Novel families of SU( N ) AKLT states with arbitrary self-conjugate edge states

Gozel, Samuel  (Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland) ; Poilblanc, Didier (Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France) (Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland) ; Affleck, Ian (Department of Physics and Astronomy, Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, B.C., V6T1Z1, Canada) ; Mila, Frédéric (Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland)

20 June 2019

Abstract: Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color SU(N) symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of SU(N) . In particular, SU(2) spin-1 AKLT states with edge states of arbitrary spin s=1/2,1,3/2,⋯ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are unique ground states. This enables us to study the stability of the edge states by interpolating between exact AKLT Hamiltonians. As an example, in the case of spin-1 physical degrees of freedom, it is shown that a quantum phase transition of central charge c=1 separates the Symmetry Protected Topological (SPT) phase with spin-1/2 edge states from a topologically trivial phase with spin-1 edge states. We also address some specificities of the generalization to SU(N) with N>2 , in particular regarding the construction of parent Hamiltonians. For the AKLT state of the SU(3) model with the 3-box symmetric representation, we prove that the edge states are in the 8-dimensional adjoint irrep, and for the SU(3) model with adjoint irrep at each site, we are able to construct two different reflection-symmetric AKLT Hamiltonians, each with a unique ground state which is either even or odd under reflection symmetry and with edge states in the adjoint irrep. Finally, examples of two-column and adjoint physical irreps for SU(N) with N even and with edge states living in the antisymmetric irrep with N/2 boxes are given, with a conjecture about the general formula for their correlation lengths.


Published in: Nuclear Physics B 945 (2019) DOI: 10.1016/j.nuclphysb.2019.114663
License: CC-BY-3.0



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