Hi,

I'm looking at the time-evolution of a product state under a time-dependent Hamiltonian using iTEBD. For certain simulation parameters, it's the case that when I measure the bond-energy, using M.bond_energies() for a model M, that energy is different for the two bonds. This is an issue in my research project, and I'm just wondering if you have any advice on how to remedy this? The issue seems to be that the state is breaking translational invariance, but it shouldn't be with exact time evolution. This leads to none smooth answers for the energy as a function of the rate at which I change the Hamiltonian coupling constants in the time evolution.

Thanks for any suggestions.

Best,

Nick

## Different Energy measures giving mixed results

### Re: Different Energy measures giving mixed results

TEBD breaks translation invariance by performing the Trotterization of the evolution operator onto even/odd bonds. The direct way to reduce this effect is to do more, but smaller timesteps.

You could also try to do higher-order evolution, but of course that doesn't make your algorithm higher-order in dt for the time-dependent case, since you would still use a first-order approximation \( U(t, 0) = \mathcal{T} \exp\left(-\frac{i}{\hbar} \int_0^t H(\tau) d\tau \right) \approx \prod_{j=0}^N \exp(-\frac{i}{\hbar} H(\frac{j t}{N}) ) \), even though you would approximate the RHS with higher order precision.

You could also try to do higher-order evolution, but of course that doesn't make your algorithm higher-order in dt for the time-dependent case, since you would still use a first-order approximation \( U(t, 0) = \mathcal{T} \exp\left(-\frac{i}{\hbar} \int_0^t H(\tau) d\tau \right) \approx \prod_{j=0}^N \exp(-\frac{i}{\hbar} H(\frac{j t}{N}) ) \), even though you would approximate the RHS with higher order precision.