Hello ITensors Team ,

Ok I decide ask a new question and close the last one.

i'm trying to investigate the spin Gap behavior of Heisenberg spin-1/2 J1-J2(J1 and J2 >0, non-frustrated) Isotropic model with interations between first pair of spins with exchange paramenter J*1 and the following pair of spins with interations mediated by J*2 an so on. This simple model presents an topological phase transitions at J1=J2 between "Haldane-Even" and "Haldane-Odd" phases.

I wrote an simple code for this model considering open boundary conditions as follows :

```
using ITensors
using DelimitedFiles
int = range(0.95, step=0.5, stop=1.5)
Data = []
Data_TW = []
for i in int
let
N = 120
sites = siteinds("S=1/2", N; conserve_qns=true)
J_1 = 1.0
J_2 = i
ampo = AutoMPO()
for j=1:2:N
ampo += 0.5*J_1,"S+",j,"S-",j+1
ampo += 0.5*J_1,"S-",j,"S+",j+1
ampo += J_1,"Sz",j,"Sz",j+1
end
for j=2:2:N-2
ampo += 0.5*J_2,"S+",j,"S-",j+1
ampo += 0.5*J_2,"S-",j,"S+",j+1
ampo += J_2,"Sz",j,"Sz",j+1
end
H = MPO(ampo,sites)
state1 = []
push!(state1, "Up", "Up")
for n=2:N/2
push!(state1, "Up", "Dn")
end
init_state = [isodd(n) ? 1 : 2 for n = 1:N]
psi0 = randomMPS(sites,init_state,32)
psi1 = randomMPS(sites,state1,32)
@show flux(psi0)
@show flux(psi1)
sweeps0 = Sweeps(10)
maxdim!(sweeps0, 160,320,600,800,1000)
cutoff!(sweeps0, 1e-11)
noise!(sweeps0, 1e-8,1e-9, 1e-10,0.0)
energy0,psi0 = dmrg(H,psi0,sweeps0)
H02 = inner(H,psi0,H,psi0)
E0 = inner(psi0,H,psi0)
var0 = H02 - E0^2
@show var0
sweeps1 = Sweeps(18)
maxdim!(sweeps1, 320,600,800,1000,1200)
cutoff!(sweeps1, 1e-11)
noise!(sweeps1, 1e-8, 1e-9,1e-10,0.0)
energy1,psi1 = dmrg(H,psi1,sweeps1)
H12 = inner(H,psi1,H,psi1)
E1 = inner(psi1,H,psi1)
var1 = H12 - E1^2
@show var1
gap = energy1-energy0
sgap = N*gap
println("The Gap is =",gap)
println("The Scaled gap is=", sgap)
println("The J_2 value is :", J_2)
push!(Data, ["$J_2" "$energy0" "$gap" "$sgap"])
end
open("Data(test).dat", "w") do f
writedlm(f, Data, '\t')
end
end
```

In my code the first loop concerning the J1 interacting spin terms i sum up to N for each 2 points and the second loop for the J2 interacting terms i sum up to N-2 terms(OBC).

The strange fact is, i cannot reproduce the literature results because my spin gap behaves like an Gapped-Gapless transition. The curious fact is, when i change the first loop summation(Up to N-2), i reproduce the correct behavior with an clear quantum phase transition at J = J*2/J*1 = 1.0 , but with the penalty of the missing interaction at last pair of spins(which shows up when i compute the local magnetization).

I already tried several initial different states , productMps vs randomMPS wavefunctions and different values of noise term .

Probably i'm missing something, i'm grateful with any help ! Thanks !