Staggered magnetization is an order parameter that describes long range order. It's value is different from 0 only for values of $D$ smaller than $D_{\text{AF}}\approx -0.33$. Below the critical value, the system has long range order and the ground state is ferromagnetic. For larger $D$, the staggered magnetization is zero and the system does not display long range order. According to Landau criterion this should be a single unique phase.
Note: a priori, before claiming that there is no long range order for $D>D_{\text{AF}}$ we should check other order parameters beyond the antiferromagnetic one. For the moment we will ignore this detail.
In the lecture notes you have seen the definition of the entanglement entroby for a generic bipartition in two systems A and B: $$S=-\text{Tr}\rho_A\text{log}\rho_A=-\sum_\alpha\lambda_\alpha\text{log}\lambda_\alpha$$ where $\lambda_\alpha$ are the Schmidt values. A divergence in the entanglement entropy signals the presence of a phase transition.
We again find the antiferromagnetic phase transtion at $D_\text{AF}$ but there now seems to be another transition that was not captured by the staggered magnetization. Such transtion occurs at $D_{\text{Hal}}=1$ and it is a topological phase transtion between two phases with the same symmetries but different topological properties. One phase is the Haldane phase for small $D$ and the other is a phase connected to $D\rightarrow\infty$, where the ground state is a product state of sites with zero magnetization.
The entanglement spectrum allows us to distinguish between topological and trivial phases. In fact, in a symmetry protected topological phase the entanglement spectrum is always doubly degenerate. In the plot each line shows a level of the entanglement spectrum and the number of dots represents its degeneracy, We therefore found that the phase $D\in [-0.33,1]$ is topological, while the other two phases are topologically trivial. This topological phase is the famous Haldane phase.
%% Cell type:markdown id: tags:
# Phase diagram for $D=0$ and varying $\lambda$
%% Cell type:markdown id: tags:
## Define the model
%% Cell type:code id: tags:
``` python
L=2#Chain length (2 for the iMPS)
J=1#Anitferromagnetic interaction
D=0
lammin=0
lammax=1/3
points=30
ll=np.linspace(lammin,lammax,points,endpoint=True)#Values considered for D
#Definition of the lattice model
site=SpinSite(S=1,conserve=None)
lat=Lattice([L],[site],order='default',bc='periodic',bc_MPS='infinite')#We choose an infinite MPS
```
%% Cell type:markdown id: tags:
## Perform iDMRG
%% Cell type:code id: tags:
``` python
entEnt=[]
entSp=[]
stagMag=[]
forlaminll:
#Define the paramters of the model
model_params={
'verbose':0,
'J':J,
'D':D,
'lambda':lam,
'lattice':lat
}
#Create the model
chain=Heisenberg(model_params)
#Define the paramters for the DMRG
dmrg_paramsGs={
'mixer':False,
'trunc_params':{
'chi_max':30,#Maximum bond dimension
'svd_min':1.e-12
},
'max_E_err':1.e-12,
'verbose':0
}
#Initialize the MPS with an arbitrary product state
