From weak to strong correlation:
A new renormalization group
approach to strongly correlated Fermi liquids
Alex Hewson , Khan Edwards, Daniel Crow, Imperial College London, U.K
Yunori Nishikawa, Osaka City University, Japan; Johannes Bauer, MPI, Stuttgart
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
free quasiparticle density of states
spin susceptibility
charge susceptibility
The low energy dynamic susceptibilities and collective excitations can be calculated by taking
account of repeated quasiparticle scattering.
Models of systems with strong electron correlations
Hubbard Model
A simplified of electrons in the 3d bands of transition metals
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
Periodic Anderson model
Essentially a lattice version of the impurity model with d or f electrons hybridized with a
conduction band – model for heavy fermions
Can we relate the parameters of Fermi liquid theory to
renormalized parameters that define these models?
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
Summary of Renormalized Perturbation Theory (RPT) approach
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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!
Kondo Limit --- only one renormalised parameter
N-fold Degenerate Anderson Model
The n-channel Anderson Model with n=2S
(renormalised
Hund’s rule term)
Relation to Fermi Liquid theory
This would be the RPA approximation for bare particles
fin the case
Renormalized Parameters calculated from the NRG energy levels
energy scales merge --- strong correlation regime
The plot shows how the renormalized parameters vary as the
impurity level
moves from below the Fermi level to above the Fermi level for a fixed value
of U with
In the Kondo regime
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
Magnetization as a function of m
magnetic field compared to NRG
results
Comparison of RPT and NRG results in the low field regime
Quantum critical points of a two impurity Anderson model
This model has two types of quantum critical points
Local singlet transition
-
Local charge order transition
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)
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)
Exact RPT results for the low energy behaviour
Predictions based on continuity of these susceptibilities at the
QCP:
Calculation of renormalized parameters by the NRG
(U12=0)
implies
and Kondo resonance at Fermi level disappears
Results for large U
Universal curves
Agreement with predictions
Convergence of energy scales for small U
Confirm predictions as J
Jc (U12=0)
Calculations for J>Jc
At J=Jc z--> 0 so we lose the Kondo resonance at the Fermi level
because the self-energy develops a singularity
At J=Jc there is a sudden change in the NRG fixed point from one for an even
(odd) chain to that for an odd (even) one
We can no longer use the RPT as we assumed the self-energy to be analytic
at the Fermi-level
We retain the equations as a description of a local Fermi liquid but in the NRG
we treat the first conduction site as an effective impurity because now the
impurities are decoupled as a local singlet.
We then calculate the renormalized parameters for J>Jc from the NRG fixed point
in a similar way as for J<Jc but we must allow for the fact that we have modified
the conduction chain.
Results through the transition (U12=0)
NFL
Fermi liquid 1
Fermi liquid 2
Local charge order transition
SU(4) point
Predictions again
confirmed
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:
Susceptibility through the transition (U12=0)
Dynamic Susceptibilities
Temperature dependence in the non-Fermi liquid regime?
We explore the idea of using temperature dependent
renormalized parameters from the NRG
Conclusions
We have demonstrated that it is possible to obtain accurate results for the low
energy behaviour of the Anderson model in the strong correlation regime by
introducing a magnetic field to suppress the low energy spin fluctuations which
lead to the large mass renormalisations, and then slowly reduce the field to zero,
renormalizing the parameters at each stage. This approach should be applicable to
a wide range of strong correlation models such as the Hubbard and periodic
Anderson model.
The results for the two impurity model support the Kondo collapse conjecture
that at the quantum critical point in some heavy fermions systems the f-states at
the Fermi level disappear and no longer contribute to a large Fermi surface (eg.
Yb2Rh2Si2). The convergence to a single energy scale T* which goes to zero at
the QCP suggest how w,T scaling can arise.