General discussions on the ideas behind the algorithm.
on0001st
Posts: 12
Joined: 27 Aug 2018, 09:30

Hi, I would like to clarify questions about DMRG. Am I right to say that DMRG works very well for gapped Hamiltonians because of the area law and the fact that entanglement in the ground states of such Hamiltonians are minimised? When we extend the method to gapless Hamiltonians, as gapless systems are considered as critical and thus do not obey the area law, it is also computationally more expensive to obtain the correct ground state.

I am asking this because when I am performing finite size effects calculations on the S=1 Heisenberg chain in a uniform magnetic field, the "gap" between the ground state and the lowest excited state increases with system size at B > \Delta{B}_c where \Delta{B}_c > 0.41. The exact parameter used is B=2.0. At this field strength, one expects the Luttinger liquid phase which is gapless. Looking at the entanglement entropy, I can see that entanglement actually increased in the Luttinger liquid phase from the Haldane phase. I suspect that this increase in entanglement coupled with the fact that we are working in the gapless regime, it makes DMRG very inefficient to calculate the ground state energies correctly and accurately with minimal chi states needed to be kept. Is this correct?

However, if the entanglement entropy were to decrease in the gapless phase due to breaking of some symmetry, can we now expect to calculate the ground states efficiently once again?

On a side note, the calculation of excited energies in iDMRG is not trivial right?
Johannes
Posts: 324
Joined: 21 Jul 2018, 12:52
Location: TU Munich

### Re: Some questions about DMRG

On general grounds:
DMRG is a variational method which looks for the best approximation of the ground state in the space of MPS with a given maximal bond dimension (assuming you put a "chi_max" in the DMRG_parameters). This class of MPS can only represent states with a entanglement entropy of at most $$S \leq \log(\chi)$$, where $$\chi$$ is the bond dimension of the MPS.

Gapped (short range) Hamiltonians obey the area law, stating $$S(L) \propto L^{D-1}$$. In 1D, this means that the entanglement entropy through a cut of the system is constant, independent of system size. Hence the class of MPS is suited well for ground states of 1D gapped systems, and hence DMRG works well in this case.

If the system is gapless (equivalently "critical"), there are corrections to the area law, namely in 1D $$S(L) \propto \log(L)$$. This means that the larger your system is, the higher bond dimension you have to choose to repesent the ground state faithfully, i.e. yes, DMRG becomes "computationally more expensive".
Of course, you can still run DMRG for the same maximal bond dimension as before, but you can expect that the approximation to the exact ground state will get worse as the system size increases.

I'm confused about your case. You say that you find the "gap" to increase with L, but expect the system to be gapless. This doesn't fit together, does it? Did you try to see what happens if you increase $$\chi$$?

Excited states can be calculated in finite DMRG by orthogonalizing against previously found ground states. In infinite systems, the overlap is not so well defined, and it is not possible calculate the excited states. (You can only talk about the energy density in infinite systems. Flipping a single spin to get an excitation is not changing the energy density in the limit $$L \rightarrow \infty$$ and thus not really a well defined state).
on0001st
Posts: 12
Joined: 27 Aug 2018, 09:30

### Re: Some questions about DMRG

Thank you for the detailed reply, Johannes. It really helps a lot.

The gap that I am talking about is a finite size gap due to finite size effects. Generally, one would expect this to decrease to the actual gap size as we approach the thermodynamic limit. In the case of what I am looking at, the system is supposed to be gapless with the given parameters at the thermodynamic limit but increasing system size actually increases the gap.

However, if I introduce some additional terms into the Hamiltonian, DMRG now gives more correct results where increasing system size decreases this finite size gap. If I look at the entanglement entropy in these two cases, the one with the additional terms has low entanglement entropy as compared to the one without. Hence, my suspicion was that the behavior of the results was related to the fact that it was gapless and with large entanglement entropy, resulting in a need for much more sweeps and much larger bond dimension to get an accurate result.