Hi ITensor Support,

I am trying to construct and extended Hubbard model Hamiltonian with next-nearest-neighbour hopping but I am having some problems with the A/C operators while using the autoMPO functions.

As a minimal example, I have just considered the usual fermion hopping term

$$ H = - t_0 \sum_{i, s} c_{i, s}^\dagger c_{i+1,s} + c_{i+1, s}^\dagger c_{i,s} $$

I have represented this three different ways using autoMPO:

-1st I just used the A operators without the Jordan-Wigner F operator (which I know should be wrong!)

-2nd I used the C operators as is standard

-3rd I used the A operators with the Jordan-Wigner F operators (this should be the same as above)

```
int L = 4;
double t0 = 5;
//Set arguments
auto args = Args("Cutoff=",1E-15,"MaxDim=",5000);
//Initialise System
auto sites = Electron(L, {"ConserveQNs=",false});
// Build Evolution Operators
auto ampo = AutoMPO(sites);
auto ampo2 = AutoMPO(sites);
auto ampo3 = AutoMPO(sites);
//Effective Hamiltonian
//NN Hopping
for(int j = 1; j < L; ++j)
{
int s1 = j;
int s2 = j + 1;
ampo += (-t0) , "Adagup", s1, "Aup", s2;
ampo += (-t0) , "Adagup", s2, "Aup", s1;
ampo += (-t0) , "Adagdn", s1, "Adn", s2;
ampo += (-t0) , "Adagdn", s2, "Adn", s1;
ampo2 += (-t0) , "Cdagup", s1, "Cup", s2;
ampo2 += (-t0) , "Cdagup", s2, "Cup", s1;
ampo2 += (-t0) , "Cdagdn", s1, "Cdn", s2;
ampo2 += (-t0) , "Cdagdn", s2, "Cdn", s1;
ampo3 += (-t0) , "Adagup", s1, "F", s1, "Aup", s2;
ampo3 += -(-t0) , "Aup", s1,"F", s1, "Adagup", s2;
ampo3 += (-t0) , "Adagdn", s1, "F", s1+1, "Adn", s2;
ampo3 += -(-t0) , "Adn", s1,"F", s1+1, "Adagdn", s2;
}
```

I then compute the hopping amplitude between a state with a single spin down on the 2nd site and a single spin down on the 3rd site. By hand I get the following results:

$$

|\psi_0 \rangle = c_{2, \downarrow}^\dagger |0\rangle, \ |\psi_1 \rangle = c_{3, \downarrow}^\dagger |0\rangle, \ \langle \psi_1 |H|\psi_0 \rangle = -t_0

$$

However, performing the same thing in ITensor (with t_0 = 5) using

```
auto Ham1 = toMPO(ampo, {"ConserveQNs", false});
auto Ham2 = toMPO(ampo2, {"ConserveQNs", false});
auto Ham3 = toMPO(ampo3, {"ConserveQNs", false});
auto states0 = InitState(sites);
states0.set(2, "Dn");
auto psi0 = MPS(states0);
psi0.orthogonalize();
psi0.normalize();
auto states1 = InitState(sites);
states1.set(3, "Dn");
auto psi1 = MPS(states1);
psi1.orthogonalize();
psi1.normalize();
cout << "Initial State Built" << endl;
auto test1 = innerC(psi1, Ham1, psi0);
auto test2 = innerC(psi1, Ham2, psi0);
auto test3 = innerC(psi1, Ham3, psi0);
Print(test1);
Print(test2);
Print(test3);
```

I get:

test1 = (-5, 0)

test2 = (5, 0)

test3 = (5, 0)

Can you understand why these calculations are not agreeing with the theory? I have also tested these results against some exact diagonalisation calculations which agrees with the theory so I am rather confused. I'm sure I have just misunderstood something. I am hoping that by sorting out this bit I might be able to get my full model working.

Many thanks,

Cameron