# 1.4. Heisenberg chain (zero temperature)¶

Let’s solve the following spin 1/2 Heisenberg model on the chain.

(1.4)$H = J \sum_{\langle i,j\rangle}{\bf S}_{i}\cdot{\bf S}_{j}$

The input file (samples/tutorial_1.4/stan1.in) for 16-site Heisenberg model is as follows:

L       = 16
model   = "Spin"
method  = "CG"
lattice = "chain"
J = 1
2Sz = 0
2S  = 1


You can execute HPhi as follows

HPhi -s stan.in


## 1.4.1. Check the energy¶

Please check whether the energies are given as follows.

(1.5)$E_{0}= -7.142296$

## 1.4.2. Obtaining the excited state¶

By adding exct=2, you can obtain the 2 low-energy states (samples/tutorial_1.4/stan2.in). Please check the energies.

(1.6)\begin{align}\begin{aligned}E_{0}= -7.142296\\E_{1}= -6.872107\end{aligned}\end{align}

## 1.4.3. Size dependence of the spin gap¶

The spin gap at finite system size is defined as $$\Delta=E_{1}-E_{0}$$. For 16-site, we obtain $$\Delta\sim 0.2701$$.

Please examine how $$\Delta$$ behaves as a function of system size L (samples/tutorial_1.4/stan3.in for L=20). (available system size on PC may be L=24)

## 1.4.4. Haldane gap¶

By performing a similar calculations for S=1 system, please examine how $$\Delta$$ behaves as a function of system size L (samples/tutorial_1.4/stan4.in). It is known that the finite spin gap exists even in the thermodynamic limit ($$L=\infty$$). This spin gap is often called Haldane gap.