient
spin susceptibility
occupation number
Impurity conductivity
quasiparticle density
of states
charge susceptibility
Phase shift
Strong correlation and the emergence of universality
Suppression of charged
fluctuations in the
localised or Kondo limit
Only one renormalized parameter
Response and thermodynamic
functions universal
What values of U, ed, and D correspond to the universal regime?
Overview of renormalized
parameters in scan with
fixed value of U
Strong correlation
regime
Derenormalization of quasiparticles in a magnetic field h
Strong correlation
regime
Strong correlation
regime
Unrenormalized
or `bare’
quasiparticles
Dynamics in the strong correlation regime
ACH JPCM 18, 1815 (2006)
Universality in more general models
eg. n-orbital model with Hund’s rule coupling JH (Yoshimori model)
Introduce renormalized parameters:
spin susceptibility
charge susceptibility
orbital susceptibility
These factors depend on the quasiparticle
interactions
Suppression of charged fluctuations:
SU(2n) symmetry
Suppression of orbital fluctuations:
Nishikawa, Crow, ACH , PRB
82, 115123 (2010)
82, 245109 (2010)
When do we expect strong correlation and universality?
Universal regime
Universal regime
Double quantum dot model with quantum critical points
This model has two types of quantum critical points
Local singlet transition
Local charge order transition
Quantum critical transitions in the symmetric model
strong J predominantly local screening
weak J predominantly Kondo screening (U12=0)
locally charged ordered state (U/D=0.05)
Nishikawa, Crow, ACH , PRL 108, 056402 (2012), PRB, 86, 125134 (2012)
Predictions of asymptotic behaviour on approach to QCPs for the symmetric model
spin susceptibility
charge susceptibility
staggered spin susceptibility
staggered charge susceptibility
At QCP:
Universality on approach to both QCPs
Renormalized parameters on the approach to the quantum critical point
Predictions confirmed
Universality for large U
NFL
Fermi liquid 1
Fermi liquid 2
Quantum Critical Points in Heavy Fermion Compounds
NFL
QCP
Candidate for the local “Kondo collapse” scenario
From a recent review by Si and Steglich -Science 329, 1161 (2010)
Emergence of symmetry for an asymmetric model
Transition in the model without particle-hole symmetry
Local charge order transition
SU(4) point
Predictions confirmed
Low energy SU(4) fixed point in capacitively coupled quantum dots?
Equal spin and pseudo-spin fluctuations?
required for complete SU(4)
Borda et al PRL 90, 026602 (2003)
Amasha et al PRL 110, 046604 (2013)
spin and pseudo-spin degenerate
Infinite dimensional lattice models
Hubbard model
quasiparticle weight
quasiparticle density
of states
Mott/Hubbard transition as a QCP
Exact equations
Occupation number per site
?
Strong Correlation regime
Renormalization of U in low density limit
In agreement with Kanamori’s estimate from
repeated particle-particle scattering
Low energy dynamics
Symmetric model in a magnetic field
Almost complete polarization but n<1
Bauer+ACH PRB 76, 030118 (2007)
Narrow quasiparticle resonance
just above the Fermi level for
the minority spin electrons
Hubbard-Holstein model
Mott/Hubbard transition
Bipolaron transition
Universality ?
Bauer+ACH PRB 81, 235113 (2010)
Summary and Conclusions
• Renormalized parameters can give a global perspective on low energy behaviour of
many strongly correlated systems
• Indicate the conditions when universal behaviour can be expected
• Specific examples of quantum critical points suggest universality may be a feature
of certain types of critical points
• They provide the parameters for more extensive calculations beyond the Fermi
liquid regime and for non-equilibrium response functions using the renormalized
perturbation theory (RPT)
• We have mainly calculated the renormalized parameters using the numerical
renormalization group (NRG) but have now successfully developed alternative less
specialized approaches based on the renormalized perturbation theory by scaling
from regimes of weak correlation to strong correlation
(see : Edwards,+ACH , JPCM 23, 045601 (2011) :Edwards,, ACH , Pandis , PRB 87
. 165128 (2013) )
Fermi Liquid Theory
The low energy single particle excitations of the system are quasiparticles with energies
in 1-1 correspondence with those of the non-interacting system
ie.
interaction between quasiparticles
We have a number of exact results at T=0:
specific heat coefficient
spin susceptibility
free quasiparticle density of states
charge susceptibility
The low energy dynamic susceptibilities and collective excitations can be calculated by taking
accout of repeated quasiparticle scattering.
Can we relate the parameters of Fermi liquid theory to renormalized
parameters for models with strong electron correlation ?
We note:
1. Quasiparticles should correspond to the low energy poles in the
single-electron Green’s function
2. The quasiparticle interactions should correspond to the low
energy vertices in a many-body perturbation theory
This enables us to interpret the Fermi liquid parameters in terms of
renormalizations of the parameters that specify these models
Renormalised Parameters: Anderson Model
Four parameters define the model
Local Green's function
Use substitution
New form of
Green’s function
renormalized parameters
quasiparticle Green’s function
Interaction interaction between quasiparticles
ACH , PRL 70, 4007 (1993)
Summary of Renormalized Perturbation Theory (RPT) approach
The renormalised parameters (RP) describe the fully dressed quasiparticles of Fermi liquid
theory.

They provide an alternative specification of the model

We can develop a renormalised perturbation theory (RPT) to calculate the behaviour of the
model under equilibrium and steady state conditions using the free quasiparticle propagator,

In powers of
conditions:

with counter terms to prevent overcounting, determined by the
Exact low temperature results for the Fermi liquid regime are obtained by working only to
second order only!

Relation to Fermi Liquid theory
This would be the RPA approximation for bare
particles fin the case
Calculation of
and
using the NRG
NRG chain
Non-interacting Green’s function
Given ed and V the excitations w=en of the noninteracting system are solution of the equation:
Interacting Case
We require the lowest single particle Ep(N) and hole Eh(N) excitations to satisfy this equation for a chain of
length N
This gives us N-dependent parameters
ACH et al Eu Phy J B 40, 177 (2004)
Renormalized Parameters from the NRG energy levels in a magnetic field
RPA regime?
Kondo regime
Strong coupling
condition
mean field regime
Can we derive these results from perturbation theory?
We calculate the parameters directly from the
definitions in four stages:
1. We use mean field theory to calculate the renormalised parameters in
extremely large field h1 (>>U)
2. Extend the calculation to include RPA diagrams in the self-energy
3. We use the renormalized parameters in field h1 to calculate the renormalized
self-energy in a reduced field h2, and calculate the renormalized parameters in
the reduced field.
4. We set up a scaling equation for the renormalized parameters to reduce the
field to zero.
Stage 1
Stage 2
?
Stages 3, 4
Weak field strong correlation regime
Strong correlation result satisfied
Further comparison of direct RPT with Bethe Ansatz and NRG results
T=0, H=0, susceptibility compared with
Bethe anasatz results as a function of U
Edwards,+ACH , JPCM 23, 045601 (2011)
Magnetization as a function of
magnetic field compared to NRG
results
Comparison of RPT and NRG results in the low field regime
Away from particle-hole symmetry
Edwards,, ACH , Pandis , PRB 87. 165128 (2013)
Overview for U>0 and U<0 as a function of the occupation nd
U>0
U<0
See Hewson, Bauer and Koller, Phys. Rev. B 73, 045117 (2006)
Leading low temperature correction terms in Fermi liquid regime 1
These corrections to the self-energy can be calculated exactly using the
second order diagrams in the RPT:
Temperature dependence of renormalized parameters?
We explore the idea of using temperature dependent
renormalized parameters from the NRG
Without particle-hole symmetry
Spin and charge dynamics in a magnetic field
Irreducible vertices
charge
U
_|_ spin
|| spin
Spin and charge irreducible Verticies
charge
spin