Abstract
The full exploitation of non-abelian symmetries in tensor network states (TNS) derived from a given lattice Hamiltonian is highly attractive in various aspects. From a theoretical perspective, it can offer deep insights into the entanglement structure and quantum information content of strongly correlated quantum many-body states. From a practical perspective, it allows one to push numerical efficiency by orders of magnitude. Physical expectation values based on TNS require the full contraction of a given tensor network, with the elementary ingredient being a pairwise contraction. While well-established for no or just abelian symmetries, this can become quickly extremely involved and cumbersome for general non-abelian symmetries. As shown in this work, however, the latter can be tackled in a transparent and efficient manner by introducing so-called X-symbols which deal with the underlying pairwise contraction of generalized Clebsch-Gordan tensors (CGTs). These X-symbols can be computed deterministically once and for all, and hence also be tabulated. Akin to 6j-symbols, X-symbols are generally much smaller than their constituting CGTs. In applications, they solely affect the tensors of reduced matrix elements, and therefore, once tabulated, allow one to completely sidestep the explicit usage of CGTs, and thus to greatly increase numerical efficiency.
Highlights
The full exploitation of non-Abelian symmetries in tensor network states (TNSs) derived from a given lattice Hamiltonian is attractive in various aspects
Starting in one dimension (1D) with matrix product states (MPSs), the numerical renormalization group (NRG) [6,7,8,9] and the density-matrix renormalization group (DMRG) [10,11,12,13] represent powerful, nonperturbative, and accurate methods to deal with strongly correlated systems at arbitrary temperature both statically and dynamically
The latter would result in a norm in Eq (7) that is equal to |S3|, i.e., the dimension of multiplet S3. This would make the normalization dependent on the direction of arrows in a given Clebsch-Gordan tensors (CGTs). This is rather inconvenient on general grounds, and so in the TNS context, since the presence or not of outer multiplicity (OM) does not depend on the direction of arrows or on the specific order of symmetry labels
Summary
Tensor network states typically describe lattice Hamiltonians, with whom they share the same lattice structure. Whereas the Hamiltonian may be longer ranged TNSs have nearestneighbor bonds only in order to minimize index loops. Physical sites may be grouped into supersites [55,64]. Each lattice site n is assigned a tensor An that maps the attached (variationally determined) auxiliary state spaces |ax to the physical state space |φσ n. The indices of a tensor are drawn as lines, referred to as the legs of a tensor. E.g., the index σ above spans the local state space of a single physical site, whereas the index x = l, r, . Spans specific named bonds, such as left (l), right (r), etc.
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