Time evolution by Trotter gate for fermions

Question

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,

Answer 1

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!

Comments

Great to hear it! Thanks for sharing your solution