1.4. Heisenberg chain (zero temperature)

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

(1.4)$$H=J\sum _{⟨i,j⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$$

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)$$\begin{array}{r}\begin{array}{c}{E}_0=-7.142296\\ E_1=-6.872107\end{array}\end{array}$$

1.4.3. Size dependence of the spin gap

The spin gap at finite system size is defined as $\mathrm{\Delta }=E_1-E_0$. For 16-site, we obtain $\mathrm{\Delta }\sim 0.2701$.

Please examine how $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ 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 ($L=\mathrm{\infty }$). This spin gap is often called Haldane gap.