# Breaking the Sz symmetry using an ITensor

+1 vote

Follow up on this question: https://itensor.org/support/2072/products-of-2-site-and-4-site-wavefunctions

The tetramers that I used to have are:$\frac{1}{\sqrt{12}}(|{\uparrow_1\uparrow_2\downarrow_3\downarrow_4}>-|{\uparrow_1\downarrow_2\uparrow_3\downarrow_4}>-2|{\uparrow_1\downarrow_2\downarrow_3\uparrow_4}>-2|{\downarrow_1\uparrow_2\uparrow_3\downarrow_4}>-|{\downarrow_1\uparrow_2\downarrow_3\uparrow_4}>+|{\downarrow_1\downarrow_2\uparrow_3\uparrow_4}>)$

Now my purpose is instead of having only 6 states for each tetramer (the Sz=0 part) is having all 2^4 = 16 states (adding the 3 ups-1 down, 1 up-3 downs, 4 ups, 4 downs cases).

I tried:

 for(int n = 1; n <= N; n += 4)
{

auto s1 = sites(n);
auto s2 = sites(n+1);
auto s3 = sites(n+2);
auto s4 = sites(n+3);

normalization = 0;
int pos = n/4*ms;
for (int i=pos; i<=pos+ms-1; i++)
normalization += pow(gsl_vector_get(cicoeff,i),2);

auto wf = ITensor(s1,s2,s3,s4);

wf.set(s1(1),s2(1),s3(2),s4(2), gsl_vector_get(cicoeff, pos)/sqrt(normalization));
wf.set(s1(1),s2(2),s3(1),s4(2), gsl_vector_get(cicoeff, pos+1)/sqrt(normalization));
wf.set(s1(1),s2(2),s3(2),s4(1), gsl_vector_get(cicoeff, pos+2)/sqrt(normalization));
wf.set(s1(2),s2(1),s3(1),s4(2), gsl_vector_get(cicoeff, pos+3)/sqrt(normalization));
wf.set(s1(2),s2(1),s3(2),s4(1), gsl_vector_get(cicoeff, pos+4)/sqrt(normalization));
wf.set(s1(2),s2(2),s3(1),s4(1), gsl_vector_get(cicoeff, pos+5)/sqrt(normalization));

wf.set(s1(2),s2(1),s3(1),s4(1), gsl_vector_get(cicoeff, pos+6)/sqrt(normalization));
wf.set(s1(1),s2(2),s3(1),s4(1), gsl_vector_get(cicoeff, pos+7)/sqrt(normalization));
wf.set(s1(1),s2(1),s3(2),s4(1), gsl_vector_get(cicoeff, pos+8)/sqrt(normalization));
wf.set(s1(1),s2(1),s3(1),s4(2), gsl_vector_get(cicoeff, pos+9)/sqrt(normalization));

wf.set(s1(2),s2(2),s3(2),s4(1), gsl_vector_get(cicoeff, pos+10)/sqrt(normalization));
wf.set(s1(2),s2(2),s3(1),s4(2), gsl_vector_get(cicoeff, pos+11)/sqrt(normalization));
wf.set(s1(2),s2(1),s3(2),s4(2), gsl_vector_get(cicoeff, pos+12)/sqrt(normalization));
wf.set(s1(1),s2(2),s3(2),s4(2), gsl_vector_get(cicoeff, pos+13)/sqrt(normalization));

wf.set(s1(2),s2(2),s3(2),s4(2), gsl_vector_get(cicoeff, pos+14)/sqrt(normalization));
wf.set(s1(1),s2(1),s3(1),s4(1), gsl_vector_get(cicoeff, pos+15)/sqrt(normalization));
psi.ref(n) = ITensor(s1);
psi.ref(n+1) = ITensor(s2);
auto [U,S,V] = svd(wf,{inds(psi.ref(n))},{"Cutoff=",1E-8});
psi.ref(n) = U;
psi.ref(n+1) = S*V;

psi.ref(n+2) = ITensor(s3);
ITensor U2(commonIndex(psi.ref(n), psi.ref(n+1)),inds(ITensor(s2))),S2,V2;
svd(S*V,U2,S2,V2);
psi.ref(n+1) = U2;
psi.ref(n+2) = S2*V2;

psi.ref(n+3) = ITensor(s4);
ITensor U3(commonIndex(psi.ref(n+1), psi.ref(n+2)),inds(ITensor(s3))),S3,V3;
svd(S2*V2,U3,S3,V3);
psi.ref(n+2) = U3;
psi.ref(n+3) = S3*V3;
}


but I am getting a "Setting Tensor element non-zero would violate its symmetry." error. The code above works perfectly if I consider only the Sz=0 part, so I am pretty sure that the problem is that I consider more Sz sectors now.

Is there a way to code that? Reading the Itensor's documentation, I found the mixedIQTensor() function, but I didn't quite understand if it does exactly what I want to do.

Any thoughts on that?

Thank you very much,
Thanos

+1 vote

Hi Thanos,
Thanks for the question. If I understand correctly, you are now trying to make ITensors which do not have a well-defined total Sz quantum number. If that's correct, it's a perfectly fine thing to do but isn't allowed by our system while the Index objects you're using (s1,s2,s3,s4 in your code) carry Sz quantum numbers.

So the solution I'd propose is making your "sites" array (site set) not have quantum numbers. The way to do that is to construct it as

auto sites = SpinHalf(N,{"ConserveQNs=",false});


for the SpinHalf site set and similar for other pre-defined site sets.

Let me know if you're looking for a different solution though.

Miles

commented by (310 points)
Once again, thank you very much. It works like a charm, it is exactly what I wanted to do.
commented by (46.8k points)
Great - thanks for that feedback!