# Time evolution by Trotter gate for fermions

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Hi Everyone,

I'm trying to simulate time-evolution of Hubbard model with Trotter gate. I start from simple tight-binding model with time-independent Hamiltonian, so the energy should be conserved. However, when I try the code, I find the energy is not conserved even with very small time steps ~ 0.001. Here is a part of my code:

using Gate = BondGate<IQTensor>;
auto gates = vector<Gate>();
for(int b = 1; b <= L-1; ++b)
{
auto hterm = -1.*sites.op("Cdagup",b)*sites.op("Cup",b+1);
hterm += -1.*sites.op("Cdagup",b+1)*sites.op("Cup",b);
hterm += -1.*sites.op("Cdagdn",b)*sites.op("Cdn",b+1);
hterm += -1.*sites.op("Cdagdn",b+1)*sites.op("Cdn",b);
auto g0 = Gate(sites,b,b+1,Gate::tReal,tau/2.,hterm);
gates.push_back(g0);
}

for(int b = L-1; b >= 1; --b)
{
auto hterm = -1.*sites.op("Cdagup",b)*sites.op("Cup",b+1);
hterm += -1.*sites.op("Cdagup",b+1)*sites.op("Cup",b);
hterm += -1.*sites.op("Cdagdn",b)*sites.op("Cdn",b+1);
hterm += -1.*sites.op("Cdagdn",b+1)*sites.op("Cdn",b);

auto g0 = Gate(sites,b,b+1,Gate::tReal,tau/2.,hterm);
gates.push_back(g0);
}

auto args = Args("Cutoff=",cutoff,"Maxm=",mmax);

gateTEvol(gates, ttotal, tau, psi, args);


Can you tell me what's wrong with this? Also, I'm confused that if I use, for example,

hterm = +1.*sites.op("Cup",b+1)*sites.op("Cdagup",b);


I get a different result. Is this due to the anti-commutation of fermions?

Many thanks,

## 1 Answer

+1 vote
answered by (180 points)
edited by

Just for anybody interested, I fixed the problem after thinking about the anti-commutations of fermions. What I should do is to replace the C-operators to combinations of A- and F-operators as:

auto hterm = -1.*sites.op("Adagup*F",b)*sites.op("Aup",b+1);
hterm += -1.*sites.op("Adagdn",b)*sites.op("F*Adn",b+1);
hterm += +1.*sites.op("Aup*F",b)*sites.op("Adagup",b+1);
hterm += +1.*sites.op("Adn",b)*sites.op("F*Adagdn",b+1);


With this I see that the energy is conserved for time-independent Hamiltonian!

commented by (30.7k points)
Great to hear it! Thanks for sharing your solution