This talk will consist of two parts. In the first part,
I will present the irreducible form of a Matrix Product States (MPS),
which is a generalization of the canonical form of an MPS in the
sense that it is also defined for states with periodicity. I will
then present a fundamental theorem for MPS in irreducible form,
namely one that specifies how two tensors in irreducible form are
related if they give rise to the same MPS. Finally, I will present
two applications of this result: an equivalence between the
refinement properties of a state and the divisibility properties
of its transfer matrix, and a more general characterisation of
tensors that give rise to matrix product states with symmetries.
In the second part, I will present a study of continuum limits
of MPS, where we show that an MPS has a continuum limit (for a
proper definition thereof) if and only if its transfer matrix
is an infinitely divisible channel. We also consider continuum
limits after a finite number of coarse graining steps, and
characterize it in terms of the divisibility properties of the
transfer matrix. I will present several examples of states with
and without the two kinds of continuum limits.
Joint work with I. Cirac, D. Perez-Garcia and N. Schuch.
Based on arXiv:1708.00880 and arxiv:1708.00029.