Truncation Problem of Density Matrix Renormalisation Group (DMRG)

Question

I am wondering that is there any restrictions for the truncation in DMRG algorithm. Currently I am using DMRG to calculate ground state energy per site of a many-body system described by on-site potential term only($H_{i} = -\mu c^{\dagger}_{i}c_{i}$). However, I got a diverging answer. This is not physical since the ground state energy per site is an intensive quantity, which is independent of the size of the system. Therefore, I re-examine my code and I found something strange there. I print out the eigenvalues of the reduced density matrix of system: \begin{equation} \text{Eigenvalues of $\rho_{sys}$} = [1.000, 6.988\text{E-31}, 8.362\text{E-32}, 6.398\text{E-33},...,0.000] \end{equation} Despite $\text{tr}(\rho_{sys}) \simeq 1$, I think that this set of eigenvalues is strange. The first element is already equal 1 and the difference between the first and second element is $10^{-30}$. This means that only the first eigenstate is most probable and other states are not. Therefore, we can truncate other states out. When I set the number of maximum states for truncation to some number(e.g. 16), I got a diverging answer. This means that I have set a wrong condition for truncation. Therefore, based on the above discussion, what truncation condition should I use if the eigenvalues of $\rho_{sys}$ like that? Besides, is there any restriction for the truncation in DMRG which I have not discussed above?

Answer 1

You target state is a product state, so the ES looks fine. However, this is not a prime application for DMRG exactly for that reason. What goes wrong depends on the details of the implementation you are using, but usually inverses of the ES appear in DMRG (e.g. for canonical forms), which become ill-conditioned. If implemented properly (e.g. discard all eigenvalues below some threshold) this should, however, not lead to instabilities of that kind.