Role of symmetries

The MPS framework allows to incorporate preserved symmetries in a transparent and

efficient way. In general, symmetries imply that many matrix elements or coefficients

are exactly zero due to selection rules. As a consequence, tensors become sparse. The essential role of abelian symmetries then is, that the non-zero coefficients can be condensed into dense blocks. Therefore while abelian symmetries must deal with the actual full state space dimension still, the efficiency derives from reducing the original problem to a sequential treatment of typically significantly smaller blocks, thus exploiting the sparsity of the problem. The numerically negligible overhead lies in an efficient book keeping of the non-zero blocks.

In the presence of abelian symmetries, one further realizes that many of the non-zero matrix elements are actually dependent on each other (e.g. consider the Wigner-Echart theorem for irreducible operators). In general, this translates into splitting off the corresponding Clebsch Gordan coefficient spaces in terms of a tensor product.

As a result, tensors can be strongly compactified. Rather than talking about the original full state spaces, the language changes to the significantly reduced multiplet spaces.

A basic introduction to abelian and non-abelian symmetries alike together with a de-tailed description of a transparent framework for their treatment in general tensor networks in terms of so-called QSpaces is given in Weichselbaum (2012). Essentially, QSpaces can be seen as powerful arbitrary-rank tensor-library that can also deal with compact non-abelian symmetries. It is based on the explicit evaluation of Clebsch-Gordan coefficient spaces from the actual generators of their Lie algebra. As such, it provides a flexible framework in particular w.r.t. to the implementation of symmetries, where not all quantities of interest are easily available analytically.

2.4.1 State symmetrization

Consider an arbitrary MPS | ψ i that originally has full A-tensors, i.e. makes no reference to any symmetry spaces whatsoever. However, assuming that | ψ i is close to a symmetry eigenstate, this state can be cast into an exact symmetry eigenstate. In practice, this procedure can be useful for testing purposes. The corresponding state symmetrization requires to reorganize all indices into state spaces with proper symmetry labels. The actual procedure is straightforward for abelian symmetries. For non-abelian symmetries, however, the procedure requires to partially recombine multiplet spaces with their Clebsch Gordan coefficient spaces into an explicit tensor product. Hence the latter is not as easy to implement while possible in principle.

The actual procedure then is as follows. Firstly, the symmetrization of a given state

requires a proper mapping of the local state spaces | σ

i for site n into proper symmetry

eigenstates. By contracting this mapping onto every local state space, the local state

space can therefore be written in proper symmetry labels. The remaining strategy then

is completely analogous to the LR-orthonormalization already discussed with Fig. 2.5. As

such, it is an iterative prescription. The starting point may again be taken as the very

left end of the MPS, i.e. site n = 1. The local state space has already been cast into a

symmetry basis. The effective basis to its left is the left-vacuums state, which transforms

like a scalar for all symmetries. Hence its symmetry labels are trivially also known, i.e.

2. Matrix Product States



(left + local) part of the system

right part of the system







Figure 2.6: Schematic depiction of orthonormalization and possibly truncation of wave function | ψ i = P

(l,σ),r

ψ

| lσ i| r i in the one-site local DMRG picture of left block (l), right block (r), and local state space (σ). Here left (l) and local (σ) state space are assumed to be orthonormal and already written in proper combined symmetry labels q

. The state space for the right (r) part of the system, however, in principle at this point can be an arbitrary coefficient space. It may already exist in terms of block labels q

(with non-zero blocks shaded in gray). his is not required, however, since only full rows are considered anyway. The latter is indicated by the slicing along the lines that separate blocks with different symmetry labels q

. Therefore in the absence of symmetry labels, for example, the entire state space r may be described by one single block with (irrelevant) block label q

= 0.

are all equal to zero. Therefore making the iterative assumption that w.r.t. to the current site n the effective basis for the left block of sites n

< n is properly written in terms of their symmetry labels, together with the local Hilbert space of site n, the tensor A

can be sliced into well-defined symmetry spaces, as schematically indicated in Fig. 2.6. There left (n

< n) and local (σ) state space are already combined into proper combined symmetry spaces q

. Since the symmetry labels for the right block are not yet known, the entire state space r can be described by one single block with irrelevant block label q

= 0. Subsequent SVD within the combined symmetry sectors q

allows to extract an orthonormal Schmidt basis for all sites up-to and now including site n, i.e. A

→ U

X

, as indicated in Fig. 2.5.

Here U

already refers to proper symmetry sectors, such that also the new index connecting U

and X

refers to proper symmetry spaces. By contracting X

onto A

, and setting n → n + 1, the iterative prescription can be repeated for site n + 1.

Once the end of the chain is reached at site n = N , the left index of X

connecting to A

will list all symmetry spaces that are contained in the original state | ψ i . Having assumed

that | ψ i is close to a symmetry eigenstate, one symmetry sector in X

will dominate in

amplitude. This allows to project the state | ψ i onto the dominant symmetry, by skipping

the contributions to X

from all other symmetries, i.e. setting them to zero while also

readjusting the global normalization of the state. Overall, now the state | ψ i itself has a

well-defined exact symmetry, while also all of its internal state spaces had been reorganized

in terms of proper symmetry labels.