Exact diagonalization of the S=1/2 Heisenberg chain
(no symmetries used)
Program: hchan_0
Input: File read.in containing:
Column 1: System size N (integer)
Example read.in (N=8)
8
Output: File eig.dat containing:
Column 1: Eigenvalue number; 0,...,2N-1
Column 2: Energy eigenvalue
Column 3: Total spin S expectation value; extracted from <S⋅S>=S(S+1)
Column 4: Magnetization (z axis) expectation value
Periodic boundary conditions are used also for N=2, resulting in energies twice the nor mal singlet/triplet energies. The output in this case is
0 -1.5000000000 0.0000000000 -0.0000000000
1 0.5000000000 1.0000000000 -0.5000000000
2 0.5000000000 1.0000000000 -0.0000000000
3 0.5000000000 1.0000000000 0.5000000000
For N=4, the following output results. Note that states 6-8 are degenerate (same energies in column 2) with the same z magneti zation (column 4). The total spin (column 3) is not diagonal, as the non-integer expectation values do not corresponding to allowed S eigenvalues. A linear combination of the 3 degenerate states with different S was produced by the diagonalization routine.
0 -2.0000000000 0.0000000000 0.0000000000
1 -1.0000000000 1.0000000000 -0.5000000000
2 -1.0000000000 1.0000000000 0.0000000000
3 -1.0000000000 1.0000000000 0.5000000000
4 -0.0000000000 1.0000000000 -0.5000000000
5 -0.0000000000 1.0000000000 0.5000000000
6 0.0000000000 0.5029870708 -0.0000000000
7 0.0000000000 0.9699263936 -0.0000000000
8 0.0000000000 0.7583057392 0.0000000000
9 0.0000000000 1.0000000000 0.5000000000
10 0.0000000000 1.0000000000 -0.5000000000
11 1.0000000000 2.0000000000 -0.5000000000
12 1.0000000000 2.0000000000 -1.0000000000
13 1.0000000000 2.0000000000 0.5000000000
14 1.0000000000 2.0000000000 0.0000000000
15 1.0000000000 2.0000000000 1.0000000000
The following are the 16 first eigenvalues for N=8. Note the many degenerate states wit h identical spin (column 2) and the fact that now the z-magnetization (column 4) values are in general not allowed magn etization eigenvalues. This is because the diagonalization subroutine (depending on exactly how it works) can mix stat es with same E, S and different magnetization if the hamiltonian matrix is not written in magnetization block-diagonal form.
0 -3.6510934089 0.0000000000 0.0000000000
1 -3.1284190638 1.0000000000 -0.4999786387
2 -3.1284190638 1.0000000000 -0.0000213613
3 -3.1284190638 1.0000000000 0.5000000000
4 -2.6996281483 0.0000000000 0.0000000000
5 -2.4587385089 1.0000000000 -0.4482606137
6 -2.4587385089 1.0000000000 0.4999793383
7 -2.4587385089 1.0000000000 -0.0427976061
8 -2.4587385089 1.0000000000 0.0059285074
9 -2.4587385089 1.0000000000 -0.4725423843
10 -2.4587385089 1.0000000000 0.4576927583
11 -2.1451483739 1.0000000000 -0.4622747971
12 -2.1451483739 1.0000000000 0.3924081181
13 -2.1451483739 1.0000000000 -0.1595269579
14 -2.1451483739 1.0000000000 0.4890986185
15 -2.1451483739 1.0000000000 -0.1851207824