11.6 Quantum renormalisation in real space 355You have now seen group theory ‘at work’. The possible improvements in perform-
ance are really impressive – for this reason, group theoretical methods are very
important in computational physics. A particular example is solid state physics; in
Chapter 6 we have already briefly mentioned the use of symmetry, in particular to
reduce the Brillouin zone to the so-called ‘irreducible wedge’. A nice discussion
ofthesymmetry-issuesinquantumsystemscanbefoundinRef. [21].
11.6 Quantum renormalisation in real space
Calculations for systems whose behaviour is characterised by large length scales
are usually time-consuming. Often, the systems have short-range interactions, but
these generate correlations over large distances. The most obvious example is the
Poisson equation for a point charge:
∇^2 V(r)=δ(r). (11.40)Discretising the Laplace operator on a grid in a first order approximation couples
only nearest neighbour lattice sites. On the other hand, the solution has long range
character. InAppendix A7.2we consider the multigrid method for treating this
problem. This method uses successive coarsenings and refinements in order to find
the solution very efficiently.
The multigrid method is reminiscent of the renormalisation method for critical
phenomena. There we perform successive coarse grainings of the system, in order
to extract its long-wavelength behaviour which is responsible for the occurrence of
critical phenomena. In renormalisation procedures, we usually tend to throw away
details relating to the short-wavelength behaviour. This causes the critical point to
be found at the wrong value, but critical exponents are still found correctly or to a
good approximation.
Often we are interested in the full solution, and not only the long-range beha-
viour. In these cases we must still treat the short-range behaviour accurately. The
multigrid method accomplishes this by the refinements which alternate with the
coarsenings. Another possible approach, which is closer to the standard renormal-
isation procedure, is to take a small sample system and extend the size of this and
see whether it is possible to find a self-consistent limit which gives us the solution
of the infinite system. Wilson has followed this approach with great success for the
Kondo problem [22]. We shall not treat the Kondo problem in detail, but emphas-
ise at the same time that the reason why this numerical renormalisation procedure
worked so well is due to the special structure of the Kondo Hamiltonian, which
contains couplings that decay exponentially with distance.
The lack of success when applying the renormalisation procedure to other prob-
lems has been nagging researchers until White and Noack [ 18 , 23 ] came up with