11.7 The density matrix renormalisation group method 363total Hamiltonian is then
HmU,sl+ 1 ,sl+ 2 ,n;m′,s′
l+ 1 ,s′l+ 2 ,n′
=HmmS ′+∑
αβJαβ[Sα,l]mm′[Sβ,l+ 1 ]sl+ 1 ,s′l+ 1+
∑
αβJαβ[Sα,l]sl+ 1 ,s′l+ 1 [Sβ,l+ 1 ]sl+ 2 ,s′l+ 2+
∑
αβJαβ[Sα,l+ 2 ]sl+ 2 ,s′l+ 2 [Sβ,l+ 3 ]n,n′+HnnE′, (11.62)where it is understood that every term in this expression must be multiplied by a
few Kronecker deltas involving all quantum numbers which do not occur in it. The
indicesαβarezz,+−and−+for the Heisenberg Hamiltonian.
Now we consider the complexity of a matrix–vector multiplication. Evaluating
the product of the first term in (11.62) with the vector|ψ〉is carried out as follows:
〈m,sl+ 1 ,sl+ 2 ,n|HS|ψ〉=∑
m′HmmS ′〈m′,sl+ 1 ,sl+ 2 ,n|ψ〉, (11.63)and this requiresM^3 M^2 Ssteps, since for each of theM^2 MS^2 elements, a sum overm′
must be carried out.
The multiplication for the second term starts by evaluating an intermediate vector
|φ〉which is|ψ〉multiplied by the term involvingSβ,l+ 1 :
〈m,sl+ 1 ,sl+ 2 ,n|φ〉=∑
s′l+ 1[Sβ,l+ 1 ]sl+ 1 ,s′l+ 1 〈m,s′l+ 1 ,sl+ 2 ,n|ψ〉 (11.64)which takesM^2 MS^3 steps. Then we multiply the intermediate state|φ〉by the term
involvingSα,l:
〈m,sl+ 1 ,sl+ 2 ,n|Sα,l|φ〉=∑
m′[Sα,l]m,m′〈m′,sl+ 1 ,sl+ 2 ,n|φ〉. (11.65)This is obviously of the same complexity (M^3 MS^2 ) as the first term.
The remaining terms can now easily be worked out analogous to this procedure.
programming exercise
Write a DMRG program for the one-dimensional Heisenberg chain.
CheckYou should be able to reproduce the ground state energies for the spin-
1/2 and spin-1 chain. The first is analytically known to be−ln 2+ 1 / 2 =
−0.443 147...per site, and the second should be found at−1.40 148...per
site. Note that the best way to calculate the energies per site is to subtract the
energies found in subsequent steps of the algorithm: this takes into account the
energy of thecentralspins and this converges much faster to a fixed value than
the total energy divided by the number of sites.