Computational Physics

358 Transfer matrix and diagonalisation of spin chains

Table 11.1. Energies for the particle in

a box after N= 10 steps.

Energies

E 0 2.3508× 10 −^6

E 1 9.4032× 10 −^6

E 2 2.1157× 10 −^5

E 3 3.7613× 10 −^5

These are the valuesn^2 π^2 /L^2 , withL=2049.

programming exercise
You have now enough information for writing a program for this ‘real-space
quantum renormalisation group’.
CheckFor the starting matrices given, you should after 10 steps arrive at the
spectral values given in Table 11.1.
There is an interesting initiative, called ALPS, to construct a C++ library for
quantum algorithms [ 24 , 25 ]. The programs for this and those in the following
sections can be found as examples on the ALPS website.

11.7 The density matrix renormalisation group method

In this section we shall describe how the ideas of the previous section can be
extended to many-body systems in one dimension. First of all, we note that the
renormalisation procedure of the previous section doubled the box size at each
step, and thereby the dimension of the Hilbert space. In the case of a quantum
chain, adding a single spin to a spin-1/2 chain already doubles the dimension of the
Hilbert space, whereas the dimension of the physical space covered increases only
by a small fraction. If we were to double the actual (physical) space, the dimension
of the Hilbert space would increase by such a large factor that we would make
gigantic steps in complexity, which is bound to fail. Therefore we add only a single
site (spin) at a time.
To illustrate how the method works, consider a finite chain of which the lower
energy states are properly described by a (small) basis set|m〉of sizeM(that is,m
runs from 1 toM). The left end of the chain does not couple to a neighbouring spin:
we consider an ‘open end’ boundary condition there. When we add a new spin, we
have states

|m,s〉 (11.45)