The Numerical Renormalization Group
Short graduate lecture course: Gothenburg, Sweden.
'Quantum Impurity Problems' are classic paradigms for strong electron correlations in condensed matter physics. They underpin the theoretical description of magnetic impurities in metals, nanodevices such as quantum dots, and appear as effective models within the dynamical mean field theory of correlated materials. Non-perturbative quantum many-body methods must be employed to solve such problems. In this course, we provide the conceptual framework of the Numerical Renormalization Group, discuss technical/practical details of the calculation, and present relevant applications.
Lecture slides in pdf format can be found below:Part 1: Quantum Impurity Problems and theoretical background.
Part 2: Kondo effect and RG. 1d chain formulation and iterative diagonalization.
Part 3: Logarithmic discretization and truncation. The RG in NRG.
Part 4: Calculation of physical quantities. Results and discussion.
ReferencesK. G. Wilson, Rev. Mod. Phys. 47, 773 (1975)
A. C. Hewson, "The Kondo problem to heavy Fermions" (CUP, 1997)
H. R. Krishnamurthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980)
R. Bulla, T. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008)
NRG tutorial code:Follow the link below to download a minimal fortran 90 code to solve the Anderson Impurity Model using NRG.
1) Compile the code. Run for impurity parameters V=0.1, epsilon=-U/2 and U=0, 0.1, 0.2, 0.3, ...
2) Examine the iterative flow of many-particle energies, in the output file energies.dat
3) Examine the temperature-dependence of the entropy, in the output file thermoav.dat
4) Determine the Kondo temperature for each data set [defined as Simp(TK)=ln(2)/2]
5) Confirm the perturbative scaling result for TK
6) Plot the entropy in terms of T/TK
7) Experiment with other impurity parameters, different Lambda, number of kept states etc
The Anderson Impurity Hamiltonian conserves total charge, Q, and spin projection, Sz. This means that the Hamiltonian has a block-diagonal structure. Label the states by these quantum numbers, and block-diagonalize the Hamiltonian. The program runs very much faster, and a lot more states can be kept per iteration! Calculate magnetic susceptibility as well as entropy.
More information on the practical implementation can be found in the PRB by Krishnamurthy et al, listed above.
Download code here.