# 1.3. Hubbard Trimer¶

Let’s solve the following the Hubbard trimer model (Hubbard model on a triangle).

(1.3)$H = -t \sum_{\sigma}(c_{0\sigma}^{\dagger}c_{1\sigma}+c_{1\sigma}^{\dagger}c_{2\sigma} +c_{2\sigma}^{\dagger}c_{0\sigma}+{\rm H.c.}) +U\sum_{i}(n_{i\uparrow}n_{i\downarrow})$

The input file (samples/tutorial_1.3/stan1.in) is as follows:

model = "Hubbard"
method = "FullDiag"
lattice = "chain"
L = 3
t = -1
U = 4
2Sz = 0
nelec = 2


Note that the filling is not half filling.

You can execute HPhi as follows

HPhi -s stan.in


## 1.3.1. Ferromagnetic ground state¶

For the Hubbard model on a triangle with one hole, it is known that the perfect ferromagnetism becomes ground state. Please check that.

If you want know the mechanism of the ferromagnetism, please see Hal Tasaki, Kotai Butsuri, Vol. 31, 173 (1996). This is one of the simplest example of the Nagaoka’s ferromagnetism.

## 1.3.2. Effects of transfer integrals¶

Please the effects of the sign of the transfer integrals. For example, what happens if you take t = 1 ?

Another interesting example is by changing the transfer integrals between site 0 and site 2. Following an example of the trans.def

========================
NTransfer      12
========================
========i_j_s_tijs======
========================
1     0     0     0         -1.000000000000000         0.000000000000000
0     0     1     0         -1.000000000000000         0.000000000000000
1     1     0     1         -1.000000000000000         0.000000000000000
0     1     1     1         -1.000000000000000         0.000000000000000
2     0     0     0         -2.000000000000000         0.000000000000000
0     0     2     0         -2.000000000000000         0.000000000000000
2     1     0     1         -2.000000000000000         0.000000000000000
0     1     2     1         -2.000000000000000         0.000000000000000
2     0     1     0         -1.000000000000000         0.000000000000000
1     0     2     0         -1.000000000000000         0.000000000000000
2     1     1     1         -1.000000000000000         0.000000000000000
1     1     2     1         -1.000000000000000         0.000000000000000


Using this transfer integrals, please examine the U dependence of the ground state. Is there phase transition between singlet ground state and the perfect ferromagnetism ?