356 Transfer matrix and diagonalisation of spin chains
Figure 11.4. Two fixed-boundary solutions for a narrow box (solid triangles) and
the ground state (open squares) for a large box. It is clear that the approximation
of the ground state will be poor.a solution for a quantum particle in a potential and for a quantum spin chain. The
particle in a potential illustrates where the standard renormalisation procedure fails
[18]. Consider a particle in a box with impenetrable walls (infinite potential outside
the box). An idea for solving this problem efficiently is first to solve the prob-
lem in a box of half the width of the full box and then solve the problem for the
full box using the ground state wave functions for the half-width boxes as basis
functions in a variational calculation. Figure 11.4 illustrates why this method fails.
The Schrödinger equation is solved for fixed boundary conditions, and building the
ground state of the full problem from two functions that vanish at the centre is not
efficient.
In order to improve the method, White and Noack built the solution of the larger
box from solutions obtained for a smaller boxwith different kinds of boundary
conditions. Consider two solutions, one of which has fixed zero boundary conditions
on both sides, whereas the other has an open boundary condition on one side. The
first has value zero, but (in general) a nonzero derivative. The second one has
nonzero value at the open boundary, but its derivative is zero.Anytype of boundary
condition can be obtained by linear combination of the two solutions. This notion
leads to the following algorithm.
- Find the solutions of a small box, with fixed and open boundary conditions on
both sides (four cases in total). Keep the lowestM/4 eigenstates of the
Hamiltonian for each case. - Construct a Hamiltonian for the large box (twice as large as the small box)
with respect to the basis found in step 1, for the same four cases as in step 1.
See below for how this is done. The dimension of these Hamiltonians is 2M.