# Measurement of local obervables

Hello,
I am using heisenberg AFM spin-\frac{1}{2} model. I calculated expectation values of all spin components. I expect <S^x_j>=<S^y_j>=<S^z_j> at j^{th} site.
Following is the part of code.

    for(int j = 1; j < N; ++j)
{
ampo += 0.5,"S+",j,"S-",j+1;
ampo += 0.5,"S-",j,"S+",j+1;
ampo +=     "Sz",j,"Sz",j+1;
}
auto H = toMPO(ampo);
auto psi0 = randomMPS(sites);

auto [energy,psi] = dmrg(H,psi0,sweeps);

    psi.position(j);

auto ket = psi(j);
auto bra = dag(prime(ket,"Site"));

auto Sxjop = op(sites,"Sx",j);
auto Syjop = op(sites,"Sy",j);
auto Szjop = op(sites,"Sz",j);

//take an inner product
auto sxj = eltC(bra*Sxjop*ket);
auto syj = eltC(bra*Syjop*ket);
auto szj = eltC(bra*Szjop*ket);


output obtained is:


site                    sxj                            syj                                   szj
1     (-0.085879786270,0.000000000000)      (0.000000000000,0.000000000000)      (0.524500286661,0.000000000000)
2     (0.052269631567,0.000000000000)      (0.000000000000,0.000000000000)      (-0.317076879182,0.000000000000)
3     (-0.061021765784,0.000000000000)      (0.000000000000,-0.000000000000)      (0.369026870114,0.000000000000)
4     (0.044076181082,0.000000000000)      (0.000000000000,0.000000000000)      (-0.263437612825,0.000000000000)
5     (-0.044070844096,0.000000000000)      (0.000000000000,-0.000000000000)      (0.261660807326,0.000000000000)
6     (0.034740648756,0.000000000000)      (0.000000000000,0.000000000000)      (-0.201093290816,0.000000000000)
7     (-0.033097508882,0.000000000000)      (0.000000000000,-0.000000000000)      (0.187972360981,0.000000000000)
8     (0.027798850184,0.000000000000)      (0.000000000000,0.000000000000)      (-0.151213984075,0.000000000000)
9     (-0.026266199085,0.000000000000)      (0.000000000000,0.000000000000)      (0.137606259785,0.000000000000)
10     (0.023422322342,0.000000000000)      (0.000000000000,0.000000000000)      (-0.114767090540,0.000000000000)
11     (-0.022604446358,0.000000000000)      (0.000000000000,-0.000000000000)      (0.103957656846,0.000000000000)
12     (0.021517400728,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.089911906053,0.000000000000)
13     (-0.021631676509,0.000000000000)      (0.000000000000,-0.000000000000)      (0.082492019101,0.000000000000)
14     (0.022014881161,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.074554770328,0.000000000000)
15     (-0.023165510846,0.000000000000)      (0.000000000000,0.000000000000)      (0.070475409087,0.000000000000)
16     (0.025001185922,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.067285420429,0.000000000000)
17     (-0.027319618034,0.000000000000)      (0.000000000000,0.000000000000)      (0.066398761413,0.000000000000)
18     (0.030833746946,0.000000000000)      (0.000000000000,0.000000000000)      (-0.067450748744,0.000000000000)
19     (-0.034497740499,0.000000000000)      (0.000000000000,0.000000000000)      (0.069689182423,0.000000000000)
20     (0.040197909615,0.000000000000)      (0.000000000000,0.000000000000)      (-0.075191674420,0.000000000000)
21     (-0.045410357418,0.000000000000)      (0.000000000000,-0.000000000000)      (0.080574504932,0.000000000000)
22     (0.054218627438,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.091524846115,0.000000000000)
23     (-0.061092524561,0.000000000000)      (0.000000000000,-0.000000000000)      (0.099997638642,0.000000000000)
24     (0.074637750447,0.000000000000)      (0.000000000000,0.000000000000)      (-0.118564227015,0.000000000000)
25     (-0.082856408373,0.000000000000)      (0.000000000000,-0.000000000000)      (0.129499434728,0.000000000000)
26     (0.104086278356,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.159896684056,0.000000000000)
27     (-0.111959278422,0.000000000000)      (0.000000000000,-0.000000000000)      (0.170666095141,0.000000000000)
28     (0.146558032998,0.000000000000)      (0.000000000000,-0.000000000000)      (-0.221241065119,0.000000000000)
29     (-0.147943534745,0.000000000000)      (0.000000000000,-0.000000000000)      (0.222623127552,0.000000000000)
30     (0.207953185334,0.000000000000)      (0.000000000000,0.000000000000)      (-0.311257802061,0.000000000000)
31     (-0.178737544373,0.000000000000)      (0.000000000000,-0.000000000000)      (0.267281948676,0.000000000000)
32     (0.296136653948,0.000000000000)      (0.000000000000,0.000000000000)      (-0.441729543539,0.000000000000)


Why <S^y> is zero(both real and imaginary part)?

Is your wave function real-valued? It looks like it may be. In that case, Sy will always have a zero expectation value.

can you please explain why <S^y> will have zero expectation value for real valued wave function?

I would suggest working through an example of a general single site wave function to see why.

I think the reason is related to the fact that Sy goes to -Sy under complex conjugation. If you write the expectation value <psi|Sy|psi> and write Sy as i times a real-valued matrix, then take the complex conjugate, this expression will go to the negative of itself. This could be non-zero if the expression was itself imaginary. But expected values of Hermitian operators are always real. So it must be zero.

Okay. Thanks a lot.