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Reduction to a one-dimensional problem: if an impurity is
isotropic, it only couples to conduction electron states that are,
likewise, isotropic around the impurity (the s waves).
Conduction states with other symmetries (p, d, ...)
are in this approximation irrelevant. The s waves form
a one parameter set (indexed by the wave number k or energy ω).
The prototype Kondo problem is thus a one-dimensional field theory.
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Logarithmic discretization of the conduction band: the effective
one-dimensional conduction band still has an infinite number of degrees of
freedom (a countable continuum for a system of finite size). For numerical
calculation, the number of states needs to be reduced to a manageable
amount. In NRG, this is achieved by logarithmically discretizing the
conduction bands: the band is divided into
intervals with geometrically decreasing widths Λ-N, where
Λ is a discretization parameter, Λ>1. Each interval is then
Fourier transformed and only the zero (constant) mode in each interval is
retained. It was shown by Wilson that the neglect of higher Fourier modes
leads to a small error, which can in principle be systematically decreased
by reducing Λ towards
1; this corresponds to going back to the continuum limit.
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Tridiagonalisation and the (Wilson) chain Hamiltonian:
the Hamiltonian that describes the impurity and the retained states in the
discretized conduction band can be cast in the form of a one-dimensional
tight-binding Hamiltonian where each site couples only to the nearest
neighbors with exponentially decreasing matrix elements. The topology of
inter-site couplings is that of a chain, and the Hamiltonian is sometimes
named the Wilson chain Hamiltonian.
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Iterative diagonalization: the resulting chain Hamiltonian is
then diagonalized iteratively. One starts by exactly diagonalizing the
impurity site and (optionally) the first few sites of the chain. One then
adds one site at a time, couples it to the eigenstates of the previous NRG
iteration and performs a new diagonalization. Due to exponentially decreasing
coupling matrix elements, this corresponds to considering excitations on a
reduced energy scale, given approximately by Λ-N/2.
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Truncation of states: since, with each added site of the Wilson chain,
the number of states grows by a factor of 4, the resulting matrices become
of unmanageable size in just a few iterations. For this reason, after each
iteration we need to truncate the number of states, i.e. we keep only a few
1000's of states with the lowest energies. Alternatively, we keep states of
(rescaled) energies below a certain arbitrary cutoff energy. Wilson had
shown that the error introduced by this procedure in the following iteration
is small for the excitations with the lowest energies, which are precisely
those states that determine the physical properties. By increasing the
number of states retained, the error can be systematically reduced. The
smaller the discretization parameter Λ is, the more states we need to
retain. In practical terms this means that a compromise must be made between
the discretization error and the truncation error, since a small Λ
would require retaining too many states, while a large Λ increases
discretization errors. With increasing performance of modern computers, the
complexity of the problems that can be efficiently computed grows. It is now
possible to accurately calculate three-impurity two-channel problems on a
modest cluster of computers.
Typical matrix structure of the Hamiltonian in some invariant subspace.
The example is taken from a calculation for a two-channel spin-1/2
Kondo model using SU(2) spin and SU(2) isospin symmetry.
The block structure of the matrix has its origin in the coupling of
16 invariant subspaces from the previous iteration step
via the newly added chain site.
If the Hamiltonian matrix is reordered so that the diagonal matrix
elements are sorted in increasing order, the previous matrix takes
a very characteristic form with matrix elements that are decreasing
as we move away from the matrix diagonal. This is an explicit
manifestation of the energy scale separation, which
is the particular property of quantum impurity problems that
makes NRG calculations successful!