1.2. Hubbard Dimer¶

Let’s solve the following the Hubbard dimer model.

(1.2)¶\[H = -t \sum_{\sigma}(c_{0\sigma}^{\dagger}c_{1\sigma}+{\rm H.c.}) +U(n_{0\uparrow}n_{0\downarrow}+n_{1\uparrow}n_{1\downarrow})\]

The input file (samples/tutorial_1.2/stan1.in) is as follows:

model = "Hubbard"
method = "FullDiag"
lattice = "chain"
L=2
t = -0.5
U = 4
2Sz = 0
nelec = 2

You can execute HPhi as follows

HPhi -s stan.in

1.2.1. Check the energy¶

For the Hubbard dimer at half filling with total Sz=0, energies are given as follows:

\(E=0,U,\frac{U}{2}\times(1\pm\sqrt{(1+(4t/U)^2)})\)

For example, by taking \(U=4,t=-1\), the energies are given as follows:

\(E=-0.828427, 0, 4, 4.828427\)

It is note that simple mathematical calculations can be done using:

bc -l

on the terminal.

The input file (samples/tutorial_1.2/stan2.in) is as follows:

model = "Hubbard"
method = "CG"
lattice = "chain"
L=2
t = -0.5
U = 4
2Sz = 0
nelec = 2
exct = 4

Please check whether LOBCG method correctly reproduces the energies including the excited states.