How to calculate the string order parameter (for Haldane phase) in density matrix renormalization group?

The ground state of the spin-1 chain is the Haldane phase, which is known to be a symmetry protected topological phase and cannot be detected by conventional order parameter (beyond the Landau-Ginzburg-wilson theory for phase transition). In contrast, the non-local string order parameter due to M. den Nijs et al. (PRB 40, 4709 (1989)) $$\label{SOP} \mathcal{O}_{\mathrm{str}}^{\alpha} = -\lim_{\vert j-i\vert\to\infty}\left<S_i^{\alpha}\exp\left(i\pi\sum_{i<k<j}S_k^{\alpha}\right)S_j^{\alpha}\right>$$ with $\alpha=x,z$ can work.

My question here is how to calculate the string order, $e.g.$ $\mathcal{O}_{\mathrm{str}}^{z}$, in the density matrix renormalization group (DMRG) method? To begin with, one shall contrust all the associated spin operators. Next we can either make the summation $\sum_{i<k<j}S_k^{\alpha}$ first and then calculate the Matrix Exponential, or calculate $e^{i\pi S_k^{\alpha}}$ for each $i<k<j$ and then calculate the product of the matrix.

The problem lies in that, in the DMRG procedure, the operators in the system block and environment block are enlarged in the different ways. Therefore, if we want to calculate the spin-spin correlation function, we must notice the involved spin operators as to whether they are in the same block or not. Especially, if the spins are in the same block, the calculation should be much more invloved. So, how can we implement the calculation of the string order parameter in the DMRG method?

• Comment to the post (v1): Consider to spell out acronyms. – Qmechanic May 26 '16 at 8:06
• Are you using DMRG based on Matrix Product States? Then it should be straightforward to see how to compute strings which are tensor products of local operators. – Norbert Schuch May 26 '16 at 8:29
• @NorbertSchuch: DMRG in its original form, in my current program. – Roger209 May 26 '16 at 12:37
• Maybe it helps if you try to understand DMRG in the MPS formulation. – Norbert Schuch May 29 '16 at 21:18