DMRG on a ring

[Chenqitrg]

Hi, guys

I am wondering whether MPSKit has any implementation of DMRG on the periodic boundary condition?

[lkdvos]

It does and doesn't, in the sense that we have the periodic_boundary_conditions function to map an infinite hamiltonian to a periodic one, but this is still implemented using open boundary MPS/MPOs, with long range interactions for the boundary crossing terms.
The main problem is that DMRG doesn't necessarily generalize to rings, because of the lack of canonical forms.

[Chenqitrg]

It does and doesn't, in the sense that we have the periodic_boundary_conditions function to map an infinite hamiltonian to a periodic one, but this is still implemented using open boundary MPS/MPOs, with long range interactions for the boundary crossing terms. The main problem is that DMRG doesn't necessarily generalize to rings, because of the lack of canonical forms.

Still thinking about this problem. Thank you. One additional gauge transformation of a ring MPS is to attach another loop of matrices with no physical bond to the existing MPS. Maybe the DMRG on a ring is to minimize the energy while also minimize the loop entanglement at the same time.

[Chenqitrg]

Oh, attaching a matrix product is not a gauge transformation when fixing the bond dimension.

Consider a matrix with dimension d on a ring (circular MPS without physical bond), its gauge group is GL(d). And the canonical form is the Jordan normal form. However, when the physical dimension is D > 1, the MPS can be viewed as a set of matrices {M_i}, i = 1,...,D. The gauge group is still GL(d), acting simultaneously on these matrices by conjugation. We can transform the matrix M_1 to the Jordan normal form but in general cannot simultaneously do that to all matrices.

[Chenqitrg]

It seems different from uniform MPS, the MPS on a ring also has gauge redundancy of redistributing eigenvalues. Still consider a matrix. If it is placed uniformly on an infinitely large translational invariant lattice, it forms a uniform MPS, then the gauge redundancy is indeed GL(d). But when one put it on a ring, the gauge redundancy enlarges because only the trace matters. Therefore, Jordan canonical forms with the same trace are all equivalent on a ring.

[lkdvos]

The main works that I know about for the specific case of periodic MPS are the following:

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.081103
https://arxiv.org/abs/1303.1333

Other, more general strategies for dealing with loopy tensor networks could be possible, but are probably outside of the scope of what I have time for right now