1,290 research outputs found
Tensor rank of the direct sum of two copies of matrix multiplication tensor is 14
The article is concerned with the problem of the additivity of the tensor
rank. That is for two independent tensors we study when the rank of their
direct sum is equal to the sum of their individual ranks. The statement saying
that additivity always holds was previously known as Strassen's conjecture
(1969) until Shitov proposed counterexamples (2019). They are not explicit and
only known to exist asymptotically for very large tensor spaces. In this
article, we show that for some small three-way tensors the additivity holds.
For instance, we give a proof that another conjecture stated by Strassen (1969)
is true. It is the particular case of the general Strassen's additivity
conjecture where tensors are a pair of matrix multiplication
tensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitution
method preserves the structure of a direct sum of tensors.Comment: 24 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1902.0658
Infinite Matrix Product States for long range SU(N) spin models
We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians
associated with infinite matrix product states. The latter are constructed from
correlators of primary fields in the SU(N) level 1 WZW model. Since the
resulting groundstates are of Gutzwiller-Jastrow type, our models can be
regarded as lattice discretizations of fractional quantum Hall systems. We then
focus on two specific types of 1D spin chains with spins located on the unit
circle, a uniform and an alternating arrangement. For an equidistant
distribution of identical spins we establish an explicit connection to the
SU(N) Haldane-Shastry model, thereby proving that the model is critical and
described by a SU(N) level 1 WZW model. In contrast, while turning out to be
critical as well, the alternating model can only be treated numerically. Our
numerical results rely on a reformulation of the original problem in terms of
loop models.Comment: 37 pages, 6 figure
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